Method of manufactured solutions code verification of elastostatic solid mechanics problems in a commercial finite element solver

Abstract

Much progress has been made in advancing and standardizing verification, validation, and uncertainty quantification practices for computational modeling in recent decades. However, examples of rigorous code verification for solid mechanics problems in the literature remain scarce, particularly for commercial software and for the non-trivial large-deformation analyses and nonlinear materials typically needed to simulate medical devices. Here, we apply the method of manufactured solutions (MMS) to verify a commercial finite element code for elastostatic solid mechanics analyses using linear-elastic, hyperelastic (neo-Hookean), and quasi-hyperelastic (Hencky) constitutive models. Analytical source terms are generated using either Python/SymPy or Mathematica and are implemented in ABAQUS/Standard without modification to the solver source code. Source terms for the three constitutive models are found to vary nearly six orders of magnitude in the number of mathematical operations they contain. Refinement studies reveal second-order displacement convergence in response to mesh refinement for all constitutive models and first-order displacement convergence in response to increment refinement for the finite-strain problems. We also investigate the sensitivity of MMS convergence order to minor coding errors using an exploratory case. Code used to generate the MMS source terms and the input files for the simulations are provided as supplemental material.

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“…Code verification Software quality assurance (SQA) (a) Very little or no SQA procedures were specified or followed (b) SQA procedures were specified and documented (c) In addition to the previously specified SQA procedures, the software anomaly list and the software development environment are fully understood and the impact on the COU is analyzed and documented; quality metrics are tracked Numerical code verification (NCV) (a) NCV was not performed (b) The numerical solution was compared to an accurate benchmark solution from another verified code (c) The numerical solution was compared to an accurate benchmark solution, either an analytical solution or using the method of manufactured solutions (MMS) [36] Calculation The calculation verification was also performed through a mesh convergence analysis of the FE model and the results are presented in Table 5. In this work, we used the fully automatic adaptive meshing convergence tool provided by ANSYS Workbench; this process provides the desired accuracy in the smallest number of runs possible.…”

“…In this paper, rigorous code verification was not performed since it out of the scope. For future studies, a rigorous verification is recommended, using, for example, the method of manufactured solutions (MMSs) that has successfully been applied to commercial finite element code for elastostatic solid mechanics analyses [36]. Although numerical error estimation was out of the scope of this paper, it is recommended the future studies involving rigorous mesh refinement using Richardson extrapolation and calculations of a grid convergence index should be performed [47].…”