Abstract The hybrid Lagrangian/Eulerian formulation of continuum shells is highly effective for producing challenging simulations of thin materials like cloth with bending resistance and frictional contact. However, existing formulations are restricted to materials that do not undergo tearing nor fracture due to the difficulties associated with incorporating strong discontinuities of field quantities like velocity via basis enrichment while maintaining continuity or regularity. We propose an extension of this formulation to simulate dynamic tearing and fracturing of thin shells using Kirchhoff–Love continuum theory. Damage, which manifests as cracks or tears, is propagated by tracking the evolution of a time‐dependent phase‐field in the co‐dimensional manifold, where a moving least‐squares (MLS) approximation then captures the strong discontinuities of interpolated field quantities near the crack. Our approach is capable of simulating challenging scenarios of this tearing and fracture, all‐the‐while harnessing the existing benefits of the hybrid Lagrangian/Eulerian formulation to expand the domain of possible effects. The method is also amenable to user‐guided control, which serves to influence the propagation of cracks or tears such that they follow prescribed paths during simulation.

Barrier functions are crucial for maintaining an intersection- and inversion-free simulation trajectory but existing methods, which directly use distance can restrict implementation design and performance. We present an approach to rewriting the barrier function for arriving at an efficient and robust approximation of its Hessian. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, from which analytic eigensystems are derived and enhanced with filtering and stiffening terms that ensure robustness with respect to the convergence of a Project-Newton solver. A further advantage of our rewriting of the barrier function is that it naturally caters to the notorious case of nearly parallel edge-edge contacts for which we also present a novel analytic eigensystem. Our approach is thus well suited for standard second-order unconstrained optimization strategies for resolving contacts, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The efficiency of our eigensystems alone yields a 3× speedup over the standard Incremental Potential Contact (IPC) barrier formulation. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration.

Anisotropic hyperelastic distortion energies are used to solve many problems in fields like computer graphics and engineering with applications in shape analysis, deformation, design, mesh parameterization, biomechanics, and more. However, formulating a robust anisotropic energy that is low order and yet sufficiently non-linear remains a challenging problem for achieving the convergence promised by Newton-type methods in numerical optimization. In this article, we propose a novel analytic formulation of an anisotropic energy that is smooth everywhere, low order, rotationally invariant, and at least twice differentiable. At its core, our approach utilizes implicit rotation factorizations with invariants of the Cauchy-Green tensor that arises from the deformation gradient. The versatility and generality of our analysis is demonstrated through a variety of examples, where we also show that the constitutive law suggested by the anisotropic version of the well-known As-Rigid-As-Possible energy is the foundational parametric description of both passive and active elastic materials. The generality of our approach means that we can systematically derive the force and force-Jacobian expressions for use in implicit and quasistatic numerical optimization schemes, and we can also use our analysis to rewrite, simplify, and speed up several existing anisotropic and isotropic distortion energies with guaranteed inversion safety.

Large-scale topological changes play a key role in capturing the fine debris of fracturing virtual brittle material. Real-world, tough brittle fractures have dynamic branching behaviour but numerical simulation of this phenomena is notoriously challenging. In order to robustly capture these visual characteristics, we simulate brittle fracture by combining elastodynamic continuum mechanical models with rigid-body methods: A continuum damage mechanics (CDM) problem is solved, following rigid-body impact, to simulate crack propagation by tracking a damage field. We combine the result of this elastostatic continuum model with a novel technique to approximate cracks as a non-manifold mid-surface, which enables accurate and robust modelling of material fragment volumes to compliment fast-and-rigid shatter effects. For enhanced realism, we add fracture detail, incorporating particle damage-time to inform localised perturbation of the crack surface with artistic control. We evaluate our method with numerous examples and comparisons, showing that it produces a breadth of brittle material fracture effects and with low simulation resolution to require much less time compared to fully elastodynamic simulations.

Isotropic As-Rigid-As-Possible (ARAP) energy has been popular for shape editing, mesh parametrisation and soft-body simulation for almost two decades. However, a formulation using Cauchy-Green (CG) invariants has always been unclear, due to a rotation-polluted trace term that cannot be directly expressed using these invariants. We show how this incongruent trace term can be understood via an implicit relationship to the CG invariants. Our analysis reveals this relationship to be a polynomial where the roots equate to the trace term, and where the derivatives also give rise to closed-form expressions of the Hessian to guarantee positive semi-definiteness for a fast and concise Newton-type implicit time integration. A consequence of this analysis is a novel analytical formulation to compute rotations and singular values of deformation-gradient tensors without explicit/numerical factorization which is significant, resulting in up-to 3.5× speedup and benefits energy function evaluation for reducing solver time. We validate our energy formulation by experiments and comparison, demonstrating that our resulting eigendecomposition using the CG invariants is equivalent to existing ARAP formulations. We thus reveal isotropic ARAP energy to be a member of the "Cauchy-Green club", meaning that it can indeed be defined using CG invariants and therefore that the closed-form expressions of the resulting Hessian are shared with other energies written in their terms.

Abstract We present a simple, efficient and low‐memory technique, targeting fast construction of bounding volume hierarchies (BVH) for broad‐phase collision detection. To achieve this, we devise a novel representation of BVH trees in memory. We develop a mapping of the implicit index representation to compact memory locations, based on simple bit‐shifts, to then construct and evaluate bounding volume test trees (BVTT) during collision detection with real‐time performance. We model the topology of the BVH tree implicitly as binary encodings which allows us to determine the nodes missing from a complete binary tree using the binary representation of the number of missing nodes. The simplicity of our technique allows for fast hierarchy construction achieving over 6× speedup over the state‐of‐the‐art. Making use of these characteristics, we show that not only it is feasible to rebuild the BVH at every frame, but that using our technique, it is actually faster than refitting and more memory efficient.

Abstract We present a remeshing‐free brittle fracture simulation method under the assumption of quasi‐static linear elastic fracture mechanics (LEFM). To achieve this, we devise two algorithms. First, we develop an approximate volumetric simulation, based on the extended Finite Element Method (XFEM), to initialize and propagate Lagrangian crack‐fronts. We model the geometry of fracture explicitly as a surface mesh, which allows us to generate high‐resolution crack surfaces that are decoupled from the resolution of the deformation mesh. Our second contribution is a mesh cutting algorithm, which produces fragments of the input mesh using the fracture surface. We do this by directly operating on the half‐edge data structures of two surface meshes, which enables us to cut general surface meshes including those of concave polyhedra and meshes with abutting concave polygons. Since we avoid triangulation for cutting, the connectivity of the resulting fragments is identical to the (uncut) input mesh except at edges introduced by the cut. We evaluate our simulation and cutting algorithms and show that they outperform state‐of‐the‐art approaches both qualitatively and quantitatively.

Simultaneous BVH traversal, as a dynamic task of pair-wise proximity tests, poses several challenges in terms of parallelization using GPUs. It is a highly dynamic and data-dependent problem which can induce control-flow divergence and inefficient data-access patterns. We present a simple solution using the bulk-synchronous parallel model to ensure a uniform mode of execution, and balanced workloads across GPU threads. The method is easy to implement, fast and operates entirely on the GPU by relying on a topology-centred work expansion scheme to ensure large concurrent workloads. We demonstrate speedups of upto 7.1x over the widely used "streams" model for GPU based parallel collision detection.