Polar Interpolants for Thin-Shell Microstructure Homogenization

TL;DR

This work tackles the challenge of conservatively homogenizing thin-shell microstructures in a six-dimensional deformation space, where prior methods struggle to simultaneously ensure conservativeness and visual fidelity. It introduces a novel eigen-strain representation for membrane and bending, paired with polar high-order RBF interpolants to construct a 6D energy function defined over the eigen-strain domain, with a 5D training subspace due to data limitations. The authors detail a complete workflow—from controlled training data generation under periodic boundaries to COBYLA-based fitting and runtime isotropic blending—to produce robust, conservative material models that align well with both simulated and real-world 3D-printed microstructures. The resulting framework delivers higher accuracy than previous approaches and enables efficient, predictive macro-scale simulations for metamaterial design and analysis, with broad practical impact in engineering and graphics.

Abstract

This paper introduces a new formulation for material homogenization of thin-shell microstructures. It addresses important challenges that limit the quality of previous approaches: methods that fit the energy response neglect visual impact, methods that fit the stress response are not conservative, and all of them are limited to a low-dimensional interplay between deformation modes. The new formulation is rooted on the following design principles: the material energy functions are conservative by definition, they are formulated on the high-dimensional membrane and bending domain to capture the complex interplay of the different deformation modes, the material function domain is maximally aligned with the training data, and the material parameters and the optimization are formulated on stress instead of energy for better correlation with visual impact. The key novelty of our formulation is a new type of high-order RBF interpolant for polar coordinates, which allows us to fulfill all the design principles. We design a material function using this novel interpolant, as well as an overall homogenization workflow. Our results demonstrate very accurate fitting of diverse microstructure behaviors, both quantitatively and qualitatively superior to previous work.

Figures (11)

• Figure 1: Conservativeness of the mechanical model is critical for simulation robustness. The same microstructure is simulated with energy-based line-search optimization (left), suitable for conservative models, and with stress-norm line-search optimization (right), the common choice for non-conservative models. The latter fails to converge.
• Figure 2: The same microstructure (left) is stretched $10\%$ (middle) and bend (right). Energy is $100\times$ higher under stretch, which indicates little correlation between energy and visual deformation.
• Figure 3: The distribution of training data is more regular and better suited for RBF interpolation on the eigen-strain domain $(\lambda_E, \mu_E, \theta_E)$ (right) than on the Green-strain domain $E$ (left).
• Figure 4: The plot shows, for Microstructure 1, the energy density wrt bending direction under uniform unaxial curvature. This microstructure reveals a clear 90-degree asymmetry, underlining the importance of fitting the material model on a range $\left[ 0, \pi \right)$ of principal directions.
• Figure 5: Same as in Fig. \ref{['fig:teaser']}, we compare twist deformations of real-world microstructures (top row), micro-scale simulations (second row), and macro-scale simulations computed with our homogenized material models (third row). But we also add simulation results with an energy-fit model (bottom row). From left to right, the images show microstructures 1, 3, 4 and 5.