Results 1 
6 of
6
Spacetime Meshing with Adaptive Refinement and Coarsening
 SCG'04
, 2004
"... We propose a new algorithm for constructing finiteelement meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain# and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain [0, T] i ..."
Abstract

Cited by 20 (10 self)
 Add to MetaCart
(Show Context)
We propose a new algorithm for constructing finiteelement meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain# and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain [0, T] in constant running time per tetrahedron in IR using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.
Spacetime Meshing for Discontinuous Galerkin Methods
 Department of Computer Science, University of Illinois at UrbanaChampaign
, 2005
"... Important applications in science and engineering, such as modeling traffic flow, seismic waves, electromagnetics, and the simulation of mechanical stresses in materials, require the highfidelity numerical solution of hyperbolic partial differential equations (PDEs) in space and time variables. Man ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
Important applications in science and engineering, such as modeling traffic flow, seismic waves, electromagnetics, and the simulation of mechanical stresses in materials, require the highfidelity numerical solution of hyperbolic partial differential equations (PDEs) in space and time variables. Many interesting physical problems involve nonlinear and anisotropic behavior, and the PDEs modeling them exhibit discontinuities in their solutions. Spacetime discontinuous Galerkin (SDG) finite element methods are used to solve such PDEs arising from wave propagation phenomena. To support an accurate and efficient solution procedure using SDG methods and to exploit the flexibility of these methods, we give a meshing algorithm to construct an unstructured simplicial spacetime mesh over an arbitrary simplicial space domain. Our algorithm is the first spacetime meshing algorithm suitable for efficient solution of nonlinear phenomena in anisotropic media using novel discontinuous Galerkin finite element methods for implicit solutions directly in spacetime. Given a triangulated ddimensional Euclidean space domain M (a simplicial complex) and initial conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured simplicial mesh of the (d + 1)dimensional spacetime domain M × [0,∞). Our algorithm uses a nearoptimal number of spacetime elements, each with bounded temporal aspect ratio for any finite prefix M × [0,T] of spacetime. Unlike Delaunay meshes, the facets of our mesh satisfy gradient constraints that allow interleaving the construction of the mesh by adding new space
The propagation problem in longestedge refinement
, 2005
"... Two asymptotic properties that arise in iterative mesh refinement of triangles are introduced and investigated. First, we provide theoretical results showing that recursive application of uniform four triangles longestedge (4TLE) partition to an arbitrary unstructured triangular mesh produces mesh ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Two asymptotic properties that arise in iterative mesh refinement of triangles are introduced and investigated. First, we provide theoretical results showing that recursive application of uniform four triangles longestedge (4TLE) partition to an arbitrary unstructured triangular mesh produces meshes in which the triangle pairings sharing a common longest edge asymptotically tend to cover the area of the whole mesh. As a consequence, we prove that for a triangle, the induced exterior conforming refinement zone extends on average to a few neighbor adjacent triangles. We determine the asymptotic extent of this propagating path and include results of supporting numerical experiments with uniform and adaptive mesh refinement. Similar behavior and LE propagation from a four triangle self similar (4TSS) local subdivision alternative is analyzed and compared numerically. Hybrid 4TLE and 4TSS LE schemes are also considered. The results are relevant to mesh refinement in finite element and finite volume calculations as well as mesh enhancement in Computer Graphics and CAGD.
ANALYSIS OF LONGESTEDGE ALGORITHMS FOR 2DIMENSIONAL MESH REFINEMENT
"... Las técnicas de generación y refinamiento de mallas no estructuradas son usadas para la descomposición de objetos geométricos. Estas técnicas son muy utilizadas en áreas como modelamiento geométrico, computación gráfica, computación cient́ıfica y aplicaciones de ingenieŕıa, entre otras, ..."
Abstract
 Add to MetaCart
(Show Context)
Las técnicas de generación y refinamiento de mallas no estructuradas son usadas para la descomposición de objetos geométricos. Estas técnicas son muy utilizadas en áreas como modelamiento geométrico, computación gráfica, computación cient́ıfica y aplicaciones de ingenieŕıa, entre otras, lo que les da un interés interdisciplinario. Trabajando con triangulaciones (mallas compuestas por triángulos), el reto es generar una descomposición precisa del objeto geométrico o dominio, y al mismo tiempo satisfacer las restricciones adicionales impuestas por la aplicación, como restricciones en la forma de los elementos, el número de elementos, o la transición entre elementos de diferentes tamaños. Los algoritmos que ofrecen garant́ıas teóricas sobre estos temas son preferidos. Los algoritmos de arista más larga fueron diseñados para el refinamiento iterativo de triangulaciones en aplicaciones de método de elementos finitos adaptativo. Estos algoritmos esta ́ basados en la estrategia de propagación por la arista más larga. Comparados a otros algoritmos de refinamiento, los algoritmos de arista más larga rápidamente producen una descomposición del dominio (o de regiones de interés) a través de operaciones locales simples. Las triangulaciones obtenidas presentan buena densidad y la calidad de los triángulos
ADAPTIVE FINITE ELEMENTS FOR VISCOELASTIC DEFORMATION PROBLEMS
"... This thesis is concerned with the theoretical and computational aspects of generating solutions to problems involving materials with fading memory, known as viscoelastic materials. Viscoelastic materials can be loosely described as those whose current stress configuration depends on their recent pa ..."
Abstract
 Add to MetaCart
(Show Context)
This thesis is concerned with the theoretical and computational aspects of generating solutions to problems involving materials with fading memory, known as viscoelastic materials. Viscoelastic materials can be loosely described as those whose current stress configuration depends on their recent past. Viscoelastic constitutive laws for stress typically take the form of a sum of an instantaneous response term and an integral over their past responses. Such laws are called hereditary integral constitutive laws. The main purpose of this study is to analyse adaptive finite element algorithms for the numerical solution of the quasistatic equations governing the small displacement of a viscoelastic body subjected to prescribed body forces and tractions. Such algorithms for the hereditary integral formulation have appeared in the literature. However the approach here is to consider an equivalent formulation based on the introduction of a set of unobservable intemal vaTiables. In the linear viscoelastic case we exploit the structure of the quasistatic problem to remove the displacement from the equations governing the internal variables. This results in an elliptic problem with right hand side dependent on the internal variables,