Thirty Years of the Finite Volume Method for Solid Mechanics

Abstract

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, which can be classified in a variety of ways, such as grid arrangement (cell-centred vs. staggered vs. vertex-centred), solution algorithm (implicit vs. explicit), and stabilisation strategy (Rhie–Chow vs. Jameson–Schmidt–Turkel vs. Godunov upwinding). This article gives an overview, historical perspective, comparison and critical analysis of the different approaches 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 more widespread acceptance.

Appendices

Appendix 1: Table of Most Cited Articles Related to the Finite Volume Method for Solid Mechanics

Table 1 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. As noted in the body of the article, 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. [193] has by far the greatest number of citations; however, a significant percentage of its received citations are related to its computational fluid mechanics developments, rather than its solid mechanics contributions.

Table 1 Most cited articles related to the finite volume method for solid mechanics from Google Scholar citations on 25th August 2018

Appendix 2: Overview of the Discretisation Used in HOTFGM/HFGMC/FVDAM Approaches

There are a variety of related methods with finite volume attributes which have been designed for the analysis of heterogenous microstructures. The related methods include the higher-order theory for functionally graded material (HOTFGM) [339], the high-fidelity generalised method of cells (HFGMC) [340,341,342,343], and the finite volume direct averaging micromechanics (FVDAM) theory [344, 345]. A brief summary of the methods is given here, and readers are referred to Aboudi et al. [339], Bansal and Pindera [345] and Cavalcante et al. [38] for further details.

The HOTFGM and HFGMC approaches start by spatially discretising the solution domain into rectangular so-called generic cells, which are further split into a second discretisation level containing four rectangular sub-cells (Fig. 32); 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. [339]. As a consequence of the assumed orthogonal Cartesian mesh, curved interfaces between material phases are approximated in a castellated staircase manner, as shown in Fig. 32; 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 displacement field can be decomposed into average and fluctuating components, $\varvec{u} = \bar{\varvec{u}} + \varvec{u}'$, where the average displacement is determined from the specified macroscopic average strains, $\bar{\varvec{u}} = \bar{\varvec{\epsilon }} \varvec{x}$. Within each sub-cell, the fluctuating displacement is then assumed to vary quadratically as a function of the local coordinates, $\bar{y}_2$ and $\bar{y}_3$:

$$\begin{aligned} \varvec{u}' (\bar{y}_2, \bar{y}_3)= & {} \varvec{W}_{00} + \bar{y}_2 \varvec{W}_{10} + \bar{y}_3 \varvec{W}_{01} \nonumber \\&+ \frac{1}{2}\left( 3 \bar{y}_2^2 - \frac{h^2}{4} \right) \varvec{W}_{20} + \frac{1}{2}\left( 3 \bar{y}_3^2 - \frac{l^2}{4} \right) \varvec{W}_{02} \end{aligned}$$

(82)

where h and l are the width and height respectively of the sub-cell; $\varvec{W}_{00}$, $\varvec{W}_{10}$, $\varvec{W}_{01}$, $\varvec{W}_{20}$, and $\varvec{W}_{02}$ are unknown vector displacement coefficients, each with three components; the $\varvec{W}_{00}$ component corresponds to the unknown displacement at the centre of the sub-cell, while the remaining coefficients correspond to higher-order displacement contributions within the sub-cell. Accordingly, there are $5 \times 3 = 15$ unknown displacement coefficients within each sub-cell and hence $4 \times 15 = 60$ within each generic cell; in three dimensions, there are 168 unknown quantities. For brevity here, the $(\gamma )$ and $(\beta )$ superscripts indicating the sub-cell have been dropped i.e. $\bar{y}_2 = \bar{y}_2^{(\beta )}$, $\bar{y}_3 = \bar{y}_3^{(\gamma )}$, 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 assuming the average displacement $\bar{\varvec{u}}$ to be zero.

Fig. 32

a Unit cell of a periodic material with microstructure discretised into rectangular building blocks. b Two-level discretisation employed by HFGMC into (q, r) generic cells further subdivided into four (b, c) subcells. c Single-level discretisation employed in the FVDAM theory into stand-alone (b, c) sub-volumes. Figure taken from Cavalcante et al. [38]. (Color figure online)

To determine the unknown displacement coefficients, the 0th, 1st and 2nd moments of momentum conservation are applied to each sub-cell, in addition to the enforcement of traction and displacement continuity between sub-cells 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 so-called moments of the governing equation. To achieve this, the governing equation (Eq. 1), where temporal and body force terms have been neglected, is written in terms of a so-called stress moment $\varvec{S}$:

$$\begin{aligned} \oint _\Gamma \varvec{n} \cdot \varvec{S} \;\text {d}\Gamma= & {} \varvec{0} \end{aligned}$$

(83)

with the stress moment defined as:

$$\begin{aligned} \varvec{S} = \frac{1}{h l} \int _{-\frac{h}{2}}^{\frac{h}{2}} \int _{-\frac{l}{2}}^{\frac{l}{2}} \left[ \varvec{\sigma } \, \bar{y}_2^m \, \bar{y}_3^n \right] \; \text {d} \bar{y}_2 \; \text {d} \bar{y}_3 \end{aligned}$$

(84)

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 time-step counter in Eq. (4).

In this way, it is possible to assemble a system of 60N algebraic equations of the standard form, $[\varvec{K}][\varvec{U}] = [\varvec{F}]$, where N is the number of generic cells in the solution domain, $[\varvec{U}]$ is a vector of unknown displacement coefficients $\varvec{W}$, the global stiffness matrix $[\varvec{K}]$ is a function of the sub-cell dimensions and mechanical properties, and the global force vector $[\varvec{F}]$ contains contributions from boundary conditions and nonlinear material stresses. The linear system is inverted to give the displacements distributions within the sub-cells.

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 two-level spatial domain decomposition (generic cells and sub-cells) of the HOTFGM/HFGMC methods is replaced by one-level of discretisation/cells;

• (b) The displacement coefficients within each cell $\varvec{W}$ are expressed in terms of surface-averaged 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 (similar to the finite element method), instead of the orthogonal Cartesian mesh of the HOTFGM/HFGMC approaches (see Fig. 32);

• (e) In the assembled system of algebraic equations $[\varvec{K}][\varvec{U}] = [\varvec{F}]$, the solution vector $[\varvec{U}]$ contains cell surface-averaged displacements, as opposed to sub-cell displacement coefficients.

Further technicals details of the HOTFGM, HFGMC and FVDAM methods can be found in Cavalcante et al. [38], Aboudi et al. [339], Aboudi [340], Aboudi et al. [341], Haj-Ali and Aboudi [342, 343], Bansal and Pindera [344, 345] and Cavalcante and Pindera [349].