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Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy

Li Zheng, Siddhant Kumar, Dennis M. Kochmann

arXiv:2012.15744v2cs.CE

TL;DR

The paper addresses the costly optimization of bodies with spatially varying cellular microstructures and tunable anisotropy. It combines spinodoid-based simultaneous two-scale topology optimization with a FEM-trained deep-neural-network surrogate for homogenized elasticity and sensitivities. The resulting designs include an 8.35% compliance improvement over SIMP in one benchmark, while automatic-differentiation sensitivities achieve R2 ≥0.999 against finite differences.

  • Problem

    Multiscale topology optimization with spatially varying microstructures requires repeated microscale homogenization, making optimization expensive and limiting design-space flexibility.

  • Method

    The framework simultaneously optimizes macroscale material distribution and spinodoid density, orientation, and rigid-body rotation using a FEM-trained DNN surrogate for homogenized elastic response.

  • Results

    The spinodoid design achieves 8.35% improvement in compliance relative to SIMP in the L-shaped benchmark, while DNN automatic-differentiation sensitivities agree with FEM finite differences at R2 ≥0.999.

  • Takeaways & Limitations

    Spinodoid architectures provide spatially variant anisotropic microstructures for multiscale compliance optimization, with the surrogate replacing nested microscale FE calculations.

  • Takeaways & Limitations

    Affine boundary conditions are used for non-periodic spinodoid RVEs, so the computed homogenized stiffness is an upper bound to the actual effective stiffness.

Abstract

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We present a two-scale topology optimization framework for the design of macroscopic bodies with an optimized elastic response, which is achieved by means of a spatially-variant cellular architecture on the microscale. The chosen spinodoid topology for the cellular network on the microscale (which is inspired by natural microstructures forming during spinodal decomposition) admits a seamless spatial grading as well as tunable elastic anisotropy, and it is parametrized by a small set of design parameters associated with the underlying Gaussian random field. The macroscale boundary value problem is discretized by finite elements, which in addition to the displacement field continuously interpolate the microscale design parameters. By assuming a separation of scales, the local constitutive behavior on the macroscale is identified as the homogenized elastic response of the microstructure based on the local design parameters. As a departure from classical FE$^2$-type approaches, we replace the costly microscale homogenization by a data-driven surrogate model, using deep neural networks, which accurately and efficiently maps design parameters onto the effective elasticity tensor. The model is trained on homogenized stiffness data obtained from numerical homogenization by finite elements. As an added benefit, the machine learning setup admits automatic differentiation, so that sensitivities (required for the optimization problem) can be computed exactly and without the need for numerical derivatives - a strategy that holds promise far beyond the elastic stiffness. Therefore, this framework presents a new opportunity for multiscale topology optimization based on data-driven surrogate models.

1. Introduction

The paper targets spatially varying cellular metamaterials whose stiffness, anisotropy, and orientation can be designed across scales. It addresses computational cost and design-space limitations by combining spinodoid microstructures with simultaneous two-scale optimization and a data-driven surrogate.

  • Motivation: Existing cellular architectures face stiffness-scaling, defect-sensitivity, and limited smooth spatial grading challenges.The introduction highlights bending-dominated trusses, symmetry-breaking sensitivity, and limited opportunities for smoothly varying structures.
  • Spinodoid metamaterials: Spinodoid topologies are inspired by spinodal decomposition and generated efficiently using anisotropic Gaussian random fields.This parametrization supports exploration of anisotropic mechanical properties without computationally expensive phase-field simulations.
  • Multiscale topology optimization: Multiscale topology optimization jointly designs macroscale material distribution and spatially varying microscale architecture, unlike multiresolution methods that only vary mesh resolution.The paper adopts simultaneous two-scale optimization rather than sequential optimization.
  • Anisotropy and orientation: Anisotropic topology optimization can align material orientations with principal stress or strain directions, but prior work often held base-material anisotropy constant or omitted spatially varying microstructural orientation.The paper positions spinodoid orientation and anisotropy as jointly designable microscale variables.
  • Computational challenge: On-the-fly FEM homogenization is expensive during optimization, while look-up tables restrict the available design space.Earlier surrogate approaches include polynomial models, Gaussian processes, NEXP, and single-layer neural networks.
  • Proposed approach: The proposed framework uses a FEM-trained deep neural network to replace nested microscale calculations and provide effective responses and sensitivities for simultaneous optimization.Automatic differentiation avoids numerical differentiation for the surrogate sensitivities.

2. Spinodoid topologies with tunable anisotropy

Spinodoid topologies are generated from Gaussian random fields whose wave-vector orientations and level set control anisotropy and density. Their effective stiffness is computed by finite-element homogenization, with affine RVE boundary conditions providing an upper-bound response.

  • Topology generation: Gaussian random fields represent spinodal phase fields as superpositions of fixed-wavelength plane waves with random directions and phase shifts.The wavenumber β sets the microstructural length scale.
  • Topology generation: Anisotropic wave-vector distributions are imposed by restricting directions to cones around the Cartesian basis, parameterized by θ1, θ2, and θ3.These cone angles determine directional preferences in the generated architecture.
  • Topology generation: A level set converts the Gaussian random field into a solid-void network, with the threshold determined by the target relative density ρ.The framework considers solid networks, while shell architectures can be generated using an isosurface.
  • Topology generation: The design parameters Θ = (ρ, θ1, θ2, θ3)T control porosity and anisotropy, producing lamellar, columnar, or cubic topologies according to the number of non-zero cone angles.The numerical bounds include ρmin = 0.3 and θmin = π/6.
  • Elastic homogenization: Finite-element computational homogenization maps each spinodoid design to an effective stiffness tensor using a cubic representative volume element.The elastic surface visualizes Young’s modulus E(d) in every direction d ∈ S^2.
  • Elastic homogenization: Affine boundary conditions yield an upper bound on the actual effective stiffness because spinodoid topologies lack periodicity, although their stretching-dominated response may reduce this discrepancy.The design space spans highly directionally soft lamellar structures, two-directionally stiff columnar structures, and approximately isotropic cubic structures.

3. Multiscale topology optimization

The framework optimizes spatially varying spinodoid microstructures through a macroscale finite-element problem coupled to locally homogenized elastic properties. Continuous parameter transformations, rotations, and surrogate-driven sensitivities support feasible anisotropic multiscale optimization.

  • 3.1. Macroscale: boundary value problem and compliance minimization: The two-scale assumption derives local macroscale elastic properties and mass density by homogenizing a microscale representative volume element defined by χ(x).For spinodoids, χ includes the design parameters Θ = (ρ, θ1, θ2, θ3)^T.
  • 3.1. Macroscale: boundary value problem and compliance minimization: The macroscale boundary value problem uses finite elements with element-wise microscale design parameters and homogenized stiffness to determine displacements and compliance.The global system is assembled from element stiffness matrices depending on local microstructural parameters.
  • 3.1. Macroscale: boundary value problem and compliance minimization: The optimization minimizes macroscale compliance under a target overall relative density while enforcing static equilibrium and filtered sensitivities.The filter cut-off radius controls the imposed minimum length scale.
  • 3.2. Microscale: spinodoid microstructures: A smooth transformation f(Θ) replaces disjoint admissible density and angle domains with continuous parameter spaces suitable for gradient-based optimization.It maps inadmissible density values toward void or the minimum spinodoid density, while λ1 and λ2 control smoothness.
  • 3.2. Microscale: spinodoid microstructures: Rigid-body rotations expand the design vector to χ = (ρ, θ1, θ2, θ3, α)^T and transform the homogenized stiffness into the rotated effective tensor passed to the macroscale.The demonstrated rotation is about the out-of-plane e3 axis, with α ∈ [−π/2, π/2].
  • 3.2. Microscale: spinodoid microstructures: Data-driven surrogate models are motivated by the repeated microscale homogenization and sensitivity computations required during iterative optimization.The optimal spatially varying parameters generate fully resolved multiscale topologies after optimization.

4. Data-driven surrogate model

The paper replaces repetitive FEM-based microscale homogenization with a deep neural-network surrogate mapping spinodoid parameters to effective stiffness. Automatic differentiation then supplies stiffness sensitivities needed for multiscale topology optimization.

  • 4.1. Bypassing FEM-based homogenization: A DNN surrogate maps spinodoid design parameters Θ to a compressed representation S of the orthotropic homogenized stiffness tensor.Orthotropy reduces the tensor representation to nine independent elastic moduli.
  • 4.1. Bypassing FEM-based homogenization: The network uses stacked linear transformations and ReLU activations to learn the nonlinear relationship between design parameters and homogenized stiffness values.The parameters are mapped through intermediate high-dimensional spaces because the stiffness space is highly disjoint.
  • 4.1. Bypassing FEM-based homogenization: The DNN is trained on randomly sampled spinodoid parameters paired with FEM-homogenized stiffnesses using back-propagation and Adam to minimize mean squared prediction error.The training dataset contains n_train parameter–stiffness pairs.
  • 4.1. Bypassing FEM-based homogenization: Once trained, the surrogate provides accurate stiffness predictions instantly, compared with several minutes for FEM-based homogenization on the study’s meshes.This speedup supports on-demand constitutive evaluation during optimization.
  • 4.2. Bypassing ND for sensitivity computations: Automatic differentiation computes stiffness sensitivities with respect to design parameters by propagating gradients through the differentiable DNN using the chain rule.This avoids the expense and stability or precision issues associated with numerical perturbations.

5. Results

The data-driven surrogate accurately predicts homogenized stiffness sensitivities and supports multiscale optimization with spatially varying spinodoid architectures. Across benchmarks, spinodoid designs improve compliance over SIMP, although gains diminish for multiple non-symmetric load cases.

  • Surrogate validation: R^2 ≥0.999 was obtained for each stiffness component when automatic-differentiation sensitivities were compared with finite-difference FEM sensitivities.The DNN was trained on 19,170 FEM-derived samples and tested on 2,130 independent samples.
  • Benchmark I: The cantilever’s optimized spinodoid design combines spatially varying density, anisotropy, and orientation, with cubic topologies dominating the final architecture.Microstructures are rotated so their preferred orientations follow the macroscale material distribution.
  • Benchmark II: The L-shaped spinodoid design improves compliance by 8.35% relative to SIMP through spatially varying anisotropy and columnar features aligned with principal stress directions.Increasing mesh resolution did not significantly affect the optimized compliance.
  • Benchmark IV: Under three non-symmetric load cases, spinodoids improve compliance by only 1.27% over SIMP, and this gain may partly reflect numerical artifacts.Different loading directions favor isotropic topologies as a compromise.

6. Conclusions

The paper presents a two-scale topology optimization framework using spatially variant spinodoid architectures and a deep-neural-network surrogate for homogenized stiffness. The approach supplies exact sensitivities through automatic differentiation and is general enough to extend beyond the studied architectures and boundary-value problems.

  • Framework: The framework jointly optimizes macroscale material distribution and spatially varying spinodoid microscale design and orientation.Spinodoids are represented by four Gaussian-random-field design parameters, while their effective response is homogenized locally.
  • Data-driven surrogate: A deep neural network maps spinodoid design parameters to the effective fourth-order stiffness tensor, replacing costly computational homogenization during optimization.The surrogate is trained using homogenized stiffness data obtained by finite elements.
  • Sensitivities: Automatic differentiation provides exact sensitivities required for gradient-based optimization without numerical derivatives.The conclusion identifies this strategy as potentially applicable beyond elastic stiffness.
  • Scope: The approach is sufficiently general to extend to other microscale architectures and more complex boundary value problems.This scope follows the framework’s formulation for macroscopic boundary-value problems with spatially variant microscale architectures.

Appendix A. Generating a spatially-variant spinodoid topology with fully resolved microstructure

The optimized quadrature-point design parameters are converted into a seamlessly varying, fully resolved spinodoid architecture by weighted superposition of local Gaussian random fields. A higher-resolution sampling mesh can resolve the resulting microscale structure without increasing FEM costs.

  • Spatial interpolation: The optimized macroscale design provides spatially varying parameters at finite-element quadrature points that must be converted into a continuous spinodoid architecture.The parameters are defined at quadrature points of the macroscale finite-element mesh.
  • Spatial interpolation: A partition-of-unity weighting function combines Gaussian random fields associated with nearby quadrature points.The weights satisfy a partition-of-unity condition, ensuring that the interpolation is properly normalized.
  • Topology construction: The spinodoid topology is obtained from the interpolated field by assigning solid material where the field is nonpositive and void where it is positive.This level-set rule converts the interpolated Gaussian field into the material distribution.
  • Spatial interpolation: Near a quadrature point, sufficiently rapid exponential decay makes the interpolated field approximate that point’s individual Gaussian random field.Away from individual points, neighboring fields are superposed to produce a smoothly varying topology across the domain.
  • Implementation: The interpolation sum can be truncated using a minimum weight cutoff, improving computational efficiency while retaining local contributions.The Gaussian-field wave number must also be sufficiently large to ensure the intended spatial behavior.
  • Implementation: A separate, much finer sampling mesh can resolve the microstructure for visualization and production while keeping the finite-element mesh coarse.Fabrication constraints and additive-manufacturing specifics are not considered.

Appendix B. Computational performance

The DNN surrogate substantially reduces stiffness-evaluation time relative to FEM homogenization, with runtimes reported as qualitative single-CPU estimates. The accompanying table documents computational resources and runtime conventions.

  • Runtime comparison: 0.001 seconds per prediction for the DNN is substantially less than the approximately 5 minutes required for FEM homogenization.The reported runtimes are intended as qualitative measures and refer to a single CPU.
  • Runtime comparison: DNN-based stiffness computation reduces computational time by several orders of magnitude compared with FEM-based computation.The comparison concerns stiffness computations using the surrogate model versus FEM homogenization.
  • Reporting conventions: Table B.1 lists computational resources and runtime conventions, including one-prediction timing for the DNN and hardware used for other computations.The table distinguishes the computational platforms used for the reported tasks.
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