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Aggressive Tetrahedral Mesh Improvement
 In Proc. of the 16th Int. Meshing Roundtable
, 2007
"... Summary. We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or “mesh cleanup. ” Our goal is to aggressively optimize the worst t ..."
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Cited by 39 (4 self)
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Summary. We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or “mesh cleanup. ” Our goal is to aggressively optimize the worst tetrahedra, with speed a secondary consideration. Mesh optimization methods often get stuck in bad local optima (poorquality meshes) because their repertoire of mesh transformations is weak. We employ a broader palette of operations than any previous mesh improvement software. Alongside the best traditional topological and smoothing operations, we introduce a topological transformation that inserts a new vertex (sometimes deleting others at the same time). We describe a schedule for applying and composing these operations that rarely gets stuck in a bad optimum. We demonstrate that all three techniques—smoothing, vertex insertion, and traditional transformations—are substantially more effective than any two alone. Our implementation usually improves meshes so that all dihedral angles are between 31 ◦ and 149 ◦ , or (with a different objective function) between 23 ◦ and 136 ◦. 1
Dynamic Local Remeshing for Elastoplastic Simulation
"... Figure 1: An elastoplastic substance slowly drips from a horizontal surface. A dynamic meshing algorithm refines the drop while maintaining highquality tetrahedra. At the narrowest part of the tendril, the mesher creates small, anisotropic tetrahedra where the strain gradient is anisotropic, so tha ..."
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Cited by 30 (3 self)
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Figure 1: An elastoplastic substance slowly drips from a horizontal surface. A dynamic meshing algorithm refines the drop while maintaining highquality tetrahedra. At the narrowest part of the tendril, the mesher creates small, anisotropic tetrahedra where the strain gradient is anisotropic, so that a modest number are adequate. Work hardening causes the tendril to become brittle, whereupon it fractures. At right, we animate a fine triangulated surface embedded in the mesh. 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 nonembeddable 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.
Simulating Liquids and SolidLiquid Interactions with Lagrangian Meshes
"... This paper describes a Lagrangian finite element method that simulates the behavior of liquids and solids in a unified framework. Local mesh improvement operations maintain a highquality tetrahedral discretization even as the mesh is advected by fluid flow. We conserve volume and momentum, locally ..."
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Cited by 13 (1 self)
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This paper describes a Lagrangian finite element method that simulates the behavior of liquids and solids in a unified framework. Local mesh improvement operations maintain a highquality tetrahedral discretization even as the mesh is advected by fluid flow. We conserve volume and momentum, locally and globally, by assigning each element an independent rest volume and adjusting it to correct for deviations during remeshing and collisions. Incompressibility is enforced with pernode 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.
T (2004) A bezierbased moving mesh framework for simulation with elastic membranes
 In Proc. 14th Int. Meshing Roundtable
"... In this paper we present an application of our Bezierbased approach to moving meshes [1] to NavierStokes simulations with several immersed elastic membranes. By a moving mesh we mean one that moves with the material and is adapted to maintain good aspect ratio triangles of minimal size. The adapta ..."
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Cited by 5 (1 self)
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In this paper we present an application of our Bezierbased approach to moving meshes [1] to NavierStokes simulations with several immersed elastic membranes. By a moving mesh we mean one that moves with the material and is adapted to maintain good aspect ratio triangles of minimal size. The adaptations we employ include point insertion and removal, as well as edge smoothing. This work is being done as part of the Sangria project [2] whose goal is to develop geometric and numerical algorithms and software for the simulation of blood
ow at the microstructural level. In our approach, we adopt the Lagrangian paradigm where domain boundaries and object interfaces move together with the
uid in which they are immersed. This approach has the advantage that boundaries and object interfaces are easy to track. A moving mesh also poses dicult geometric problems since very distorted elements can be created as the simulation evolves. This can lead to several undesirable or catastrophic situations such as inverted or overlapping elements. From the computational geometry perspective, the challenge presented by the Lagrangian paradigm is the ability to maintain a good quality mesh as the simulation evolves in time. We tackle this problem by
Dynamic mesh refinement with quad trees and offcenters
, 2008
"... Many algorithms exist for producing quality meshes when the input point cloud is known a priori. However, modern finite element simulations and graphics applications need to change the input set during the simulation dynamically. In this paper, we show a dynamic algorithm for building and maintainin ..."
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Cited by 3 (2 self)
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Many algorithms exist for producing quality meshes when the input point cloud is known a priori. However, modern finite element simulations and graphics applications need to change the input set during the simulation dynamically. In this paper, we show a dynamic algorithm for building and maintaining a quadtree under insertions into and deletions from an input point set in any fixed dimension. This algorithm runs in O(lg L/s) time per update, where L/s is the spread of the input. The result of the dynamic quadtree can be combined with a postprocessing step to generate and maintain a simplicial mesh under dynamic changes in the same asymptotic runtime. The mesh output by the dynamic algorithm is of good quality (it has no small dihedral angle), and is optimal in size. This gives the first timeoptimal dynamic algorithm that outputs good quality meshes in any dimension. As a second result, we dynamize the quadtree postprocessing technique of HarPeled and Üngör for generating meshes in two dimensions. When composed with the dynamic quadtree algorithm, the resulting algorithm yields quality meshes that are the smallest known in practice, while guaranteeing the same asymptotic optimality guarantees.
Dynamic Mesh Refinement
, 2007
"... Mesh refinement is the problem to produce a triangulation (typically Delaunay) of an input set of points augmented by Steiner points, such that every triangle or tetrahedron has good quality (no small angles). The requirement arises from the applications: in scientific computing and in graphics, mes ..."
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Cited by 1 (1 self)
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Mesh refinement is the problem to produce a triangulation (typically Delaunay) of an input set of points augmented by Steiner points, such that every triangle or tetrahedron has good quality (no small angles). The requirement arises from the applications: in scientific computing and in graphics, meshes are often used to discretely represent the value of a function over space. In addition to the quality requirement, the user often has input segments or polygons (generally, a piecewise linear complex) they would like see retained in the mesh; the mesh must respect these constraints. Finally, the mesh should be sizeconforming: the size of mesh elements should be related to a particular sizing function based on the distance between input features. The static meshing problem is increasingly wellunderstood: one can download software with provable guarantees that on reasonable input, the meshes will have good quality, will respect the input, and will be sizeconforming; more recently, these algorithms have started to come with optimal runtimes of O(n lg(L/s) +m), where L/s is the spread of the input. As a first result, I
Representing Topological Structures Using CellChains
, 2006
"... A new topological representation of surfaces in higher dimensions, “cellchains ” is developed. The representation is a generalization of Brisson’s celltuple data structure. Cellchains are identical to celltuples when there are no degeneracies: cells or simplices with identified vertices. The pr ..."
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A new topological representation of surfaces in higher dimensions, “cellchains ” is developed. The representation is a generalization of Brisson’s celltuple data structure. Cellchains are identical to celltuples when there are no degeneracies: cells or simplices with identified vertices. The proof of correctness is based on axioms true for maps, such as those in Brisson’s celltuple representation. A critical new condition (axiom) is added to those of Lienhardt’s nGmaps to give “cellmaps”. We show that cellmaps and cellchains characterize the same topological representations.