Submitted for review Submitted for review 0000; 00:1–83
Thirty years of the finite volume method for solid mechanics P. Cardiff1∗ , I. Demirdˇzi´c2
arXiv:1810.02105v1 [math.NA] 4 Oct 2018
1 University
College Dublin, Bekaert University Technology Centre, School of Mechanical and Materials Engineering, Belfield, Ireland 2 Maˇ sinski fakultet Sarajevo, Vilsonovo sˇetaliˇste 9, 71000 Sarajevo, BosniaHerzegovina
SUMMARY Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, including “cellcentred”, “staggered”, “vertexcentred”, “periodic heterogenous microstructural”, “Godunovtype”, “matrixfree”, “meshless”, as well as others. This article gives an overview, historical perspective, comparison and critical analysis of the different approaches, including their relative strengths, weaknesses, similarities and dissimilarities, where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finishes with a look towards future research directions and steps required for finite volume solid mechanics to achieve widespread acceptance. Received . . .
KEY WORDS: finite volume methods; solid mechanics; stress analysis; finite element methods
1. INTRODUCTION “Now however I recognise the FVM/FEM dichotomy as being comparable with those between Protestant and Catholic, or Sunni and Shia. That is to say that it promotes needless conflict; and expense; and loss of opportunity.” [1] With these words, Brian Spalding highlighted the, at times, unproductive nature of the debate around the relative merits of the finite volume (FVM) and finite element (FEM) methods. Although accepted in the Computational Fluid Dynamics (CFD) field, there remains a reluctance and general confusion around the application of the finite volume method to solid mechanics. The aim of this article is to clarify this confusion, by: providing an overview of the significant developments within the field; linking variants of the finite volume method for solid mechanics analyses; comparing finite volume methods with standard finite element methods; and, finally, identifying future directions for the field. Building on the foundations of the finite difference method, the finite volume method is a generalisation in terms of geometry and topology: simple finite volume schemes reduce to finite ∗ Correspondence to: P. Cardiff, University College Dublin, Bekaert University Technology Centre, School of Mechanical
and Materials Engineering, Belfield, Ireland. Email:
[email protected]
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´ ˇ C CARDIFF & DEMIRDZI
difference schemes. Whereas the finite difference method is based on nodal relations for differential equations, the finite volume method balances fluxes through control volumes, directly discretising the integral form of the conservation laws. A number of prior developments within CFD provided the foundation for the seminal paper on the cellcentred finite volume method for solid mechanics by Demirdˇzi´c et al. [2]. In the past three decades, the finite volume method for solid mechanics has developed in a number of directions, differing in terms of discretisation, solution methodology and overall philosophy, including: (a) implicit cellcentred methods, based on the original work of Demirdˇzi´c et al. [3–132]; (b) staggeredgrid methods [133–161]; (c) vertexcentred methods [162–251]; (d) methods for periodic heterogenous microstructures [252–315]; (e) Godunovtype approaches [316–347]; (f) matrixfree approaches [226, 227, 229, 236, 239, 348–365]; (h) meshless approaches [366–388]; as well as several others. The implicit cellcentred approach, stemming from Demirdˇzi´c et al. [2], closely resembles the procedures commonly used in fluid dynamics – for example, see Ferziger and Peric [389] – where memoryefficient segregated solution algorithms are used in conjunction with iterative linear solvers. The staggered approaches are similar in philosophy, but store dependent variables, displacement/velocity/pressure, on different grids. A third approach, the vertexbased finite volume method, differs greatly in terms of philosophy: as noted by Baliga and Atabaki [390], the cellcentred method can be thought of as a controlvolume finite difference method, combining ideas borrowed from finite volume and finite difference methods; whereas, the vertexcentred approach may be viewed as a controlvolume finite element method, formulated by amalgamating concepts native to finite volume and finite element methods. A fourth variant of the finite volume method has exclusively focussed on the analysis of heterogeneous periodic microstructures, where local quadratic displacement distributions are employed. An additional distinct flavour of the finite volume method for solid mechanics is the explicit Godunovtype mixed formulation, which has its roots in the solution of compressible gas flow Euler equations; in these methods, the linear momentum, deformation gradient tensor and total energy are the conservation variables, and, as such, are well suited to capturing shocks and solution discontinuities. The matrixfree approach is closely related to the cellcentred and vertexcentred approaches, where it differ in terms of solution methodology as well as intended application. A final major variant, meshless approaches, aim to overcome drawbacks of meshbased approaches for large deformations and crack problems. From afar, each of these main variants of finite volume method may seem quite distinct; however, upon closer inspective, they share many similarities in terms of discretisation and solution algorithm. Nevertheless, the fragmented nature of the finite volume solid mechanics community is evident from many recent publications in the area, where authors, reviewers and editors tend to be unaware of developments within the field. In addition, and more generally, there is a lack of awareness in the computational mechanics community around the capabilities of the finite volume method for solid mechanics. Accordingly, a comparative and critical review of the finite volume method for solid mechanics is timely, relevant and essential for the future progress of the field. Within this domain, there are a number of open questions: What are the strengths and weaknesses of the various approaches? Which approaches show the greatest potential and widest applicability? Are there possibilities for the various methods to be combined to produce superior methods? How do the finite volume approaches relate to finite element methods? Are there directions of development which are missing? This article will attempt to provide answers to these questions, as well as Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
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providing a unifying framework for the discussion of finite volume methods for solid mechanics, and their relationship with finite element methods. The remainder of the article is constructed as follows: Section 2 provides a chronological overview of the prominent finite volume developments for solid mechanics, where each major variant is discussed in turn. Section 3 compares the cellcentred finite volume approach for solid mechanics with the other flavours of the method. In Section 4, the “standard” cellcentred finite volume method for solid mechanics is closely compared to the “standard” finite element method, highlighting similarities and differences in terms of discretisation, solution methodology and overall philosophy. Section 5 reviews the variety of structural applications to which the finite volume method has been applied in its thirty year history. The penultimate section briefly reviews software that use, or have previously used, the finite volume method for solid mechanics simulations. Finally, Section 7 summaries the main conclusions of the article and considers the current challenges facing the field.
2. HISTORY OF THE FINITE VOLUME METHOD FOR COMPUTATIONAL SOLID MECHANICS The development of the finite volume method for solid mechanics has occurred independently in a number of forms, with finite difference methods, computational fluid mechanics algorithms, and finite element methods providing much of the inspiration. This section gives an overview of the historical development of these finite volume methods, where each variant is considered in turn; indepth dissections of the technical details are left to the subsequent section. Although the field is small and fragmented, there are a number of notable partial reviews, including those of Maneeratana [29], Aboudi [265], Pindera et al. [293], Vaz et al. [230], Charalambakis [296], Cavalcante et al. [306, 312], HajAli and Aboudi [307], and Gong et al. [238]. Before delving into details of influential publications, it is insightful to first consider the literature landscape as a whole. To this end, Figure 1 presents a histogram of the publications in the area todate, separated into journal articles, conferences, Ph.D. theses, Masters theses and books. To complement this, Table I gives the finite volume solid mechanics references chronologically and separated into distinct variants, while Figure 2 lists the most popular international journals for publishing finite volume solid mechanics articles. Both peerreviewed and nonpeerreviewed publications have been included in Table I; whereas, in Figure 2 only journals with greater than five articles have been included. Furthermore, Table II lists the most cited articles related to the finite volume method for solid mechanics; the references have been listed in order of decreasing number of citations, and only articles with greater than fifty citations have been included, according to Google Scholar citations on 25th August 2018. Care should be taken when interpreting the data, as the number of citations may not be directly proportional to impact on the field; for example, Weller et al. [18] has by far the greatest number of citations; however, a significant percentage of its received citations are not related to its solid mechanics contributions. Submitted for review (0000)
´ ˇ C CARDIFF & DEMIRDZI
4 35 Books 30
MScs PhDs
25
Conferences 20
Journals
15 10 5
20 18
20 16
20 14
20 12
20 10
20 08
20 06
20 04
20 02
20 00
19 98
19 96
19 94
19 92
19 90
19 88
0
Figure 1. Histogram of the finite volume method for solid mechanics publications todate
1988 to 1989
1990 to 1999
2000 to 2009
2010 to 2018
cellcentred
[2]
[3–23]
[24–74]
[75–132]
staggeredgrid

[133–152]
[153–160]
[161]
vertexcentred

[163–192]
[193–230]
[231–251]
HOTFGM/HFGMC/FVDAM

[252–256]
[257–293]
[294–315]
Godunovtype

[316–320]
[321–325]
[326–347]
matrixfree


[226, 227, 229, 348–351]
[236, 239, 352–365]
meshless

[366]
[367–376]
[377–388]
other

[391]
[47, 207, 392– 395]
[232, 404]
396–
Table I. Chronological overview of prominent publications in the finite volume method for solid mechanics, separated into distinct variants
A final perspective on the literature landscape is gained from the coauthorship network presented in Figure 3, which has been generated using the VOSviewer software [406]. A coauthorship network is a visual method to assess research collaborations within a field, as well as observe detached regions of research. Referring to Figure 3, the three larger subnetworks broadly correspond to: cellcentred approaches indicated by the red subnetwork centred on demirdˇzi´c and ivankovi´c; vertexcentred approaches indicated by the green subnetwork centred on cross, bailey Submitted for review (0000)
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THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
Int. J. Numer. Methods Eng. Journal of Computational Physics Applied Mathematical Modelling Comput. Methods Appl. Mech. Eng. Int. J. Solids Struct. ASME Journal of Applied Mechanics Computers & Structures Computational Materials Science Numerical Heat Transfer, Part B International Journal of Plasticity Engineering Fracture Mechanics Computer Modeling in Eng. and Sci. Composites Part B: Engineering International Journal of Fracture Composite Structures
5 5 0
5
6 6
7
8
9 9 9
12 11
15
17
20
10 15 20 Number of related articles
24
25
Figure 2. Number of publications per journal related to computational solid mechanics using the finite volume method, based on the cited references, where only journals with greater than five articles have been counted
and fallah; and methods for periodic heterogenous microstructures indicated by the blue subnetwork centred on aboudi, pindera and cavalcante. Although the coauthorship graph is not a direct measure of contribution to the field, it does provide insight into the collaboration between authors and subdivisions within the domain. In the image, individual authors are represented by filled circles, and a coauthored publication between two individual authors is symbolised by a line connecting the two filled circles; the line thickness increases with increasing number of coauthored publications; the size of an author’s filled circle is directly proportional to the number of their related publications. To maintain interpretability, only authors with five or more related publications have been included in the network, and only peerreviewed publications have been incorporated. Finally, before considering individual contributions, we will clarify the meaning of a number of terms in the current context to avoid any confusion. In terms of the domain spatial discretisation, this will be referred to interchangeably as the “mesh” or “grid”; each subdomain in a finite volume mesh will be equivalently referred to as a “control volume”, a “cell”, or even an “element”; similarly, each subdomain in the finite element mesh will be referred to as an “element” or “cell’. 2.1. “Standard” cellcentred approaches Thirty years ago, Demirdˇzi´c et al. [2] proposed the first application of the cellcentred finite volume method, in its modern form, to solid mechanics. This original method was applied to the simulation of thermal deformations in welded workpieces on a 2D structured quadrilateral grid. Small strains and rotations were assumed and the material behaviour was described by the so called DuhamelNeumann form of Hooke’s law. This cellcentred approach takes its name from the dependent variable, in this case displacement, residing at the cell centres (control volume centroids); equivalently, the approach has been termed the colocated, colocated or collocated finite volume method, as the dependent variables share their location at the cell centres/centroids. In the initial approach of Demirdˇzi´c et al. [2], the displacement was assumed to locally vary linearly. A distinguishing feature of the proposed solution algorithm was the partitioning of the surface force Submitted for review (0000)
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6 Reference
Formulationtype
Number of citations
Weller et al. [18]
cellcentred
1988
Demirdˇzi´c and Muzaferija [10]
cellcentred
408
Idelsohn and O˜nate [169]
vertexcentred and cellcentred
221
Demirdˇzi´c and Muzaferija [8]
cellcentred
189
Jasak and Weller [27]
cellcentred
185
Pindera et al. [293]
HFGMC/HOTFGM/FVDAM
185
Demirdˇzi´c and Martinovi´c [3]
cellcentred
172
Bailey and Cross [171]
vertexcentred
143
O˜nate et al. [170]
vertexcentred and cellcentred
128
Fryer et al. [163]
vertexcentred
103
Slone et al. [204]
vertexcentred
101
Slone et al. [210]
vertexcentred
97
Bansal and Pindera [273]
HFGMC/HOTFGM/FVDAM
94
Voller [405]
vertexcentred
91
Bijelonja et al. [53]
cellcentred
87
Taylor et al. [174]
vertexcentred
86
Cavalcante et al. [278]
HFGMC/HOTFGM/FVDAM
80
Taylor et al. [211]
vertexcentred
70
Khatam and Pindera [291]
HFGMC/HOTFGM/FVDAM
70
Fallah et al. [195]
vertexcentred
64
Ivankovi´c et al. [9]
cellcentred
61
Karaˇc et al. [82]
cellcentred
59
Taylor et al. [205]
vertexcentred
58
Bailey et al. [187]
vertexcentred
57
Slone et al. [216]
vertexcentred
56
Murphy and Ivankovi´c [51]
cellcentred
55
Tukovi´c and Jasak [67]
cellcentred
54
Demirdˇzi´c et al. [15]
cellcentred
52
Bailey et al. [176]
vertexcentred
51
Table II. Most cited articles related to the finite volume method for solid mechanics from Google Scholar citations on 25th August 2018. Approaches for heterogeneous periodic microstructures are indicated by HFGMC/HOTFGM/FVDAM.
term into a compactstencil implicit term and a larger stencil explicit term; as a result, the linear momentum vector equation was temporarily decoupled into three scalar component equations that were independently solved, where outer fixedpoint/GaussSeidel/Picard iterations provided the required coupling. This form of solution methodology is known as a segregated approach, as the governing conservation of linear momentum equation is segregated into three scalar equations during solution. Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
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Figure 3. Coauthorship network of the finite volume method for solid mechanics, generated using the VOSviewer software [406]
The original 2D structured grid method of Demirdˇzi´c et al. [2] (Fig. 4(a)) was subsequently generalised to 3D [8], where significant freedom in valid mesh definition was provided through the assumption of general convex polyhedral cells (Fig. 4(b)). The cellcentred approach has since been extended to deal with a wide variety of solid and multiphysics phenomena, including: • elastoplasticity [3, 7, 29, 30, 47, 75, 77, 87, 107, 113, 118, 119, 125] and viscoelasticity • • • • • • •
[23, 25, 31, 49, 115, 407]; thermoelasticity [2, 8, 83, 92, 93] and hygrothermoelasticity [17, 24, 34, 71, 73, 76, 86]; poroelasticity [99–102, 104, 110, 116, 124]; anisotropy [11, 17, 24, 34, 71, 73, 76, 86, 94, 121] and heterogeneous material properties [85, 88, 90, 99, 124]. incompressibility and quasiincompressibility [19, 20, 48, 50, 53, 54, 70, 117]; contact mechanics [28, 84, 95, 107]; finite strains and rotations [29, 48, 67, 68, 94, 107, 131]; fracture mechanics [6, 9, 12, 16, 21, 32, 39, 40, 45, 51, 52, 57, 63, 78, 82, 90, 99, 100, 102, 115, 124]; Submitted for review (0000)
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(a) 2D structured quadrilateral mesh from Demirdˇzi´c (b) 3D general convex polyhedral control volume et al. [2] from Demirdˇzi´c and Muzaferija [8]
Figure 4. Original 2D structured quadrilateral mesh of Demirdˇzi´c et al. [2] (left) and the subsequent generalisation to 3D unstructured convex polyhedra [8] (right)
• casting, melting, and solidification [41, 59, 105, 106];
• fluidsolid interaction [4, 5, 10, 19, 20, 26, 33, 36–38, 42, 43, 46, 50, 54, 56, 60–62, 64–
66, 68, 70, 72, 74, 81, 89, 91, 97, 98, 111, 114, 120, 122, 130]; • beams, plates and shells [14, 42, 43, 55, 62, 112, 121, 126, 127, 214, 220–223, 237, 241, 242, 249–251, 407, 408]; • solidelectrostatic interaction [79, 80]; Furthermore, alternative solution methodologies have been developed, including geometric multigrid procedures [11, 12, 15], blockcoupled algorithms [79, 80, 108, 128], hybrid/mixed pressuredisplacement formulations [44, 47, 48, 53, 117, 409], and Aitken acceleration [98, 104, 128], while extension to fourthorder accuracy [109] and inclusion of novel gradient calculation methods have provided improvements in accuracy and robustness [69, 81, 96, 98, 103, 107, 123, 128, 132]. Apart from the continuum approaches adopted by the majority of cellcentred formulations, a number of authors have also proposed cellcentred finite volume methods for beams, plates and shells [14, 42, 43, 55, 62, 112, 121, 126, 127, 407, 408]. Of particular note are the developments of Weller et al. [18] and Jasak and Weller [27], where the cellcentred form has been demonstrated wellsuited to running on distributed memory clusters/supercomputers; the domain decomposition method was used where the solution domain is decomposed into a number of subdomains, each solved on a separate Central Processing Unit (CPU) core; necessary coupling between the subdomains is performed using a messagepassing protocol (initially Parallel Virtual Machine, but Message Passing Interface in later publications). The Weller et al. [18] and Jasak and Weller [27] implementations form a component of the opensource C++ library OpenFOAM (formerly commercial software FOAM). Many of the subsequent developments in the implicit cellcentred field have been based on the OpenFOAM platform, for example, [88, 94, 104, 107, 108, 110, 129]. It should, however, be noted that the standard cellcentred form of the finite volume method is not the only approach that has built on the OpenFOAM Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
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library; Godunovtype methods, for example, have been implemented by Haider et al. [342, 344]; these are discussed later. In addition to extensions of the original Demirdˇzi´c et al. [2] approach, there have been a number of related cellcentred approaches that have differed in terms of formulation, discretisation, and solution procedure. Henry and Collins [4, 5] were one of the first to outline a cellcentred procedure for interaction between incompressible fluids and incompressible solids; the authors proposed a 2D axisymmetric approach, where displacement and pressure were the primary unknowns and the SIMPLE solution algorithm of Patankar and Spalding [410, 411] was employed. A curvilinear formulation was proposed by Oliveira and Rente [22] for static and transient analyses, which was subsequently extended to include elastoplasticity by Rente and Oliveira [30]. As an alternative to unstructured grid formulations for complex geometry and multiple material regions, Sch¨afer and coworkers [36, 38] proposed a novel blockstructured grid storage approach, which they combined with a multigrid solution procedure to analyse fluidsolid interaction problems. Whereas Ni and Murthy [58] developed a novel oneway multiscale formulation coupling a macroscale finite volume stress analysis procedure with a nanoscale molecular dynamics procedures. Considering the calculation of tractions at the control volume faces, Nordbotten and Keilegavlen [96, 103, 123] proposed socalled multipoint stress approximation methods. For the analysis of functionally graded beam vibration, Jing et al. [112] proposed a cellcentred finite volume method using Timoshenko beam theory. Independently of the Demirdˇzi´c et al. [2] lineage and that of other cellcentred approaches, a number of alternative forms of the finite volume method for solid mechanics have been developed: these are discussed in the following paragraphs. 2.2. Staggeredgrid approaches The distinguishing characteristic of staggeredgrid approaches, originally proposed by Harlow and Welch for CFD [412], is their grid arrangement: the components of the primary solution variable, for example, the x and y components of velocity, are stored at different locations; in addition, different sets of control volumes may be used when integrating the discretised governing equation in each Cartesian direction, for example, the xmomentum component equation may employ a different grid to the y momentum component equation. The primary motivation for such staggeredgrid approaches is the avoidance of the “checkerboarding” numerical phenomenon, where high frequency variations appear in the solution variables that are unobservable to the discretisation. The extension of staggeredgrid approaches to general unstructured 3D meshes is, however, not trivial; consequently, the popularity of such approaches is declining. In one of the first applications of a staggeredgrid formulation to solid mechanics, Beale and Elias [133, 134] showed how the finite volume CFD code PHOENICS could be applied to stress analysis problems; in their method, Beale and Elias used the analogy between the stream functions in creepingfluidflow and the Airy stress functions in solid mechanics. This method was later generalised by Spalding, Bukhari, Qin, Hamill and coworkers [135, 137, 144, 149, 151, 152, 157, 158, 160], where a displacement, rather than a force analogy, was adopted, having the major advantage of being more general in three dimensions. Spalding noted that a CFD solution procedure designed for computing velocities is suitable for computing displacements if the convection terms are set to zero and the volume/dilatation stress term is introduced by inclusion of a pressureSubmitted for review (0000)
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´ ˇ C CARDIFF & DEMIRDZI
and temperaturedependent source term. As a consequence, the resulting solution method used a SIMPLE algorithm [410, 411], still popular in modern CFD codes, with displacement and pressure being the primitive variables, as opposed to velocity and pressure in standard fluid flow analyses. The use of mixed displacementpressure approaches, however, is not a requirement of staggeredgrid approaches, as shown by Hattel, Hansen and collaborators [136, 138–143, 145–148, 150, 153– 156] where the sole primary variable was displacement. Hattel and Hansen [136] initially proposed their staggeredgrid approach for the analysis of thermally induced stresses in casting problems, and subsequently extended it for other applications including thermal stresses in concrete aging and thermoelastoplasticity problems. They termed their approach “a control volume based finite difference method”, further indicating the closerelationship with finite difference methods. The authors noted that their method resulted in an elegant formulation for nonconstant material properties, a benefit of the staggered grid approach. In contrast, the method of Demirdˇzi´c et al. [2] was first extended to nonconstant material properties by Tukovi´c et al. [88], following a somewhat more involved procedure. In addition to the displacementpressure and pure displacement approaches, Spalding [157] proposed a staggered formulation where rotation and displacement were the primitive variables; Spalding surmised that the rotationbased method may provide a more efficient solution algorithm in certain situations. The approach is described in documentation from an early version of the PHOENICS software; however, issues with boundary conditions are noted and no further articles appear on the formulation. A similar idea was subsequently proposed by Wenke, Wheel, Pan and Qin [213, 231, 234] in the framework of vertexbased finite volume methods, where both displacements and rotations are considered the primitive variables. A number of alternative staggered grid formulations have also been proposed: Wang and Melnik [159] put forward a staggered finite volume approach for analysis of shape memory alloys, defined on 2D structured rectangular grids; in addition, Rajagopal et al. [161] investigated the response of a layered viscoelastic plate using a onestep explicit staggered grid finite volume approach on structured meshes. 2.3. Vertexcentred approaches In addition to the cellcentred and staggered methods discussed above, Fryer et al. [163] presented a vertexbased finite volume method for static elastic stress analysis. Vertexcentred approaches stored the primary unknowns, for example, displacement, at the mesh vertices and integrated the governing equation over control volumes surrounding each mesh vertex. The original method of Fryer et al. [163], initially termed a control volumeunstructured mesh procedure, could analyse complex 2D geometry using both quadrilateral and triangular cells (Fig. 5(a)). The method of Fryer et al. [163] followed closely the approach of Baliga and Patankar [413] from a decade earlier, who developed a socalled controlvolumebased finite element method for convectiondiffusion equations on unstructured triangular grids; they noted that the controlvolume formulation possessed the property of local conservation and lends itself to easy physical interpretation. Vertexbased methods stem from the finiteelement variational methods philosophy, where instead of taking the weighting functions equal to the shape functions, as in the standard Galerkin finite element method, they assume the weighting functions to be the identity matrix, thus breaking the association between the element and the weighting function. Consequently, vertexbased finite volume methods more closely relate to standard finite element formulations than the cellcentred Submitted for review (0000)
solved for via corrections, 
(14)
(SCV) contributes two surfaces to the control volume boundary so that a typical integral in equations (8) and (9) may be evaluated as follows:
f c
Nr,,"{
a¢
f
a¢
}
ady = 2: YEARSaOF  'THE dy FINITE VOLUME METHOD FOR SOLID MECHANICS THIRTY
ax
'_
11
512
{2 [
M aM :::: Nr,," 2: 2: af 2:  . i=1
vector whose elements are f the linear solver. The tolthere is no need to achieve ner loops; this is guaranteed
11
ax
s=1
ax
Control Volute
'J}
G. A. TAYLOR, C. BAILEY AND M. CROSS (16) Control volume
y x
Subcontrol volumes
z
4) are solved by a convenechnique with simple Jacobi ved to yield much quicker rocedures. Also to ensure ource term, b.; in each inoverrelaxed so that
r w lies between 1 and 2. ng the overall solution procompute
III
Vertex Quadrilateral c.lb
Integration points
Trianlular eell.
Figure 2. 3D(b)assembly of FV subcontrol volumes vertex. Figure Control volume unstructured vertex based (a) 2D1.unstructured quadrilateral andmesh trianglular mesh 3D unstructured mesh from Tayloratetaal. [211] (CVUMV) discretization from Fryer et al. [163]
3.2. Vertexbased FV method
Figure Vertexbased formulation641 of Fryer et al. [163] (left) showing the construction of Appl. 5. Math. Modelling,finite 1991,volume Vol. 15, Nov./Dec. In thevertex vertexbased FV method weighting2D functions associated withet aal.vertex control volumes around a mesh for quadrilateral andthetriangular cells, and of Taylor [211]are equal to = [I ], and zero elsewhere. This de!nition is equivalent unity within the control volume, [W ] i showing the extension to 3D cells.
to that for the subdomain collocation method as de!ned in the standard texts [2, 4]. However, it is important to note that weighting functions de!ned in this manner permit a variety of possibilities with regard to the control volume de!nition [6]. This is because the weighting or staggeredgrid finite volume methods. Typically vertexcentred approaches, as in Fryer et al. functions are not restricted to a direct association with the cell or element as in the Galerkin [163], use shape functions to This describe the localconsideration variation ofanddisplacement, similar toof standard case. is an important requires the recognition the vertexbased FV method as a discretization technique in its own right [4]. finite element methods; however, it is possible to formulate vertexcentred approaches without the For the vertexbased FV method the contributions as described by Equations (8) and (9) are use of shape functions, for example, [226, 227, 229]. In addition to the designation vertexbased ! T = − [R] d" (12) [K] ij finite volume method, the formulation is referred to by a number ofi [D][B] other jnames, including vertex"ui
centred finite volume method, cellvertex finite volume method, control volume procedure, controland ! ! ! volume finite element method, controlvolumebased finiteelement method, and elementbased finite {f }i = {b} d# − [R]Ti [D]{$!vp } d" + {tp } d" (13) #i "ui the governing equation "ti volume method. Although vertexbased approaches often take weak form as a starting point, for example, Idelsohn toand O˜nthat ate the [169] and Bailey andconditions Cross [171], finite directly as It is important note traction boundary can beallapplied another surface integral, but strong in the previous an additional surface element volume methods can in fact directly start from the integralGalerkin form ofapproach the governing equation, is generally included on the domain boundary. A nonoverlapping control volume de!nifor example, in the original of Fryer al. [163]; this Sections 3 tionwork suitable for a etvertexbased FV point methodis isdiscussed illustratedfurther in twoinand threedimensions in Figures 1(b) and 2, respectively. The !gures illustrate the assembling of vertexbased control and 4. volumes from their related subcontrol volumes [7]. Additionally, the asymmetric nature of The original 2D approach of Fryer et toal.the [163] wassti%ness subsequently dimensions the contributions overall matrix asextended described to by three Equation (12) does not ensure thatofsymmetry always be preserved. For this reason FV methods were initially argued as (Fig. 5(b)) and a wide range physicalwill phenomena:
being inferior, but in the light of recent research where di%erent control volume de!nitions have been proposed, the extent of this inferiority has come into question [5, 6, 14].
• compressible and incompressible elasticity [171, 179, 180, 185, 190, 194, 218, 227, 239, 240];
• elastoplasticity, elastoviscoplasticity, creep andLtd. material nonlinearity Copyright ? 2002 John Wiley & Sons, Int. J.[174, Numer.183, Meth.211, Engng233, 2003; 56:507–529 • • • • • • •
241, 250]; finite strains and geometric nonlinearity [188, 195–197, 212, 225]; beams, plates and shells [182, 214]; multiphysics and fluidsolid interaction [176, 178, 181, 186, 187, 189, 191, 199, 204, 210, 216, 226, 229, 239, 248]; casting [162, 165, 167, 168, 175–177, 181, 184, 186, 187, 193], extrusion and forging [202, 207, 232]; welding [200, 205, 206]; contact mechanics [201]; functionally graded solids [238, 244] and wood drying [173, 235]; Submitted for review (0000)
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• micropolar elasticity (shear stress may not be symmetric) [209, 228]; • vibrations, acoustics and wave propagation [236, 237, 245, 246].
In addition, alternative solution methodologies have been examined, including mixed displacementrotation approaches [213, 231, 234], and addressing parallelisation on distributed memory supercomputers [198, 203]. Also of note is the novel method of Tsui et al. [239], where control volumes were constructed about each cell vertex by joining the mesh cell centres together rather than joining the cellcentres to the facecentres as per the classic vertexbased method. Within the class of vertexcentred methods, there exists a subclass called dualtimestepping matrixfree methods, where the solution is calculated in an explicit manner and a linear system is not directly formed [226, 227, 229, 236, 239]; these dualtimestepping matrixfree formulations, which have also employed cellcentred frameworks, are considered further in a separate subsequent section below. A number of authors have performed detailed analyses of the vertexbased finite volume method and compared it with the finite element, cellcentred finite volume, and finite difference methods. Employing the finite element variational framework, Filippini et al. [243] and Vaz et al. [230] compared the vertexbased finite volume method with the standard Galerkin finite element method; they found that both methods share the same data structure and general strategy to assemble the corresponding characteristic matrices, with the main conceptual difference being in the local integration domain. Selmin [414] made similar conclusions when comparing the vertexcentred finite volume, finite element, and finite difference methods for solution of the Euler equations; they noted that the vertexcentred approach led to a discretisation equivalent to those resulting from both classical finite volume and finite elements theories, and depended only on the method used to build the control volume surrounding a vertex of the grid. When comparing cellcentred and vertexcentred finite volume methods, Hejranfar and Azampour [247] argued for the superiority of the vertexcentred method in terms of efficiency (CPU time and memory) and accuracy; they concluded that the vertexcentred method was less sensitive to grid irregularity than the cellcentred version. When examining the stresses generated in wood due to drying, Perr´e and Passard [173] showed how to choose interpolation coefficients in a vertexbased procedure in order to obtain equivalent results as a standard Galerkin finite element method. Further analysis of control volume methods can be found in Voller [405]. The vertexbased methods come in a number of forms; however, it is worth noting the distinction between overlapping and nonoverlapping vertexbased methods: the more common approach involves nonoverlapping control volumes, where the control volumes about each mesh vertex do not overlap with the control volumes of neighbouring vertices. In the less popular overlapping control volumes approaches, for example, as discussed by O˜nate et al. [170], vertices partly share control volumes with neighbouring vertices, resulting in the governing equations being integrated more than once over regions of the domain. In this and earlier publications [164, 170], Zienkiewicz and O˜nate mention “an early attempt to use FV concepts in CSM” [170] to be in the Lawrence Livermore Laboratory ‘hydrocodes’ of the 1960s [415, 416]; this link is, however, tenuous; the Gauss divergence cell gradient calculation approach described by Wilkins has indeed been used, along with some other methods (for example, the least square method was used in [10]), for calculating cell gradients in later finite volume publications; however, it is not a core postulate of the finite volume method and is in fact not required, as shown in Section 3.2. Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
13
An additional form of vertexcentred approach is the socalled grid method, where an explicit solution algorithm is employed. The approach was initially proposed by Zhang and Liu [192], where the main intended application was geomechanical wave propagation problems; subsequently, the method has been extended to three dimensions, anisotropy, fluidsolid interaction and multiCPU parallelisation [208, 215, 217, 219, 224]. Although the originators of the method, Zhang and Liu [192], argue for the distinction between the grid method and the vertexbased finite volume method, the authors of the current article believe this distinction is unwarranted: like other finite volume methods, the grid method starts from the governing momentum equation in strong integral form and approximates the forces over the boundary of control volumes; although there are minor differences in the techniques used to approximate the surface forces, for example, comparing Dormy and Tarantola [172] and Zhang and Liu [208], the grid method still remains a form of vertexcentred finite volume method. 2.4. Approaches for periodic heterogenous microstructures The analysis of periodic heterogenous microstructures has given rise to specialised versions of the finite volume method. Depending on the formulation, the related methods can be referred to by a number of names, including: the higherorder theory for functionally graded material (HOTFGM) [256], the highfidelity generalised method of cells (HFGMC) [257, 258, 290, 307], the finite volume direct averaging micromechanics (FVDAM) theory [272, 273], or some variant thereof. Although there is some disagreement over the naming convention [273, 290, 307], these methods have a common origin in the method of cells and the generalised method of cells developed by Aboudi and Paley [252, 254, 417]. In the initial approach, Aboudi [417–421] employed a finite differencebased technique to analyse the periodic microstructure of a unidirectional fibrereinforced composite. The formulation was based on a twostep volume discretisation, with rectangular cells split into four subcells (Figure 6); an averaging approach was then used to enforce the governing field equations in each subcell. Subsequently, the socalled higherorder theory for functionally graded A continuum theory for fiberreinforced elasticviscoplastic composites
607
4
t
I
(I’
’ X2 X’i’
(2’ X2
/
* X3 Fig. 2. A representative cell.
Figure 6. Representative cell in the original method of Aboudi [417]
material linear to
based on the expansion of the displacement vector in each subcell in terms of the distances from the center. This expansion can be expressed in terms of the Legendre polynomials permitting the modeling of increasing complex deformation patternssubcell within the displacement subcell. For an [255, 256] was constructed by extending the assumed Nth order theory the displacement components at any point within the subcell can be expressed quadratic. as Building further on this, the highfidelity generalised method of
zp) = U~~~~“)(X~, xy,xp,t)Pm(zp)Pn(zp) i= 1,2,3
(1)
variation from cells provided
Submitted for review (0000)
where zy’ = @‘/(4/2), z y’ = f$B’/(h$2), t is the time, P,,, is the Legendre polynomial of order m and summation is implied by the repeated indices m,n = 0 , 1, . . . , N. The displacement distribution in the composite is given in terms of the higher order displacement components U!‘@) r(m,njand these are defined only on the discrete lines x2 = x:“’ and x3 = xiB’. The derivation of the continuum theory requires a smoothing operation by which the
´ ˇ C CARDIFF & DEMIRDZI
14
a homogenised counterpart of the approach [257–260, 262, 264–267, 270, 271, 276, 277, 283– 285, 290, 294, 295, 297, 299]. As outlined by Cavalcante, HajAli and Aboudi [306, 307], the HOTFGM and HFGMC approaches were simplified and generalised to arbitrary quadrilateral subcells using parametric mapping (Figure 7) by Bansal, Pindera, Zhong, Cavalcante, and coworkers [253, 261, 263, 266, 268, 269, 272–275, 278–282, 286–289, 291–293, 298, 300–306, 308–315]. The reformulated approach employed surfaceaveraged displacements as the sole independent variables, and, as discussed by Cavalcante et al. [306], revealed the approach to be a form of the finite volume method. This motivated the proposed name change to finite volume direct averaging micromechanics (FVDAM) theory; however, as noted previously, this is disputed [307]. Recent discussions around the development of these methods can be found in HajAli and Aboudi [307], Cavalcante et al. [306], Gong et al. [238], and Cavalcante and Pindera [312], along with reviews of the high fidelity generalised method of cells approaches by Aboudi [265] and of microstructural analysis approaches 2054 R. HajAli, J. Aboudi / International Journal of Solids and Structures 49 (2012) 2051–2058 by Pindera et al. [293] and Charalambakis [296].
Figure 7. Orthogonal grid for the original HFGMC (left), and an unstructured quadrilateral grid for the parametric HFGMC (right)
Since its initial development, the HOTFGM, HFGMC, FVDAM, or reformulations thereof, have been applied to a range of physical phenomena related to analysis of heterogenous microstructures, including: • Electromagnetothermoelastic and thermoelastic analysis of composites [257–259]; • Inelastic and viscoelasticviscoplastic micromechanical effects [260, 262, 270]; • Interfacial damage and fibre loss effects [267, 283];
Fig. 2. (a) A schematic RUC representation using orthogonal HFGMC; (b) an orthogonal subcell; (c) a schematic RUC representation using parametric HFGMC; (d) a general quadrilateral subcell.
• Periodic lattice blocks [271];
• Pressurised foam insulation cell microstructures [285];
Z 1 lk=2 r % ndymemory • Shape k lk $ lk =2
ð14Þ alloys [294]; • traction Optimisation porousformicrostructures [277]; The two continuityof conditions h i ðbÞ [295, 299]; Tðb Þ ¼ Nðb•Þ CDamage !0 þ Aðb Þmechanics WðbÞ ð15Þ • Total and incremental formulations for nonlinear materials [290]; ðb Þ where the matrix N is composed of the components of the norðb Þ mal vector, , for each side of[259, the subcell. It has the general form: • nFinite strains 284]; Tðbk Þ ¼
k
k
k
k
k
2
n1 0 0 0 6 N ¼ 4 0 n2 0 n3 Further 0 0reviews n3 n2 ðbk Þ
n3 0
n2
3ðbk Þ
7 n1 5
ð16Þ
ofn1applications can be found in Aboudi et al. [276] and Aboudi [284]. 0
The periodicity relations can be satisfied by mirroring and extending the cells near the periodic interfaces as shown in Fig. 2. Therefore Eqs. (12), (13) and (14), (15) are equivalent to the periodic relations by using the proposed mirroring technique, (HajAli and Aboudi, 2009). The contribution RðbÞ of subcell (b) to the overall residual vector can be arranged in the order of traction continuity, internal equi
Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
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2.5. Godunovtype approaches Godunovtype finite volume methods were first proposed for the solution of hyperbolic partial differential equations characterised by waves and shocks [324, 422, 423], and have been popularised for the solution of Euler compressible gas flow equations. The typical approach, which casts the conservation laws as a system of firstorder hyperbolic equations, is characterised by the approximate solution of a Riemann problem at the control volume faces to determine fluxes. The resulting discretisation evaluates the flux at a control volume face as a weighted average of the flux evaluated at each side of the face. Godunovtype methods were first applied to structural problems by Trangenstein and Colella, when they modelled the 1D propagation of waves in elastoplastic solids [316–318]; in their approach, the primitive conservation variables were the linear momentum vector and the deformation gradient tensor. Subsequently, the method has been extended to unstructured 3D grids in a variety of forms, differing in terms of grid arrangements, discretisations, and primitive variables [319–347], for example, see Figure 8. Although cellcentred formulations are the most common form of Godunovtype method, vertexcentred [330, 331] and, recently, facecentred approaches [346, 347] have also been explored. A distinctive characteristic of Godunovtype methods is the adoption of fully explicit
(a) 2D unstructured cellcentred polygonal mesh from Kluth and Despr´es [326]
(b) 2D/3D unstructured vertexcentred polygonal/polyhedral mesh from Aguirre et al. [331]
Figure 8. Forms of mesh employed by Kluth and Despr´es [326] and Aguirre et al. [331]
solution algorithms, where the time increment size is restricted by the standard CourantFriedrichsLewy constraint [424]. Consequently, the approach is efficient for transient hyperbolictype systems, and somewhat less so for elliptic ones, for example, quasistatic stress analysis. To date, Godunovtype approaches have used to model a wide range of physical mechanisms: • linear elasticity [316, 317, 343, 347];
• material nonlinearity [320, 322, 326, 328, 329, 332, 334–345]; • fracture and cavitation [320, 334];
• finite strains [328, 329, 332, 334–342, 344, 345]; • material heterogeneity [321, 323];
• wave propagation and impacts [316–324, 328, 329, 332–341, 343, 345]. Submitted for review (0000)
16
´ ˇ C CARDIFF & DEMIRDZI
2.6. Matrixfree approaches Of the previously discussed variants, all apart from the Godunovtype approaches employ implicit solution methodologies, where a system of algebraic linear equations is solved via iterative or direct methods. These implicit methods require the formation and inversion of a sparse matrix, where the surface force term within the momentum equation is discretised implicitly in terms of the primitive variables. In contrast, socalled matrixfree approaches evaluate the surface force term using the old timestep values of the primitive variable, and so avoid the construction of a linear system matrix. Given the distinction in solution methodology, matrixfree methods merit separate consideration; however, these approaches typically adopt vertexcentred, and in some cases cellcentred, spatial discretisations and could equally be considered under those headings. It should also be noted that Godunovtype methods employ matrixfree solution methodologies, but are considered separate given their distinct temporalspatial discretisation. Lv et al. [226] and Xia et al. [227, 229] developed a vertexbased finite volume approach with nonoverlapping control volumes that does not use shape functions. A socalled dual time stepping solution procedure was adopted, where a pseudo time term is added to the governing momentum equation; this pseudo time term is eliminated within each physical timestep by performing subiterations. The resulting procedure is implicit in physical time, i.e. no physical timestep restriction, but is limited by the standard CourantFriedrichsLewy constraint [424] in pseudo time. SabbaghYazdi and coworkers [348–354, 356–359, 365] developed a similar vertexcentred approach, which has been used to analyse 2D elastic thermoelastic problems including crack propagation in concrete; their approach, termed the Galerkin finite volume method, has been implemented in an inhouse software called NASIR: a Numerical Analyzer for Scientific and Industrial Requirements; the expression Galerkin finite volume method here indicates that a piecewise constant distribution of the primary variable is assumed. Similar approaches were developed by Chen and Yu [355] for impact analysis of thin plates, by Zhu et al. [236] for stress wave propagation, by Tsui et al. [239] for fluidsolid interactions, and by Suliman et al. [361] for bending beams. Alagappan et al. [360, 363] developed a novel staggered grid finite volume technique for wave propagation problem in viscoelastic solids, where the displacement is calculated at alternating nodes to the stress. Selim et al. [362, 364] implemented a cellcentred matrixfree dual timestepping method for fluidsolid interaction problems, where the solid spatial discretisation closely resembles the standard Demirdˇzi´c and Muzaferija [8] method. 2.7. Meshless finite volume methods Meshless methods have been proposed in order to overcome the drawbacks of meshbased finite element and finite volume methods, particularly related to large deformations and cracking. Atluri and Shen [367] were the first to proposed a meshless finite volume formulation, based on the earlier generalised Meshless Local PetrovGalerkin (MLPG) method by Atluri and Zhu [366]. The approach has subsequently been extended and applied to a variety of problems in elastostatics, elastodynamics and fracture mechanics [368–382]. Whereas the Atluri and Shen [367] approach uses overlapping control volumes, the meshless finite volume method developed by Ebrahimnejad et al. [383, 384, 386] employs nonoverlapping control volumes. The methods has been applied to 2D elasticity with adaptive mesh refinement, and Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
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was later extended to problems with material discontinuities by Fallah and Delzendeh [388], Fallah [425], and to free vibration analysis of laminated composite plates by DavoudiKia and Fallah [385, 387].
2.8. Other approaches Across the spectrum of finite volume methods for solid mechanics, not all procedures align with the previously discussed divisions. Although it is not possible to mention all approaches, the following are a selection of notable contributions. Teng et al. [391] and Chen et al. [392] developed a finite volume method to simulate the draping of woven fabrics, where the governing nonlinear equations were solved using a singlestep full NewtonRaphson method. Martin and Pascal [397, 400] proposed a novel discrete duality finite volume method for solving linear elasticity problems on unstructured meshes; the main characteristic of the discretisation is the integration of the governing equations over two meshes: the given primal mesh and also over a dual mesh built from the primal one. Pietro et al. [398] proposed a novel discretisation scheme for linear elasticity with only one degree of freedom per controlvolume face, corresponding to the normal component of the displacement. There are many solid mechanics problems which can equivalently considered from the fluid mechanics perspective, for example, in the analysis of extrusion and drawing. Eulerian “fluid” approaches used to solve such problems, for example, [47, 207, 232, 393–396, 399, 401–404], fall more closely in the domain of fluid mechanics and not discussed here further.
3. COMPARING VARIANTS OF THE FINITE VOLUME METHOD FOR COMPUTATIONAL SOLID MECHANICS The finite volume method, like other related numerical approaches, consists of the following main components: a) Discretisation of space and time; b) Discretisation of the mathematical model equations; c) Solution algorithm. In this section, the “standard” cellcentred finite volume method is reviewed, and subsequently compared with the different variants of the finite volume method in terms of each of these core parts.
3.1. Mathematical model for dynamic linear elasticity To allow clear comparison of methods, the dynamic behaviour of a body (Fig. 9) with volume Ω and surface Γ is analysed, where part of its boundary is subjected to a specified displacement, ub , and the remainder is subjected to a specified traction, T b . Assuming the relationship between stress and strain to be described by Hooke’s law, the governing conservation of linear momentum equation, a Submitted for review (0000)
´ ˇ C CARDIFF & DEMIRDZI
18
Tb
AAACRXicbVDLSgMxFM3UV63P6tLNYBF0U2ZE0KXoQt1VtA9oh3InvdNGk8yYZMQy9B/c6g/5DX6EO3Gr6WNhqwcCh3NfJydMONPG896d3Nz8wuJSfrmwsrq2vrFZ3KrpOFUUqzTmsWqEoJEziVXDDMdGohBEyLEe3p8P6/VHVJrF8tb0EwwEdCWLGAVjpVrrAoSA9mbJK3sjuH+JPyElMkGlXXQOWp2YpgKloRy0bvpeYoIMlGGU46DQSjUmQO+hi01LJQjUQTayO3D3rNJxo1jZJ407Un9PZCC07ovQdgowPT1bG4r/1ihIinzqepZKZvSMIROdBBmTSWpQ0rGfKOWuid1hRG6HKaSG9y0Bqpj9kkt7oIAaG+TU8qfIqtPL9UMKCmWQ3VyNT9tw/dko/5LaYdn3yv71Uen0bBJznuyQXbJPfHJMTsklqZAqoeSOPJMX8uq8OR/Op/M1bs05k5ltMgXn+wfJq7ML
AAACUXicbVDLTsJAFL2tLwQfoEs3jcREN6Q1JrokutEdRl4JNGQ63MKE6bTOTImk4U/c6g+58lPcOTwWAt7kZk7OfZ05QcKZ0q77bdlb2zu7e7n9fOHg8Oi4WDppqjiVFBs05rFsB0QhZwIbmmmO7UQiiQKOrWD0MKu3xigVi0VdTxL0IzIQLGSUaEP1isVuEPO+mkTmyerTXtArlt2KOw9nE3hLUIZl1Hol66rbj2kaodCUE6U6nptoPyNSM8pxmu+mChNCR2SAHQMFiVD52Vz61LkwTN8JY2lSaGfO/p3ISKRm6kxnRPRQrddm5L81SgRFvnI9SwXTak2QDu/8jIkk1SjoQk+YckfHzswup88kUs0nBhAqmfmSQ4dEEqqNqSvL30LDri5XrymRKPzs5Wlx2pjrrVu5CZrXFc+teM835er90uYcnME5XIIHt1CFR6hBAyiM4R0+4NP6sn5ssO1Fq20tZ05hJezCL82wtQQ=
⌦
AAACRXicbVDLTsJAFJ3iC/EFunTTSEx0Q1pjokuiG12JUR4JNOR2uIWR6bTOTI2k4R/c6g/5DX6EO+NWh8dCwJNMcnLu68zxY86UdpwPK7O0vLK6ll3PbWxube/kC7s1FSWSYpVGPJINHxRyJrCqmebYiCVC6HOs+/3LUb3+hFKxSNzrQYxeCF3BAkZBG6nWugmxC+180Sk5Y9iLxJ2SIpmi0i5Yx61ORJMQhaYclGq6Tqy9FKRmlOMw10oUxkD70MWmoQJCVF46tju0D43SsYNImie0PVb/TqQQKjUIfdMZgu6p+dpI/LdGQVDkM9fTRDCt5gzp4NxLmYgTjYJO/AQJt3VkjyKyO0wi1XxgCFDJzJds2gMJVJsgZ5Y/B0adXa4eE5AovPTuenLahOvOR7lIaicl1ym5t6fF8sU05izZJwfkiLjkjJTJFamQKqHkgbyQV/JmvVuf1pf1PWnNWNOZPTID6+cX1QGzEQ==
ub AAACUXicbVDLSsNAFL2Jr9r6SHXpJlgE3ZREBF0W3eiuon1AG8pketMOTiZxZlIsoX/iVn/IlZ/izuljYVsvXOZw7uvMCVPOlPa8b8ve2Nza3insFkt7+weHTvmoqZJMUmzQhCeyHRKFnAlsaKY5tlOJJA45tsKXu2m9NUKpWCKe9TjFICYDwSJGiTZUz3G6YcL7ahybJ88mvbDnVLyqNwt3HfgLUIFF1Htl66LbT2gWo9CUE6U6vpfqICdSM8pxUuxmClNCX8gAOwYKEqMK8pn0iXtmmL4bJdKk0O6M/TuRk1hN1ZnOmOihWq1NyX9rlAiKfOl6ngmm1YogHd0EORNpplHQuZ4o465O3Kldbp9JpJqPDSBUMvMllw6JJFQbU5eWv0WGXV6uXjMiUQT508P8tDHXX7VyHTQvq75X9R+vKrXbhc0FOIFTOAcfrqEG91CHBlAYwTt8wKf1Zf3YYNvzVttazBzDUtilXwt+tSU=
Figure 9. Generic solid body, with volume Ω and surface Γ, subjected to boundary displacements, ub , and boundary tractions, T b
generalisation of Newton’s second law, can be given in strong integral form as: Z 
ρ
Ω
∂2u dΩ ∂t2 {z }
Inertia forces
=
I
Z n · µ∇u + µ(∇u)T + λ tr(∇u)I dΓ + ρf b dΩ Γ Ω   {z } {z } Surface forces
(1)
Body forces
where small deformations are assumed i.e. no distinction is made between the initial and deformed configurations; ρ is the density, u is the unknown total displacement vector, n is the outwardpointing surface unit normal vector, ∇ is the del operator, λ is the first Lam´e parameter, µ is the second Lam´e parameter, synonymous with the shear modulus, I is the secondorder identity tensor, and f b is a body force acceleration. To avoid unnecessary complexity, the material properties (ρ, µ and λ) are assumed to be uniform and isotropic; however, this assumption is not required. It should be noted that the finite volume method directly discretises this strong integral form of the governing equation, without requiring weighting functions and/or the weak form of the equation. This point will be discussed further when comparing the “standard” finite volume method with the “standard” finite element method in Section 4. 3.2. “Standard” cellcentred finite volume method Discretisation of time and space Discretisation of the solution domain comprises time discretisation and space discretisation. The total specified simulation time is divided into a finite number of time increments, ∆t, and the discretised governing equations are solved in a timemarching manner. The solution spatial domain is divided into a finite number of contiguous convex polyhedral cells bounded by polygonal faces that do not overlap and fill the space completely. A typical control volume is shown in Figure 10, with the computational node P located at the cell centre/centroid, and the cell volume is ΩP ; Nf is the centroid of a neighbouring control volume, which shares face f with the current control volume; Γf is the area vector of face f , vector df joins P to Nf , and x is a positional vector. No distinction is made between different cell volume shapes, as all general convex polyhedra are discretised in the same general fashion. Discretisation of the mathematical model equations The governing conservation law (Equation 1) is applied to each polyhedral cell in the mesh. The discretisation of each of the three terms within the equation will now be discussed in turn. Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
N fi
z x
fi
AAACWnicbVDLSgMxFE3Ht/XRqhtxM1gE3ZQZEXRZdKHuFG0V2mG4k95pQ5PMmGTEMhS/xq1+j+DHmD4WtvVCyOHc17knSjnTxvO+C87C4tLyyuraenFjc2u7VN5p6CRTFOs04Yl6jkAjZxLrhhmOz6lCEBHHp6h3Ncw/vaLSLJGPpp9iIKAjWcwoGEuFpf1WlPC27gv75a1rEAIGYR6HbBCWKl7VG4U7D/wJqJBJ3IXlwkmrndBMoDSUg9ZN30tNkIMyjHIcrLcyjSnQHnSwaaEEgTrIRzcM3CPLtN04UfZJ447Yvx05CD2UaSsFmK6ezQ3Jf3MUJEU+tT3PJDN6RpCJL4KcyTQzKOlYT5xx1yTu0De3zRRSw/sWAFXMnuTSLiigxro7Nfwttuz0cP2SgUIZ5A+349XWXH/WynnQOK36XtW/P6vULic2r5IDckiOiU/OSY3ckDtSJ5S8kw/ySb4KP47jrDnFcalTmPTskqlw9n4BHYy4Dg==
19
AAACRnicbVDLTsJAFL3FF+ILdOmmkZjohrTGRJdEN7oxGOWRQEOmwy1MmE7rzNRIGj7Crf6Qv+BPuDNuHaALAU8yycm5rzPHjzlT2nE+rdzK6tr6Rn6zsLW9s7tXLO03VJRIinUa8Ui2fKKQM4F1zTTHViyRhD7Hpj+8ntSbzygVi8SjHsXohaQvWMAo0UZq3nXToMvG3WLZqThT2MvEzUgZMtS6Jeu004toEqLQlBOl2q4Tay8lUjPKcVzoJApjQoekj21DBQlReenU79g+NkrPDiJpntD2VP07kZJQqVHom86Q6IFarE3Ef2uUCIp87nqaCKbVgiEdXHopE3GiUdCZnyDhto7sSUZ2j0mkmo8MIVQy8yWbDogkVJsk55a/BEadX66eEiJReOnD7ey0CdddjHKZNM4qrlNx78/L1ass5jwcwhGcgAsXUIUbqEEdKAzhFd7g3fqwvqxv62fWmrOymQOYQw5+AQPjsqo=
dfi AAACVXicbVDLSsNAFJ3EWmt9tFV3boJF0E1JRNBl0Y3uKtoHtCFMJjft0MkkzkzEGvIvbvWHxI8RnD4WtvXCZQ7nvs4cP2FUKtv+NsyNwmZxq7Rd3tnd269UawcdGaeCQJvELBY9H0tglENbUcWglwjAkc+g649vp/XuCwhJY/6kJgm4ER5yGlKClaa86tHAj1kgJ5F+siD3stCjuVet2w17FtY6cBagjhbR8mrG+SCISRoBV4RhKfuOnSg3w0JRwiAvD1IJCSZjPIS+hhxHIN1sJj+3TjUTWGEsdHJlzdi/ExmO5FSh7oywGsnV2pT8t0YwJ8CWrmcpp0quCFLhtZtRnqQKOJnrCVNmqdiaWmYFVABRbKIBJoLqL1lkhAUmShu7tPw11OzycvmcYgHczR7v56e1uc6qleugc9Fw7IbzcFlv3ixsLqFjdILOkIOuUBPdoRZqI4Le0Dv6QJ/Gl/FjFszivNU0FjOHaCnMyi/nubcA
y
r
x AAACTXicbVDLTsJAFJ3iA8QX6NJNIzHRDWmNiS6JbnSHUR4JVDKd3sKE6bTOTA2k4T/c6g+59kPcGeMUurDgTW7m5NzXmeNGjEplWZ9GYW19Y7NY2ipv7+zu7VeqB20ZxoJAi4QsFF0XS2CUQ0tRxaAbCcCBy6Djjm/SeucFhKQhf1TTCJwADzn1KcFKU099N2SenAb6SSazQaVm1a15mKvAzkANZdEcVI2zvheSOACuCMNS9mwrUk6ChaKEwazcjyVEmIzxEHoachyAdJK57Jl5ohnP9EOhkytzzv6dSHAgU226M8BqJJdrKflvjWBOgOWuJzGnSi4JUv6Vk1AexQo4WejxY2aq0EytMj0qgCg21QATQfWXTDLCAhOlDc0tn/iazS+XzzEWwJ3k4W5xWptrL1u5Ctrndduq2/cXtcZ1ZnMJHaFjdIpsdIka6BY1UQsRJNArekPvxofxZXwbP4vWgpHNHKJcFIq/FEy2IA==
dfi
fi
AAACVXicbVDLSsNAFJ3EWmt9tFV3boJF0E1JRNBl0Y3uKtoHtCFMJjft0MkkzkzEGvIvbvWHxI8RnD4WtvXCZQ7nvs4cP2FUKtv+NsyNwmZxq7Rd3tnd269UawcdGaeCQJvELBY9H0tglENbUcWglwjAkc+g649vp/XuCwhJY/6kJgm4ER5yGlKClaa86tHAj1kgJ5F+siD3stCjuVet2w17FtY6cBagjhbR8mrG+SCISRoBV4RhKfuOnSg3w0JRwiAvD1IJCSZjPIS+hhxHIN1sJj+3TjUTWGEsdHJlzdi/ExmO5FSh7oywGsnV2pT8t0YwJ8CWrmcpp0quCFLhtZtRnqQKOJnrCVNmqdiaWmYFVABRbKIBJoLqL1lkhAUmShu7tPw11OzycvmcYgHczR7v56e1uc6qleugc9Fw7IbzcFlv3ixsLqFjdILOkIOuUBPdoRZqI4Le0Dv6QJ/Gl/FjFszivNU0FjOHaCnMyi/nubcA
AAACQnicbVDLTgIxFG19Ir5Al24aiYluyIwx0SXRje4wyCOBCemUO9DQ6Yxtx0gmfIJb/SF/wl9wZ9y6sMAsBDxJk5NzX6fHjwXXxnE+8Mrq2vrGZm4rv72zu7dfKB40dJQoBnUWiUi1fKpBcAl1w42AVqyAhr6Apj+8mdSbT6A0j+SDGcXghbQvecAZNVaqBV3eLZScsjMFWSZuRkooQ7VbxGedXsSSEKRhgmrddp3YeClVhjMB43wn0RBTNqR9aFsqaQjaS6dex+TEKj0SRMo+achU/TuR0lDrUejbzpCagV6sTcR/a4xKBmLueppIbvSCIRNceSmXcWJAspmfIBHERGSSD+lxBcyIkSWUKW6/RNiAKsqMTXFu+XNg1fnl+jGhCqSX1u5mp2247mKUy6RxXnadsnt/UapcZzHn0BE6RqfIRZeogm5RFdURQ330gl7RG37Hn/gLf89aV3A2c4jmgH9+AVoEsdw=
P
⌦P
Figure 10. General convex polyhedral control volume (adapted from [10, 64, 107])
Considering first the spatial discretisation of the inertia term in Equation 1: the term can be approximated by making an assumption about the variation of the displacement vector u over the cell; in the standard cellcentred approach, a linear variation is assumed; this linear variation can be expressed in terms of a truncated Taylor series expansion about the cell centre: (2)
u(x) = uP + (x − xP ) · (∇u)P
which says that the displacement u(x) at any point in the cell can be calculated using the cellcentre displacement uP and the gradient of displacement at the cellcentre (∇u)P . This approximation is secondorder order in space i.e. as the mesh spacing is reduced, the error in this approximation reduces at secondorder rate. It should, however, be noted that, in principle, any other distribution could be used; for example, the fourthorder cubic distribution proposed by Demirdˇzi´c [109]. Using the approximation in Equation 2, the temporal term can be calculated in terms of the cellcentred values using the midpoint rule as: Z
∂2u ρ 2 dΩ Ω ∂t
≈
ρ
∂2u ∂t2
ΩP
(3)
P
where the subscript P on the density has been dropped as a uniform density field is assumed i.e. ρP = ρ. The acceleration ∂ 2 u/∂t2 may then be discretised in time using any appropriate finite difference scheme; in the original cellcentred approach of Demirdˇzi´c et al. [2], the bounded firstorder Euler method was used:
∂2u ∂t2
[m−1]
P
≈
[m−2]
uP − 2uP + uP ∆t2
(4)
where m is the timestep counter; the superscript on the unknown current time value of displacement has been dropped for brevity. Here it is assumed that the timestep size ∆t is constant; however, the method can be generalised to changing timestep sizes. There are numerous other temporal discretisations that may be used; for example, the unbounded secondorder backward Euler scheme analysed by Jasak and Weller [27]. Submitted for review (0000)
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The final discretised temporal term is: Z
∂2u ρ 2 dΩ Ω ∂t
[m−1]
≈ ρ
[m−2]
uP − 2uP + uP 2 ∆t
!
ΩP
(5)
In a similar fashion, to approximate the body force term (second term on the righthand side of Equation 1), we assume that the body force f b varies over the cell according to Equation 2; consequently, the term can be approximated in terms of the cellcentre values using the midpoint rule as: Z (6) ρf b dΩ ≈ ρf bP ΩP Ω
Towards the discretisation of the surface force term, the closed surface integral is converted into a sum of surface integrals for each polygonal face: I
Γ
n · µ∇u + µ(∇u)T + λ tr(∇u)I dΓ
=
nF aces Z X f =1
Γi
n · µ∇u + µ(∇u)T + λ tr(∇u)I dΓ
(7)
where nF aces is the number of faces in the cell. To approximate the stress term [µ∇u + µ(∇u)T + λ tr(∇u)I] at each cell face, we will once again make use of the assumed variation of displacement in Equation 2; accordingly, the stress can be calculated in terms of the values at the face centre (centroid): nF aces Z X f =1
nF aces X f =1
Γi
n · µ∇u + µ(∇u)T + λ tr(∇u)I dΓ ≈
nf · µ(∇u)f + µ(∇u)Tf + λ tr [(∇u)f ] I Γf 
(8)
where subscript f indicates a quantity at a face centre, for example, (∇u)f is the gradient of displacement tensor at the centre of face f . As a consequence of the assumed variation (Equation 2), the gradient of displacement ∇u is constant within each cell and so the displacement gradient at a face, between two cells, is discontinuous; to resolve this, the standard cellcentred approach expresses the displacement gradient at a face (∇u)f as a weighted mean of the displacement gradients at the two adjacent cellcentres: (∇u)f
≈ γf (∇u)P + (1 − γf )(∇u)Nf
(9)
where subscript P indicates a quantity at the current cellcentre, subscript Nf indicates a quantity at the neighbour cellcentre adjacent to face f , and 0 < γf < 1 is the interpolation weight. Typically,
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THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
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the interpolation weight is calculated using an inverse distance method: γf =
xNf − xf  (xf − xP ) + (xNf − xf )
(10)
In addition to the standard cellcentred method for approximating the face centre displacement gradient given in Equation 9, a number of alternative methods have also been examined, including, for example, decomposing the gradient into normal and tangential components [98, 107, 132] or using temporary elements with isoparametric formulations [69]. At this point, it is worth noting the locally conservative nature of the discretisation: adjacent cells share integration points at the face centres, resulting in the force at cell faces being locally and hence globally conserved. To complete the discretisation of the surface force in terms of displacement u, an expression is required for the the cellcentred displacement gradient; a variety of methods have been proposed, where, for its accuracy on unstructured grids and ease of implementation, the leastsquares method is the most popular:
≈
(∇u)P
nF aces X
f =1
−1
wf2 df df
·
nF aces X f =1
2 wf df uNf − uP
(11)
As shown previously in Figure 10, vector df joins cellcentre P to the neighbour cellcentre Nf . The scalar weighting function can be taken as unity (wf = 1) [10] or as the inverse distance (wf = 1/df ) [27]. Although the discretisation of the surface force term in terms of displacement u is complete, the standard discretisation presented is known to suffer from socalled checkerboarding errors, where oscillations in the displacement field twice the period of the cell size go unnoticed, similar to zeroenergy modes in finite element discretisations. To overcome this issue, a thirdorder diffusion term (force) is added at each cell face: nF aces Z X f =1
nF aces X f =1
+
where:

nf · µ(∇u)f + µ(∇u)Tf + λ tr [(∇u)f ] I Γf 
uNf − uP + kf · (∇u)f − Γf · [Kf (∇u)f ] Kf ∆f  df  {z }
nF aces X f =1
Γ
n · µ∇u + µ(∇u)T + λ tr(∇u)I dΓ ≈
(12)
thirdorder diffusion term
∆f =
df Γf 2 df · Γf
and kf = Γf − ∆f
(13)
This thirdorder diffusion term corresponds to the difference between two ways of calculating the normal gradient of displacement at a face, resulting in an ability to ‘sense’ highfrequency
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oscillations in u; a similar method was first proposed by Rhie and Chow [426] in the context of cellcentred finite volume methods for incompressible fluid flow, and is commonly used in cellcentred finite volume fluid formulations. The Kf coefficient controls the magnitude of the smoothing effect and is taken as Kf = 2µ + λ in the standard cellcentred method. The thirdorder diffusion term also serves a purpose towards choice of implicit components within the segregated solution algorithm: this is discussed further below. Alternative forms of diffusion/smoothing terms have been also been proposed, for example, the fourthorder JamesonSchmidtTurkel [427] term employed in Godunovtype approaches [330, 331]. The final discretised form of the governing momentum equation is expressed as: [m−1]
ρ
[m−2]
uP − 2uP + uP 2 ∆t +
nF aces X f =1
!
ΩP =
nF aces X f =1
nf · µ(∇u)f + µ(∇u)Tf + λ tr [(∇u)f ] I Γf 
uN − uP (2µ + λ) ∆f  f + kf · (∇u)f − Γf · [(2µ + λ)(∇u)f ] df 
+ ρf bP ΩP (14)
where the face displacement gradients (∇u)f are calculated using Equations 9, 10 and 11, and the primitive unknown variables are the cellcentre displacement vectors uP . Boundary conditions are incorporated through appropriate modification of the surface force term discretisation at faces coinciding with the boundary of the solution domain; in the case of a displacement condition ub (Dirichlet boundary condition), the face displacement gradients are calculated at the face, while in the case of a traction condition T b (Neumann boundary condition), the specified traction directly replaces the surface stress expression. As the mathematical model is timedependent, the displacement field must be specified at t = 0, t = −∆t, and t = −2∆t.
Solution algorithm In order to solve the discretised governing equation (Equation 14) for the unknown displacement vector, a decision must be made regards the implicit or explicit treatment of terms. In this context, implicit indicates contribution to the matrix of a linear system and explicit indicates contribution to the source of the linear system. To this end, the standard cellcentre approach uses a segregated solution procedure, where the centraldifferencing component uNf −uP , and temporal terms are treated implicitly; all other terms in Equation 14, (2µ + λ)∆f  d f are calculated explicitly using the latest available displacement field. The purpose of the segregated approach is to temporarily decouple/segregate the three scalar components of the vector momentum equation so that they can be solved sequentially; outer fixedpoint/GaussSeidel/Picard iterations provide the necessary coupling, where the displacement gradient terms are explicitly updated each outer iteration using the latest available displacement field. Employing this implicitexplicit split, the discretised equation for each cell can be written in the form of an algebraic equation: aP uP −
nF aces X
aNf uNf
=
bP
(15)
f =1
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THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
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where aP
=
nF aces X ρΩP + aNf ∆t2
(16)
f =1
aNf bP
∆f  (2µ + λ) df  # " [m−1] [m−2] 2uP − uP ΩP = ρ ∆t2
(17)
=
+
nF aces X f =1
+
nF aces X f =1
nf · µ(∇u)f + µ(∇u)Tf + λ tr [(∇u)f ] I Γf 
(2µ + λ)(kf − Γf ) · (∇u)f + ρf bP ΩP
(18)
The algebraic equations (Equation 15) can be assembled for all M cells in the domain into the form: [K] [U ] = [F ]
(19)
where [K] is a M × M sparse matrix with diagonal coefficients aP and offdiagonal coefficients aNf , [U ] is a vector of the unknown cellcentre displacement vectors, and [F ] is the source vector containing contributions from bP . In finite element parlance, [K] is the global stiffness matrix and [F ] is the global force vector. The standard cellcentred discretisation ensures that matrix [K], has the following desirable properties [10]: • It is sparse with the number of nonzero elements in each row equal to the number of nearest
neighbours cells plus one;
• It is symmetric;
• It is diagonally dominant (aP  ≥
PnF aces f =1
aNf ), which makes the linear system (Equation
19) solvable by a number of iterative methods, which retain the sparsity of matrix [K]: this results in significantly lower memory requirements than equivalent direct linear solvers, for example, see [107]. The most common iterative method used is the conjugate gradient method with incomplete Cholesky preconditioning [27, 428].
The linear system (Equation 19) need not be solved to a tight tolerance as coefficients and source terms are approximated from the previous outer iteration; instead, a reduction in the residuals of one order of magnitude is typically sufficient. Outer iterations are performed until the predefined solution tolerance has been achieved. Underrelaxation of the displacement field and/or the linear system may improve convergence, depending on the boundary conditions and mesh. Acceleration of the approach can be achieved through geometric multigrid procedures [11, 12, 15], blockcoupled algorithms [79, 80, 108, 128], Aitken acceleration [98, 104, 128], and/or parallelisation on distributed memory clusters [27, 94]. A favourable characteristic of the solution procedure is the straightforward extension to nonlinearity: nonlinear terms (material, geometric or boundary conditions) are resolved on the fly; after each outer iteration the coefficients and the source terms are updated and the procedure continues as in the linear case, for example, compare the linear elasticity Submitted for review (0000)
´ ˇ C CARDIFF & DEMIRDZI
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approach of Jasak and Weller [27] with the finite strain elastoplastic frictional contact procedure of Cardiff et al. [108]. In addition to the displacementbased approach described above, alternative solution algorithms, where pressure and displacement are the primary variables, have been proposed by Bijelonja et al. [48, 53, 117] and Fowler and Yee [44]; the benefit of such approaches is their ability to deal with incompressible and quasiincompressible solids in a straightforward manner. The standard cellcentred finite volume discretisation and solution procedure has been presented above; in the subsequent sections, the other variants of the finite volume method for solid mechanics will be discussed in turn, where they will be compared with the standard cellcentred approach in terms of: discretisation of space and time; discretisation of the equations of the mathematical model; and solution procedure. 3.3. Staggeredgrid approaches The staggeredgrid approach closely resembles the cellcentred approach, with the primary distinction being the grid arrangement. In the cellcentred method the unknown displacements are stored at the cellcentres, while in the the staggeredgrid approach, the displacements (or velocities) are stored at the facecentres. As described in Hattel and Hansen [146] for a structured 2D quadrilateral grid (Figure 11), the x component of displacement is stored at the cell faces with normals in the x direction, whereas the y component of displacement is stored at the cell faces with normals in the y direction. Extension to 3D structured grids is straightforward; however, it is unclear how the approach might be extended to general unstructured meshes: for this reason, staggered grid approaches are in decline. In the solution of the NavierStokes equations, the principal motivation for the staggered grid was to eliminate decoupling between the components of velocity and pressure that caused numerical oscillations in the solution; in contrast, as discussed previously, the cellcentred approach eliminates this decoupling by adding a thirdorder diffusion term to the surface force term, first proposed for CFD by Rhie and Chow [426]. Discretisation of time in the staggeredgrid approaches follows the standard timemarching approach. Solving
equilibrium
equations
in rerms of displacement:
J. H. Harrel and P. N. Hansen
As with the fluid dynamics case, the three equi equations introduces three staggered control v The u control volume corresponding to t equilibrium equation is shown in Figure 5. It is imagine the u and w control volumes.
3. Comparing thermoelasticity
to heat condu
Since this work aims at adopting the volumebased numerical solution of the heat con equation, in the solution of the thermoelastic eq it seems appropriate to outline similarities and dif between the two kinds of equations. Due to the Figure 11. Staggeredgrid 2D orthogonal grid, is stored Figurestructured 4. Staggered locations for where u and the V, x+ component = u, t = v, of displacement volume solution of the heat conduction equation l = the normal = shear at the cellfaces designed with → stresses, symbol,x and thestresses. y component of displacement is stored based at onthe thecellfaces use of thermal resistances, it is vital similar resistances in the case thermoelasticity, designed with the ↑ symbol. The blue dashed box represents a control volume for the xmomentum equation, adopting the control and the orange dashed boxdisplacement represents a incontrol volume for thedisplacement ymomentumin equation. Figure adapted from volume formulation. As i the xdirection, v the the seen, this resistance formulation is not similar and Hansen [146] ydirection, and Hattel w the displacement in the zdirection. A wellknown, classical resistance definition i corresponding grid pattern to the one of Figure 1 is shown mechanics involving springs. in Figure 4.
When discretising the Note governing the tostaggeredgrid performs the that the momentum normal stressesequation, are attached the middle of the central volumes, whereas the shear stresses ID heat conduction discretisation over different sets of control volumes for each component direction; the x components
are attached to the corners. This is due to the definition The basic law in heat conduction is Fourier’s of the normal stresses and the shear stresses in an elastic Submitted for review (0000) model. The normal stresses are given by firstorder q= derivatives of the displacements differentiated with ax respect to their “corresponding” space parameter, i.e., u This law is rewritten in a form more suita corresponds to x, v to y, and w to z, as in gXX: numerical treatment, by replacing the derivative difference quotient,
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THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
25
of displacement are located at the centre of the control volumes used for the xmomentum integration (shown as blue dashed boxes in Figure 11) and y components of displacement are located at the centre of the control volumes used for the ymomentum integration (shown as orange dashed boxes in Figure 11). As a result, the control volumes used in the staggeredgrid approach overlap, whereas they do not in a cellcentred approach. When calculating local gradients of displacement, centraldifferencing can be conveniently used; however, implementation of the boundary conditions is, in the words of Hattel and Hansen [146], not at all straightforward, and a special procedure is required. The discretised governing equations are then assembled for all control volumes to produce a system a linear equations, which are solved in a segregated manner, similar to that of the standard cellcentred approach. Given the structured nature of the grid and resulting linear system matrix, the efficient Tridiagonal Matrix Algorithm direct solver can be used to solve the inner linear system, where outer fixedpoint/GaussSeidel/Picard iterations provide the necessary coupling between the momentum components. In addition to a purely displacementbased approach, pressure can be included as an additional unknown, in lieu of calculating the pressure directly from the displacement field. In the displacementpressure (or velocitypressure) approach, the unknown pressure variables are stored at the cellcentres of the original grid, similar to the staggeredgrid approaches used to solve the NavierStokes equations. As noted previous, the benefit of this hybrid displacementpressure (or velocitypressure) approach is that incompressibility can be easily handled, and similar hybrid displacementpressure approaches have been implemented in the cellcentred context by Bijelona et al. [48, 53, 117]. Todate, the staggeredgrid finite volume method has been shown suitable for solving complex multiphysics problems; however, when compared with the other approaches, including the cellcentred variant, a significant limitation of the staggeredgrid formulation is the requirement of a structured Cartesian grid; as a consequence, only a small subset of general problems of stress analysis may be tractable.
3.4. Vertexcentred approaches The vertexcentred finite volume method comes in a number of forms. Here, the vertexcentred approaches with nonoverlapping control volumes are considered, for example, as proposed originally by Fryer et al. [163]. The primary unknown, displacement, is stored at the mesh vertices/points/nodes, while material properties are stored within each cell. Rather than integrating the governing equation over cells in the primary mesh, as in the cellcentred approach, the vertexcentred approach integrates the governing equation over subvolumes/subcells, which are constructed around each mesh vertex by partitioning the surrounding primary mesh cells (Figure 5). Cells can take the shape of quadrilaterals and triangles in 2D and hexahedra, tetrahedra, pyramids and wedges in 3D; unlike the cellcentred approach, general polyhedra are not directly accommodated. In addition to this form of the vertexcentred approach, a variant exists where the primary mesh cells around a vertex are used to perform the integration; this produces overlapping regions of integration i.e. neighbouring vertices share part of their integrated volume, for example, as discussed by O˜nate et al. [170]. As regards discretisation of time, the vertexcentred variant follows the standard timemarching approach. Submitted for review (0000)
26
´ ˇ C CARDIFF & DEMIRDZI
Similar to the cellcentred approach, the vertexcentred approach starts from the strong integral form of the governing momentum equation [163]; the momentum balance is applied to the boundaries of each subvolume, as opposed to the boundaries of the primary mesh cells as is the case in the cellcentred approach. As explored by, for example, Bailey and Cross [171] and Idelsohn and O˜nate [169], all finite volume methods have also be shown to be derivable from the weak form of the governing momentum equation; in that case, to recover a finite volume method, the weighting functions are taken as unity within the control volumes and zero elsewhere; thus, the weak form of the governing equation condenses back to the strong form of the equation; this point is discussed further in Section 4; however, it should be noted that all finite volume methods directly apply a momentum balance to each control volumes and hence do not require the use of a weak formulation during their derivation. Another distinction with the cellcentred approach is that the most vertexcentred approaches explicitly uses shape functions to describe the variation of the displacement field within each cell of the primary mesh, following the standard notation: u(x) =
nV X ertices
N i (x) U i
(20)
i=1
where nV ertices is the number of vertices (nodes) in the (primary mesh) cell of interest, N i (x) is the shape function associated with vertex i, and U i is the displacement associated with vertex i. Using these shape functions, the vertexcentred approach calculates the displacement gradient at the subvolume boundaries when applying the momentum balance to the control volume around each vertex. As is standard in the finite element method, the shape functions, which are defined in the reference domain and mapped to the physical domain, approximate the continuous displacement field as a piecewise distribution that is continuous between cells. When comparing with the cellcentred approach, the explicit use of shape functions is a clear distinction between the variants; however, although the cellcentred approach does not explicitly use shape functions, the assumed truncated Taylor series expansion about the cell centres fills this role; this point will be discussed further in Section 4. In addition, it is possible to develop a vertexcentred finite volume approach that does not directly use shape functions, as shown by, for example, Tsui et al. [239]; in that case, the vertexcentred approach more closely resembles the cellcentred variant. In earlier publications, for example, Fryer et al. [163], the vertexcentred approach follows a similar solution algorithm to the standard cellcentred approach, where the surface force term is partitioned into implicit and explicit components; in that case, the implicit component contributes to the matrix coefficient of a linear system and the explicit component contributes to the source. The linear system is solved using a conjugate gradient method, where Jacobi/diagonal preconditioning, as opposed to incomplete Cholesky preconditioning, is used. Outer fixedpoint/GaussSeidel/Picard iterations are performed until the system has converged. In later publications, for example, Bailey and Cross [171], a coupled rather than segregated solution procedure is used, where the entire surface force term is discretised implicitly; similar coupled solution algorithms were later proposed for the cellcentred variant by Das et al. [79] and Cardiff et al. [107].
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THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
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3.5. Approaches for periodic heterogenous microstructures As discussed in Section 2, there are a variety of related finite volume methods specifically designed for the analysis of heterogenous microstructures. The related methods include the higherorder theory for functionally graded material (HOTFGM) [256], the highfidelity generalised method of cells (HFGMC) [257, 258, 290, 307], and the finite volume direct averaging micromechanics (FVDAM) theory [272, 273]. A brief summary of the methods is given here, and readers are referred to Aboudi et al. [256], Bansal and Pindera [273] and Cavalcante et al. [306] for further details. The HOTFGM and HFGMC approaches start by spatially discretising the solution domain into rectangular socalled generic cells, which are further split into a second discretisation level containing four rectangular subcells (Figure 12); for brevity and clarity, the description here has been limited to two dimensions; however, the approaches have been extended to three dimensions, as described in Aboudi et al. [256]. As a consequence of the assumed orthogonal Cartesian mesh, Author's personal copy curved interfaces between material phases are approximated in a castellated staircase manner, as shown in Figure 12; this limitation was later removed by the FVDAM approach with extension to unstructured quadrilateral meshes. By considering the unit cell of a periodic material, the M.A.A. Cavalcante et al. / Composites: Part B 43 (2012) 2521–2543
2525
Fig. 4. (a) Unit cell of a periodic material with microstructure discretized into rectangular building blocks. (b) Twolevel discretization employed by HFGMC into (q, r) generic cells further subdivided into four (b, c) subcells. (c) Simplified singlelevel discretization employed in the FVDAM theory into standalone (b, c) subvolumes.
Figure 12. (a) Unit cell of a periodic material with microstructure discretised into rectangular building blocks. (b) Twolevel discretisation employed by HFGMC into (q, r) generic cells further subdivided into four (b, c) subcells. (c) Singlelevel discretisation employed in the FVDAM theory into standalone (b, c) introduced as intermediate variables in the course of generating continuity of displacement and tractions between individual subsubvolumes. Figure taken from Cavalcante et al. [306]
the final system of 60NqNr algebraic equations in the unknown cells within generic cells and between generic cells in a surfaceðb;cÞ average sense, following the original idea proposed by Achenbach coefficients W iðmnÞ of the form: [5]. For periodic materials with continuous phases along the y1 KU fþ g ¯ + u0 , where ð13Þ displacement fieldfield canisbe decomposed into average and¼ fluctuating components, u = u direction, the displacement decomposed into average and fluctuating components The displacement vector U average contains the unknown displacement ¯= ¯x. the average displacement is determined from the specified macroscopic strains, u coefficients, ðb;cÞ ðq;rÞ 0ðb;cÞ ðq;rÞ
¼ !eij xj þ each ui ðysubcell, i ¼the 1; 2; 3 ð10Þ 2 ; y3 Þj i Within fluctuating displacement ish then assumed to vary quadratically as a ð1;1Þ ð2;2Þ cÞ and Uðb; U ¼ U11 ; . . . ; UN N qr using global and local coordinates, (x1, x2, x3) and 2, y3),y function of the local coordinates, y¯2(yand ¯respec3: ðb;cÞ ui
j
q
tively, with the latter associated with the unit cell following the 0thorder homogenization theory framework. The fluctuating com0ðb;cÞ 1 ponents ui j0ðq;rÞ induced by the heterogeneous microstructure are u (¯ y , y¯3 ) = Wexpansion ¯2 W y¯3 W 01 + 00 + y 10 +coordinates !2ðbÞ , approximated by2a quadratic in local (y 2 !3ðcÞ ) centered in each subcell as follows: y
¼
"
j
ð14Þ
2 and of 1 functions l2the subcell dimensions and 2 thehelements of K are 2 3¯ ymechanical W 20 + 3¯ y3 − contained W 02 within(21) these sub2 − 4 properties of2the materials 4 cells resident in the generic cells comprising the multiphase periodic composite. The mechanical force f contains information on
!e, and the applied average strains W vector where h and l are the width and height respectively of the subcell; Winelastic 00 ,the 10 , Wforce 01 , W 20 ,g con# ! ! tains integrals of inelastic strain distributions within individual lc h 1 1 c c c c c c c ! W ! are þ y þ y W unknown þ þ displacement W W 3y! % vector 3y! % and W 02 coefficients, eachintegrals with three components; theelements W 00 of U, subcells. These depend implicitly on the 4 4 2 2
0ðb;cÞ ðq;rÞ
ui
r
¼ ½W ið00Þ ; W ið10Þ ; W ið01Þ ; W ið20Þ ; W ið02Þ 'qr
ðb; Þ
W ið00Þ
ðbÞ 2
ðb; Þ ið10Þ
ð Þ 3
ðb; Þ ið01Þ
ðbÞ2 2
2 b
2
ðb; Þ ið20Þ
ð Þ2 3
ðb; Þ ið02Þ
ðq;rÞ
ð11Þ For structural analyses, the average displacement components represented by !eij xj are set equal to zero. The introduction of the twolevel domain discretization reðb;cÞ quires that the 60 unknown coefficients W iðmnÞ jðq;rÞ ði ¼ 1; 2; 3Þ in each generic cell be determined by satisfying the 0th, 1st and
requiring an incremental solution procedure of Eq. (13) at each point along the loading path. The elements of K, f and g have not Submitted for review (0000) been provided in closed form due to the complicated algebraic structure of these equations. Solution of this system of equations yields the localization relations given by Eq. (1) used in construction of the homogenized constitutive equations, Eq. (3). As previously mentioned, spatially uniform temperature change is readily accom
´ ˇ C CARDIFF & DEMIRDZI
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component corresponds to the unknown displacement at the centre of the subcell, while the remaining coefficients correspond to higherorder displacement contributions within the subcell. Accordingly, there are 5 × 3 = 15 unknown displacement coefficients within each subcell and hence 4 × 15 = 60 within each generic cell; in three dimensions, there are 168 unknown quantities. For brevity here, the (γ) and (β) superscripts indicating the subcell have been dropped i.e. (β) (γ) y¯2 = y¯2 , y¯3 = y¯3 , etc. It is also worth pointing out that although the approach has been developed for periodic microstructures, the method can also be used for general structural stress analysis by ¯ to be zero. assuming the average displacement u To determine the unknown displacement coefficients, the 0th , 1st and 2nd moments of momentum conservation are applied to each subcell, in addition to the enforcement of traction and displacement continuity between subcells and generic cells, and inclusion of boundary conditions. A characteristic of the method, which is not possessed by the other finite volume variants, is the enforcement of these socalled moments of the governing equation. To achieve this, the governing equation (Equation 1), where temporal and body force terms have been neglected, is written in terms of a socalled stress moment S : I n · S dΓ = 0 (22) Γ
with the stress moment is defined as: S=
1 hl
Z
h 2
−h 2
Z
l 2
− 2l
[σ y¯2m y¯3n ] dy¯2 dy¯3
(23)
The exponents m and n indicate the order of the equation; for example, when m = n = 0, the relation reduces to conservation of force; when m = 1 and n = 1 the relation represents conservation of angular momentum; while for m > 1 and n > 1 the relation represents conservation of higher stress moments. Note: m is not related to the timestep counter in Equation 4. In this way, it is possible to assemble a system of 60N algebraic equations of the standard form, [K][U ] = [F ], where N is the number of generic cells in the solution domain, [U ] is a vector of unknown displacement coefficients W , the global stiffness matrix [K] is a function of the subcell dimensions and mechanical properties, and the global force vector [F ] contains contributions from boundary condition and nonlinear material stresses. The linear system is inverted to give the displacements distributions within the subcells. The HOTFGM and HFGMC approaches described above provide the basis for the subsequent FVDAM approach; the FVDAM approach differs from the HOTFGM and HFGMC methods in a number of ways: (a) the twolevel spatial domain decomposition (generic cells and subcells) of the HOTFGM/HFGMC methods are replaced by onelevel of discretisation/cells; (b) the displacement coefficients within each cell W are expressed in terms of surfaceaveraged displacements i.e. displacement averaged at each cell surface; (c) higher order moments of the equilibrium equation are not used; (d) in the parametric form of the FVDAM, the use of parametric mapping with a parent/reference cell allows the use of an unstructured mesh, instead of the orthogonal Cartesian mesh of the HOTFGM/HFGMC approaches (see Figure 7); Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
29
(e) in the assembled system of algebraic equations [K][U ] = [F ], the solution vector [U ] contains cell surfaceaveraged displacements, as opposed to subcell displacement coefficients. Further technicals details of the HOTFGM, HFGMC and FVDAM methods can be found in Aboudi et al. [256], Aboudi [257], Aboudi et al. [258], Bansal and Pindera [272, 273], HajAli and Aboudi [290], Cavalcante et al. [306], HajAli and Aboudi [307] and Cavalcante and Pindera [312].
3.6. Godunovtype approaches Godunovtype procedures are specialised approaches for analysis of problems including wave propagation, shocks and solution discontinuities. The majority of Godunovtype approaches are spatially discretised using cellcentred approaches, for example, [325–329, 332–334, 336–345]; consequently, this section will focus on such approaches; however, Godunovtype approaches have also used vertexcentred [330, 331], staggered grid [335, 337], and facecentred [346, 347] formulations. Todate, many of the developments have been limited to one and two dimensions, but in general the procedures can be extended to three dimensions. A distinguishing feature of Godunovtype methods is the acknowledgement and capturing of discontinuities at cell boundaries using Riemann solvers. An additional characteristic of Godunovtype approaches is the use of explicit time marching solution algorithms; in this way, there is no need to assemble and invert a linear system matrix. As a result, such explicit methods can perform calculations for one timestep in a small fraction of the time required by an implicit procedure; however, this advantage is offset by the standard CourantFriedrichsLewy timestep constraint [424]. Where the other variants of the finite volume method for solid mechanics were first applied to small strain problems, the first Godunovtype methods proposed for solid mechanics were finite strain formulations [316–318]. Consequently, many of the proposed procedures contain moving meshes and require explicit enforcement of the space/geometric conservation law, i.e. conservation of volume; in addition, to control curl errors/modes over longterm responses, techniques are used to enforce the curlfree deformation gradient. By assuming the displacement varies linearly over each cell by a truncated Taylor series expansion (Equation 2), as in the standard cellcentred variant, the Lagrangian form of the governing equation (Equation 1), neglecting the body force term, can be written as: I ∂v m = n · σ dΓ (24) ∂t P Γ where m is the mass of the cell and v is the cellcentred velocity, In the methods of, for example, Kluth and Despr´es [326], Maire et al. [328] and Sambasivan et al. [329], the cell is then divided into subcells about each vertex in the cell; this decomposition allows the surface force term in the governing equation to be rewritten as a summation of contributions coming from each subcell boundary. In this way, the governing equation can be expressed as: m
∂v ∂t
= P
nPX oints
Fi
(25)
i=1
Submitted for review (0000)
´ ˇ C CARDIFF & DEMIRDZI
30
R where nP oints indicates the number of points/vertices in the cell, and F i = Γ0 n · σ dΓ0 is the force associated with the subcell around point/vertex i; Γ0 = Γ ∩ Γsubcell is the region of the subcell boundary shared with the primary cell boundary. The subcell ‘nodal forces’ can be approximated as: F i = li ni · σ + M i · (v i − v), where li is a scalar related to the area of Γ0 , σ is the piecewise constant stress tensor within the cell, ni is the unit normal vector associated with vertex i within the cell, M i is a matrix with physical dimensions of a length times density times velocity, v i is the velocity of vertex i within the cell, and v is the cellcentre velocity. Discretising Equation 25 in time using a secondorder method leads to: m
v − v [m−1] = ∆t
nPX oints
[m− 21 ]
Fi
(26)
i=1
where v = v [m] is the unknown cellcentred velocity at timestep m, v [m−1] is the cellcentred [m− 1 ] velocity at the old timestep m − 1, and F i 2 are the ‘nodal forces’ evaluated at the midpoint between the current timestep and previous timestep. As the only unknown, v can be explicitly calculated. Further details of the discretisation and solution procedure can be found in, for example, Kluth and Despr´es [326] and Maire et al. [328]. The above procedure, based on socalled ‘nodal forces’, is just one Godunovtype approach; alternatively, Godunovtype approaches can be expressed in terms of face forces, for example, see Lee et al. [327] and Haider et al. [344]; in this case, the governing momentum equation for each cell reads: 1 ∂(ρo v) = ∂t ΩP
nF aces nGauss X X f =1
[tig Wig ] + ρo f b
(27)
g=1
where ρo is the original density, nF aces indicates the number of faces within the cell, nGauss indicates the number of Gauss integration points on face i, tig is the traction associated with face i and Gauss integration point g , and Wig is the area associated with the g th quadrature point on face i. The temporal derivative can be approximated using a number of approaches, such as secondorder twostage Total Variation Diminishing RungeKutta time integration scheme [327, 344]. To achieve secondorder spatial accuracy, a single quadrature point, located at the face centre, is sufficient; in which case the balance equation, when multiplied by the cell volume ΩP becomes: nF aces X ∂(ρo v) ΩP = ti Γi  + ρo f b ΩP ∂t
(28)
f =1
The Godunovapproach in this case calculates the traction ti at face i as a weighted mean of stresses within the current cell and the neighbouring cell adjacent to face i, where a Riemann solver is used to account for the discontinuity in the solution variable at the boundary between cells; in addition, a stabilising diffusion term is added to ti to damp numerical instabilities. When comparing the ‘nodal force’ and ‘face force’ Godunovtype approaches with that of the standard cellcentred approach, the approaches bear many similarities, in terms of the assumed distribution of displacement within each cell, as well as the use of a diffusion term to quell numerical oscillations. The greatest differences lie in the calculation of the tractions/forces through the use of Riemann solvers, and, in particular, the Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
31
use of ‘nodal forces’; in addition, Godunovtype approaches employ a fully explicit time integration technique, in contrast to the implicit technique of the standard cellcentred approach.
3.7. Matrixfree approaches Matrixfree formulations proceed in a manner similar to standard vertexcentred and cellcentred approaches; however, the distinguishing features are that: (a) the surface force term is treated in an entirely explicit manner; and (b) a local timestepping procedure is employed and/or a pseudotemporal term is added to the governing equation to circumnavigate the physical CourantFriedrichsLewy timestep restriction [424]. Although the matrixfree approach, which is a solution algorithm technique, can be applied to any grid arrangement, todate it has predominantly been developed using vertexcentred formulations, for example, [226, 227, 229, 348–354, 356–359, 365]; however, cellcentred formulations have also been proposed [362, 364]. The matrixfree algorithms begin by adding a pseudotime term to the governing momentum (Equation 1): Z
∂v ∂v ρ + ∂t ∂τ Ω
dΩ =
I
Γ
n · σ dΓ
(29)
where τ is the pseudotime. In addition, many approaches will also add a physical damping term ρc ∂v ∂t to the governing equation. After rearranging and spatially discretising using the finite volume method (vertexcentred or other), the equation can be expressed as: v [m,n] − v [m,n−1] ˜ [m,n−1] =R ∆τ
(30)
where m indicates the physical time step counter, as before, and n represents the pseudo timestep ˜ [m,n−1] encompasses the contributions from the physical temporal term, counter. The vector term R the surface force term and the body force term. Explicit pseudo time stepping is then performed using, for example, a fivestage RungeKutta time integration algorithm [226], to march in pseudo time until convergence is reached. At convergence, the lefthand side of Equation 30 becomes ˜ [m,n−1] becomes zero; as a consequence, the velocity vector v satisfies the original zero and so R governing equation at the new physical time. Although the solution algorithm employs an explicit integration procedure to progress in pseudo time, the method is implicit in physical time, in the sense that the velocity v satisfies the governing equation at the new physical time, as opposed to the old physical time. The pseudo timestep size ∆τ is limited by the CourantFriedrichsLewy timeconstraint, whereas the method is stable for any size of physical timestep ∆t. The primary difference between these matrixfree methods and standard cellcentred and vertexcentred approaches is in terms of the explicit solution algorithm; accordingly, it would be expected that matrixfree approaches may prove computationally more efficient for highrate dynamic problems; in contrast, the implicit methods would be expected to be more efficient for quasistatic (elliptic/quasielliptic) problems. It should be noted that unlike Godunovtype approaches, these matrixfree methods have not been developed for analysis of wave propagation problems; instead their application todate has been to dynamic and quasistatic stress analysis problems. Submitted for review (0000)
ûi
ui
u x1
x2
xi
Boundary node
32
CARDIFF &
Fig. 1. The distinction between ui and u^i
point yi preserves the local character of the Moving Least Squares approximation. The smoothness of the shape functions /i ÖxÜ is determined by that of the basis functions and of the weight functions. Let C k ÖXÜ be the space of kth continuously x differentiable functions. If wi ÖxÜ 2 C k ÖXÜ and pj ÖxÜ 2 C l ÖXÜ, i à 1; 2; . . . ; n; j à 1; 2; . . . ; m, then ´ /i ÖxÜ 2 Cr ÖXÜ with r à min Ök; lÜ. ˇ C DEMIRDZI The partial derivatives of /i ÖxÜ are obtained as (Belytschko et al., 1994)
119
X not be arranged in a special pattern such as on a straight /i;k à âpj;k ÖA 1 BÜji á pj ÖA 1 B;k á A;k1 BÜji ä Ö14Ü line. Here a sample point may be a nodal point under jà1 consideration or a quadrature point. Solving for aÖxÜ from Eq. (8) and substituting it into Eq. in which A;k1 à ÖA 1 Ü;k represents the derivative of the (1) gives a relation which may be written as the form of an inverse of A with respect to xk , which is given by interpolation function similar to that used in FEM, as 1 1 1 m
3.8. Meshless approaches
There are two main varieties of meshless finite volume method that have been appliedÖ15Üto solid A à A A A X ^ ^ à from u ÖxÜstemming à U ÖxÜ u / ÖxÜ^ u ; the u Öx Ügeneral ⌘ u 6à u ; Meshless where, Ö Ü Local denotes oÖ PetrovGalerkin Ü=ox . mechanics problems: (a) those (MLPG) In implementing the MLS approximation for the present local symmetric weak form, the basis functions and Ö11Ü from X framework of Atluri and Zhu [366]; and (b) thosex 2originating the work Ebrahimnejad et al. and weight functions should be chosen at Ærst. Both Gaussian where and spline weight functions with compact supports can be coworkers [383, 384, 386] Uthat employ nonoverlappingÖ12Ü control volumes. considered in the present work. The Gaussian weight ÖxÜ à p ÖxÜA ÖxÜBÖxÜ function corresponding to node i may be written as ( or The general framework proposed by Atluri and Zhu [366] was first formulated in a finite volume 0d r w ÖxÜ à Ö16Ü â ä X / ÖxÜ à p ÖxÜâA ÖxÜBÖxÜä : Ö13Ü 0 d r form for elastostatics by Atluri et al. [370]. In this approach, the domain is spatially discretised where d à jx x j is the distance from node x to point / ÖxÜ is usually called the shape function of the MLS ap x; c is a constant controlling the shape of the weight with a finite number of points within and along the boundary of the solution domain (Figure 13). proximation corresponding to nodal point y . From Eqs. function w and therefore the relative weights; and r is the size of the support for the weight function w and deter(10) and (13), it may be seen that / ÖxÜ à 0 when The governing strong integral of applications, the momentum is then to computation, subdomains mines the support of nodeapplied x . In the present w ÖxÜ àform 0. In practical w ÖxÜ is generallyequation T
;k
;k
n
h
i
i
h
i
i
i
i
;i
ià1
x
T
T
1
m
j
i
1
expâ Ödi =ci Ü2k ä expâ Öri =ci Ü2k ä 1 exp Öri =ci Ü2k
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
ji
jà1
i
i
Fig. 2. The shapes of weight functions wi ÖxÜ
Figure 13. Distribution of points within the interior and on the boundary of the solution domain, showing moving least squares shape functions (adapted from Atluri and Zhu [366])
around each point in the domain. As such, no elements/cells or background grid is required. It should be noted that although Atluri et al. [370] employs the weak integral form of the governing momentum equation during the method derivation, the strong integral form can be taken as the starting point, given the choice of Heaviside function as the test/weight function, for example, as shown in Moosavi and Khelil [375]. The strains are assumed to vary accordingly to moving least squares functions, where the least squares weights are calculated using radial basis functions. The governing momentum equation (Equation 1), neglecting the temporal term, then becomes: −
nPX oints Z i=1
Ls
Φi (x) · E · n dΓ · i −
nPX oints i=1
"Z
Γsu
Z
#
Φi (x) · E · n dΓ · i =
ΓsT
T b dΓ +
Z
ρf b dΩ
(31)
Ωs
where nP oints is the number of points in the local subdomain, Φi (x) are the moving least squares shape functions, E is the fourth order stiffness tensor, i is the strain associated with neighbour point i, Γsu is the part of the subdomain boundary shared with the global domain boundary that has a displacement condition specified, ΓsT is the part of the subdomain boundary shared with the global domain boundary that has a traction condition specified, and Ls is the remainder of the subdomain boundary. The displacement is assumed to vary locally according to the same shape functions as the strains, and the point strains are related to the point displacements by a linear transformation i = 12 [(∇u)i + (∇u)Ti ] = H i · ui . Combining this transformation with a quadrature method to evaluate the integrals, Equation 31 can be written as a linear system of algebraic equations in the Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
33
standard notation: [K][U ] = [F ]. Inversion of this system provides the displacements at the domain points. In contrast to the method of Atluri and Zhu [366], the meshless method of Ebrahimnejad et al. and coworkers [383, 384, 386] employs nonoverlapping control volumes, which are automatically generated about each point using Delaunay triangulation.
4. COMPARING OF THE FINITE VOLUME METHOD FOR COMPUTATIONAL SOLID MECHANICS WITH THE FINITE ELEMENT METHOD Within this section, the “standard” cellcentred finite volume method for solids mechanics is compared with the “standard” finite element method. Here, the “standard” finite volume method refers to the cellcentred approach, as described in Section 3.2; whereas, the “standard” finite element method refers to the continuous Galerkin finite element method, as described by, for example, Bathe [429], Zienkiewicz and Taylor [430], and Belytschko et al. [431]. Following the approach from the previous section, both finite volume and finite element methods for solid mechanics will be examined in terms of: (a) discretisation of space and time; (b) discretisation of the mathematical model equations; and (c) solution algorithm. 4.1. Discretisation of time and space Like the finite volume method, the finite element method follows the standard timemarching discretisation approach. Similarly, the solution domain space is divided into a finite number of convex cells (or elements) that do not overlap and fill the space completely. A distinguishing difference between the methods is that the standard finite volume method is formulated in terms of general polyhedral cells bounded by polygonal faces, with the discretisation independent of the cell shape. In contrast, the finite element discretisation is directly linked to the element shape through the employed shape functions; for example, tetrahedral elements require a different discretisation to hexahedral elements; as a consequence, the use of ‘hanging nodes’ (Fig. 14) is common within the finite volume method and not directly possible with the standard finite element method.
Figure 14. 2D quadrilateral mesh of a beam, where ‘hanging nodes’ are shown as black dots
4.2. Discretisation of the mathematical model equations In Section 3.1, the conservation of linear momentum (Equation 1) in strong integral form was taken as the starting point for the finite volume discretisation. The finite element method instead starts from the strong differential form of the governing equation, signifying momentum conservation at Submitted for review (0000)
´ ˇ C CARDIFF & DEMIRDZI
34 each infinitesimal point: ρ
∂2u ∂t2
=
∇ · µ∇u + µ(∇u)T + λ tr(∇u)I + ρf b
(32)
The strong differential form is then multiplied by an arbitrary weighting function ω and integrated over the material volume to give the conservation of momentum in weak form: 2 ∂ u T ω · ρ 2 − ∇ · µ∇u + µ(∇u) + λ tr(∇u)I − ρf b dΩ = 0 ∂t Ω
Z
(33)
By employing integration by parts combined with the Gauss divergence theorem, the weak form can be rewritten in the principle of virtual work form: Z
Ω
ω·ρ
∂2u dΩ + ∂t2
Z
µ∇u + µ(∇u)T + λ tr(∇u)I : ∇ω dΩ = Ω I Z ω · T Γ dΓ + ω · ρf b dΩ Γ
(34)
Ω
where T Γ are tractions applied on the domain boundary. The finite element method then assumes the displacement u within each mesh element to vary according to socalled shape functions: u(x) =
nN odes X i
N i (x) · U i
(35)
where nN odes is the number of nodes/vertices/points in the element, N i is the shape function in the element associated with vertex i, and U i is the as yet unknown displacement at vertex i. For direct comparison with the standard finite volume method, we will consider only linear shape functions; however, higher order shape functions are common. The Galerkin form of the finite element method is achieved by assuming the weighting functions ω in Equation 34 to be equal to the shape functions N .
4.3. Solution algorithm Combining the shape functions (Equation 35) with Equation 34 for each element in the mesh, and employing an appropriate quadrature rule to evaluate the integrals, a system of algebraic equations can be assembled of the form: [K] [U ] = [F ]
(36)
where [K] is the socalled global stiffness matrix, [U ] is a vector containing the unknown nodal/vertex/point displacements, and [F ] is the global force vector. The local stiffness matrix [K] is a sparse Nn × Nn matrix, where Nn is the number of nodes/vertices/points in the mesh, and the matrix coefficients are a function of local element geometry and material properties. The global force vector [F ] contains contributions from the boundary conditions, the body forces and the inertia term. Matrix [K] is then inverted to give the nodal displacements, where typically a direct linear Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
35
solver approach is used, for example, Gaussian elimination, LU decomposition or multifrontal methods. 4.4. Discussion The fact that the finite volume method starts from the strong integral form of the governing equation while the finite element method start from the weak form highlights the differences in philosophy between the two methods; where the finite volume method balances physical fluxes through the surfaces of a control volume, the finite element balances the governing equation is a volumeaveraged sense over the domain. A consequence of this, as discussed by Bailey and Cross [171], is that the standard Galerkin finite element method guarantees the global residual to be zero, but not necessarily the local one i.e. local conversation is not necessarily assured. On the other hand, the finite volume method directly discretises the surface stress integral, ensuring local and hence global conservation. This is a result of the inherent local conservation property of the finite volume method: as the integration points for adjacent control volumes are at the same location (at the faces), the fluxes/forces are exactly equal and opposite; consequently, no artificial creation or destruction of the conserved variable (momentum) occurs at the local level. The errors that do appear are therefore due to the approximation of the fluxes/forces at the cell faces, which stems from the assumed local distribution of the displacement field. Related to this point is the enforcement of traction boundary conditions (i.e. Neumann conditions): in the finite element method traction conditions are enforced in an average sense, and as the mesh is refined they approach strong enforcement. In contrast, the finite volume method strongly enforces traction conditions for any level of mesh refinement. As noted by a number of authors, for example, [169–171], by choosing unity weighting functions in the weak form of the governing equation (Equation 34), it is possible to recover the finite volume method. Spalding [160] alludes to this point by referring to finite volume approaches as unityweighting function methods and to finite element approaches as nonunity weighting function methods. It should be noted, however, that although the finite volume method can be derived from the weighted residual weak form, it is not required. As seen above, the finite element method employs shape functions to describe the variation of the displacement within each cell/element; the standard finite volume method, on the other hand, does not explicitly use shape functions; instead the displacement is assumed to vary according to a truncated Taylor series expansion about the cell centre. In this way, the discrete representation of the displacement field contains discontinuous jumps across cell boundaries, in contrast to the continuous representation employed by the standard Galerkin finite element method (see Figure 15). Given the discontinuous nature of the discrete displacement field, the finite volume method bears some resemblance to the discontinuous Galerkin finite element method (as opposed to the standard continuous Galerkin method) as discussed by, for example, Hesthaven and Warburton [432]. Given the close relationship between the finite volume and finite element methods, it is not surprising that both have been compared previously: some notable dissections include those from O˜nate, Zienkiewicz, Idelsohn [164, 166, 169, 170, 433–436], Lahrmann [437], Perr´e and Passard [173], Harrild and Henriquez [438], Zarrabi and Basu [23], Fang et al. [439], Yamamoto et al. [440], Jacquemet and Henriquez [441, 442], Vaz et al. [230], and Filippini et al. [243]. The general consensus is that both methods share the same data structure and general strategy to assemble the corresponding characteristic matrices, with the main conceptual difference being in the local Submitted for review (0000)
´ ˇ C CARDIFF & DEMIRDZI
36
(a) Discontinuous field representation within the finite (b) Continuous field representation within the finite volume method element method
Figure 15. A comparison between the representation of displacement field in the standard finite volume and standard finite element methods (adapted from Lee et al. [327])
integration domain and local integration method. One of the primary appeals of the finite volume method is, however, the ease with which it can be followed by its target audience, engineers; the ! finite volume method is simply based on balancing forces/fluxes within a volume, requiring no ! ! knowledge of advanced mathematical frameworks. As regards differences in accuracy, some authors ! ! method can give more accurate results than similar [230, 437, 438] have argued that the finite volume ! ! of authors have found both methods to produce finite element procedures; however, the majority ! ! [173, 439, 440, 442]. similar predictions with no significant differences ! ! As a specific example of a comparison between the standard cellcentred finite volume method ! ! and the standard finite element method, the outofplane bending of an elliptic plate NAFEMS ! ! by Cardiff et al. [107], is briefly discussed; the case benchmark case [443] (Figure 16), as analysed ! ! consists of a thick elliptic plate that is fully constrained around its outer face, and is subjected to a ! ! segregated and coupled cellcentred finite volume pressure on its upper face. In Cardiff et al. [107],
! ! ! ! ! ! ! !
!
! ! ! ! ! ! ! ! ! !
! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !
! ! ! ! ! !
! !
xx
yy
!
xx
! !
yy
abaqus hexahedral ! ! yz xz yz xz ! ! ! ! ! ! ! 1904 1904 ! ! (a) Cellcentred finite volume method (coupled (b) Finite element method (Abaqus) ! algorithm) ! ! ! Figure 16. Stress component distributions on a cutplane through an elliptic plate in bending. Adapted from ! Cardiff ! et al. [107] ! yy xx ! 1904
== S. ?? C, I.I. DEMIRD AND M. M. PERI C === M. PERI PERIC C DEMIRDZI ZI C, S. MUZAFERIJA MUZAFERIJA AND
!
xz
yz
Submitted for review (0000) 7.5 MPa
11.5 MPa
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS
37
methods were compared with the finite element method (linear elements with reduced integration) as implemented in commercial software Abaqus [444]. Both segregated and coupled finite volume approaches employed iterative linear solvers, while the finite element approach used a direct linear solver. When the execution time and memory requirements of all three methods were analysed Number of
Segregated
Coupled
Finite Element
cells
Time
Memory
Time
Memory
Time
72
0.5
6
0.03
7
4
24
576
1
8
0.15
13
5
31
4 608
6.5
20
1.6
51
6
107
36 864
102
80
11
300
34
1 197
294 912
1 474
500
242
2 200
1 375
17 900
Memory
Table III. Wallclock time (in s) and maximum memory usage (in MB) of Segregated and Coupled finite volume approaches compared with a finite element method for the elliptic plate in bending test case with varying mesh densities. Adapted from Cardiff et al. [107]
(shown in Table III), the following observations were made: • Both finite volume approaches used significantly less memory than the finite element
approach: this can be primarily attributed to the finite volume approaches using iterative linear solvers, as opposed to the direct linear solver used by the finite element approach; in addition, the implicitexplicit split of the segregated approaches reduced the memory requirement further; • The Coupled method was found to be faster than the segregated finite volume method and the finite element approach in all cases; while the Coupled method used more memory than the segregated finite volume approach, it used less memory than the finite element method in all cases. For the largest mesh, the Coupled finite volume method was found to be almost 6 times faster and use 8 times less memory than the finite element approach. As noted above, the reduced memory usage may be chiefly attributed to the use of an iterative linear solver, as opposed to the direct linear solver used by the finite element approach. • For the smaller meshes, the Segregated approach was faster than the finite element approach, whereas for the two larger meshes the finite element approach was 1.13 times faster;
5. APPLICATIONS OF THE FINITE VOLUME METHOD FOR COMPUTATIONAL SOLID MECHANICS Some of the main areas where finite volume methods have been applied to solid mechanics applications are summarised below, including example images. 5.1. Fluidsolid interaction Example cases are shown in Figure 17, and examples references include: [4, 5, 10, 12, 16, 20, 26, 36–38, 42, 46, 50, 54, 61, 64–66, 68, 70, 72, 74, 81, 89, 91, 98, 114, 120, 128, 132, 189, 204, 216, 217, 226, 226, 229, 239, 248, 320, 364, 445–455]; Submitted for review (0000)
fixed zero displacement at its root and symmetry planes for the remaining domain boundaries. At the start of the full DFSI simulation, the wing planform was subject only to the fixed displacement condition at its root. The full DFSI problem was run from the static flow only solution for 30 time steps, making a total simulation time of 0.155 s and the average run time on a Dec Alpha processor was in excess ´for FluidSolid Interaction ˇ C OpenFOAM Finite Volume Solver 38of oscillation. The maximum displacements areCARDIFF & DEMIRD for 460 h per cycle at the trailing edge of theZIwing 346
ˇ Tukovi´c, A. Karaˇc, P. Cardiff, Z. H. Jasak, A. Ivankovi´c
Y.Y. TSUI ET AL.
1030
D
W Wiedemair et al
2.230e02 0.016727 0.011151 0.0055755 0.000e+00
Downloaded by [University Of Pittsburgh] at 07:01 09 October 2013
U 0.000e+00
Fig.fluid 10. Fluid and wing mesh.for the (a) Wing and domain meshes prediction of flutter [216]
0.091
0.18
0.27
3.636e01
Fig. 28 Fluid velocity magnitude and solid displacement magnitude field for the channel flow over an (b) solidpeakdisplacement results are obtainedand for increased inlet velocity (0.3 m/s) and reelasticFluid thick platevelocity test case. Thesemagnitude 4 N/m2 ) in order to cause larger beam deformation. Calculated displacement duced magnitude Young’s modulus (10 distribution for channel flow over an elastic of point A in steady state is (0.01463, 0.005, −0.000447). thick plate [132]
0.00102 0.001015 0.01 Relative error
ux, m
0.00101 0.001005 0.001
ope
0.001 nd
2
ord
er sl
0.000995 0.00099 0
0.001
0.002
0.003
0.004
0.005
0.0001 0.001
t, s
(a) Calculated displacement of point A as a function of time step size
0.01 t, s
(b) Relative error as a function of time step size
Fig. 29 Calculating temporal accuracy for the channel flow over an elastic thick plate test case.
TRANSACTIONS OF FAMENA XLIx (2017)
25
(c) Pressure (top) velocity (d)velocity Pressure field(lower for vessel flowsection) over for a rigid Figureand 6. Pressure (upper(bottom) vessel section) and distribution case 1 square with a thin flexible column) and the respective configuration RBC (right column). The top row compares Figurewithout 14. Flow field for Re ¼[239] 290: (a) streamlines and (b) pressure distribution (color figure available online). distribution(leftaround a red blood cell plate attached conditions during the expansion phase, where the presence of the RBC has little impact on the [89] rﬃﬃﬃﬃﬃﬃﬃﬃﬃ pressure distribution and almost no impact on the flow field. The bottom row depicts the same comparison at the onset of the contraction phase when the flow is reversed. Here,xthe field is EI ¼flow 22:035 significantly altered due to the inertia of the RBC and the pressure distribution is less homogeneousmL4 than without RBC.
ð46Þ
Figure 17.AsFluidsolid interaction examples shown in Figure 13, the plate is deformed into a bowlike shape, reflecting the fact
that the second mode dominates the vibration. The flow field behind the square
The RBCs are free to move in axial direction. Their displacement amplitudes are 0.08 µm
5.2. Fracture adhesive joints (case 1) and and 0.18 µm (case 2). These small values are conceivable despite the relatively high velocities, as the very small time scale of oscillation renders the overall fluid flux small.
defined in examples section 2.10 varies between −2 and intraluminal pressureinlevel Pi as 18, ExampleThe cases are shown Figure and references include: [9, 12, 16, 21, 26, 32, 39, 8.3 kPa (case 1), and −17.5 and 17.2 kPa (case 2). The pressure distribution induced by 40, 45, 57, unaffected 63, 78, 82, 365, 448, 456–470]; the51, MB 52, is mostly by the124, presence of 445, the RBC, with449, the exception of the phases where the MB is close to its maximum and minimum radius, respectively, as can be seen by comparison to a case without RBC (figure 6).
5.3. Microstructure analysis Example cases are shown in Figure 19, and examples references include: [87, 252–315, 468, 472, 473]; 5.4. Metal forming and casting Example cases are shown in Figure 20, and examples references include: [41, 47, 59, 77, 105, 106, 108, 113, 113, 118, 119, 125, 148, 154, 156, 175, 202, 207, 232, 394, 395, 399, 401, 403, 474–477];
5.5. Biomechanics Example cases are shown in Figure 21, and examples references include: [4, 74, 89, 95, 115, 159, 212, 344, 438, 441, 452, 453, 478–486]. Submitted for review (0000)
39
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS Fig. 7. Effect of mesh density on the final crack paths for Case 1.
Fig. 8. Crack path predictions using a trigonal cell based mesh.
Fig. 11. Crack path and damage evolution for (a) 0.5 mm, (b) 0.16 mm, and (c) 0.1 mm notches. The sizes of the images in (b) and (c) are 25 by 20 mm, while that of (a) is 25 by 4 mm. Here, the colour of a given cohesive cell represents its status. Fully separated cells are shown in red, whilst those undergoing elastic unloading are shown in blue and yellow. Z
SPE Production & Operations Page 6 of 22 Dynamic crack path predictions in (a) Crack path predictions through a material interface [90] (b) Pressure (Pa) PMMA [57] 6 SPE179126MS unknown in order to determine a unique mesh independent and finally penetration into material 2 at two locations along the
interface. However, the variation in crack path between the two cases demonstrates the fact that the allowable crack propagation path is predetermined by the mesh. Therefore care is still needed when meshing complex geometries to ensure that the direction of crack propagation should be as independent of the underlying mesh as possible. A systematic mesh refinement would be required for complex geometries where the possible crack paths are completely
depth increases, as observed experimentally by Ivankovic and Hillmansen [20]. The 0.1 mm and 0.16 mm notches are predicted to reach a similar mean terminal velocity of about 820 m/s, whilst the 0.5 mm notch
solution.
3.2. Case 2: E1 = 210 GPa, E2 = 420 GPa In this case the Dundur’s parameters are nonzero and one can calculate, a = 1/3, and b = 2/15. The corresponding critical ratio for cohesive strengths is approximately 0.4. Therefore, one would expect to observe some level of crack deflection into the interface be
Gas velocity (m/s)
b)
Fo
Pressure (Pa)
rR
Z
Fig. 3 Impact of number of clusters per stage on fracture trajectory and width distribution. Left figure shows the resulting fracture trajectory for 3 clusters per stage and the right figure shows the resulting fracture trajectory for 5 clusters per stage.
(c) Crack path predictions during hydraulic
ev
(d) Predicting rapid crack propagation in pipes [471]
Impact of perforation parameters. As shown in Eq. (6) there are various parameters that control the fracturing [124] gaspressurised perforation pressure drop. At the high injection rates used in fracturing operations, perforation parameters can lead to a large perforation pressure drop. The Base Case perforation pressure drop coefficient chosen to be 1x109, is equivalent to a cluster with 12 active perforations, 1 cm perforation diameter, 0.9 damage factor, and 1500 kg/m3 slurry density. The comparison case perforation pressure drop coefficient was chosen to be 4x109, which can be obtained by reducing the active number of perforations to 6 or decreasing the perforation diameter to 0.707 cm. c) Fig. 4 shows the impact of perforation coefficient on the observed pressure trends. Fig. 4(a) shows the wellbore pressure to be higher for the case of larger perforation coefficient. Fig. 4(b) shows that the perforation pressure drop for the larger perforation coefficient case is approximately 4 times the perforation pressure drop observed for the smaller perforation coefficient case. Thus, the fracture pressure is much smaller for the second case. This suggests that the perforation variables can potentially be varied for the various clusters in a stage to promote multiple fracture growth.
ie
Figure 18. Fracture and adhesive joint examples
On
5.6. Screw compressors
Gas velocity (m/s)
w
Example cases are shown in Figure 22, and examples references include: [46, 56, 60, 65, 66, 97, Pressure (Pa) 487–493];
ly
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Gas velocity (m/s)
a)
Z
5.7. Geomechanics and poroelasticity
Figure 5: Predicted RCP along gas pressurised pipe at times: a) 3 ms, b) 6 ms, c) 9 ms.
Example cases are shown in Figure 23, and examples references include: [61, 100–102, 104, 124, 172, 192, 208, 215, 217, 219, 224]. Fig. 4 Impact of perforation coefficient on (a) wellbore pressure, and (b) perforation pressure drop.
6. SOFTWARES EMPLOYING THE FINITE VOLUME METHOD FOR SOLID MECHANICS A number of software have, or previously have, implemented versions of the finite volume method for solid mechanics; these software, in alphabetical order, include: • COMET / STARCD (commercial software) [8, 10, 12, 13, 15, 25, 41, 46–49, 53, 56, 60, 65,
66, 97, 105, 106, 117, 409, 487–492, 494]; • FOAM / OpenFOAM (opensource software) [18, 20, 26–28, 50, 54, 64, 68, 70, 74, 84, 89– 91, 94, 95, 98, 100, 104, 107, 108, 110, 114, 120, 124, 129, 132, 342, 344, 445–451, 454, 455, 464–470, 473, 482]; • GTEA (inhouse software) [236]; Submitted for review (0000)
M.A.A. Cavalcante, M.J. Pindera / International Journal of Plasticity 77 (2016) 90e117
40
!( ¼ ð29Þ
¼
1þm m r~ ' trð~rÞI E E
ð30Þ
~ is given in terms of the stress r e the effective stress tensor r lows
!
" 1 htrðrÞi ev ½Hdev ðrÞH & þ ' h'trðrÞi I 3 1 ' trðDÞ
ð31Þ
e initiation of damage in the present case is determined by ollowing condition
¼ ! ' jðtrðDÞÞ > 0 (
ð32Þ
e the damage equivalent strain !⁄ is defined in terms of the ipal values !A, A = 1, 2, 3, of the strain tensor in the form
813
2107
J. Aboudi / International Journal of Solids and Structures 48 (2011) 2102–2119
e hxi = x U(x), with U(x) being the Heaviside unit step function H is given by Eq. (18). From Eq. (29), the following expression ~ is obtained e strain tensor ! in terms of the effective stress r
G
´ ˇ C CARDIFF & DEMIRDZI Y. Bansal, M.J. Pindera / International Journal of Plasticity 22 (2006) 775–825
1þm tr½Hdev ðrÞHdev ðrÞ& 2E " # 1 ' 2m htrðrÞi2 þ þ h'trðrÞi2 6E 1 ' trðDÞ
r
101
obtained from these three meshes are very small, and the FVDAMgenerated homogenized response compares very well with the ﬁniteelement results. Comparison of s22(y2,y3 ) distributions in the elastic and elasticeplastic regions at the applied strains ε22 ¼ 0:1 and 1.0%, respectively, is given in Figs. 5 and 6 . In the elastic region, Fig. 5, the generalized FVDAM theory produces superior results relative to the 0thorder theory when both linear and quadratic mappings are employed. However, the generalized FVDAM stress ﬁeld based on the linear mapping exhibits disturbances at the ﬁber/matrix interface for insufﬁciently reﬁned meshes
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h!1 i2 þ h!2 i2 þ h!3 i2
ð33Þ
and the function j(trD) is given by
jðtrDÞ ¼ a tan
! $j %" trD 0 þ arctan aa( a
ð34Þ
where a0 , a are material constants and j0 is the damage threshold. The damage evolution is controlled by
" & ( '2 #'1 ( ! !_ D_ A ¼ a0 1 þ h! i2 ; a !(2 A
A ¼ 1; 2; 3
ð35Þ
(a) Stress predictions for a hexagonal array of cylindrical 8 8 a 9 inclusions 9 boron/aluminum composite [312] 2 3in ~ Expansion of Eq. (31) yields the following relation expressed in terms of the principal values r~ 1 ; r~ 2 ; r~ 3 of the effective stress and the principal values r 1, r 2, r 3 of the stress tensors
K 11 K 12 r1 > = K 22 r~ 2 ¼ 6 4 > : ~ > sym: r3 ; > <
K 13 < > 7 K 23 5 > :
K 33
r1 > = r2 > r3 ;
ð36Þ
Fig. 14. Eﬀective plastic strain distributions within diﬀerently rotated unit cells of a boron/aluminum unidirectional composite at the applied average transverse shear strain !e23 ¼ 0.50% obtained from the reformulated HFGMC analysis.
(b) Effective plastic strain distributions for a boron/aluminium unidirectional linearity, andcomposite a substantiallywith greateran increase in theaverage strain hardening rate which applied produces an overall strengthening eﬀect. Figs. 19 and 20 compare the eﬀective transverse shear strain [273] and hydrostatic stress distributions, respectively, of both fiber arrays at the ap
plied macroscopic transverse strain of 1%. These distributions explain the mechanism that leads to the increase in hardening rate of the clustered array. Specifically, clustering reduces the eﬀective stress in the matrix phase within the fiber cluster, Fig. 19, thereby eﬀectively increasing fiber volume fraction from 19% to around 29%. That is, the entire fiber cluster with a portion of the matrix Fig. 5. Comparison of the stress ﬁelds s22(y2,y3 ) obtained from numerical solutions based on different mesh discretizations for uniaxial loading by s22 s0 at the macroscopic strain ε22 ¼ 0:1%. contained therein responds in an elastic manner due to the eﬀective stress suppression in the sheltered matrix phase. As expected, this is accompanied by an increase in the hydrostatic stress magnitude in the matrix phase within the cluster. In addition, the fiber cluster responds like a square fiber, enhancing hydrostatic
(c) Transverse stress distribution (d) Microstructural analysis of advanced ceramics [469] in the repeating unit cell of Figure a 1: Initial 3D microstructure with 2D cuts and resulting microstructure carbon/aluminium composite loaded by a longitudinal uniaxial stress [299] the initial material properties used in the study.
Figure 19. Microstructure analysis examples
Table 1: Material properties for advanced ceramic used in this study
Young’s modulus (GP a) • MulPhys (inhouse software) [225]; Grain 800950 • NASIR (inhouse software) [348, 349, 351–354, 356, 357, 365];
Surface plots of the internal spatial transverse stress distributions r 22 in the RUC, generated by applying a longitudinal uniaxial stress loading of the carbon/aluminum site at !11 ¼ 0:02 at room temperature. (a) r 22 distribution generated by the anisotropic damage law, (b) r 22 distribution generated by the isotropic damage law, (c) r 22 ution generated in the absence of damage.
Density (kg/m3 )
⌫
4000
0.1
• PHOENICS (commercial 158, 160]; Secondsoftware) Phase [133–135, 137, 200 144, 149, 151, 152, 157, 8500
0.33
• PHYSICA (inhouse software) [181, 186, 187, 211, 495]; • TRANSPORE (inhouse software) [173].
To examine the role of the second phase material on the overall strength,
a number of microstructures with a second phase ranging from 3%  8% 7. CONCLUSIONS AND CHALLENGES area fraction were tested. A recently developed arbitrary crack solver by [1] wasapproaches implemented, andmechanics a linear traction separation law was specifiedinfor the The various finite volume to solid can be seen to share many similarities, terms of constructing control volumes, localstrength, solution variable approximating cohesive zone.assuming A cohesive 1000 MPa was chosen based on max = distributions, tractions/forces, performing momentum balances, and assembling/solving systems of equations. previous experimental work on the material under investigation ([18]). The The primary differences relate to: fracture energy used in this study was G = 100 J/m2 . (a) How the integration control volumes are constructed: It was found that the failure loads for all the microstructures were comWhile cellcentred approaches directly use the primary mesh cells, vertexcentred approaches parable, failing between 60  70% of the maximum max value. The data was Submitted for review (0000)
subsequently fitted to a Weibull distribution for further analysis.The fitted strength distribution was found to have a Weibull modulus of 13.73 (Fig. 2).
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS CALCULATION OF CASTING PROCESSES—PART II: APPLICATION
Fig. 18. Casting on an existing tunnel element. Temperatures after 24 h. 325
A.J. Williams et al. / Computer Methods in Applied Mechanics and Engineering 199 (2010) 2123–2134
2131
41
Downloaded by [FU Berlin] at 02:28 25 June 2015
Fig. 19. Casting on an existing tunnel element. Crack criterion after 24 h.
Figure 10.(a) Shape of the and cast and effective stresses “crashing” the for part TE model (left) and(b) for Shape effective stressbefore predictions in mold a cast [106] TEVP model (right). 1796 8. Conclusion
Casting of a tunnel section showing the crack criterion [155] P. CARDIFF distribution ET AL.
Fig. 11. Mesh for extended air region.
the major stress component for the point at the edge. After the contact forcemodel is for the thermomechanical conditions during hydration of earlyage conA numerical presented. Both 2D (plane stress and strain) and 3D analyses have been carried out. The removed, one can see the jump in the value of the effective stress (left)crete andis the change of sign of the normal stress (right). This peak in the effective stress is understandable, since the solid is treated according to the TEVP formulation. The change in sign of the normal stress is also expected. Namely, when the solid is in the mold, the compressive stress is coming from the mold. After this force is removed, the forces at the contact places have not been of such intensity as to lead the body toward the stressfree condition. The residual deformation and the residual stress are locked within the body. Figures 10 and 11 show the cast shape and the effective stresses after the cast is cooled, before and after “crashing” the mold, respectively. The black contour at the corner of the cast where the concentration of stresses is highest (Figure 10, right) marks the position where the effective stress is equal to the yield stress, and the
Figure 16. Flat rolling of wire: predicted deformed geometry, showing equivalent plastic strain and
Fig. 12. Temperatures in workpiece — extended air region. (c) Temperature distribution during extrusion [232] (d) Predicted geometry, plasticof strain and hydrostatic pressure distributions. (a) Equivalent equivalent plastic strain: comparison vertexwise (left) and celldistributions. Hydrostatic pressure: comparison of vertexwise (left) and cellwise (right) modelled. However an extension towise this (right) approach would be to(b) take hydrostatic pressuredistributions. distributions in flat wire the mesh associated with the deformed die and reextrude the rolling workpiece, again only solving for ﬂow and heat transfer, through [108] the
upstream of the die exit is ﬁxed in space. In this ﬁgure the deformation of the die is magniﬁed by a factor of 50, and it can be seen that the maximum deformation occurs in the region of the die bearing. To demonstrate how this deformation occurs over time the resultant displacement of a node in the region of the die bearing has been tracked. Fig. 16 shows the location of the node, node A, on the die, and the displacement is plotted in Fig. 17. From this graph it can be seen that once the leading edge of the extrudate has exited the die, after 0.35 s, the displacement of node A remains fairly constant with an average value of 3.34 × 10−4 m. This indicates that, as expected, the structural behaviour of the die reaches a steadystate solution once the startup phase of the extrusion has been completed.
Figure 20. Metal forming and casting examples
after “crashing” the mold for TE model (left) build subvolumes aboutstresses the primary mesh vertices, and staggeredgrids create subvolumes about the primary mesh faces. On the other hand, HFGMC/HOTGFM methods use the Figure 17. Flat rolling of wire: error reduction in predicted force as the mesh is refined. primary mesh cells in combination with an additional layer of subcells, and meshless methods contact, the shear traction vectors are aligned with the roller surface velocity direction and have use regions of neighbouring points; the greatest magnitude. Near the exit region of the contact, the shear traction vectors flip direction; consequently, there is a region of noslip located near the contact exit. This noslip point can be theoretically explained by considering that the downstream wire velocity is greater than the roller (b) Limitations on the cell shapes in the primary mesh: surface velocity, whereas the upstream wire velocity is less than roller surface velocity; hence, at some point in the contact region, the wire surface velocity and roller surface velocity must be equal, Meshless methods potentially offer resulting the ingreatest freedom, as no cells are required, a noslip point. only points; cellcentred approaches allow convex polyhedra; vertexcentred and 4.6. Impact of cylinder against a rigid wall The final test consists of a cylindrical bar of copper impacting wall, where methods the initial radius HFGMC/HOTGFM/FVDAM methods rbear similarities to standard finitea rigid element D 3:2 mm, the initial length l D 32:4 mm and the initial velocity v D 227 m/s. The case in their use of shape functions, and hence these approaches are limited to “standard” cell Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2017; 109:1777–1803 DOI: 10.1002/nme shapes; (c) The location where the solution variables are stored: Cellcentred approaches store the displacement solution variable at the centres/centroids of the primary mesh cells; the vertexcentred approaches actually also use storage locations at the centre of the integration control volumes, which corresponds to the primary mesh vertices. Similarly, staggeredgrids are constructed such that the storage locations are at the centres of the employed control volumes, coinciding with the primary mesh faces, while meshless methods store their solution variables at discrete points centred about integration subdomains;
4.2.1. DiscussionFigure 11. Shape of the cast and effective (residual) It is clear that using the ﬂuid–structure interaction approach to and for TEVP model (right). model this problem is compute intensive, with a simulation on a single processor taking nearly 8 h to complete. Since the deformation of the die reaches a steadystate solution after 0.3 s it might be suggested that it is not necessary to carry out a ﬂuid–structure interaction approach to model this problem. One alternative would be to solve the ﬂow and heat transfer equations up to t = 0.35 s, and then to solve the structural mechanics equations at the end of the simulation to obtain the stress state and deformation of the die. Unfortunately this approach alone would not allow the impact of the die deformation on the resulting shape of the workpiece to be
Fig. 13. Dimensions of die.
0
0
Submitted for review (0000)
three distinct contact regions are discernible, occurring in anterior superior, posterior superior, and superior regions of the acetabulum. The maximum predicted contact pressure is 26 MPa occurring in the most superior contact region. The predicted contact ´ ˇ C & DEMIRD and has been calculated by summing the areaCARDIFF is 3:96 ! 10"4 m2 ZI
42 J. Haider et al. / Comput. Methods Appl. Mech. Engrg. 340 (2018) 684–727
723
(c)
(d)
gure 6.14: Mesh generation of an idealised carotid artery; (a) geometry centerline
h, (b) volume split into sections and meshed, and (c) generated fluid and (d) solid
mains of the idealised carotid artery.
ure 6.15 shows the computational domains for the idealised and patient specific
otid arterial geometries, showing the fluid and solid domains. Since the geome
s are assumed symmetrical about about the XY plane, only one half is modelled XY plane specified as a symmetry plane. (a) Pressure distribution in a loaded stentlike structure [344]
(b) Von Mises stress distribution in the natural hip joint during stance [95]
Teran et al. / FVM for Skeletal Muscle
re: Mesh refinement of deformed shapes plotted with pressure distribution at time t = 400 µs using the {p, F, H, J } s obtained with a discretisation of 6912 and 43 648 hexahedral elements using traction loading tb = [0, 0, 100]T kPa. used with ⇢ = 1100 kg/m3 , E = 17 MPa, ⌫ = 0.45 and ↵CFL = 0.3.
Hookean model (i.e. by imposing the values of ↵ = µ2 and = 0), both the pressure and shear d at the initial undeformed configuration (F = H = I and J = 1) become
µ( 4 + 1) ⇢0
r
4: Deformable ; torus simulated c = c with=FVM. 3,4
5,6
µ , ⇢0
(A.8)
ess parameter is defined as := ˜ and ˜ is a userdefined material constant, usually taken in the bulk modulus of the material. The wave speeds c1,2 presented in (A.8a) reduces to
e limit of incompressibility (
(A.9) µ).
5: Muscle fiber active and passive behavior.
Figure 7: Simulation of isometric contraction. A posterior (c) Simulation contraction (from behind) view ofof theisometric upper arm shows contraction of in themuscle triceps and biceps muscles the triceps and the partially occluded biceps muscle (a) from[212] passive (left) to full activation (right).
(d) Mesh of an idealised carotid artery wall [481]
Fig. 8 Midstance model
P where f refers lar surface and C Inspecting the anterior superio negative contact head. When the von strike models ar in the bone supe midstance mode ular directly ab around the iliosa Inspecting the models, shown s once again visib regions occur i predicted conta 3:83 ! 10"4 m2 two contact regi mum predicted 4:62 ! 10"4 m2 , To ensure the tional midstanc dense globally re of 769,529 cells average articular predicted a ma is within 3% distributions clo independence.
3.4 Effect o have approxima thickness layer, been performed 0.72 mm, and de For the thinne tact pressure is 9.30 MPa, and th cartilage model sure is 21.59 MP the contact are results for the th
Journal of Biomechanical Figure 21. Biomechanics examplesEngineering
165
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/21/2014 Terms of Use: http
(d) The representation of the local displacement field: While cellcentred and staggeredgrid approaches assume linear (or cubic [109]) variations of displacement across cells (truncated Taylor series) or cell faces, the vertexcentred approaches Figure 8: Muscle contraction with skeletal motion. Inverse followcalculations in the footsteps ofsequence finite element methods by using shape functions. Similarly, dynamics were used with the motion to compute muscle activations. These activations influence the amount of tension in the muscle during the animation HFGMC/HOTGFM/FVDAM employ explicit local quadratic representations. and hence cause the muscle to deform in a realistic manner. (e) The use of implicit or explicit solution algorithms: ctivations computed from three different poses. ment muscles are colorcoded along the specIn principle, all finite volume variants could be implementation using either explicit (matrixactivation (blue) to full activation (red). free) or implicit (solution of a linear system) methods; however, traditionally cellcentred and c The Eurographics Association 2003. ⃝
Submitted for review (0000)
THIRTY YEARS OF THE FINITE VOLUME METHOD FOR SOLID MECHANICS 156
43
A. KOVACEVIC, N. STOSIC AND I.K. SMITH
SPE174818MS
(a)
Figure 9 Rotor displacement vectors and temperature distribution for an oil free compressor
Figure 6 Numerical mesh for CCM calculation (a) Displacement and temperature (b) Mesh of screw compressor [65] consumption. distribution of a power rotor in an In the case of high working pressures as, for example, in CO a refrigeration application, the rotors oilfree compressor thisandnumber of cells, the numerical solution of fluid flow and rotor deformation was obtai deform[46] more and the decrease With in the delivery rise in
z (m)
x (m)
Case 3 represents a high pressure oil injected compressor. The example is given here for a CO2 compressor with suction conditions of 30 bar and 0°C and a discharge pressure of 90 bar. The discharge temperature was 40°C. In this case, the large pressure difference caused higher rotor deflections than in Case 1, as shown in Figure 10. The highest deformation was in excess of 15 µm, which is the same order of magnitude as it was found in the case where temperature deformation was dominant. The deformation pattern of the rotors is similar to that in the Case 1 but only slightly enlarged at the discharge side. y (m) 525 350 the175 0 The influence of the rotor deformation on integral 525 screw compressor parameters caused z=0 by the changet=0.26s in clearance is given in Figure 11. The reduction of rotor 350 clearances due to the enlargement of the rotors caused by temperature dilatations results in an increase in both, the 175 R compressor flow and power input. However the flow increase is relatively larger than that of the power and hence 0 results in a decrease in specific0 power, or more conventionally, an increase in efficiency, as shown in the 175 diagram. However, the rotor deflections caused by the S pressure enlarge the clearances. For a moderate compressor 350 pressure, the clearances gap is enlarged only slightly and hence has only a negligible influence on the delivery and 520 x=0
2
specific power becomes more pronounced.
use of the COMET CCM solver in less than 30 hours. Here, it is important to emphasise the fa numerical simulation ofexamples fluid flow and structure analysis is used as a check rather then as an op Figure 22. Screw compressor 5 CONCLUSIONS because of the time required to obtain results for all the required 240 time steps. It is therefore 3D elastic wave modelling 793their rotors are Screw compressor elements, especially once at the end of the design process, before the prototyping of heavily loaded and are subjected to (b)by pressure forces (c)the machine. 350 525distortions. These cause the rotors to deform temperature during normal operation. The working clearances therefore vary and become smaller or larger, thereby affecting the internal leakage. This usually leads to deterioration in the compressor performance. A full 3D calculation has been performed to quantify the interaction of the compressor structure and compressor fluid flow. The effects of the change in working clearances are compared for different compressor applications and presented through distortion diagrams and flowpower charts. As is shown in the results, the P rotor deformation is dependent on the compressor application. It must, therefore, be taken into account in every specific compressor design.
175
The pressure and velocity distribution field obtained from the CCM calculation are presented Figure 7. The deformation of the male rotor is presented on the right of the same figure. The ro which do not exceed 6 micrometers in this case, are magnified 20,000 times in order to be visibl distribution on the rotor surface is also given in the same figure. It is clear that the highest defor critical area of the machine but since it is not substantial, the leakage flow is practically unch performance is hardly affected by it.
y (m)
350 525 525 z=0
175
y (m) 0
175
350
525
t=0.4s D
x (m)
350
R
175
0
0
z (m)
175 S
350 P
520 x=0
y (m) y (m) 0
(a) Elastic waves in a (b) Porepressure predictions near a hydraulic fracture [102] Figure 14 Base case for the fluid leakoff results: heterogenous (a) Fluid pressure insideFigure the fracture, medium [208] 7 Distribution of pressure and velocity in the cross sectional view of the machine (b) Fluid pressure inside the fracture and in the reservoir, (c) Displacement in y direction in the reservoir. Figure 23. Geomechanics and poroelasticity examples solid interaction Measured 5.2. at theFluid time 40 sec and displacement is magnified to visualize the fracture propagation. 525 525 z=0
350
175
175
350
525
t=0.45s
x (m)
350
175
R
0
0
z (m)
175
S
350
Fluid and solid interaction is presented here for three common applications of screw compres the same geometry, as shown in Figure 8. These are: an oilinjected air compressor of moderate vertexcentred methods (excluding Godunovtype approaches) haveratio, used implicit algorithms; a dry air compressor, of low pressure and a high pressure oil flooded compressor. In all c are of ‘N’ type with a 5/6 lobe configuration. The rotor outer diameters are 128 and 101 mm fo in the case of the “standard” cellcentred approach, segregated algorithms have prevailed. female rotors respectively, and their centre distance is 90 mm. The rotor length to diameter ra numerical mesh for the test case in this study comprises 513,617 cells of which 162,283 describ (a) (b) the rotors, with 189,144 cellselement map the element fluid partsapproach, between thethe rotors while the rest specify When comparing the finite volumeofapproach theother finite discharge ports and oil openings. A cross section through the mesh for the rotors and their likenesses are clear: both approachespresented adopt the same general strategy to discretise space into in Figure 9. 520 x=0
y (m)
Figure 13. Snapshots of ydirection component of the displacement at propagation times of 0.26, 0.4 and 0.45 s. The top of each figure is related to the free surface, and the bottom is related to a vertical plane (x = 0) that is orthogonal with the trench on the surface. A vertical Ricker wavelet point source is positioned at the midpoint of the intersection line of two planes (x = 0, y = 0, z = 0). The trench can be seen in z = 0 surface at y = 280 m. ⃝ C
2002 RAS, GJI, 150, 780–799
Downloaded from https://academic.oup.com/gji/articleabstract/150/3/780/614602 by University College Dublin user on 06 June 2018
cells/elements, both use similar data storage structures, and both follow similar approaches to assemble their corresponding characteristic matrices. The main differences lie in their fundamental philosophy: finite volume approaches are rooted in balance laws, where the governing7 equation is enforced by summing forces/fluxes for all faces of a control volume; in contrast, finite element methods adopt a more mathematical approach, based on variational methods, where the weak form of the governing equation is imposed in a volumetricallyaveraged sense. Submitted for review (0000)
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To end this article, we state three main challenges we see for the development of finite volume solid mechanics, such that its strengths and weaknesses may be rigorously explored in the context of solid mechanics, and its merits, relative to other similar approaches, may become clear to the computational solid mechanics community at large:
1. Awareness: Three decades after the first contributions to the field, there is still a general lack of awareness around the capabilities of the finite volume method in the sphere of solid mechanics; consequently, in the worst case, prestigious journals assign inappropriate reviewers, reviewers who, having only limited knowledge of the area, accept poor or reject good articles, for example, see [496], and authors look no further than past publications of their own research group when surveying the literature. 2. Benchmarking: As should be clear from this review, numerous differing finite volume formulations are possible; although some variants have been developed specifically for specialised applications, the comparison of the differing approaches on standard benchmarks is rare. Without such comparisons, in terms of efficiency, accuracy and robustness, it will not be possible to determine which approaches are optimal for certain classes of problems. Furthermore, given the trends in modern computing, the suitability of proposed approaches for execution on largescale distributed memory clusters (>1000s CPU cores) should be further explored. Simulations using hundreds of millions of cells are commonplace at CFD conferences: clearly finite volumebased solid mechanics procedures have great potential. Similarly, finite volume variants should be benchmarked against alternative approaches, such as the finite element method, to determine relative merits, not just on academic standard cases but on complex industrial cases to test robustness. Processes such as round robin benchmarking series, for example [443], may offer one solution. 3. Code dissemination: Where possible, code for published procedures should be shared for academic scrutiny: such distribution has the potential to: (a) accelerate academic progress, as others learn from and build on methods, as well as aiding in the discovery and resolution of errors; (b) facilitate ease of understanding and ease of implementation; (c) allow direct comparison of methods; (d) provide insight into the algorithm intricacies that may not be clear from academic articles; and (e) allow faster integration into commercial software and industrial use.
ACKNOWLEDGEMENTS The first author gratefully acknowledges financial support from Bekaert through the University Technology Centre (UTC), and from the Irish Composites Centre (IComp). Submitted for review (0000)
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