Barrier potentials gained popularity as a means for robust contact handling in physical modeling and for modeling self-avoiding shapes. The key to the success of these approaches is adherence to geometric constraints, i.e., avoiding intersections, which are the cause of most robustness problems in complex deformation simulation with contact. However, existing barrier-potential methods may lead to spurious forces and imperfect satisfaction of the geometric constraints. They may have strong resolution dependence, requiring careful adaptation of the potential parameters to the object discretizations. We present a systematic derivation of a continuum potential defined for smooth and piecewise smooth surfaces, starting from identifying a set of natural requirements for contact potentials, including the barrier property, locality, differentiable dependence on shape, and absence of forces in rest configurations. Our potential is formulated independently of surface discretization and addresses the shortcomings of existing potential-based methods while retaining their advantages. We present a discretization of our potential that is a drop-in replacement for the potential used in the incremental potential contact formulation [Li et al. 2020], and compare its behavior to other potential formulations, demonstrating that it has the expected behavior. The presented formulation connects existing barrier approaches, as all recent existing methods can be viewed as a variation of the presented potential, and lays a foundation for developing alternative (e.g., higher-order) versions.

We introduce a general differentiable solver for time-dependent deformation problems with contact and friction. Our approach uses a finite element discretization with a high-order time integrator coupled with the recently proposed incremental potential contact method for handling contact and friction forces to solve ODE- and PDE-constrained optimization problems on scenes with complex geometry. It supports static and dynamic problems and differentiation with respect to all physical parameters involved in the physical problem description, which include shape, material parameters, friction parameters, and initial conditions. Our analytically derived adjoint formulation is efficient, with a small overhead (typically less than 10% for nonlinear problems) over the forward simulation, and shares many similarities with the forward problem, allowing the reuse of large parts of existing forward simulator code. We implement our approach on top of the open-source PolyFEM library and demonstrate the applicability of our solver to shape design, initial condition optimization, and material estimation on both simulated results and physical validations.

We propose In-Timestep Remeshing, a fully coupled, adaptive meshing algorithm for contacting elastodynamics where remeshing steps are tightly integrated, implicitly, within the timestep solve. Our algorithm refines and coarsens the domain automatically by measuring physical energy changes within each ongoing timestep solve. This provides consistent, degree-of-freedom-efficient, productive remeshing that, by construction, is physics-aware and so avoids the errors, over-refinements, artifacts, per-example hand-tuning, and instabilities commonly encountered when remeshing with timestepping methods. Our in-timestep computation then ensures that each simulation step's output is both a converged stable solution on the updated mesh and a temporally consistent trajectory with respect to the model and solution of the last timestep. At the same time, the output is guaranteed safe (intersection- and inversion-free) across all operations. We demonstrate applications across a wide range of extreme stress tests with challenging contacts, sharp geometries, extreme compressions, large timesteps, and wide material stiffness ranges - all scenarios well-appreciated to challenge existing remeshing methods.

Abstract We introduce the firstexactroot parity counter for continuous collision detection (CCD). That is, our algorithm computes the parity (even or odd) of the number of roots of the cubic polynomial arising from a CCD query. We note that the parity is unable to differentiate between zero (no collisions) and the rare case of two roots (collisions). Our method does not have numerical parameters to tune, has a performance comparable to efficient approximate algorithms, and is exact. We test our approach on a large collection of synthetic tests and real simulations, and we demonstrate that it can be easily integrated into existing simulators.

We introduce the first implicit time-stepping algorithm for rigid body dynamics, with contact and friction, that guarantees intersection-free configurations at every time step. Our algorithm explicitly models the curved trajectories traced by rigid bodies in both collision detection and response. For collision detection, we propose a conservative narrow phase collision detection algorithm for curved trajectories, which reduces the problem to a sequence of linear CCD queries with minimal separation. For time integration and contact response, we extend the recently proposed incremental potential contact framework to reduced coordinates and rigid body dynamics. We introduce a benchmark for rigid body simulation and show that our approach, while less efficient than alternatives, can robustly handle a wide array of complex scenes, which cannot be simulated with competing methods, without requiring per-scene parameter tuning.

We introduce a large-scale benchmark for continuous collision detection (CCD) algorithms, composed of queries manually constructed to highlight challenging degenerate cases and automatically generated using existing simulators to cover common cases. We use the benchmark to evaluate the accuracy, correctness, and efficiency of state-of-the-art continuous collision detection algorithms, both with and without minimal separation. We discover that, despite the widespread use of CCD algorithms, existing algorithms are (1) correct but impractically slow; (2) efficient but incorrect, introducing false negatives that will lead to interpenetration; or (3) correct but over conservative, reporting a large number of false positives that might lead to inaccuracies when integrated in a simulator. By combining the seminal interval root finding algorithm introduced by Snyder in 1992 with modern predicate design techniques, we propose a simple and efficient CCD algorithm. This algorithm is competitive with state-of-the-art methods in terms of runtime while conservatively reporting the time of impact and allowing explicit tradeoff between runtime efficiency and number of false positives reported.

Contacts weave through every aspect of our physical world, from daily household chores to acts of nature. Modeling and predictive computation of these phenomena for solid mechanics is important to every discipline concerned with the motion of mechanical systems, including engineering and animation. Nevertheless, efficiently time-stepping accurate and consistent simulations of real-world contacting elastica remains an outstanding computational challenge. To model the complex interaction of deforming solids in contact we propose Incremental Potential Contact (IPC) - a new model and algorithm for variationally solving implicitly time-stepped nonlinear elastodynamics. IPC maintains an intersection- and inversion-free trajectory regardless of material parameters, time step sizes, impact velocities, severity of deformation, or boundary conditions enforced. Constructed with a custom nonlinear solver, IPC enables efficient resolution of time-stepping problems with separate, user-exposed accuracy tolerances that allow independent specification of the physical accuracy of the dynamics and the geometric accuracy of surface-to-surface conformation. This enables users to decouple, as needed per application, desired accuracies for a simulation's dynamics and geometry. The resulting time stepper solves contact problems that are intersection-free (and thus robust), inversion-free, efficient (at speeds comparable to or faster than available methods that lack both convergence and feasibility), and accurate (solved to user-specified accuracies). To our knowledge this is the first implicit time-stepping method, across both the engineering and graphics literature that can consistently enforce these guarantees as we vary simulation parameters. In an extensive comparison of available simulation methods, research libraries and commercial codes we confirm that available engineering and computer graphics methods, while each succeeding admirably in custom-tuned regimes, often fail with instabilities, egregious constraint violations and/or inaccurate and implausible solutions, as we vary input materials, contact numbers and time step. We also exercise IPC across a wide range of existing and new benchmark tests and demonstrate its accurate solution over a broad sweep of reasonable time-step sizes and beyond (up to h = 2s) across challenging large-deformation, large-contact stress-test scenarios with meshes composed of up to 2.3M tetrahedra and processing up to 498K contacts per time step. For applications requiring high-accuracy we demonstrate tight convergence on all measures. While, for applications requiring lower accuracies, e.g. animation, we confirm IPC can ensure feasibility and plausibility even when specified tolerances are lowered for efficiency.

3-axis milling enables cheap and precise fabrication of target objects from precut slabs of materials such as wood or stone. However, the space of directly millable shapes is limited since a 3-axis mill can only carve a height-field (HF) surface during each milling and their size is bounded by the slab dimensions, one of which, the height , is typically significantly smaller than the other two for many typical materials. Extending 3-axis milling of precut slabs to general arbitrarily-sized shapes requires decomposing them into bounded-height 3-axis millable parts, or slices , which can be individually milled and then assembled to form the target object. We present DHFSlicer , a novel decomposition method that satisfies the above constraints and significantly reduces both milling time and material waste compared to alternative approaches. We satisfy the fabrication constraints by partitioning target objects into double height-field (DHF) slices, which can be fabricated using two milling passes: the HF surface accessible from one side is milled first, the slice is then flipped using appropriate fixtures, and then the second, remaining, HF surface is milled. DHFSlicer uses an efficient coarse-to-fine decomposition process: It first partitions the inputs into maximally coarse blocks that satisfy a local DHF criterion with respect to per-block milling axes, and then cuts each block into well-sized DHF slices. It minimizes milling time and material waste by keeping the slice count small, and maximizing slice height. We validate our method by embedding it within an end-to-end DHF milling pipeline and fabricating objects from slabs of foam, wood, and MDF; demonstrate that using the obtained slices reduces milling time and material waste by 42% on average compared to existing automatic alternatives; and highlight the benefits of DHFSlicer via extensive ablation studies.

We introduce the first fully automatic pipeline to convert arbitrary 3D shapes into knit models. Our pipeline is based on a global parametrization remeshing pipeline to produce an isotropic quad-dominant mesh aligned with a 2-RoSy field. The knitting directions over the surface are determined using a set of custom topological operations and a two-step global optimization that minimizes the number of irregularities. The resulting mesh is converted into a valid stitch mesh that represents the knit model. The yarn curves are generated from the stitch mesh and the final yarn geometry is computed using a yarn-level relaxation process. Thus, we produce topologically valid models that can be used with a yarn-level simulation. We validate our algorithm by automatically generating knit models from complex 3D shapes and processing over a hundred models with various shapes without any user input or parameter tuning. We also demonstrate applications of our approach for custom knit model generation using fabrication via 3D printing.

A parameterization decouples the resolution of a signal on a surface from the resolution of the surface geometry. In practice, parameterized signals are conveniently and efficiently stored as texture images. Unfortunately, seams are inevitable when parametrizing most surfaces. Their visual artifacts are well known for color signals, but become even more egregious when geometry or displacement signals are used: cracks or gaps may appear in the surface. To make matters worse, parameterizations and their seams are frequently ignored during mesh processing. Carefully accounting for seams in one phase may be nullified by the next. The existing literature on seam-elimination requires non-standard rendering algorithms or else overly restricts the parameterization and signal. We present seam-aware mesh processing techniques. For a given fixed mesh, we analytically characterize the space of seam-free textures as the null space of a linear operator. Assuming seam-free textures, we describe topological and geometric conditions for seam-free edge-collapse operations. Our algorithms eliminate seam artifacts in parameterized signals and decimate a mesh---including its seams---while preserving its parameterization and seam-free appearance. This allows the artifact-free display of surface signals---color, normals, positions, displacements, linear blend skinning weights---with the standard GPU rendering pipeline. In particular, our techniques enable crack-free use of the tessellation stage of modern GPU's for dynamic level-of-detail. This decouples the shape signal from mesh resolution in a manner compatible with existing workflows.