A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models

Guan，Jiashen1; Yuan，Hongyan1; Liu，Ju1,2

2023-09-01

10.1016/j.cma.2023.116145

Computer Methods in Applied Mechanics and Engineering   Impact Factor & Quartile

0045-7825

1879-2138

We present a structure-preserving scheme based on a recently-proposed mixed formulation for incompressible hyperelasticity formulated in principal stretches. Although there exist several different Hamiltonians introduced for quasi-incompressible elastodynamics based on different multifield variational formulations, there is not much study on the fully incompressible materials in the literature. The adopted mixed formulation can be viewed as a finite-strain generalization of Herrmann variational formulation, and it naturally provides a new Hamiltonian for fully incompressible elastodynamics. Invoking the discrete gradient and scaled mid-point formulas, we are able to design fully-discrete schemes that preserve the Hamiltonian and momenta. Our analysis and numerical evidence also reveal that the scaled mid-point formula is non-robust numerically. The generalized Taylor–Hood element based on the spline technology conveniently provides a higher-order, robust, and inf-sup stable spatial discretization option for finite strain analysis. To enhance the element performance in volume conservation, the grad-div stabilization, a technique initially developed in computational fluid dynamics, is introduced here for elastodynamics. It is shown that the stabilization term does not impose additional restrictions for the algorithmic stress to respect the invariants, leading to an energy-decaying and momentum-conserving fully discrete scheme. A set of numerical examples is provided to justify the claimed properties. The grad-div stabilization is found to enhance the discrete mass conservation effectively. Furthermore, in contrast to conventional algorithms based on Cardano's formula and perturbation techniques, the spectral decomposition algorithm developed by Scherzinger and Dohrmann is robust and accurate to ensure the discrete conservation laws and is thus recommended for stretch-based material modeling.

Discrete mass conservation Elastodynamics Grad-div stabilization Isogeometric analysis Stretch-based model Structure-preserving scheme

SCI

English

First ; Corresponding

National Natural Science Foundation of China[12072143];National Natural Science Foundation of China[12172160];Guangdong Science and Technology Department[2020B1212030001];Guangdong Science and Technology Department[2021QN020642];Southern University of Science and Technology[Y01326127];

Engineering ; Mathematics ; Mechanics

Engineering, Multidisciplinary ; Mathematics, Interdisciplinary Applications ; Mechanics

WOS:001022517000001

ELSEVIER SCIENCE SA

COMPUTER SCIENCE

2-s2.0-85160540669

Scopus

1.Department of Mechanics and Aerospace Engineering,Southern University of Science and Technology,Guangdong,1088 Xueyuan Avenue, Shenzhen,518055,China
2.Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications,Southern University of Science and Technology,Guangdong,1088 Xueyuan Avenue, Shenzhen,518055,China

Guan，Jiashen,Yuan，Hongyan,Liu，Ju. A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models[J]. Computer Methods in Applied Mechanics and Engineering,2023,414.

Guan，Jiashen,Yuan，Hongyan,&Liu，Ju.(2023).A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models.Computer Methods in Applied Mechanics and Engineering,414.

Guan，Jiashen,et al."A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models".Computer Methods in Applied Mechanics and Engineering 414(2023).