Versatile stabilized finite element formulations for nearly and fully incompressible solid mechanics

Abstract

Computational formulations for large strain, polyconvex, nearly incompressible elasticity have been extensively studied, but research on enhancing solution schemes that offer better tradeoffs between accuracy, robustness, and computational efficiency remains to be highly relevant.In this paper, we present two methods to overcome locking phenomena, one based on a displacement-pressure formulation using a stable finite element pairing with bubble functions, and another one using a simple pressureprojection stabilized P 1 − P 1 finite element pair. A key advantage is the versatility of the proposed methods: with minor adjustments they are applicable to all kinds of finite elements and generalize easily to transient dynamics. The proposed methods are compared to and verified with standard benchmarks previously reported in the literature. Benchmark results demonstrate that both approaches provide a robust and computationally efficient way of simulating nearly and fully incompressible materials.

A computationally efficient physiologically comprehensive 3D–0D closed-loop model of the heart and circulation

“…Optimal function depends on matching the coupling between these two systems [ 8 ]. From a physics point of view, coupling poses a fluid–structure interaction (FSI) problem, with pressure and blood flow velocity fields as coupling variables [ 9 , 10 ]. These are relevant for investigating flow patterns or wall shear stresses, but are less suitable for systems level investigations.…”

Mixed Kirchhoff stress–displacement–pressure formulations for incompressible hyperelasticity

“…However, the size of the systems involved leads to rather high computational costs, thus making the adoption of robust and efficient iterative solvers important. Apart from the references mentioned above, we stress that within the context of cardiac mechanics, the present contribution complements the works defining strain-based formulations [61], stabilised formulations [44,66], displacement-pressure discontinuous Galerkin [12], and pure displacement [26,36,38,76] methods. In clinically-oriented applications, the performance of numerical solvers is particularly crucial.…”

“…Existence of infimisers of the elastic energy in a given product space E is not guaranteed in general. One can however assume growth conditions on Ψ and that the initial data maintains the material in the hyperelastic range, in such a way that the product space L 4 sym (Ω) × W 1,4 D (Ω) × L 4 (Ω) is contained in E [9] (see also [44]). This is discussed further in what follows.…”

“…While the present family of methods exhibits appealing features as mentioned above, a drawback, shared with other stabilised methods for hyperelasticity (see, e.g., [12,66,44]), is that the stabilisation constant required for the lowest-order method needs to be chosen heuristically. For this we could explore different formulations using Kirchhoff stress, for instance applying integration by parts differently in the derivation of the variational formulation, and leading to divergence-conforming approximations of stress.…”

Efficient identification of myocardial material parameters and the stress-free reference configuration for patient-specific human heart models

“…It is well known that simple P1-P0-elements may suffer from locking effects and hence other FE formulations are required for certain applications where accurate stresses are essential. To show the capabilities of the MFF method in this scenario, we applied the algorithm to cases 06-CoA and 03-AS using stabilized, locking-free P1-P1-elements [68] and an incompressible material, i.e., 1/κ = 0. Normalized fitting results for the two example cases are presented in relation to results obtained above using P1-P0-elements in Figure 6 and Table 5.…”