Consistent machine learning for topology optimization with microstructure-dependent neural network material models

TL;DR

This work presents a framework for consistent machine-learning-driven topology optimization of multiscale hyperelastic structures, integrating a homogenization-based TO loop with a polyconvex ICNN-based constitutive model that depends on microstructural descriptors such as the inclusion volume fraction $\alpha$. By enforcing key physical principles (objectivity, natural state, volumetric growth, and polyconvexity) and incorporating microstructure through expanded inputs, the approach yields a differentiable constitutive mapping suitable for analytic sensitivities in optimization. The authors demonstrate, across single- and two-scale problems, that the ML material models closely reproduce ground-truth phenomenological responses and that two-scale optimization with variable microstructure offers improved performance over fixed-microstructure baselines. The framework is positioned to enable functionally graded designs with controllable microstructures and points toward extensions to include orientation effects and history-dependent behavior, with practical implications for additive manufacturing of heterogeneous hyperelastic devices.

Abstract

Additive manufacturing methods together with topology optimization have enabled the creation of multiscale structures with controlled spatially-varying material microstructure. However, topology optimization or inverse design of such structures in the presence of nonlinearities remains a challenge due to the expense of computational homogenization methods and the complexity of differentiably parameterizing the microstructural response. A solution to this challenge lies in machine learning techniques that offer efficient, differentiable mappings between the material response and its microstructural descriptors. This work presents a framework for designing multiscale heterogeneous structures with spatially varying microstructures by merging a homogenization-based topology optimization strategy with a consistent machine learning approach grounded in hyperelasticity theory. We leverage neural architectures that adhere to critical physical principles such as polyconvexity, objectivity, material symmetry, and thermodynamic consistency to supply the framework with a reliable constitutive model that is dependent on material microstructural descriptors. Our findings highlight the potential of integrating consistent machine learning models with density-based topology optimization for enhancing design optimization of heterogeneous hyperelastic structures under finite deformations.

Figures (50)

• Figure 1: An overview of the proposed consistent machine learning-driven topology optimization framework for multiscale hyperelastic structures. The consistent ML block learns the mapping between microstructural descriptors $d_m$ (set as the inclusion volume fraction ${\alpha}$) and the homogenized constitutive response in an offline phase adhering to the hyperelastic principles. The multiscale TO block treats ${\alpha}$ as an additional design variable using its filtered counterpart to approximate the spatially varying material response. By employing a differentiable ML material model in the forward analyses, the optimizer can efficiently update the design variables using analytically computed sensitivities of the objective/constraint functions obtained through automatic differentiation. The magnified regions in the multiscale TO block illustrate the filtered and projected pseudo-density field $\bar{{\rho}}$ and the filtered microstructural inclusion volume fraction field $\hat{{\alpha}}$ in the initial design state (left) and the final optimized state (right).
• Figure 2: Schematic illustration of the consistent machine learning model for isotropic hyperelasticity. The input to the neural network is ${\boldsymbol{E}}$, which passes through a fixed transformation layer ${\mathcal{T}}$ to arrive at $\tilde{{\boldsymbol \Lambda}}$ that subsequently goes through input convex neural layers as in chen2018opti to arrive at polyconvex output ${\mathcal{N}}$. This output is regularized for the natural state condition \ref{['eq:natural_state']} and passed through a gradient layer which obtains the second Piola-Kirchhoff stress ${\boldsymbol{S}}$ as in \ref{['eq:overall_stress_strain_relationship']}. Extension of the model to include microstructural descriptors $\mathrm{d_m}$ is achieved by appending the input space of the ICNN. The treatment of the microstructural descriptor $\mathrm{d_m}$ is such that the ICNN output (strain energy density) is convex with respect to the microstructural descriptor, which is a reasonable assumption for the volume fraction ${\alpha}$ of a stiff inclusion in a soft matrix.
• Figure 3: Stress response of the single scale consistent ML model compared to the ground truth phenomenological model for a uniaxial strain loading case. The plot is split into two subplots to better visualize the compressive and tensile loading ranges.
• Figure 4: Strain energy density contour plot of the single scale consistent ML model and the corresponding difference plot to the ground truth phenomenological model. The region within the red box corresponds to the trained domain.
• Figure 5: Stress response of the microstructure-dependent consistent ML model compared to the ground truth obtained through RVE simulations for a uniaxial strain loading scenario for (left) compressive loading, and (right) tension loading range. The top and bottom plots show the stress components $S_{11}$ and $S_{22}$ separately.