Pragmatism and the Finite Element Method in the Modern Era

Shakeri, Rezgar

Graduate Thesis Or Dissertation

Pragmatism and the Finite Element Method in the Modern Era Public Deposited

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/6m311r08t

Abstract

Since numerical modeling of hyperelastic materials is ubiquitous in many engineering fields, a main effort in computational mechanics is to develop stable, accurate and efficient tools for their analysis. A stable numerical calculation of a well-conditioned function will have a relative accuracy of ϵmachine. Many open source finite element analysis packages contain numerically unstable formulations at some point, even for simple material models. We trace the source of this instability and show how to formulate various compressible/incompressible hyperelastic constitutive models in a stable way.

For an incompressible material, the hydrostatic stress becomes an independent variable. Thus, it can be discretized independently of the displacement field. One of the common approaches of simulating the incompressible model is mixed finite element discretization. Mixed finite element method is an efficient approach to overcome locking that is observed in the numerical treatment of nearly incompressible materials when a pure displacement method with linear element is used. In order to guarantee the stability and optimal convergence of a mixed formulation, the inf-sup conditions must be satisfied. The inf-sup constants, which are computable through a set of eigenvalue problems for a given mixed discretization, determines the stability of the chosen FE space for displacement and pressure fields.

Despite the accuracy gained by high-order mixed finite element method, most of the industrial finite element problems are constructed within the framework of matrix-based approach by using at most quadratic/linear order of interpolation for displacement/pressure fields. A matrix-free approach yields better performance, both in terms of storage and solve time over the assembled sparse matrices method. We introduce a matrix-free p-multigrid mixed finite element discretization with Newton-Krylov iterative solvers for the hyperelasticity problem. We implemented our problem in Ratel which is a performance portable solid mechanics library that uses matrix-free operators from libCEED and solvers from PETSc to provide fast, efficient, and accurate simulations on next generation architectures. To demonstrate the reliability and efficiency of the mixed high-order finite element implementation, a series of numerical examples are investigated with p-multigrid preconditioning and different interpolation orders.

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