cs.LGSep 5, 2026

Nonlinear elliptic homogenization with the parametric Deep Ritz method

Authors: Conor Rowan

Organizations: Smead Aerospace Engineering Sciences University of Colorado Boulder 3775 Discovery Drive Boulder, CO 80309

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Abstract

Elliptic homogenization is used to determine coarse-grained properties of materials with features on small scales. When these small scale features have rapid, periodic fluctuations, the solution field corresponding to a homogenized constitutive relation closely resembles the true solution based on the heterogeneous material. This homogenized behavior of the material is computed from a cell problem, where a cell is defined to be one period of the fluctuating material. In the context of linear elliptic partial differential equations, the homogenized constitutive relation is defined simply by a constant coefficient tensor, but for nonlinear problems, the homogenized response depends on the macroscopic state and/or its gradient, thus requiring solutions to parametric cell problems. When computing a numerical solution with the homogenized constitutive relation, it is useful to have a differentiable representation of the solution to the cell problem, as derivatives of the homogenized constitutive relation are required in Newton iterations for the macroscopic state field. In this work, we use the Deep Ritz method to solve the parametric cell problems that arise from nonlinear homogenization. First, we exploit the variational structure of the cell problem, then we discretize the dependence of the cell response on both space and the macroscopic state with a neural network. Enforcing boundary conditions on the cell response strongly, we next use the parametric Deep Ritz method to simultaneously solve the cell problem over a range of macroscopic states. We show that this method is accurate, efficient, and offers a continuous and differentiable representation of the cell response over the macroscopic state and gradient. We then show that our parametric representation of the cell response significantly expedites macroscale solutions when compared to a traditional FE2\text{FE}^2 scheme.

Figures & tables

Figure 1: As the scale η\eta of the material goes to zero, and the material property fluctuates at increasingly high frequencies, the oscillations in the state die out. When η\eta is sufficiently small, the system behaves approximately as if the material property were constant.
Figure 2: Extracting a cell of microstructure from a macroscopic domain with a periodic material property. The material property in the cell is indexed by y\mathbf{y} , and represents one period of the fluctuation.
Figure 3: As the scale parameter η\eta gets small, the fluctuations in the material become higher frequency and the cell appears to occupy a single point in the macroscopic domain. The idea of elliptic homogenization is to treat the constitutive relation at a point to be determined by the response of the cell to the temperature gradient at that point. This approach turns out to give good approximations of the system’s response even when the cells are not infinitesimally small. We motivate the notion of the fictitious spatially constant material by showing a material field for η1>η2>η3\eta_{1}>\eta_{2}>\eta_{3} . As η→0\eta\rightarrow 0 , the oscillations in the temperature die out, and the system response is accurately captured by a fictitious material with no spatial fluctuations.
Figure 4: Visualizing the temperature-dependent multiscale conductivity and the cell material properties at different temperatures. Both the amplitude and the shape of the fluctuation in the material property changes with the macroscopic temperature.
Figure 5: Sweeping over different resolutions of the basis expansion of the multiscale temperature, we see that coarse bases cannot capture even the macroscopic component of the solution.
Figure 6: Results for the one-dimensional Fourier-type heat conduction problem solved with the parametric Deep Ritz method. The energy converges in the allotted 50005000 epochs, and we compare the homogenized coefficients from the two numerical approximations of the parametric corrector field.
Figure 7: The homogenized response accurately captures the low-frequency part of the multiscale solution, which dominates the response. We use 20×20\times fewer basis functions in computing the homogenized response. The homogenized material property only depends on the temperature, and thus exhibits no rapid fluctuations.
Figure 8: The two-phase composite microstructure is built up of a petal-shaped inclusion and a surrounding matrix material. The conductivity of both phases depends on the temperature in different ways, and the cell has uniform conductivity at ∣u∣=1|u|=1 .
Figure 9: We compute the corrector corresponding to a unit y1y_{1} temperature gradient at each macroscopic temperature on the integration grid. We see that qualitative behavior of the corrector is similar at different macroscopic temperatures, but the sign flips when the more conductive phase swaps at u=1u=1 .
Figure 10: The corrector corresponding to a unit y2y_{2} temperature gradient is computed at each macroscopic temperature integration point. Note that when the corrector is computed in this brute force way, it is not represented as a continuous function of the macroscopic temperature. The unit corrector response to a vertical temperature gradient is a rigid rotation of the cell’s response to a horizontal temperature gradient.
Figure 11: The corrector response to a unit y1y_{1} heat flux computed with the neural network discretization. While we show the corrector field on integration points of the macroscopic temperature grid, the neural network provides the corrector as a continuous function of the macroscopic temperature.
Figure 12: The corrector response to a unit y2y_{2} heat flux computed with the neural network discretization.
Figure 13: Comparing the components of the homogenized conductivity tensor computed from both the Fourier and neural network discretizations of the solution. We visualize the homogenized tensor as a curve parameterized by the macroscopic temperature u0u_{0} . Because the off-diagonal term of the homogenized conductivity is negligibly small, the parameterized curve is plotted in the κ^11−κ^22\hat{\kappa}_{11}-\hat{\kappa}_{22} plane.
Figure 14: The metamaterial cell problem, made up of two material phases with different temperature dependence. Here, the conductivity of the cell is shown over a range of macroscopic temperatures.
Figure 15: Reference solutions obtained from the Fourier discretization of the pp -Laplace cell problem. Because we cannot exploit linearity of the cell problem, it is necessary to solve for the corrector at all settings of the macroscopic temperature and temperature gradient. We show the computed corrector χ(u0,∇xu0)\chi(u_{0},\nabla_{x}u_{0}) at 99 randomly sampled parameter settings.
Figure 16: Solutions obtained from the neural network at the same random settings of the macroscopic parameters as the reference solutions shown in Figure 15 .
Figure 17: The geometry of the inclusion in the cell changes with the macroscopic temperature. In particular, an elliptical inclusion rotates with the macroscopic temperature u0u_{0} . An interesting consequence of this choice of nonlinearity is that the anisotropy of the material depends on the macroscopic temperature.
Figure 18: Visualizing corrector fields χ(u0,∇xu0)\chi(u_{0},\nabla_{x}u_{0}) at randomly sampled macroscopic parameters. Recall that the elliptical inclusion rotates with the macroscopic temperature.
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 19: We consider a two-dimensional nonlinear heat transfer problem to motivate periodic boundary conditions on the corrector field χ(y)\chi(\mathbf{y}) . Because the cell problem states that the heat flux is divergence free, the equality of integrated heat fluxes through different parallel cross-sections can be guaranteed by boundary properties of the corrector.
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