State-of-the-art cloth simulations rely on linear triangular elements in mass-spring or continuum based finite element formulations. These methods typically decompose the surface energy density into in-plane (shearing and stretching) and out-of-plane (bending) components, with bending energies modeled using discrete mean curvature measures. While effective, they are prone to mesh-dependent behavior and locking. Higher-order formulations can mitigate these issues, but their adoption poses significant challenges due to the requirement for continuity of basis functions’ derivatives across element boundaries to accurately represent surface curvature. We introduce a novel continuum-based approach that addresses the limitations of existing methods without requiring globally smooth (H2-continuous) basis functions. Our method uses non-conforming function spaces and weakly enforces the continuity of tangent basis through carefully derived interface terms. In fact, the proposed method builds on Interior Penalty methods, which we adapt to effectively handle simulations of curved surfaces. Our approach uses standard Lagrangian basis functions, and supports straightforward extension to high-order bases, while adhering to the in-plane/out-of-plane decoupling paradigm widely adopted in cloth simulation. We demonstrate the robustness and versatility of our method through garment simulations, illustrating its ability to handle complex deformations and a variety of bending behaviors with high fidelity.

We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems, but only when equipped with powerful preconditioners. Unfortunately, effective preconditioners like multigrid require costly setup phases that must be re-executed every time domain shapes or boundary conditions change, forming a severe bottleneck for problems with evolving boundaries. In contrast, we present a neural preconditioner trained to efficiently approximate the inverse of the discrete Laplacian in the presence of such changes. Our approach generalizes to domain shapes, boundary conditions, and grid sizes outside the training set. The key to our preconditioner's success is a novel, lightweight neural network architecture featuring spatially varying convolution kernels and supporting fast inference. We demonstrate that our solver outperforms state-of-the-art methods like algebraic multigrid as well as recently proposed neural preconditioners on challenging test cases arising from incompressible fluid simulations.

We present a novel Material Point Method (MPM) discretization of surface tension forces that arise from spatially varying surface energies. These variations typically arise from surface energy dependence on temperature and/or concentration. Furthermore, since the surface energy is an interfacial property depending on the types of materials on either side of an interface, spatial variation is required for modeling the contact angle at the triple junction between a liquid, solid and surrounding air. Our discretization is based on the surface energy itself, rather than on the associated traction condition most commonly used for discretization with particle methods. Our energy based approach automatically captures surface gradients without the explicit need to resolve them as in traction condition based approaches. We include an implicit discretization of thermomechanical material coupling with a novel particle-based enforcement of Robin boundary conditions associated with convective heating. Lastly, we design a particle resampling approach needed to achieve perfect conservation of linear and angular momentum with Affine-Particle-In-Cell (APIC) [Jiang et al. 2015]. We show that our approach enables implicit time stepping for complex behaviors like the Marangoni effect and hydrophobicity/hydrophilicity. We demonstrate the robustness and utility of our method by simulating materials that exhibit highly diverse degrees of surface tension and thermomechanical effects, such as water, wine and wax.