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 propose a conservative algorithm to test the geometrical validity of simplicial (triangles, tetrahedra), tensor product (quadrilaterals, hexahedra), and mixed (prisms) elements of arbitrary polynomial order as they deform linearly within a time interval. Our algorithm uses a combination of adaptive Bézier refinement and bisection search to determine if, when, and where the Jacobian determinant of an element’s polynomial geometric map becomes negative in the transition from one configuration to another. In elastodynamic simulation, our algorithm guarantees that the system remains physically valid during the entire trajectory, not only at discrete time steps. Unlike previous approaches, physical validity is preserved even when our method is implemented using floating point arithmetic. Hence, our algorithm is only slightly slower than existing non-conservative methods while providing guarantees and while being an easy drop-in replacement for current validity tests. To prove the practical effectiveness of our algorithm, we demonstrate its use in a high-order Incremental Potential Contact (IPC) elastodynamic simulator and experimentally show that it prevents invalid, simulation-breaking configurations that would otherwise occur using non-conservative methods.

Many problems in computer graphics can be formulated as finding the global minimum of a function subject to a set of non-linear constraints (Minimize), or finding all solutions of a system of non-linear constraints (Solve). We introduce MiSo, a domain-specific language and compiler for generating efficient C++ code for low-dimensional Minimize and Solve problems, that uses interval methods to guarantee conservative results while using floating point arithmetic. We demonstrate that MiSo-generated code shows competitive performance compared to hand-optimized codes for several computer graphics problems, including high-order collision detection with non-linear trajectories, surface-surface intersection, and geometrical validity checks for finite element simulation.

Abstract Two‐scale topology optimization, combined with the design of microstructure families with a broad range of effective material parameters, is widely used in many fabrication applications to achieve a target deformation behavior for a variety of objects. The main idea of this approach is to optimize the distribution of material properties in the object partitioned into relatively coarse cells, and then replace each cell with microstructure geometry that mimics these material properties. In this paper, we focus on adapting this approach to complex shapes in situations when preserving the shape's surface is essential. Our approach extends any regular (i.e. defined on a regular lattice grid) microstructure family to complex shapes, by enriching it with tiles adapted to the geometry of the cut‐cell. We propose a fully automated and robust pipeline based on this approach, and we show that the performance of the regular microstructure family is only minimally affected by our extension while allowing its use on 2D and 3D shapes of high complexity.

Mechanical shock is a common occurrence in various settings, there are two different scenarios for shock protection: catastrophic protection (e.g. car collisions and falls) and routine protection (e.g. shoe soles and mattresses). The former protects against one-time events, the latter against periodic shocks and loads. Common shock absorbers based on plasticity and fracturing materials are suitable for the former, while our focus is on the latter, where elastic structures are useful. Further, we optimize the effective elastic material properties which control the critical shock parameter, maximal stress, with energy dissipation by viscous forces assumed adequate. Improved elastic materials protecting against shock can be used in applications such as automotive suspension, furniture like sofas and mattresses, landing gear systems, etc. Materials offering optimal protection against shock have a highly non-linear elastic response: their reaction force needs to be as close as possible to constant with respect to deformation. In this paper, we use shape optimization and topology search to design 2D families of microstructures approximating the ideal behavior across a range of deformations, leading to superior shock protection. We present an algorithmic pipeline for the optimal design of such families combining differentiable nonlinear homogenization with self-contact and an optimization algorithm. We validate the effectiveness of our advanced 2D designs by extruding and fabricating them with 3D printing technologies and performing material and drop testing.

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.