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Lightweight, mesh-level models of knit fabric behavior are useful for both interactive pattern editing and initialization of yarn-level simulations. However, existing mesh-level simulation methods abstract knitting as a homogeneous material, which prevents them from capturing more complicated mixed structures. Furthermore, these methods require different simulation parameters depending on the knit pattern, or arrangement of stitches within the knit. Thus, fitting these parameters to physical examples must be done for each new pattern, even when the same types of stitches are used. To address this, we observe that physical behavior of a stitch is determined not only by its individual structure but also by the stitch types that surround it. In our work, we extend the stitch mesh model to allow for neighbor-aware material properties at the stitch level. Using structural analysis of stitch connections, we derive a finite set of four-way kernels that combine to create general knit-purl patterns for relaxation. From this, we generate a set of reference patterns that can be measured to infer the rest-lengths of the kernels using a linear model. After knitting and measuring these reference patterns, we used the derived kernel rest lengths to run relaxation on our stitch mesh models with mixtures of knits and purls that we then validated against physical examples. Our results show that the 4 neighbors of each stitch is sufficient to account for much of the neighborhood-dependent deformation, while remaining simple enough to directly fit to measured data with a set of 11 basis swatches. This allows our relaxation method to efficiently estimate the rest shape of mixed knit-purl patterns, which enables fast fabric preview and more accurate yarn-level simulation. 

Cosserat rods have become an increasingly popular framework for simulating complex bending and twisting in thin elastic rods, used for hair, tree, and yarn-level cloth models. However, traditional approaches often encounter significant challenges in robustly and efficiently solving for valid quaternion orientations, even when employing small time steps or computationally expensive global solvers. We introduce stable Cosserat rods, a new solver that can achieve high accuracy with high stiffness levels and maintain stability under large time steps. It is also inherently suitable for parallelization. Our key contribution is a split position and rotation optimization scheme with a closed-form Gauss-Seidel quasi-static orientation update. This solver significantly improves the numerical stability with Cosserat rods, allowing faster computation and larger time steps. We validate our method across a wide range of applications, including simulations of hair, trees, yarn-level cloth, slingshots, and bridges, demonstrating its ability to handle diverse material behaviors and complex geometries. Furthermore, we show that our method is orders of magnitude faster and more stable than alternative rod solvers, such as extended position-based dynamics and discrete elastic rods. 

We present a novel contact model, termed Offset Geometric Contact (OGC), for guaranteed penetration-free simulation of codimensional objects with minimal computational overhead. Our method is based on constructing a volumetric shape by offsetting each face along its normal direction, ensuring orthogonal contact forces, thus allows large contact radius without artifacts. We compute vertex-specific displacement bounds to guarantee penetration-free simulation, which improves convergence and avoids the need for expensive continuous collision detection. Our method relies solely on massively parallel local operations, avoiding global synchronization and enabling efficient GPU implementation. Experiments demonstrate real-time, large-scale simulations with performance more than two orders of magnitude faster than prior methods while maintaining consistent computational budgets. 

We introduce vertex block descent, a block coordinate descent solution for the variational form of implicit Euler through vertex-level Gauss-Seidel iterations. It operates with local vertex position updates that achieve reductions in global variational energy with maximized parallelism. This forms a physics solver that can achieve numerical convergence with unconditional stability and exceptional computation performance. It can also fit in a given computation budget by simply limiting the iteration count while maintaining its stability and superior convergence rate. We present and evaluate our method in the context of elastic body dynamics, providing details of all essential components and showing that it outperforms alternative techniques. In addition, we discuss and show examples of how our method can be used for other simulation systems, including particle-based simulations and rigid bodies. 

Recent advancements in spatiotemporal reservoir resampling (ReSTIR) leverage sample reuse from neighbors to efficiently evaluate the path integral. Like rasterization, ReSTIR methods implicitly assume a pinhole camera and evaluate the light arriving at a pixel through a single predetermined subpixel location at a time (e.g., the pixel center). This prevents efficient path reuse in and near pixels with high-frequency details. We introduceArea ReSTIR, extending ReSTIR reservoirs to also integrate each pixel's 4D ray space, including 2D areas on the film and lens. We design novel subpixel-tracking temporal reuse and shift mappings that maximize resampling quality in such regions. This robustifies ReSTIR against high-frequency content, letting us importance sample subpixel and lens coordinates and efficiently render antialiasing and depth of field. 

Strand-based hair simulations have recently become increasingly popular for a range of real-time applications. However, accurately simulating the full number of hair strands remains challenging. A commonly employed technique involves simulating a subset of guide hairs to capture the overall behavior of the hairstyle. Details are then enriched by interpolation using linear skinning. Hair interpolation enables fast real-time simulations but frequently leads to various artifacts during runtime. As the skinning weights are often pre-computed, substantial variations between the initial and deformed shapes of the hair can cause severe deviations in fine hair geometry. Straight hairs may become kinked, and curly hairs may become zigzags. This work introduces a novel physical-driven hair interpolation scheme that utilizes existing simulated guide hair data. Instead of directly operating on positions, we interpolate the internal forces from the guide hairs before efficiently reconstructing the rendered hairs based on their material model. We formulate our problem as a constraint satisfaction problem for which we present an efficient solution. Further practical considerations are addressed using regularization terms that regulate penetration avoidance and drift correction. We have tested various hairstyles to illustrate that our approach can generate visually plausible rendered hairs with only a few guide hairs and minimal computational overhead, amounting to only about 20% of conventional linear hair interpolation. This efficiency underscores the practical viability of our method for real-time applications. 

We introduce a method for efficiently computing the exact shortest path to the boundary of a mesh from a given internal point in the presence of self-intersections. We provide a formal definition of shortest boundary paths for self-intersecting objects and present a robust algorithm for computing the actual shortest boundary path. The resulting method offers an effective solution for collision and self-collision handling while simulating deformable volumetric objects, using fast simulation techniques that provide no guarantees on collision resolution. Our evaluation includes complex self-collision scenarios with a large number of active contacts, showing that our method can successfully handle them by introducing a relatively minor computational overhead. 

Lagrangian/Eulerian hybrid strand-based hair simulation techniques have quickly become a popular approach in VFX and real-time graphics applications. With Lagrangian hair dynamics, the inter-hair contacts are resolved in the Eulerian grid using the continuum method, i.e., the MPM scheme with the granular Drucker-Prager rheology, to avoid expensive collision detection and handling. This fuzzy collision handling makes the authoring process significantly easier. However, although current hair grooming tools provide a wide range of strand-based modeling tools for this simulation approach, the crucial sag-free initialization functionality remains often ignored. Thus, when the simulation starts, gravity would cause any artistic hairstyle to sag and deform into unintended and undesirable shapes. This paper proposes a novel four-stage sag-free initialization framework to solve stable quasistatic configurations for hybrid strand-based hair dynamic systems. These four stages are split into two global-local pairs. The first one ensures static equilibrium at every Eulerian grid node with additional inequality constraints to prevent stress from exiting the yielding surface. We then derive several associated closed-form solutions in the local stage to compute segment rest lengths, orientations, and particle deformation gradients in parallel. The second global-local step solves along each hair strand to ensure all the bend and twist constraints produce zero net torque on every hair segment, followed by a local step to adjust the rest Darboux vectors to a unit quaternion. We also introduce an essential modification for the Darboux vector to eliminate the ambiguity of the Cosserat rod rest pose in both initialization and simulation. We evaluate our method on a wide range of hairstyles, and our approach can only take a few seconds to minutes to get the rest quasistatic configurations for hundreds of hair strands. Our results show that our method successfully prevents sagging and has minimal impact on the hair motion during simulation. 

In recent years, reservoir-based spatiotemporal importance resampling (ReSTIR) algorithms appeared out of nowhere to take parts of the realtime rendering community by storm, with sample reuse speeding direct lighting from millions of dynamic lights [1], diffuse multi-bounce lighting [2], participating media [3], and even complex global illumination paths [4]. Highly optimized variants (e.g. [5]) can give 100x efficiency improvement over traditional ray- and path-tracing methods; this is key to achieve 30 or 60 Hz framerates. In production engines, tracing even one ray or path per pixel may only be feasible on the highest-end systems, so maximizing image quality per sample is vital. ReSTIR builds on the math in Talbot et al.'s [6] resampled importance sampling (RIS), which previously was not widely used or taught, leaving many practitioners missing key intuitions and theoretical grounding. A firm grounding is vital, as seemingly obvious "optimizations" arising during ReSTIR engine integration can silently introduce conditional probabilities and dependencies that, left ignored, add uncontrollable bias to the results. In this course, we plan to: 1. Provide concrete motivation and intuition for why ReSTIR works, where it applies, what assumptions it makes, and the limitations of today's theory and implementations; 2. Gently develop the theory, targeting attendees with basic Monte Carlo sampling experience but without prior knowledge of resampling algorithms (e.g., Talbot et al. [6]); 3. Give explicit algorithmic samples and pseudocode, pointing out easily-encountered pitfalls when implementing ReSTIR; 4. Discuss actual game integrations, highlighting the gotchas, challenges, and corner cases we encountered along the way, and highlighting ReSTIR's practical benefits.