An enhanced single Gaussian point continuum finite element formulation using automatic differentiation

TL;DR

The study introduces Q1STc+, an enhanced single Gaussian-point continuum finite element with hourglass stabilization, where automatic differentiation yields an approximation-free computation of the inverse Jacobian ($m{J}^{-1}$). This replaces the prior Taylor-series-based approximation in Q1STc, improving accuracy on distorted meshes and in elasto-plastic regimes. Numerical tests show that Q1STc+ passes the membrane patch test and converges reliably on distorted meshes, closely matching a volumetric-locking-free reference (U-P-SBFEM) in elastic and plastic scenarios, while solids patch tests remain challenging. Public code availability on Zenodo ensures reproducibility and broader adoption.

Abstract

This contribution presents an improved low-order 3D finite element formulation with hourglass stabilization using automatic differentiation (AD). Here, the former Q1STc formulation is enhanced by an approximation-free computation of the inverse Jacobian. To this end, AD tools automate the computation and allow a direct evaluation of the inverse Jacobian, bypassing the need for a Taylor series expansion. Thus, the enhanced version, Q1STc+, is introduced. Numerical examples are conducted to compare the performance of both element formulations for finite strain applications, with particular focus on distorted meshes. Moreover, the performance of the new element formulation for an elasto-plastic material is investigated. To validate the obtained results, a volumetric locking-free element based on scaled boundary parametrization is used. Both the implementation of the element routine Q1STc+ and the corresponding material subroutine are made accessible to the public at https://doi.org/10.5281/zenodo.14259791

Figures (15)

• Figure 1: Membrane patch test - Geometry of the five-element patch test with $t = \text{0.001}\, \text{mm}$ with element numbers I - V.
• Figure 2: Membrane patch test - Stress distribution for Q1, Q1STc and Q1STc+, where Q1 denotes the low-order finite element formulation using full integration, and serves as an additional reference solution to the analytical solution in Table \ref{['membrane_table']}.
• Figure 3: Membrane patch test - Force-displacement curves for an elasto-plastic material, see Table \ref{['mat_param_plast']}. Comparison between Q1STc, Q1STc+, the element U-P-SBFEM sauren_mesh_2024 and the conventional low-order formulation using full integration (Q1). Here, the results of the proposed Q1STc+ element coincide with those of U-P-SBFEM and Q1.
• Figure 4: Patch test for solids - Geometry of the seven-element patch test with element numbers I - VII.
• Figure 5: Patch test for solids - Stress distribution at $z = \text{0.5}\,\text{mm}$ for Q1, Q1STc and Q1STc+, where Q1 denotes the low-order finite element formulation using full integration, and serves as an additional reference solution to the analytical solution in Table \ref{['tab_solid']}.