Multi-material Multi-physics Topology Optimization with Physics-informed Gaussian Process Priors

TL;DR

The results demonstrate that the proposed PIGP framework can effectively solve coupled multi-physics and design problems simultaneously -- generating super-resolution topologies with sharp interfaces and physically interpretable material distributions.

Abstract

Machine learning (ML) has been increasingly used for topology optimization (TO). However, most existing ML-based approaches focus on simplified benchmark problems due to their high computational cost, spectral bias, and difficulty in handling complex physics. These limitations become more pronounced in multi-material, multi-physics problems whose objective or constraint functions are not self-adjoint. To address these challenges, we propose a framework based on physics-informed Gaussian processes (PIGPs). In our approach, the primary, adjoint, and design variables are represented by independent GP priors whose mean functions are parametrized via neural networks whose architectures are particularly beneficial for surrogate modeling of PDE solutions. We estimate all parameters of our model simultaneously by minimizing a loss that is based on the objective function, multi-physics potential energy functionals, and design-constraints. We demonstrate the capability of the proposed framework on benchmark TO problems such as compliance minimization, heat conduction optimization, and compliant mechanism design under single- and multi-material settings. Additionally, we leverage thermo-mechanical TO with single- and multi-material options as a representative multi-physics problem. We also introduce differentiation and integration schemes that dramatically accelerate the training process. Our results demonstrate that the proposed PIGP framework can effectively solve coupled multi-physics and design problems simultaneously -- generating super-resolution topologies with sharp interfaces and physically interpretable material distributions. We validate these results using open-source codes and the commercial software package COMSOL.

Figures (16)

• Figure 1: PIGP framework for multi-physics topology optimization: The mean functions for the design variable, as well as for the primary and adjoint fields, are parameterized using the multi-output PGCAN architecture, which consists of three modules: (i) feature-space encoding via a convolutional neural network, (ii) feature interpolation at all collocation points, and (iii) decoding through a shallow MLP. The design variable $\boldsymbol{\rho}(\boldsymbol{x})$ is a vector of volume fractions for the void and all candidate materials at each domain point $\boldsymbol{x}$. The final-layer multi-output from PGCAN for $\boldsymbol{\rho}(\boldsymbol{x})$ is passed through a $\mathrm{softmax}$ function to enforce the partition of unity of $\boldsymbol{\rho}(\boldsymbol{x})$. Gaussian processes are employed on these mean functions to impose the BCs for the primary and adjoint fields, and to enforce the design constraints.
• Figure 2: Benchmark examples for CM: Four canonical benchmark problems are considered, including the Messerschmitt--Bölkow--Blohm (MBB) beam, cantilever beam, bridge beam, and L-shaped beam. The configurations of the 2D examples are shown in (a)–(d) and their 3D counterparts are presented in (e)–(h).
• Figure 3: Designed multi-material topologies for 2D CM: Representative optimized topologies are shown for each method under different constraints. PolyMat accommodates only mass or cost constraint but not both.
• Figure 4: Representative designs for 3D CM: For each benchmark, the topologies corresponding to the maximum, median, and minimum compliance values are shown, comparing SIMP 3D in the single-material setting with our PIGP 3D framework under both single- and multi-material settings. The three colors denote the three candidate artificial materials defined in \ref{['tab cm mat']}. The multi-material designs are subject to both mass and cost constraints.
• Figure 5: Topology evolution during multi-material CM training: Intermediate topologies at selected epochs are shown along with the history of design objective $L_C(\cdot)$ for (a) 2D MBB beam, (b) 2D cantilever beam, (c) 2D bridge beam, and (d) 2D L-shaped beam. Panels (e)--(h) show the corresponding topology evolution for the same four examples in the 3D setting.