A Matlab code for analysis and topology optimization with Third Medium Contact

TL;DR

The paper introduces an open-source Matlab implementation of the Third Medium Contact (TMC) model for density-based topology optimization of hyperelastic structures. It combines TMC with HuHu-regularization to stabilize void-element distortions, enabling differentiable, fully implicit contact within optimization loops. The framework employs a PDE-based density filter, Heaviside projection (with beta continuation), and RAMP interpolation to drive end-compliance minimization under volume constraints, demonstrated on C-shape and TO scenarios. The contribution is an accessible, educational codebase that reproduces results and serves as a platform for extending TMC TO to more complex problems and higher dimensions.

Abstract

We present a Matlab code for modelling and topology optimization of hyperelastic structures, including contact modelled by the Third Medium Contact (TMC) approach. By using the so-called HuHu-regularization we penalize the skew distortion of the bilinear finite elements discretizing void regions, thus promoting convergence of the nonlinear solver. First, we show how this method is implemented in a compact code, allowing to simulate contact and force transfer in hyperelastic structures. Then, we solve a topology optimization problem for minimum end-compliance of a structure exhibiting contact. The Matlab scripts that replicate the results are included, and we discuss some possible extensions to more general problems.

Figures (8)

• Figure 1: (a) Illustration of two solid bodies, collectively occupying the region $\Omega_{s}$, interacting through the Third Medium (TM), which fills the region $\Omega_{v}$. (b) Qualitative illustration of the stiffening behaviour of the material law \ref{['eq:sedNeoHookean']}, as the local volume shrinks to zero under uniaxial compression. The arrows show the normal force for a reference volume (dashed) compressed to a current volume (gray) for higher compression levels
• Figure 2: Illustrations of the deformations that can be represented by bilinear $\mathcal{Q}_{1}$ elements. The HuHu regularization only penalizes the skew deformation
• Figure 3: Bulk modulus $K$, and longitudinal modulus $M = K + \frac{4}{3}G$ as functions of the Poisson's ratio $\nu$ ($E=1$). The two moduli become coincident for $\nu \rightarrow 0.5$. However, for low $\nu$ values $K < 1$, whereas $M$ is always $> 1$
• Figure 4: Geometry and mechanical setup for the C-shape example. The thickness of the solid region $\Omega_{s}$ is $t = 0.1L$, and the void region $\Omega_{v}$ extends of $t/2$ to the right of the solid part
• Figure 5: Deformed configurations (top row), and distribution of the SED (bottom row) for the C-shape example at three load steps. The SED is normalized with respect to the maximum domain value, and plotted in log-scale