Inverse designing surface curvatures by deep learning

TL;DR

This work establishes curvature as a controllable design modality for porous microstructures by coupling a curvature-based energy functional with a phase-field topology generator and a dual neural network framework. A forward NN learns the mapping from design parameters to curvature encodings, while an inverse NN retrieves design parameters that realize targeted curvature profiles, addressing ill-posedness by training through the forward surrogate. The method demonstrates strong generalization beyond the training space, successfully matching curvature targets derived from trabecular bone, spinodoid, and periodic nodal surface geometries, and links curvature to mechanics via a membrane-versus-bending energy decomposition that favors stretching-dominated responses near zero mean curvature. Overall, the framework enables rapid, robust inverse design of smooth curved microstructures with potential applications in mechanical metamaterials and bio-scaffolds, and it highlights curvature as a powerful lever to tailor mechanical resilience.

Abstract

Smooth and curved microstructural topologies found in nature - from soap films to trabecular bone - have inspired several mimetic design spaces for architected metamaterials and bio-scaffolds. However, the design approaches so far have been ad hoc, raising the challenge: how to systematically and efficiently inverse design such artificial microstructures with targeted topological features? Here, we explore surface curvature as a design modality and present a deep learning framework to produce topologies with as-desired curvature profiles. The inverse design framework can generalize to diverse topological features such as tubular, membranous, and particulate features. Moreover, we demonstrate successful generalization beyond both the design and data space by inverse designing topologies that mimic the curvature profile of trabecular bone, spinodoid topologies, and periodic nodal surfaces for application in bio-scaffolds and implants. Lastly, we bridge curvature and mechanics by showing how topological curvature can be designed to promote mechanically beneficial stretching-dominated deformation over bending-dominated deformation.

Figures (14)

• Figure 1: Natural materials and structures such as trabecular bone, soap films, porous ceramics, and vascular systems tend to have smooth curved topologies. Borrowing inspiration from those topologies, the ML-based inverse design framework provides a unified solution for on-demand rational design or mimicry of various smooth topologies with targeted curvature profiles. Since the curvature profile of a topology is directly related to its mechanical properties, this approach provides an efficient pathway to designing mechanical metamaterials with superior properties for applications in light-weight engineering materials, bio-mimetic structures, bio-implants, and more. Images adapted: trabecular bone by Laboratoires Servier, CC BY-SA 3.0, via Wikimedia Commons;bonesoap film by Blinking Spirit, CC0, via Wikimedia Commons;soapfilmporous ceramics by Onnovisser1979, CC BY-SA 3.0, via Wikimedia Commons;ceramicvasculature by I'm in the garden, CC BY-SA 3.0, via Wikimedia Commons.vessels
• Figure 1: Distribution of components of $\boldsymbol{\Theta}$ in the dataset before scaling.
• Figure 2: (a) Schematic of topology generation using computationally expensive phase-field simulation based on curvature-driven energy functional. (b) Representative selection of diverse topologies and their corresponding curvature profiles for different design parameters $\boldsymbol{\Theta}$. Typical microstructural features include spheres, tubules, or membranes. The curvature profile is visualized as the density scatter of the (surface) element-wise principal curvatures ($\kappa_1$ and $\kappa_2$); the density color of each scatter point is proportional to the cumulative surface area of the elements with similar principal curvatures in the mesh. Two elements are said to have the similar principal curvatures if they lie in the same cell of a finely-gridded $\kappa_1$-$\kappa_2$ plane.
• Figure 2: Distribution of maximal value of each component of curvature encodings in the dataset: (a) before and (b) after scaling.
• Figure 3: Geometric interpretation of the design space. For the cases when (a)$\alpha>0$, (b)$\alpha=0$, and (c)$\alpha<0$, the surface energy density $f[S]$ can be visualized as different kinds of quadratic surfaces (left column) and the corresponding contours (middle column) of $\kappa_1$-$\kappa_2$. For each case, a representative example (right column) shows the qualitative correspondence between relevant contours of $f[S]$ and the resulting curvature profile.