Coarsening Procedures Research Articles

Adaptive refinement techniques based on tetrahedral and hexahedral grids for finite element analysis of mold filling in casting processes

A 3-D finite element model combined with a volume tracking method is presented in this work to simulate the mold filling for casting processes. Especially, the analysis involves an adaptive grid method that is created under a criterion of element categorization of filling states and locations in the total region at each time step. By using an adaptive grid wherein the elements, finer than those in internal and external regions, are distributed at the surface region through refinement and coarsening procedures, a more efficient analysis of transient fluid flow with free surface is achieved. Adaptive grid based on VOF method is developed in tetrahedral and hexahedral elements. Through a 3-D analysis of the benchmark test of the casting process using two types of elements, the efficiency of the proposed adaptive grid method is verified.

Ignorability in Statistical and Probabilistic Inference

When dealing with incomplete data in statistical learning, or incomplete observations in probabilistic inference, one needs to distinguish the fact that a certain event is observed from the fact that the observed event has happened. Since the modeling and computational complexities entailed by maintaining this proper distinction are often prohibitive, one asks for conditions under which it can be safely ignored. Such conditions are given by the missing at random (mar) and coarsened at random (car) assumptions. In this paper we provide an in-depth analysis of several questions relating to mar/car assumptions. Main purpose of our study is to provide criteria by which one may evaluate whether a car assumption is reasonable for a particular data collecting or observational process. This question is complicated by the fact that several distinct versions of mar/car assumptions exist. We therefore first provide an overview over these different versions, in which we highlight the distinction between distributional and coarsening variable induced versions. We show that distributional versions are less restrictive and sufficient for most applications. We then address from two different perspectives the question of when the mar/car assumption is warranted. First we provide a ''static'' analysis that characterizes the admissibility of the car assumption in terms of the support structure of the joint probability distribution of complete data and incomplete observations. Here we obtain an equivalence characterization that improves and extends a recent result by Grunwald and Halpern. We then turn to a ''procedural'' analysis that characterizes the admissibility of the car assumption in terms of procedural models for the actual data (or observation) generating process. The main result of this analysis is that the stronger coarsened completely at random (ccar) condition is arguably the most reasonable assumption, as it alone corresponds to data coarsening procedures that satisfy a natural robustness property.

Evaluation of adaptive mesh refinement and coarsening for the computation of compressible flows on unstructured meshes

AbstractThe compressible gas flows of interest to aerospace applications often involve situations where shock and expansion waves are present. Decreasing the characteristic dimension of the computational cells in the vicinity of shock waves improves the quality of the computed flows. This reduction in size may be accomplished by the use of mesh adaption procedures. In this paper an analysis is presented of an adaptive mesh scheme developed for an unstructured mesh finite volume upwind computer code. This scheme is tailored to refine or coarsen the computational mesh where gradients of the flow properties are respectively high or low. The refinement and coarsening procedures are applied to the classical gas dynamic problems of the stabilization of shock waves by solid bodies. In particular, situations where oblique shock waves interact with an expansion fan and where bow shocks arise around solid bodies are considered. The effectiveness of the scheme in reducing the computational time, while increasing the solution accuracy, is assessed. It is shown that the refinement procedure alone leads to a number of computational cells which is 20% larger than when alternate passes of refinement and coarsening are used. Accordingly, a reduction of computational time of the same order of magnitude is obtained. Copyright © 2005 John Wiley & Sons, Ltd.

Conformal Refinement and Coarsening of Unstructured Hexahedral Meshes

This paper describes recently developed procedures for local conformal refinement and coarsening of all-hexahedral unstructured meshes. Both refinement and coarsening procedures take advantage of properties found in the dual or “twist planes” of the mesh. A twist plane manifests itself as a conformal layer or sheet of hex elements within the global mesh. We suggest coarsening techniques that will identify and remove sheets to satisfy local mesh density criteria while not seriously degrading element quality after deletion. A two-dimensional local coarsening algorithm is introduced. We also explain local hexahedral refinement procedures that involve both the placement of new sheets, either between existing hex layers or within an individual layer. Hex elements earmarked for refinement may be defined to be as small as a single node or as large as a major group of existing elements. Combining both refinement and coarsening techniques allows for significant control over the density and quality of the resulting modified mesh.

Adaptive modification for time evolving meshes

A mesh adaptation scheme is presented that employs edge collapse to coarsen the mesh and selective point placement combined with an incremental Delaunay algorithm for mesh enrichment. The coarsening and enrichment procedures can be used in isolation to achieve a solution adapted mesh for time varying computations whose boundaries remain fixed. These procedures can also be combined with a mesh movement technique to form an adaptation algorithm that will modify the mesh of a domain whose shape is evolving in time. The adaptation techniques are described and examples are presented to illustrate the method for time dependent problems arising from crack nucleation studies and vortex shedding of incompressible flow over a cylinder.

An algebraic 2‐level domain decomposition preconditioner with applications to the compressible Euler equations

AbstractA domain decomposition technique to solve large‐scale aerodynamics problems on unstructured grids is investigated. The linear system, arising when an implicit time‐advancing scheme is used, is preconditioned using a Schwarz‐based method. The key idea of the Schwarz preconditioner is to solve (approximately) a linear problem on each subdomain, then to exchange information only with neighbouring subdomains. Unfortunately, the performance of the Schwarz‐type preconditioner deteriorates as the number of processors grows. So, in this case, a key element for obtaining a scalable preconditioner is to provide a coarse level operator. Since many of the coarse operators proposed in literature are difficult to implement on unstructured 2D and 3D meshes, a purely algebraic procedure, that requires the entries of the matrix only, has been developed. This procedure may be seen as an Algebraic MultiGrid (AMG) method applied as a coarse grid correction operator. The key idea is to take advantage of the use of local data of domain decomposition preconditioners, and of the automatic coarsening procedures of AMG methods.Two possible schemes to introduce the coarse grid operator will be described. Both cases have been implemented and tested in a distributed parallel environment, using the MPI library. It will be shown that for suitable values of the rank of the coarse grid operator it is possible to obtain a considerable reduction in the number of iterations compared to the Schwarz preconditioner without coarse operator. Copyright © 2002 John Wiley & Sons, Ltd.

Application of an adaptive grid refinement technique to three-dimensional finite element analysis of the filling stage in the die-casting process

In order to obtain a higher resolution of the free surface configuration, an adaptive grid refinement technique is incorporated into the volume of fluid (VOF) and the marker surface methods for three-dimensional finite element analysis of the filling stage in the die-casting process. The adaptive grid is created at each time step through the grid refinement and coarsening procedures, as opposed to employing the fixed grid in the classical VOF and marker particle methods. In the adaptive grid, the elements finer than those of the internal and external regions are distributed over the surface region. The adaptive grid is used for generating the fluid region grid that better describes the shape of the predicted free surface at each time step, and for advancing the predicted free surface more precisely. To verify the efficiency of the adaptive grid refinement technique rigorously, a benchmark problem of the conference on the “Modeling of Casting, Welding and Advanced Solidification Processes VII” (MCWASP VII) has been analyzed and the numerical results are found to be in good agreement with the experimental results. A typical industrial problem of the die-casting process such as a cover of transmission case has been analyzed and it has been shown that the adaptive grid incorporated into the VOF and marker surface methods can be effectively applied to general die-casting process analysis.

Three-dimensional finite element analysis of transient fluid flow with free-surface using marker surface method and adaptive grid refinement

For three-dimensional finite element analysis of transient fluid flow with free-surface, a new marker surface method is proposed, in which the fluid flow is represented by the marker surface composed of marker elements instead of marker particles used in the marker particle method. This also involves an adaptive grid that is created under a new criterion of element categorization of filling states and the locations in the total region at each time step. The marker surface is used in order to represent the free-surface accurately, as well as to decrease the memory and computation time, and to effectively display the predicted three-dimensional free-surface. By using the adaptive grid in which the elements, finer than those in internal and external regions, are distributed in the surface region through refinement and coarsening procedures, the analysis of three-dimensional transient fluid flow with free-surface is achieved more efficiently. Through three-dimensional analysis of two kinds of problems using several grids, the efficiency of the proposed marker surface method and the adaptive grid are verified. Copyright © 1999 John Wiley & Sons, Ltd.

Spatial adaptation of unstructured meshes for unsteady aerodynamic flow computations

Spatial adaptation procedures for the accurate and efficient solution of steady and unsteady inviscid flow problems are described. The adaptation procedures were developed and implemented within a two-dimensional unstructured-grid upwind-type Euler code. These procedures involve mesh enrichment and mesh coarsening to either add points in high gradient regions of the flow or remove points where they are not needed, respectively, to produce solutions of high spatial accuracy at minimal computational cost. The paper describes a new approach to detect high gradient regions of the flow that uses the substantial derivative of density. Additionally, the paper gives a detailed description of the enrichment and coarsening procedures and further describes a new weighted averaging procedure and the interpolation of flow variables that improves the overall accuracy of the flow solver. Presented are comparisons with alternative results and experimental data to provide an assessment of the accuracy and efficiency of the capability. Steady and unsteady transonic results, obtained using spatial adaptation for the NAG A 0012 airfoil, are shown to be of high spatial accuracy, primarily in that the shock waves are very sharply captured. The results were obtained with large computational savings when compared to results for a globally enriched mesh with cells subdivided as many times as the finest cells of the adapted grid.

An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates

A simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two-phase Stefan problems in 2D. A typical triangulation is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas it is locally refined in the vicinity of the discrete interface so that the relation becomes hyperbolic. Several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations. Mesh selection is based upon equidistributing pointwise interpolation errors between consecutive meshes and imposing that discrete interfaces belong to the so-called refined region. Consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated. A general theory for interpolation between noncompatible meshes is set up in ${L^p}$-based norms. The resulting scheme is stable in various Sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods on quasi-uniform meshes, namely $O({\tau ^{ - 3/2}})$ as opposed to $O({\tau ^{ - 2}})$, to achieve the same global asymptotic accuracy; here $\tau > 0$ is the (uniform) time step. A rate of convergence of essentially $O({\tau ^{1/2}})$ is derived in the natural energy spaces provided the total number of mesh changes is restricted to $O({\tau ^{ - 1/2}})$, which in turn is compatible with the mesh selection procedure. An auxiliary quasi-optimal pointwise error estimate for the Laplace operator is proved as well. Numerical results illustrate the scheme’s efficiency in approximating both solutions and interfaces.