Abstract Modeling contact between deformable solids is a fundamental problem in computer animation, mechanical design, and robotics. Existing methods based on C 0 ‐discretizations—piece‐wise linear or polynomial surfaces—suffer from discontinuities and irregularities in tangential contact forces, which can significantly affect simulation outcomes and even prevent convergence. In this work, we show that these limitations can be overcome with a smooth surface representation based on Implicit Moving Least Squares (IMLS). In particular, we propose a self collision detection scheme tailored to IMLS surfaces that enables robust and efficient handling of challenging self contacts. Through a series of test cases, we show that our approach offers advantages over existing methods in terms of accuracy and robustness for both forward and inverse problems.

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

Navigating topological transitions in cellular mechanical systems is a significant challenge for existing simulation methods. While abstract models lack predictive capabilities at the cellular level, explicit network representations struggle with topology changes, and per-cell representations are computationally too demanding for large-scale simulations. To address these challenges, we propose a novel cell-centered approach based on differentiable Voronoi diagrams. Representing each cell with a Voronoi site, our method defines shape and topology of the interface network implicitly. In this way, we substantially reduce the number of problem variables, eliminate the need for explicit contact handling, and ensure continuous geometry changes during topological transitions. Closed-form derivatives of network positions facilitate simulation with Newton-type methods for a wide range of per-cell energies. Finally, we extend our differentiable Voronoi diagrams to enable coupling with arbitrary rigid and deformable boundaries. We apply our approach to a diverse set of examples, highlighting splitting and merging of cells as well as neighborhood changes. We illustrate applications to inverse problems by matching soap foam simulations to real-world images. Comparative analysis with explicit cell models reveals that our method achieves qualitatively comparable results at significantly faster computation times.

Nonlinear metamaterials with tailored mechanical properties have applications in engineering, medicine, robotics, and beyond. While modeling their macromechanical behavior is challenging in itself, finding structure parameters that lead to ideal approximation of high-level performance goals is a challenging task. In this work, we propose Neural Metamaterial Networks (NMN)---smooth neural representations that encode the nonlinear mechanics of entire metamaterial families. Given structure parameters as input, NMN return continuously differentiable strain energy density functions, thus guaranteeing conservative forces by construction. Though trained on simulation data, NMN do not inherit the discontinuities resulting from topo-logical changes in finite element meshes. They instead provide a smooth map from parameter to performance space that is fully differentiable and thus well-suited for gradient-based optimization. On this basis, we formulate inverse material design as a nonlinear programming problem that leverages neural networks for both objective functions and constraints. We use this approach to automatically design materials with desired strain-stress curves, prescribed directional stiffness and Poisson ratio profiles. We furthermore conduct ablation studies on network nonlinearities and show the advantages of our approach compared to native-scale optimization.

Recent monocular human performance capture approaches have shown compelling dense tracking results of the full body from a single RGB camera. However, existing methods either do not estimate clothing at all or model cloth deformation with simple geometric priors instead of taking into account the underlying physical principles. This leads to noticeable artifacts in their reconstructions, e.g. baked-in wrinkles, implausible deformations that seemingly defy gravity, and intersections between cloth and body. To address these problems, we propose a person-specific, learning-based method that integrates a simulation layer into the training process to provide for the first time physics supervision in the context of weakly supervised deep monocular human performance capture. We show how integrating physics into the training process improves the learned cloth deformations, allows modeling clothing as a separate piece of geometry, and largely reduces cloth-body intersections. Relying only on weak 2D multi-view supervision during training, our approach leads to a significant improvement over current state-of-the-art methods and is thus a clear step towards realistic monocular capture of the entire deforming surface of a clothed human.

Recent advances in implicit neural representations show great promise when it comes to generating numerical solutions to partial differential equations. Compared to conventional alternatives, such representations employ parameterized neural networks to define, in a mesh-free manner, signals that are highly-detailed, continuous, and fully differentiable. In this work, we present a novel machine learning approach for topology optimization---an important class of inverse problems with high-dimensional parameter spaces and highly nonlinear objective landscapes. To effectively leverage neural representations in the context of mesh-free topology optimization, we use multilayer perceptrons to parameterize both density and displacement fields. Our experiments indicate that our method is highly competitive for minimizing structural compliance objectives, and it enables self-supervised learning of continuous solution spaces for topology optimization problems.