Computational efficiency of numerical approximations of tangent moduli for finite element implementation of a fiber-reinforced hyperelastic material model

Abstract

In this study, we evaluated computational efficiency of finite element (FE) simulations when a numerical approximation method was used to obtain the tangent moduli. A fiber-reinforced hyperelastic material model for nearly incompressible soft tissues was implemented for 3D solid elements using both the approximation method and the closed-form analytical method, and validated by comparing the components of the tangent modulus tensor (also referred to as the material Jacobian) between the two methods. The computational efficiency of the approximation method was evaluated with different perturbation parameters and approximation schemes, and quantified by the number of iteration steps and CPU time required to complete these simulations. From the simulation results, it can be seen that the overall accuracy of the approximation method is improved by adopting the central difference approximation scheme compared to the forward Euler approximation scheme. For small-scale simulations with about 10,000 DOFs, the approximation schemes could reduce the CPU time substantially compared to the closed-form solution, due to the fact that fewer calculation steps are needed at each integration point. However, for a large-scale simulation with about 300,000 DOFs, the advantages of the approximation schemes diminish because the factorization of the stiffness matrix will dominate the solution time. Overall, as it is material model independent, the approximation method simplifies the FE implementation of a complex constitutive model with comparable accuracy and computational efficiency to the closed-form solution, which makes it attractive in FE simulations with complex material models.

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“…The parameter h defines the angle between one of the mean local fiber direction and the circumferential axis of the local coordinate system. The anisotropic material model was implemented into ABAQUS with a user subroutine VUANISOHYPER [27][28][29].…”

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“…The fiber orientation was defined by structural tensor, M i = m 0 i ⊗ m 0 i with m 0 1 = [ cosθ , sinθ , 0] and m 0 2 = [ cosθ , − sinθ ,0]. The anisotropic hyperelastic material model was implemented into Abaqus 6.14 (SIMULIA, Providence, RI) with a user sub-routine VUMAT 16, 33 . Local coordinate systems were defined for each leaflet to include fiber orientations for each region.…”

Higher‐order and higher floating‐point precision numerical approximations of finite strain elasticity moduli

“…However, the lower CPU times are likely due to the somewhat random nature of low duration simulations that was found. Using a greater number of mesh refinements than previous studies, the normalized CPU times are shown to generally increase as the mesh is refined until it reaches a maximum before then decreasing with further refinements. However, the quadruple precision approximations all require more than twice the CPU time for all meshes.…”

“…While the numerical approximations implemented here with optimal perturbation magnitude show quadratic convergence, it is known from the first implementation by Miehe that approximation methods have higher computation times. However, previous studies have shown that, with increased mesh refinement, the difference between analytical and approximate methods' solve times is decreased. This is due to the increased solution time required for the global matrix iterations in a more complex model, such that the computational effort of the stress and tangent components becomes less significant.…”

“…The central difference method is a second‐order approximation that has been investigated in several studies . In related works, it is shown that second‐order approximations give more accurate tangent moduli; however, their optimal perturbation magnitude is larger due to the influence of the subtractive round‐off error. Kiran and Khandelwal used numerical approximations of tangent moduli up to the eighth order.…”