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This article describes a Lagrangian finite element method that simulates the behavior of liquids and solids in a unified framework. Local mesh improvement operations maintain a high-quality tetrahedral discretization even as the mesh is advected by fluid flow. We conserve volume and momentum, locally and globally, by assigning to each element an independent rest volume and adjusting it to correct for deviations during remeshing and collisions. Incompressibility is enforced with per-node pressure values, and extra degrees of freedom are selectively inserted to prevent pressure locking. Topological changes in the domain are explicitly treated with local mesh splitting and merging. Our method models surface tension with an implicit formulation based on surface energies computed on the boundary of the volume mesh. With this method we can model elastic, plastic, and liquid materials in a single mesh, with no need for explicit coupling. We also model heat diffusion and thermoelastic effects, which allow us to simulate phase changes. We demonstrate these capabilities in several fluid simulations at scales from millimeters to meters, including simulations of melting caused by external or thermoelastic heating. Copyright © 2013 ACM.

This article describes a Lagrangian finite element method that simulates the behavior of liquids and solids in a unified framework. Local mesh improvement operations maintain a high-quality tetrahedral discretization even as the mesh is advected by fluid flow. We conserve volume and momentum, locally and globally, by assigning to each element an independent rest volume and adjusting it to correct for deviations during remeshing and collisions. Incompressibility is enforced with per-node pressure values, and extra degrees of freedom are selectively inserted to prevent pressure locking. Topological changes in the domain are explicitly treated with local mesh splitting and merging. Our method models surface tension with an implicit formulation based on surface energies computed on the boundary of the volume mesh. With this method we can model elastic, plastic, and liquid materials in a single mesh, with no need for explicit coupling. We also model heat diffusion and thermoelastic effects, which allow us to simulate phase changes. We demonstrate these capabilities in several fluid simulations at scales from millimeters to meters, including simulations of melting caused by external or thermoelastic heating. Copyright © 2013 ACM.

We present warp-canceling corotation, a nonlinear finite element formulation for elastodynamic simulation that achieves fast performance by making only partial or delayed changes to the simulation's linearized system matrices. Coupled with an algorithm for incremental updates to a sparse Cholesky factorization, the method realizes the stability and scalability of a sparse direct method without the need for expensive refactorization at each time step. This finite element formulation combines the widely used corotational method with stiffness warping so that changes in the per-element rotations are initially approximated by inexpensive per-node rotations. When the errors of this approximation grow too large, the per-element rotations are selectively corrected by updating parts of the matrix chosen according to locally measured errors. These changes to the system matrix are propagated to its Cholesky factor by incremental updates that are much faster than refactoring the matrix from scratch. A nested dissection ordering of the system matrix gives rise to a hierarchical factorization in which changes to the system matrix cause limited, well-structured changes to the Cholesky factor. We show examples of simulations that demonstrate that the proposed formulation produces results that are visually comparable to those produced by a standard corotational formulation. Because our method requires computing only partial updates of the Cholesky factor, it is substantially faster than full refactorization and outperforms widely used iterative methods such as preconditioned conjugate gradients. Our method supports a controlled trade-off between accuracy and speed, and unlike most iterative methods its performance does not slow for stiffer materials but rather it actually improves. © 2012 ACM.

We present warp-canceling corotation, a nonlinear finite element formulation for elastodynamic simulation that achieves fast performance by making only partial or delayed changes to the simulation's linearized system matrices. Coupled with an algorithm for incremental updates to a sparse Cholesky factorization, the method realizes the stability and scalability of a sparse direct method without the need for expensive refactorization at each time step. This finite element formulation combines the widely used corotational method with stiffness warping so that changes in the per-element rotations are initially approximated by inexpensive per-node rotations. When the errors of this approximation grow too large, the per-element rotations are selectively corrected by updating parts of the matrix chosen according to locally measured errors. These changes to the system matrix are propagated to its Cholesky factor by incremental updates that are much faster than refactoring the matrix from scratch. A nested dissection ordering of the system matrix gives rise to a hierarchical factorization in which changes to the system matrix cause limited, well-structured changes to the Cholesky factor. We show examples of simulations that demonstrate that the proposed formulation produces results that are visually comparable to those produced by a standard corotational formulation. Because our method requires computing only partial updates of the Cholesky factor, it is substantially faster than full refactorization and outperforms widely used iterative methods such as preconditioned conjugate gradients. Our method supports a controlled trade-off between accuracy and speed, and unlike most iterative methods its performance does not slow for stiffer materials but rather it actually improves. © 2012 ACM.

We propose a finite element simulation method that addresses the full range of material behavior, from purely elastic to highly plastic, for physical domains that are substantially reshaped by plastic flow, fracture, or large elastic deformations. To mitigate artificial plasticity, we maintain a simulation mesh in both the current state and the rest shape, and store plastic offsets only to represent the non-embeddable portion of the plastic deformation. To maintain high element quality in a tetrahedral mesh undergoing gross changes, we use a dynamic meshing algorithm that attempts to replace as few tetrahedra as possible, and thereby limits the visual artifacts and artificial diffusion that would otherwise be introduced by repeatedly remeshing the domain from scratch. Our dynamic mesher also locally refines and coarsens a mesh, and even creates anisotropic tetrahedra, wherever a simulation requests it. We illustrate these features with animations of elastic and plastic behavior, extreme deformations, and fracture.

We propose a finite element simulation method that addresses the full range of material behavior, from purely elastic to highly plastic, for physical domains that are substantially reshaped by plastic flow, fracture, or large elastic deformations. To mitigate artificial plasticity, we maintain a simulation mesh in both the current state and the rest shape, and store plastic offsets only to represent the non-embeddable portion of the plastic deformation. To maintain high element quality in a tetrahedral mesh undergoing gross changes, we use a dynamic meshing algorithm that attempts to replace as few tetrahedra as possible, and thereby limits the visual artifacts and artificial diffusion that would otherwise be introduced by repeatedly remeshing the domain from scratch. Our dynamic mesher also locally refines and coarsens a mesh, and even creates anisotropic tetrahedra, wherever a simulation requests it. We illustrate these features with animations of elastic and plastic behavior, extreme deformations, and fracture.

We describe a method for animating incompressible liquids with detailed free surfaces. For each time step, semi- Lagrangian contouring computes a new fluid boundary (represented as a fine surface triangulation) from the previous time step's fluid boundary and velocity field. Then a mesh generation algorithm called isosurface stuffing discretizes the region enclosed by the new fluid boundary, creating a tetrahedral mesh that grades from a fine resolution at the surface to a coarser resolution in the interior. The mesh has a structure, based on the body centered cubic lattice, that accommodates graded tetrahedron sizes but is regular enough to aid efficient point location and to save memory used to store geometric properties of identical tetrahedra. Although the mesh is warped to conform to the liquid boundary, it has a mathematical guarantee on tetrahedron quality, and is generated very rapidly. Each successive time step entails creating a new triangulated liquid surface and a new tetrahedral mesh. Semi-Lagrangian advection computes velocities at the current time step on the new mesh. We use a finite volume discretization to perform pressure projection required to enforce the fluid's incompressibility, and we solve the linear system with algebraic multigrid. A novel thickening scheme prevents thin sheets and droplets of liquid from vanishing when their thicknesses drop below the mesh resolution. Examples demonstrate that the method captures complex liquid motions and fine details on the free surfaces without suffering from excessive volume loss or artificial damping.

We describe a method for animating incompressible liquids with detailed free surfaces. For each time step, semi- Lagrangian contouring computes a new fluid boundary (represented as a fine surface triangulation) from the previous time step's fluid boundary and velocity field. Then a mesh generation algorithm called isosurface stuffing discretizes the region enclosed by the new fluid boundary, creating a tetrahedral mesh that grades from a fine resolution at the surface to a coarser resolution in the interior. The mesh has a structure, based on the body centered cubic lattice, that accommodates graded tetrahedron sizes but is regular enough to aid efficient point location and to save memory used to store geometric properties of identical tetrahedra. Although the mesh is warped to conform to the liquid boundary, it has a mathematical guarantee on tetrahedron quality, and is generated very rapidly. Each successive time step entails creating a new triangulated liquid surface and a new tetrahedral mesh. Semi-Lagrangian advection computes velocities at the current time step on the new mesh. We use a finite volume discretization to perform pressure projection required to enforce the fluid's incompressibility, and we solve the linear system with algebraic multigrid. A novel thickening scheme prevents thin sheets and droplets of liquid from vanishing when their thicknesses drop below the mesh resolution. Examples demonstrate that the method captures complex liquid motions and fine details on the free surfaces without suffering from excessive volume loss or artificial damping.

This paper describes a method for building interpolating or approximating implicit surfaces from polygonal data. The user can choose to generate a surface that exactly interpolates the polygons, or a surface that approximates the input by smoothing away features smaller than some user-specified size. The implicit functions are represented using a moving least-squares formulation with constraints integrated over the polygons. The paper also presents an improved method for enforcing normal constraints and an iterative procedure for ensuring that the implicit surface tightly encloses the input vertices.

This paper describes a method for building interpolating or approximating implicit surfaces from polygonal data. The user can choose to generate a surface that exactly interpolates the polygons, or a surface that approximates the input by smoothing away features smaller than some user-specified size. The implicit functions are represented using a moving least-squares formulation with constraints integrated over the polygons. The paper also presents an improved method for enforcing normal constraints and an iterative procedure for ensuring that the implicit surface tightly encloses the input vertices.

We introduce a noise-resistant algorithm for reconstructing a watertight surface from point cloud data. It forms a Delaunay tetrahedralization, then uses a variant of spectral graph partitioning to decide whether each tetrahedron is inside or outside the original object. The reconstructed surface triangulation is the set of triangular faces where inside and outside tetrahedra meet. Because the spectral partitioner makes local decisions based on a global view of the model, it can ignore outliers, patch holes and undersampled regions, and surmount ambiguity due to measurement errors. Our algorithm can optionally produce a manifold surface. We present empirical evidence that our implementation is substantially more robust than several closely related surface reconstruction programs. © The Eurographics Association 2004.

This paper describes a method for building interpolating or approximating implicit surfaces from polygonal data. The user can choose to generate a surface that exactly interpolates the polygons, or a surface that approximates the input by smoothing away features smaller than some user-specified size. The implicit functions are represented using a moving least-squares formulation with constraints integrated over the polygons. The paper also presents an improved method for enforcing normal constraints and an iterative procedure for ensuring that the implicit surface tightly encloses the input vertices.

This paper describes a method for building interpolating or approximating implicit surfaces from polygonal data. The user can choose to generate a surface that exactly interpolates the polygons, or a surface that approximates the input by smoothing away features smaller than some user-specified size. The implicit functions are represented using a moving least-squares formulation with constraints integrated over the polygons. The paper also presents an improved method for enforcing normal constraints and an iterative procedure for ensuring that the implicit surface tightly encloses the input vertices.

We introduce a noise-resistant algorithm for reconstructing a watertight surface from point cloud data. It forms a Delaunay tetrahedralization, then uses a variant of spectral graph partitioning to decide whether each tetrahedron is inside or outside the original object. The reconstructed surface triangulation is the set of triangular faces where inside and outside tetrahedra meet. Because the spectral partitioner makes local decisions based on a global view of the model, it can ignore outliers, patch holes and undersampled regions, and surmount ambiguity due to measurement errors. Our algorithm can optionally produce a manifold surface. We present empirical evidence that our implementation is substantially more robust than several closely related surface reconstruction programs. © The Eurographics Association 2004.