Simulation of non-linear structural elastodynamic and impact problems using minimum energy and simultaneous diagonalization high-order bases

TL;DR

This paper introduces a novel high-order finite element basis using simultaneous diagonalization and minimum energy principles, significantly improving computational efficiency in simulating non-linear impact elastodynamics with hyperelastic materials.

Contribution

The authors develop and implement SDME high-order bases for non-linear elastodynamic simulations, achieving up to 41 times speedup over standard bases.

Findings

Speedup of up to 41 times in computational time

Reduced number of iterations in conjugate gradient methods

Effective handling of large deformation impact problems

Abstract

We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with neo-hookean hyperelastic materials. The bases are constructed using procedures for simultaneous diagonalization of the internal modes and Schur complement of the boundary modes from the standard nodal and modal bases, constructed using Lagrange and Jacobi polynomials, respectively. The implementation of these bases in a high-order finite element code is straightforward, since the procedure is applied only to the one-dimensional expansion bases. Non-linear transient structural problems with large deformation, hyperelastic materials and impact are solved using the obtained bases with explicit and implicit time integration procedures. Iterative solutions based on preconditioned conjugate gradient methods are considered. The performance of the proposed bases in terms of the number of iterations of pre-conditioned conjugate gradient methods and computational time are compared with the standard nodal and modal bases. Our numerical tests obtained speedups up to 41 using the considered bases when compared to the standard ones.