Boundary element methods (BEM) for solving linear elliptic partial differential equations have gained traction in a wide range of graphics applications: they eliminate the need for volumetric meshing by solving for variables exclusively on the domain boundary through a linear boundary integral equation (BIE). However, BEM often generate dense and ill-conditioned linear systems that lead to poor computational scalability and substantial memory demands for large-scale problems, limiting their applicability and efficiency in practice. In this paper, we address these limitations by generalizing the Kaporin-based approach to asymmetric preconditioning: we construct a sparse approximation of the inverse-LU factorization of arbitrary BIE matrices in a massively parallel manner. Our sparse inverse-LU factorization, when employed as a preconditioner for the generalized minimal residual (GMRES) method, significantly enhances the efficiency of BIE solves, often yielding orders-of-magnitude speedups in solving times.

Many 3D mesh processing tasks revolve around generating and manipulating curves on surface meshes. While it is intuitive to explicitly model these curves using mesh edges or parametric curves in the ambient space, these methods often suffer from numerical instability or inaccuracy due to the projection operation. Another natural strategy is to adapt spline based tools, these methods are quite fast but are hard to be extended to more versatile constraints and need heavy manual interactions. In this article, we present an efficient and versatile approach to curve design based on an implicit representation known as the level set. While previous works have explored the use of the level set to generate curves with minimal length, they typically have limitations in accommodating additional conditions for rich and robust control. To address these challenges, we formulate curve editing with constraints like smoothness, interpolation, tangent control, etc., via a level set based variational problem by constraining the values or derivatives of the level set function. However, the widely used gradient flow strategy converges very slowly for this complicated variational problem compared to the classical geodesic one. Thus, we propose to solve it via Newton's method enhanced by local Hessian correction and a trust-region strategy. As a result, our method not only enables versatile control, but also excels in terms of performance due to nearly quadratic convergence and almost linear complexity in each iteration via narrow band acceleration. In practice, these advantages effectively benefit various applications, such as interactive curve manipulation, boundary smoothing for surface segmentation and path planning with obstacles as demonstrated.

We propose a simple method for the intuitive authoring and efficient animation of virtual tornadoes. Users control the tornado kinematics by sketching two types of curves to specify the initial geometry of the tornado’s core and the profile of the surrounding swirling air, known as the funnel. The first input, a 3D curve, initializes the core as a vortex filament. This filament induces a swirl flow and advects according to its initial curvature, resulting in progressive bending and twisting. The second input consists of one or multiple 2D profile curves that parameterize the Stokes stream function, governing the radial and axial motion of the air around the core and thereby dictate the funnel shape over time. The core and funnel profile are coupled in local frames through closed-form velocities, which together describe the rotation, sliding and uplift within the tornado’s air volume. As shown in our case studies, our method provides a controllable and efficient way to animate visually plausible tornadoes capable of tearing off infrastructure and transporting debris, as well as interacting with uneven terrain.

Abstract Explosive volcanic eruptions rank among the most terrifying natural phenomena, and are thus frequently depicted in films, games, and other media, usually with a bespoke once‐off solution. In this paper, we introduce the first general‐purpose model for bi‐directional interaction between the atmosphere and a volcano plume. In line with recent interactive volcano models, we approximate the plume dynamics with Lagrangian disks and spheres and the atmosphere with sparse layers of 2D Eulerian grids, enabling us to focus on the transfer of physical quantities such as temperature, ash, moisture, and wind velocity between these sub‐models. We subsequently generate volumetric animations by noise‐based procedural upsampling keyed to aspects of advection, convection, moisture, and ash content to generate a fully‐realized volcanic skyscape. Our model captures most of the visually salient features emerging from volcano‐sky interaction, such as windswept plumes, enmeshed cap, bell and skirt clouds, shockwave effects, ash rain, and sheathes of lightning visible in the dark.

Abstract Interactive deformation via control handles is essential in computer graphics for the modeling of 3D geometry. Deformation control structures include lattices for free‐form deformation and skeletons for character articulation, but this report focuses on cage‐based deformation. Cages for deformation control are coarse polygonal meshes that encase the to‐be‐deformed geometry, enabling high‐resolution deformation. Cage‐based deformation enables users to quickly manipulate 3D geometry by deforming the cage. Due to their utility, cage‐based deformation techniques increasingly appear in many geometry modeling applications. For this reason, the computer graphics community has invested a great deal of effort in the past decade and beyond into improving automatic cage generation and cage‐based deformation. Recent advances have significantly extended the practical capabilities of cage‐based deformation methods. As a result, there is a large body of research on cage‐based deformation. In this report, we provide a comprehensive overview of the current state of the art in cage‐based deformation of 3D geometry. We discuss current methods in terms of deformation quality, practicality, and precomputation demands. In addition, we highlight potential future research directions that overcome current issues and extend the set of practical applications. In conjunction with this survey, we publish an application to unify the most relevant deformation methods. Our report is intended for computer graphics researchers, developers of interactive geometry modeling applications, and 3D modeling and character animation artists.

The method of fundamental solutions (MFS) and its associated boundary element method (BEM) have gained popularity in computer graphics due to the reduced dimensionality they offer: for three-dimensional linear problems, they only require variables on the domain boundary to solve and evaluate the solution throughout space, making them a valuable tool in a wide variety of applications. However, MFS and BEM have poor computational scalability and huge memory requirements for large-scale problems, limiting their applicability and efficiency in practice. By leveraging connections with Gaussian Processes and exploiting the sparse structure of the inverses of boundary integral matrices, we introduce a variational preconditioner that can be computed via a sparse inverse-Cholesky factorization in a massively parallel manner. We show that applying our preconditioner to the Preconditioned Conjugate Gradient algorithm greatly improves the efficiency of MFS or BEM solves, up to four orders of magnitude in our series of tests.

As a natural extension to the harmonic coordinates, the biharmonic coordinates have been found superior for planar shape and image manipulation with an enriched deformation space. However, the 3D biharmonic coordinates and their derivatives have remained unexplored. In this work, we derive closed-form expressions for biharmonic coordinates and their derivatives for 3D triangular cages. The core of our derivation lies in computing the closed-form expressions for the integral of the Euclidean distance over a triangle and its derivatives. The derived 3D biharmonic coordinates not only fill a missing component in methods of generalized barycentric coordinates but also pave the way for various interesting applications in practice, including producing a family of biharmonic deformations, solving variational shape deformations, and even unlocking the closed-form expressions for recently-introduced Somigliana coordinates for both fast and accurate evaluations.

Abstract Denoising is a common, yet critical operation in geometry processing aiming at recovering high‐fidelity models of piecewise‐smooth objects from noise‐corrupted pointsets. Despite a sizable literature on the topic, there is a dearth of approaches capable of processing very noisy and outlier‐ridden input pointsets for which no normal estimates and no assumptions on the underlying geometric features or noise type are provided. In this paper, we propose a new robust‐statistics approach to denoising pointsets based on line processes to offer robustness to noise and outliers while preserving sharp features possibly present in the data. While the use of robust statistics in denoising is hardly new, most approaches rely on prescribed filtering using data‐independent blending expressions based on the spatial and normal closeness of samples. Instead, our approach deduces a geometric denoising strategy through robust and regularized tangent plane fitting of the initial pointset, obtained numerically via alternating minimizations for efficiency and reliability. Key to our variational approach is the use of line processes to identify inliers vs. outliers, as well as the presence of sharp features. We demonstrate that our method can denoise sampled piecewise‐smooth surfaces for levels of noise and outliers at which previous works fall short.

Step-and-project is a popular method to simulate non-penetrating deformable bodies in physically based animation. The strategy is to first integrate the system in time without considering contacts and then resolve potential intersections, striking a good balance between plausibility and efficiency. However, existing methods can be defective and unsafe when using large time steps, taking risks of failure or demanding repetitive collision testing and resolving that severely degrade performance. In this article, we propose a novel two-way method for fast and reliable continuous collision handling. Our method launches an optimization from both ends of the intermediate time-integrated state and the previous intersection-free state. It progressively generates a piecewise linear path and eventually obtains a feasible solution for the next time step. The algorithm efficiently alternates between a forward step and a backward step until the result is conditionally converged. Thanks to a set of unified volume-based contact constraints, our method offers flexible and reliable handling of various codimensional deformable bodies, including volumetric bodies, cloth, hair, and sand. Experimental results demonstrate the safety, robustness, physical fidelity, and numerical efficiency of our method, making it particularly suitable for scenarios involving large deformations or large time steps.

Many computer graphics applications boil down to solving sparse systems of linear equations. While the current arsenal of numerical solvers available in various specialized libraries and for different computer architectures often allow efficient and scalable solutions to image processing, modeling and simulation applications, an increasing number of graphics problems face large-scale and ill-conditioned sparse linear systems --- a numerical challenge which typically chokes both direct factorizations (due to high memory requirements) and iterative solvers (because of slow convergence). We propose a novel approach to the efficient preconditioning of such problems which often emerge from the discretization over unstructured meshes of partial differential equations with heterogeneous and anisotropic coefficients. Our numerical approach consists in simply performing a fine-to-coarse ordering and a multiscale sparsity pattern of the degrees of freedom, using which we apply an incomplete Cholesky factorization. By further leveraging supernodes for cache coherence, graph coloring to improve parallelism and partial diagonal shifting to remedy negative pivots, we obtain a preconditioner which, combined with a conjugate gradient solver, far exceeds the performance of existing carefully-engineered libraries for graphics problems involving bad mesh elements and/or high contrast of coefficients. We also back the core concepts behind our simple solver with theoretical foundations linking the recent method of operator-adapted wavelets used in numerical homogenization to the traditional Cholesky factorization of a matrix, providing us with a clear bridge between incomplete Cholesky factorization and multiscale analysis that we leverage numerically.

Abstract Rod‐like one‐dimensional elastic objects often exhibit complex behaviors which pose great challenges to the discretization method for pursuing a faithful simulation. By only moving a small portion of material points, the Eulerian‐on‐Lagrangian (EoL) method already shows great adaptivity to handle sharp contact, but it is still far from enough to reproduce rich and complex geometry details arising in simulations. In this paper, we extend the discrete configuration space by unifying all Lagrangian and EoL nodes in representation for even more adaptivity with every sample being assigned with a dynamic material coordinate. However, this great extension will immediately bring in much more redundancy in the dynamic system. Therefore, we propose additional energy to control the spatial distribution of all material points, seeking to equally space them with respect to a curvature‐based density field as a monitor. This flexible approach can effectively constrain the motion of material points to resolve numerical degeneracy, while simultaneously enables them to notably slide inside the parametric domain to account for the shape parameterization. Besides, to accurately respond to sharp contact, our method can also insert or remove nodes online and adjust the energy stiffness to suppress possible jittering artifacts that could be excited in a stiff system. As a result of this hybrid rh‐adaption, our proposed method is capable of reproducing many realistic rod dynamics, such as excessive bending, twisting and knotting while only using a limited number of elements.

In this paper, we introduce a hierarchical construction of material-adapted refinable basis functions and associated wavelets to offer efficient coarse-graining of linear elastic objects. While spectral methods rely on global basis functions to restrict the number of degrees of freedom, our basis functions are locally supported; yet, unlike typical polynomial basis functions, they are adapted to the material inhomogeneity of the elastic object to better capture its physical properties and behavior. In particular, they share spectral approximation properties with eigenfunctions, offering a good compromise between computational complexity and accuracy. Their construction involves only linear algebra and follows a fine-to-coarse approach, leading to a block-diagonalization of the stiffness matrix where each block corresponds to an intermediate scale space of the elastic object. Once this hierarchy has been precomputed, we can simulate an object at runtime on very coarse resolution grids and still capture the correct physical behavior, with orders of magnitude speedup compared to a fine simulation. We show on a variety of heterogeneous materials that our approach outperforms all previous coarse-graining methods for elasticity.

In this paper, an efficient and scalable approach for simulating inhomogeneous and non-linear elastic objects is introduced. Our numerical coarsening approach consists in optimizing non-conforming and matrix-valued shape functions to allow for predictive simulation of heterogeneous materials with non-linear constitutive laws even on coarse grids, thus saving orders of magnitude in computational time compared to traditional finite element computations. The set of local shape functions over coarse elements is carefully tailored in a preprocessing step to balance geometric continuity and local material stiffness. In particular, we do not impose continuity of our material-aware shape functions between neighboring elements to significantly reduce the fictitious numerical stiffness that conforming bases induce; however, we enforce crucial geometric and physical properties such as partition of unity and exact reproduction of representative fine displacements to eschew the use of discontinuous Galerkin methods. We demonstrate that we can simulate, with no parameter tuning, inhomogeneous and non-linear materials significantly better than previous approaches that traditionally try to homogenize the constitutive model instead.