Stabilized finite element method for incompressible solid dynamics using an updated Lagrangian formulation

TL;DR

This paper introduces a stabilized finite element method within an updated Lagrangian framework for incompressible solid dynamics, effectively handling nearly incompressible materials and complex geometries with improved accuracy and robustness.

Contribution

It develops a mixed displacement-pressure formulation combined with a Variational Multi-Scale method to suppress spurious pressure modes in tetrahedral elements.

Findings

Successfully handles incompressibility in solid dynamics.

Demonstrates robustness in bending dominated problems.

Accurately models complex geometries.

Abstract

This paper proposes a novel way to solve transient linear, and non-linear solid dynamics for compressible, nearly incompressible, and incompressible material in the updated Lagrangian framework for tetrahedral unstructured finite elements. It consists of a mixed formulation in both displacement and pressure, where the momentum equation of the continuum is complemented with a pressure equation that handles incompresibility inherently. It is obtained through the deviatoric and volumetric split of the stress, that enables us to solve the problem in the incompressible limit. The Varitaional Multi-Scale method (VMS) is developed based on the orthogonal decomposition of the variables, which damps out spurious pressure fields for piece wise linear tetrahedral elements. Various numerical examples are presented to assess the robustness, accuracy and capabilities of our scheme in bending dominated problems, and for complex geometries.