gProximity: Hierarchical GPU‐based Operations for Collision and Distance Queries

We present novel parallel algorithms for collision detection and separation distance computation for rigid and deformable models that exploit the computational capabilities of many‐core GPUs. Our approach uses thread and data parallelism to perform fast hierarchy construction, updating, and traversal using tight‐fitting bounding volumes such as oriented bounding boxes (OBB) and rectangular swept spheres (RSS). We also describe efficient algorithms to compute a linear bounding volume hierarchy (LBVH) and update them using refitting methods. Moreover, we show that tight‐fitting bounding volume hierarchies offer improved performance on GPU‐like throughput architectures. We use our algorithms to perform discrete and continuous collision detection including self‐collisions, as well as separation distance computation between non‐overlapping models. In practice, our approach (gProximity) can perform these queries in a few milliseconds on a PC with NVIDIA GTX 285 card on models composed of tens or hundreds of thousands of triangles used in cloth simulation, surgical simulation, virtual prototyping and N‐body simulation. Moreover, we observe more than an order of magnitude performance improvement over prior GPU‐based algorithms.     

Hybrid Simulation of Miscible Mixing with Viscous Fingering

By modeling mass transfer phenomena, we simulate solids and liquids dissolving or changing to other substances. We also deal with the very small‐scale phenomena that occur when a fluid spreads out at the interface of another fluid. We model the pressure at the interfaces between fluids with Darcy's Law and represent the viscous fingering phenomenon in which a fluid interface spreads out with a fractal‐like shape. We use hybrid grid‐based simulation and smoothed particle hydrodynamics (SPH) to simulate intermolecular diffusion and attraction using particles at a computable scale. We have produced animations showing fluids mixing and objects dissolving.     

Realtime Two‐Way Coupling of Meshless Fluids and Nonlinear FEM

In this paper, we present a novel method to couple Smoothed Particle Hydrodynamics (SPH) and nonlinear FEM to animate the interaction of fluids and deformable solids in real time. To accurately model the coupling, we generate proxy particles over the boundary of deformable solids to facilitate the interaction with fluid particles, and develop an efficient method to distribute the coupling forces of proxy particles to FEM nodal points. Specifically, we employ the Total Lagrangian Explicit Dynamics (TLED) finite element algorithm for nonlinear FEM because of many of its attractive properties such as supporting massive parallelism, avoiding dynamic update of stiffness matrix computation, and efficient solver. Based on a predictor‐corrector scheme for both velocity and position, different normal and tangential conditions can be realized even for shell‐like thin solids. Our coupling method is entirely implemented on modern GPUs using CUDA. We demonstrate the advantage of our two‐way coupling method in computer animation via various virtual scenarios.     

Fast and Robust Normal Estimation for Point Clouds with Sharp Features

This paper presents a new method for estimating normals on unorganized point clouds that preserves sharp features. It is based on a robust version of the Randomized Hough Transform (RHT). We consider the filled Hough transform accumulator as an image of the discrete probability distribution of possible normals. The normals we estimate corresponds to the maximum of this distribution. We use a fixed‐size accumulator for speed, statistical exploration bounds for robustness, and randomized accumulators to prevent discretization effects. We also propose various sampling strategies to deal with anisotropy, as produced by laser scans due to differences of incidence. Our experiments show that our approach offers an ideal compromise between precision, speed, and robustness: it is at least as precise and noise‐resistant as state‐of‐the‐art methods that preserve sharp features, while being almost an order of magnitude faster. Besides, it can handle anisotropy with minor speed and precision losses.     

Incompressible SPH using the Divergence‐Free Condition

In this paper, we present a novel SPH framework to simulate incompressible fluid that satisfies both the divergence‐ free condition and the density‐invariant condition. In our framework, the two conditions are applied separately. First, the divergence‐free condition is enforced when solving the momentum equation. Later, the density‐invariant condition is applied after the time integration of the particle positions. Our framework is a purely Lagrangian approach so that no auxiliary grid is required. Compared to the previous density‐invariant based SPH methods, the proposed method is more accurate due to the explicit satisfaction of the divergence‐free condition. We also propose a modified boundary particle method for handling the free‐slip condition. In addition, two simple but effective methods are proposed to reduce the particle clumping artifact induced by the density‐invariant condition.     

Multi‐Resolution Cloth Simulation

We propose a novel, multi‐resolution method to efficiently perform large‐scale cloth simulation. Our cloth simulation method is based on a triangle‐based energy model constructed from a cloth mesh. We identify that solutions of the linear system of cloth simulation are smooth in certain regions of the cloth mesh and solve the linear system on those regions in a reduced solution space. Then we reconstruct the original solutions by performing a simple interpolation from solutions computed in the reduced space. In order to identify regions where solutions are smooth, we propose simplification metrics that consider stretching, shear, and bending forces, as well as geometric collisions. Our multi‐resolution method can be applied to many existing cloth simulation methods, since our method works on a general linear system. In order to demonstrate benefits of our method, we apply our method into four large‐scale cloth benchmarks that consist of tens or hundreds of thousands of triangles. Because of the reduced computations, we achieve a performance improvement by a factor of up to one order of magnitude, with a little loss of simulation quality.     

Tetrahedral Embedded Boundary Methods for Accurate and Flexible Adaptive Fluids

When simulating fluids, tetrahedral methods provide flexibility and ease of adaptivity that Cartesian grids find difficult to match. However, this approach has so far been limited by two conflicting requirements. First, accurate simulation requires quality Delaunay meshes and the use of circumcentric pressures. Second, meshes must align with potentially complex moving surfaces and boundaries, necessitating continuous remeshing. Unfortunately, sacrificing mesh quality in favour of speed yields inaccurate velocities and simulation artifacts. We describe how to eliminate the boundary‐matching constraint by adapting recent embedded boundary techniques to tetrahedra, so that neither air nor solid boundaries need to align with mesh geometry. This enables the use of high quality, arbitrarily graded, non‐conforming Delaunay meshes, which are simpler and faster to generate. Temporal coherence can also be exploited by reusing meshes over adjacent timesteps to further reduce meshing costs. Lastly, our free surface boundary condition eliminates the spurious currents that previous methods exhibited for slow or static scenarios. We provide several examples demonstrating that our efficient tetrahedral embedded boundary method can substantially increase the flexibility and accuracy of adaptive Eulerian fluid simulation.     

Scales and Scale‐like Structures

We present a method for generating scales and scale‐like structures on a polygonal mesh through surface replacement. As input, we require a triangular mesh that will be covered with scales and one or more proxy‐models to be used as the scale's shape. A user begins scale generation by drawing a lateral line on the model to control the distribution and orientation of scales on the surface. We then create a vector field over the surface to control an anisotropic Voronoi tessellation, which represents the region occupied by each scale. Next we replace these regions by cutting the proxy model to match the boundary of the Voronoi region and deform the cut model onto the surface. The result is a fully connected 2‐manifold that is suitable for subsequent post‐processing applications like surface subdivision.     

Hybrid Particle‐grid Modeling for Multi‐scale Droplet/Spray Simulation

This paper presents a novel hybrid particle‐grid method that tightly couples Lagrangian particle approach with Eulerian grid approach to simulate multi‐scale diffuse materials varying from disperse droplets to dissipating spray and their natural mixture and transition, originated from a violent (high‐speed) liquid stream. Despite the fact that Lagrangian particles are widely employed for representing individual droplets and Eulerian grid‐based method is ideal for volumetric spray modeling, using either one alone has encountered tremendous difficulties when effectively simulating droplet/spray mixture phenomena with high fidelity. To ameliorate, we propose a new hybrid model to tackle such challenges with many novel technical elements. At the geometric level, we employ the particle and density field to represent droplet and spray respectively, modeling their creation from liquid as well as their seamless transition. At the physical level, we introduce a drag force model to couple droplets and spray, and specifically, we employ Eulerian method to model the interaction among droplets and marry it with the widely‐used Lagrangian model. Moreover, we implement our entire hybrid model on CUDA to guarantee the interactive performance for high‐effective physics‐based graphics applications. The comprehensive experiments have shown that our hybrid approach takes advantages of both particle and grid methods, with convincing graphics effects for disperse droplets and spray simulation.     

A Global Parity Measure for Incomplete Point Cloud Data

Shapes with complex geometric and topological features such as tunnels, neighboring sheets, and cavities are susceptible to undersampling and continue to challenge existing reconstruction techniques. In this work we introduce a new measure for point clouds to determine the likely interior and exterior regions of an object. Specifically, we adapt the concept of parity to point clouds with missing data and introduce the parity map, a global measure of parity over the volume. We first examine how parity changes over the volume with respect to missing data and develop a method for extracting topologically correct interior and exterior crusts for estimating a signed distance field and performing surface reconstruction. We evaluate our approach on real scan data representing complex shapes with missing data. Our parity measure is not only able to identify highly confident interior and exterior regions but also localizes regions of missing data. Our reconstruction results are compared to existing methods and we show that our method faithfully captures the topology and geometry of complex shapes in the presence of missing data.     

On the shape of a set of points and lines in the plane

Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct urban models. In urban scenes, the individual surfaces can be reconstructed and connected to form the scene geometry. There are various methods for reconstructing the free‐form shape of a point sample on a single surface. However, these methods do not take the context of the surface into account. We present the guided α‐shape: an extension of the well known α‐shape that uses lines (guides) to indicate preferred locations for the boundary of the shape. The guided α‐shape uses (parts of) these lines as boundary where the points suggest that this is appropriate. We prove that the guided α‐shape can be constructed in O((n + m) log (n + m)) time, from an input of n points and m guides. We apply guided α‐shapes to urban reconstruction from LiDAR, where neighboring surfaces can be connected conveniently along their intersection lines into adjacent surfaces of a 3D model. We analyze guided α‐shapes of both synthetic and real data and show they are consistently better than α‐shapes for this application.     

Stream Surface Parametrization by Flow‐Orthogonal Front Lines

The generation of discrete stream surfaces is an important and challenging task in scientific visualization, which can be considered a particular instance of geometric modeling. The quality of numerically integrated stream surfaces depends on a number of parameters that can be controlled locally, such as time step or distance of adjacent vertices on the front line. In addition there is a parameter that cannot be controlled locally: stream surface meshes tend to show high quality, well‐shaped elements only if the current front line is “globally” approximately perpendicular to the flow direction. We analyze the impact of this geometric property and present a novel solution – a stream surface integrator that forces the front line to be perpendicular to the flow and that generates quad‐dominant meshes with well‐shaped and well‐aligned elements. It is based on the integration of a scaled version of the flow field, and requires repeated minimization of an error functional along the current front line. We show that this leads to computing the 1‐dimensional kernel of a bidiagonal matrix: a linear problem that can be solved efficiently. We compare our method with existing stream surface integrators and apply it to a number of synthetic and real world data sets.     

Noise‐Adaptive Shape Reconstruction from Raw Point Sets

We propose a noise‐adaptive shape reconstruction method specialized to smooth, closed shapes. Our algorithm takes as input a defect‐laden point set with variable noise and outliers, and comprises three main steps. First, we compute a novel noise‐adaptive distance function to the inferred shape, which relies on the assumption that the inferred shape is a smooth submanifold of known dimension. Second, we estimate the sign and confidence of the function at a set of seed points, through minimizing a quadratic energy expressed on the edges of a uniform random graph. Third, we compute a signed implicit function through a random walker approach with soft constraints chosen as the most confident seed points computed in previous step.     

Isotropic Surface Remeshing Using Constrained Centroidal Delaunay Mesh

We develop a novel isotropic remeshing method based on constrained centroidal Delaunay mesh (CCDM), a generalization of centroidal patch triangulation from 2D to mesh surface. Our method starts with resampling an input mesh with a vertex distribution according to a user‐defined density function. The initial remeshing result is then progressively optimized by alternatively recovering the Delaunay mesh and moving each vertex to the centroid of its 1‐ring neighborhood. The key to making such simple iterations work is an efficient optimization framework that combines both local and global optimization methods. Our method is parameterization‐free, thus avoiding the metric distortion introduced by parameterization, and generating more well‐shaped triangles. Our method guarantees that the topology of surface is preserved without requiring geodesic information. We conduct various experiments to demonstrate the simplicity, efficacy, and robustness of the presented method.     

An Efficient Hybrid Incompressible SPH Solver with Interface Handling for Boundary Conditions

We propose a hybrid smoothed particle hydrodynamics solver for efficientlysimulating incompressible fluids using an interface handling method for boundary conditions in the pressure Poisson equation. We blend particle density computed with one smooth and one spiky kernel to improve the robustness against both fluid–fluid and fluid–solid collisions. To further improve the robustness and efficiency, we present a new interface handling method consisting of two components: free surface handling for Dirichlet boundary conditions and solid boundary handling for Neumann boundary conditions. Our free surface handling appropriately determines particles for Dirichlet boundary conditions using Jacobi‐based pressure prediction while our solid boundary handling introduces a new term to ensure the solvability of the linear system. We demonstrate that our method outperforms the state‐of‐the‐art particle‐based fluid solvers.     

Isosurfaces Over Simplicial Partitions of Multiresolution Grids

We provide a simple method that extracts an isosurface that is manifold and intersection‐free from a function over an arbitrary octree. Our method samples the function dual to minimal edges, faces, and cells, and we show how to position those samples to reconstruct sharp and thin features of the surface. Moreover, we describe an error metric designed to guide octree expansion such that flat regions of the function are tiled with fewer polygons than curved regions to create an adaptive polygonalization of the isosurface. We then show how to improve the quality of the triangulation by moving dual vertices to the isosurface and provide a topological test that guarantees we maintain the topology of the surface. While we describe our algorithm in terms of extracting surfaces from volumetric functions, we also show that our algorithm extends to generating manifold level sets of co‐dimension 1 of functions of arbitrary dimension.     

Fitting Polynomial Volumes to Surface Meshes with Voronoï Squared Distance Minimization

We propose a method for mapping polynomial volumes. Given a closed surface and an initial template volume grid, our method deforms the template grid by fitting its boundary to the input surface while minimizing a volume distortion criterion. The result is a point‐to‐point map distorting linear cells into curved ones. Our method is based on several extensions of Voronoi Squared Distance Minimization (VSDM) combined with a higher‐order finite element formulation of the deformation energy. This allows us to globally optimize the mapping without prior parameterization. The anisotropic VSDM formulation allows for sharp and semi‐sharp features to be implicitly preserved without tagging. We use a hierarchical finite element function basis that selectively adapts to the geometric details. This makes both the method more efficient and the representation more compact. We apply our method to geometric modeling applications in computer‐aided design and computer graphics, including mixed‐element meshing, mesh optimization, subdivision volume fitting, and shell meshing.     

As‐Rigid‐AsPossible Distance Field Metamorphosis

Widely used for morphing between objects with arbitrary topology, distance field interpolation (DFI) handles topological transition naturally without the need for correspondence or remeshing, unlike surface‐based interpolation approaches. However, lack of correspondence in DFI also leads to ineffective control over the morphing process. In particular, unless the user specifies a dense set of landmarks, it is not even possible to measure the distortion of intermediate shapes during interpolation, let alone control it. To remedy such issues, we introduce an approach for establishing correspondence between the interior of two arbitrary objects, formulated as an optimal mass transport problem with a sparse set of landmarks. This correspondence enables us to compute non‐rigid warping functions that better align the source and target objects as well as to incorporate local rigidity constraints to perform as‐rigid‐aspossible DFI. We demonstrate how our approach helps achieve flexible morphing results with a small number of landmarks.     

Polyhedral Finite Elements Using Harmonic Basis Functions

Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efficient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive refinement. We propose a flexible finite element method for arbitrary polyhedral elements, thereby effectively avoiding the need for remeshing. Our polyhedral finite elements are based on harmonic basis functions, which satisfy all necessary conditions for FEM simulations and seamlessly generalize both linear tetrahedral and trilinear hexahedral elements. We discretize harmonic basis functions using the method of fundamental solutions, which enables their flexible computation and efficient evaluation. The versatility of our approach is demonstrated on cutting and adaptive refinement within a simulation framework for corotated linear elasticity.     

Moving Least Squares Coordinates

We propose a new family of barycentric coordinates that have closed‐forms for arbitrary 2D polygons. These coordinates are easy to compute and have linear precision even for open polygons. Not only do these coordinates have linear precision, but we can create coordinates that reproduce polynomials of a set degree m as long as degree m polynomials are specified along the boundary of the polygon. We also show how to extend these coordinates to interpolate derivatives specified on the boundary.