A key step in the finite element method is to generate a high quality mesh that is as small as possible for an input domain. Several meshing methods and heuristics have been developed and implemented. Methods based on advancing front, Delaunay triangulations, and quadtrees/octrees are among the most popular ones. Advancing front uses simple data structures and is efficient. Unfortunately, in general, it does not provide any guarantee on the size and quality of the mesh it produces. On the other hand, the circle-packing based Delaunay methods generate a well-shaped mesh whose size is within a constant factor of the optimal. In this paper, we present a new meshing algorithm, the biting method, which combines the strengths of advancing front and circle packing. We prove that it generates a high quality 2D mesh, and the size of the mesh is within a constant factor of the optimal.

In this paper, we analyze the complexity of natural parallelizations of Delaunay refinement methods for mesh generation. The parallelizations employ a simple strategy: at each iteration, they choose a set of "independent" points to insert into the domain, and then update the Delaunay triangulation. We show that such a set of independent points can be constructed efficiently in parallel and that the number of iterations needed is O(log²(L/s)), where L is the diameter of the domain, and s is the smallest edge in the output mesh. In addition, we show that the insertion of each independent set of points can be realized sequentially by Ruppert's method in two dimensions and Shewchuk's in three dimensions. Therefore, our parallel Delaunay refinement methods provide the same element quality and mesh size guarantees as the sequential algorithms in both two and three dimensions. For quasi-uniform meshes, such as those produced by Chew's method, we show that the number of iterations can be reduced to O(log(L/s)). To the best of our knowledge, these are the first provably polylog(L/s) parallel time Delaunay meshing algorithms that generate well-shaped meshes of size optimal to within a constant.

In the numerical simulation of the combustion process and microstructural evolution, we need to consider the adaptive meshing problem for a domain with a moving boundary, in which, the submesh in the region behind the moving boundary needs to be coarsened while the submesh in the region ahead of the moving boundary needs to be refined. In this paper, we present a unified scheme for simultaneously refining and coarsening a mesh. Our method guarantees that the resulting mesh is well-shaped and is of a size that is within a constant factor of the optimal possible. We also present several practical variations of our provably good algorithm.

Abstract. Traditional load balancing algorithms for data-intensive iterative routines can successfully load balance relatively small problems. We demonstrate that they may fail for large problem sizes on computational clusters with memory heterogeneity. Traditional algorithms use too simplistic models of processors ’ performance which cannot reflect many aspects of heterogeneity. This paper presents a new dynamic load balancing algorithm based on the advanced functional performance model. The model consists of speed functions of problem size, which are built adaptively from a history of load measurements. Experimental results demonstrate that our algorithm can successfully balance data-intensive iterative routines on parallel platforms with memory heterogeneity.

In numerical simulation of the combustion process and microstructural evolution, we need to consider the adaptive meshing problem for a domain that has a moving boundary. During the simulation, the region ahead of the moving boundary needs to be refined (to satisfy stronger numerical conditions), and the submesh in the region behind the moving boundary should be coarsened (to reduce the mesh size). We present a unified scheme for simultaneously refining and coarsening a mesh. Our method uses sphere packings and guarantees that the resulting mesh is well-shaped and is within a constant factor of the optimal possible in the number of mesh elements. We also present several practical variations of our provably good algorithm.

Computational Science and Engineering (CSE) applications often exhibit the pattern of adaptive mesh applications. Adaptive mesh algorithm starts with a coarse base-level grid structure covering entire computational domain. As the computation intensified, individual grid points are tagged for refinement. Such tagged grid points are dynamically overlayed with finer grid points. Similarly if the level of refinement in a cell is greater than required, all such regions are replaced with coarser grids. These refinements proceed recursively. We have developed an object-oriented framework enabling time-stepped adaptive mesh application developers to convert their sequential applications to MPI applications in few easy steps. We present in this thesis our positive experience converting such application using our framework. In addition to the MPI support, framework does the grid expansion/contraction and load balancing making the application developer’s life easier.

Received (received date) Revised (revised date) Communicated by (Name)

To conduct numerical simulations by finite element methods, we often need to generate a high quality mesh, yet with a smaller number of elements. Moreover, the size of each of the elements in the mesh should be approximately equal to a given size requirement. Li et al. recently proposed a new method, named biting, which combines the strengths of advancing front and sphere packing. It generates high quality meshes with a theoretical guarantee. In this paper, we show that biting squares instead of circles not only generates high quality meshes but also has the following advantages. It is easier to generate high quality elements near the boundary with theoretical guarantee; it is very efficient time-wise; in addition, it is easier to implement. Furthermore, it provides simple and straightforward boundary protections in three dimensions.

by CHARUKA SILVA Under the Direction of Sushil K. Prasad Computational Science and Engineering (CSE) applications often exhibit the pattern of adaptive mesh applications. Adaptive mesh algorithm starts with a coarse base-level grid structure covering entire computational domain. As the computation intensified, individual grid points are tagged for refinement. Such tagged grid points are dynamically overlayed with finer grid points. Similarly if the level of refinement in a cell is greater than required, all such regions are replaced with coarser grids. These refinements proceed recursively. We have developed an object-oriented framework enabling time-stepped adaptive mesh application developers to convert their sequential applications to MPI applications in few easy steps. We present in this thesis our positive experience converting such application using our framework. In addition to the MPI support, framework does the grid expansion/contraction and load balancing making the application developer’s life easier. INDEX WORDS: