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

Technical Summary: Consistent Machine Learning for Topology Optimization with Microstructure-Dependent Neural Network Material Models

Problem Statement
The integration of additive manufacturing (AM) and topology optimization (TO) has enabled the design of multiscale structures with spatially varying material microstructures. However, optimizing such structures under nonlinear conditions (geometric and material) remains a significant challenge. Traditional approaches relying on computational homogenization (e.g., FE2) are computationally prohibitive when performed concurrently with TO. Conversely, standard machine learning (ML) constitutive models often fail to adhere to fundamental physical principles—such as thermodynamic consistency, objectivity, and polyconvexity—leading to numerical instabilities and inaccurate predictions, particularly in data-scarce regimes or near singularities (e.g., zero volume). While ML has been applied to linear elastic multiscale TO, its application to geometrically and materially nonlinear hyperelastic responses is still in its infancy.

Methodology
The authors propose a framework that merges a homogenization-based TO strategy with a "consistent" machine learning approach grounded in hyperelasticity theory. The methodology consists of two primary components:

1. Consistent ML Constitutive Modeling:

   • Architecture: The framework utilizes Input Convex Neural Networks (ICNNs) to learn the mapping between microstructural descriptors and the homogenized constitutive response.
   • Physical Constraints: To ensure numerical stability and physical validity, the neural network architecture enforces critical hyperelasticity conditions:
     • Polyconvexity: The strain energy density is formulated as a convex function of the deformation gradient, cofactor, and determinant, ensuring the existence of minimizers for the variational problem.
     • Objectivity and Symmetry: The model respects material frame indifference and isotropy through symmetric weight sharing and specific input transformations based on principal stretches.
     • Thermodynamic Consistency: The model satisfies natural state conditions (zero stress and energy at zero strain) and volumetric growth conditions (energy approaches infinity as volume approaches zero or infinity).
   • Microstructure Dependence: The ICNN input space is extended to include microstructural descriptors, specifically the volume fraction of stiff inclusions ($\alpha$) within a soft matrix. The model assumes a convex relationship between the strain energy density and the inclusion volume fraction.
   • Differentiability: The entire constitutive model is differentiable, allowing for the efficient computation of tangent stiffness and sensitivities via automatic differentiation.

2. Multiscale Topology Optimization:

   • Formulation: The TO problem is formulated as a work maximization problem subject to volume fraction constraints for both the macroscopic material distribution ($\rho$) and the microscopic inclusion volume fraction ($\alpha$).
   • Implementation: A density-based approach with SIMP interpolation and smooth Heaviside projection is used to handle intermediate densities and ensure geometric nonlinearity in solid elements while maintaining small-strain kinematics in void elements.
   • Optimization Loop: The optimizer (MMA) updates both the macroscopic pseudo-densities and the microstructural volume fractions simultaneously. The consistent ML model provides the homogenized stress response and its derivatives for the forward analysis and sensitivity calculations.

Key Contributions

• Consistent ML Framework: The paper introduces a novel framework for hyperelastic TO that integrates ICNNs with rigorous physical constraints (polyconvexity, objectivity, natural state), addressing the stability issues of "black-box" neural networks in nonlinear optimization.
• Microstructure-Dependent Modeling: The authors demonstrate a method to embed microstructural descriptors (volume fraction) directly into the ICNN input, enabling the prediction of homogenized responses that vary spatially across the design domain.
• Two-Scale Concurrent Optimization: The work extends TO to a two-scale concurrent design problem for hyperelastic materials, simultaneously optimizing the macroscopic layout and the local microstructural composition.
• Validation: The study validates the ML model against classical phenomenological models (Arruda-Boyce and Neo-Hookean) and demonstrates its efficacy in single-scale and multi-scale optimization benchmarks (T-bracket and cantilever beam).

Results

• Model Accuracy: The trained consistent ML models ($M_s$ for single-scale and $M_m$ for multi-scale) achieved high accuracy ($R^2 > 0.999$) in predicting stress responses and strain energy densities compared to ground truth phenomenological models and RVE simulations. The models demonstrated robust interpolation and reasonable extrapolation capabilities, particularly near singularities, due to the enforced physical constraints.
• Single-Scale TO: In single-scale optimization examples (T-bracket and cantilever), the designs generated using the ML material model were nearly identical to those obtained using the ground truth phenomenological models, confirming the reliability of the ML approach for standard TO tasks.
• Multi-Scale TO: In concurrent two-scale optimization, allowing the inclusion volume fraction ($\alpha$) to vary spatially resulted in lower (better) objective function values (maximized external work) compared to baseline cases with fixed microstructures. The optimizer naturally distributed material such that regions requiring higher compliance utilized lower inclusion volume fractions, while stiffer regions utilized higher fractions.
• Convergence: The optimization process converged smoothly, with the differentiable nature of the ML model facilitating efficient sensitivity analysis without the need for expensive numerical homogenization at every iteration.

Significance and Claims
The paper claims that integrating consistent machine learning models with density-based topology optimization significantly enhances the design optimization of heterogeneous hyperelastic structures under finite deformations. The primary significance lies in the ability to perform efficient, stable, and physically consistent multiscale optimization without the computational cost of concurrent FE2 simulations. By ensuring the ML model adheres to polyconvexity and other hyperelastic principles, the framework guarantees the existence of solutions in the forward problem and produces reliable final designs. The authors posit that this approach provides a viable path for designing future functionally graded materials and structures with complex, spatially varying microstructural details, and note that the framework is generalizable to additional descriptors (e.g., orientation) and potentially to history-dependent materials.