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Simultaneous and Meshfree Topology Optimization with Physics-informed Gaussian Processes

Amin Yousefpour, Shirin Hosseinmardi, Carlos Mora, Ramin Bostanabad

TL;DR

The paper addresses topology optimization without meshes or nested solvers by formulating a simultaneous, meshfree approach for minimizing dissipated power in Stokes/Brinkman flow. It introduces Gaussian processes on both design and state variables with a shared deep neural network mean, and solves a single penalized objective $\mathcal{L}(\boldsymbol{\zeta})$ that couples PDE constraints and design constraints, while BCs/ICs are satisfied by GP conditioning. Key findings show discretization-invariant designs, elimination of density filtering, and consistent computational costs with robustness to initialization, achieving results comparable to a commercial SIMP-based solver across four problems. This framework advances TO by delivering an end-to-end, meshfree method capable of handling complex domains and multi-physics scenarios.

Abstract

Topology optimization (TO) provides a principled mathematical approach for optimizing the performance of a structure by designing its material spatial distribution in a pre-defined domain and subject to a set of constraints. The majority of existing TO approaches leverage numerical solvers for design evaluations during the optimization and hence have a nested nature and rely on discretizing the design variables. Contrary to these approaches, herein we develop a new class of TO methods based on the framework of Gaussian processes (GPs) whose mean functions are parameterized via deep neural networks. Specifically, we place GP priors on all design and state variables to represent them via parameterized continuous functions. These GPs share a deep neural network as their mean function but have as many independent kernels as there are state and design variables. We estimate all the parameters of our model in a single for loop that optimizes a penalized version of the performance metric where the penalty terms correspond to the state equations and design constraints. Attractive features of our approach include $(1)$ having a built-in continuation nature since the performance metric is optimized at the same time that the state equations are solved, and $(2)$ being discretization-invariant and accommodating complex domains and topologies. To test our method against conventional TO approaches implemented in commercial software, we evaluate it on four problems involving the minimization of dissipated power in Stokes flow. The results indicate that our approach does not need filtering techniques, has consistent computational costs, and is highly robust against random initializations and problem setup.

Simultaneous and Meshfree Topology Optimization with Physics-informed Gaussian Processes

TL;DR

The paper addresses topology optimization without meshes or nested solvers by formulating a simultaneous, meshfree approach for minimizing dissipated power in Stokes/Brinkman flow. It introduces Gaussian processes on both design and state variables with a shared deep neural network mean, and solves a single penalized objective that couples PDE constraints and design constraints, while BCs/ICs are satisfied by GP conditioning. Key findings show discretization-invariant designs, elimination of density filtering, and consistent computational costs with robustness to initialization, achieving results comparable to a commercial SIMP-based solver across four problems. This framework advances TO by delivering an end-to-end, meshfree method capable of handling complex domains and multi-physics scenarios.

Abstract

Topology optimization (TO) provides a principled mathematical approach for optimizing the performance of a structure by designing its material spatial distribution in a pre-defined domain and subject to a set of constraints. The majority of existing TO approaches leverage numerical solvers for design evaluations during the optimization and hence have a nested nature and rely on discretizing the design variables. Contrary to these approaches, herein we develop a new class of TO methods based on the framework of Gaussian processes (GPs) whose mean functions are parameterized via deep neural networks. Specifically, we place GP priors on all design and state variables to represent them via parameterized continuous functions. These GPs share a deep neural network as their mean function but have as many independent kernels as there are state and design variables. We estimate all the parameters of our model in a single for loop that optimizes a penalized version of the performance metric where the penalty terms correspond to the state equations and design constraints. Attractive features of our approach include having a built-in continuation nature since the performance metric is optimized at the same time that the state equations are solved, and being discretization-invariant and accommodating complex domains and topologies. To test our method against conventional TO approaches implemented in commercial software, we evaluate it on four problems involving the minimization of dissipated power in Stokes flow. The results indicate that our approach does not need filtering techniques, has consistent computational costs, and is highly robust against random initializations and problem setup.
Paper Structure (15 sections, 27 equations, 12 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 27 equations, 12 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nomenclature
  3. Background and Related Works
  4. Proposed Methodology
  5. Parameterization of Design Space and State Variables
  6. Efficient Parameter Estimation
  7. High-Order Finite Difference
  8. Dynamic Loss Weighting
  9. Results and Discussions
  10. Summary of Comparative Studies
  11. Evolution of Topology, Objective Function, and Design Constraint
  12. Evolution of PDE Residuals
  13. Conclusions and Future Directions
  14. Solving PDE Systems with Neural Networks
  15. Loss Histories

Figures (12)

  • Figure 1: Nested and discretized vs simultaneous and mesh-free topology optimization: Rough flowcharts of the SIMP method (a) and our approach (b) for topology optimization. Algorithmic details are intentionally excluded from these flowcharts.
  • Figure 2: Simultaneous and meshfree topology optimization:$\rho(\boldsymbol{\mathrm{x}})$ denotes the design variable and it is assumed that the structure has two state variables $\boldsymbol{\mathrm{u}}(\boldsymbol{\mathrm{x}}) = \mathopen{}\left[u_1(\boldsymbol{\mathrm{x}}), u_2(\boldsymbol{\mathrm{x}})\right]\mathclose{}$. The covariance matrices ensure that the variables satisfy the boundary conditions while the parameters of the mean function are optimized to minimize \ref{['eq penalty opt generic']}. In practice, we fix the length-scale parameters of all the kernels to $10^2$ and only optimize $\boldsymbol{\theta}$ via \ref{['eq penalty opt generic 2']}.
  • Figure 3: Design domain and the imposed boundary conditions: The domain is a unit square in all cases and the boundary conditions are shown in each case. Pressure is known at only one point in all cases.
  • Figure 4: Topology evolution across epochs: The evolution of $\rho(\boldsymbol{\mathrm{x}})$ is visualized with respect to the objective function over $50,000$ epochs. Optimal topology is obtained as $\mathcal{J}$ converges to its minimum. All quantities are the median across $10$ training repetitions.
  • Figure 5: Volume loss history: Each curve represents the median values across $10$ training repetitions. For comparison, in each case we also provide the error obtained by COMSOL (with random initialization) in satisfying the volume fraction constraint, see the markers at epoch $50k$.
  • ...and 7 more figures