Publications

Newton's Method is widely used to find the solution of complex non-linear simulation problems. To guarantee a descent direction, it is common practice to clamp the negative eigenvalues of each element Hessian prior to assembly — a strategy known as Projected Newton (PN) — but this perturbation often hinders convergence. In this work, we observe that projecting only a small subset of element Hessians is sufficient to secure a descent direction. Building on this insight, we introduce Progressively Projected Newton (PPN), a novel variant of Newton's Method that uses the current iterate's residual to cheaply determine the subset of element Hessians to project. The benefit is twofold: most eigendecompositions are avoided and the global Hessian remains closer to its original form, reducing the number of Newton iterations. We compare PPN with PN and Project-on-Demand Newton (PDN) in a comprehensive set of experiments covering contact-free and contact-rich deformables, co-dimensional and rigid-body simulations, and a range of time step sizes, tolerances and resolutions. PPN reduces the amount of element projections in dynamic simulations by one order of magnitude while simultaneously improving convergence, consistently being the fastest solver in our benchmark.

Neural shape representation generally refers to representing 3D geometry using neural networks, e.g., computing a signed distance or occupancy value at a specific spatial position. In this paper we present a neural-network architecture suitable for accurate encoding of 3D shapes in a single forward pass. Our architecture is based on a multi-scale hybrid system incorporating graph-based and voxel-based components, as well as a continuously differentiable decoder. The hybrid system includes a novel way of voxelizing point-based features in neural networks by projecting the point "feature-field" onto a grid. This projection is insensitive to local point density, and we show that it can be used to obtain smoother and more detailed reconstructions, in particular when combined with oriented point clouds as input. Our architecture also requires only a single forward pass, instead of the latent-code optimization used in auto-decoder methods. Furthermore, our network is trained to solve the well-established eikonal equation and only requires knowledge of the zero-level set for training and inference. We additionally propose a modification to the aforementioned loss function for the case that surface normals are not well defined, e.g., in the context of non-watertight surfaces and non-manifold geometry. Overall, our method consistently outperforms other baselines on the surface reconstruction task across a wide variety of datasets, while being more computationally efficient and requiring fewer parameters.