Applications of Graph Embedding in Mesh Untangling

The subject of this thesis is mesh untangling through graph embedding, a method of laying out graphs on a planar surface, using an algorithm based on the work of Fruchterman and Reingold[1]. Meshes are a variety of graph used to represent surfaces with a wide number of applications, particularly in simulation and modelling. In the process of simulation, simulated forces can tangle the mesh through deformation and stress. The goal of this thesis was to create a tool to untangle structured meshes of complicated shapes and surfaces, including meshes with holes or concave sides. The goals of graph embedding, such as minimizing edge crossings align very well with the objectives of mesh untangling. I have designed and tested a tool which I named MUT (Mesh Untangling Tool) on meshes of various types including triangular, polygonal, and hybrid meshes. Previous methods of mesh untangling have largely been numeric or optimizationbased. Additionally, most untangling methods produce low quality graphs which must be smoothed separately to produce good meshes. Currently graph embedding techniques have only been used for smoothing of untangled meshes. I have developed a tool based on the Fruchterman-Reingold algorithm for force-directed layout[1] that effectively untangles and smooths meshes simultaneously using graph embedding techniques. It can untangle complicated meshes with irregular polygonal frames, internal holes, and other complications that previous methods struggle with. The MUT does this by using several different approaches: untangling the mesh in stages from the frame in and anchoring the mesh at corner points to stabilize the untangling

Local Optimization-Based Untangling Algorithms for Quadrilateral Meshes

The generation of a valid computational mesh is an essential step in the solution of many complex scientific and engineering applications. In this paper we present a new, robust algorithm, and several variants, for untangling invalid quadrilateral meshes. The primary computational aspect of the algorithm is the solution of a sequence of local linear programs, one for each interior mesh vertex. We show that the optimal solution to these local subproblems can be guaranteed and found efficiently. We present experimental results showing the effectiveness of this approach for problems where invalid, or negative area, elements can arise near highly concave domain boundaries

Development and implementation of adaptive mesh refinement methods for numerical simulations of metal forming and machining

In metal forming or cutting simulations, inelastic processes in the work piece, as well as complex building component geometries or production process boundary conditions, may result in extreme deformation of the mesh and the development of large gradients in the stress or other fields. In the context of standard finite element formulations, this often leads to a loss of robustness and efficiency in the numerical simulation, and even to its failure. One method to improve the efficiency and robustness of the numerical solution under such circumstances is to automatically remesh the deformed workpiece while required. In addition, error control is required in order to achieve optimal graded meshes and maintain discretization errors within prescribed limits. The current work is focused on the issues in adaptive remeshing, which consists of error estimation, mesh refinement and coarsening, mesh optimization and application to metal forming simulations. The accuracy of a finite element solution is an important issue in finite element simulations. The main study in Chapter 1 is concentrated on the discretization error which is due to the finite element approximation of the solution. Based on the pioneer work on recovery based error estimation (Zienkiewicz and Zhu, 1987, 1992a,b), several modified versions of the SPR recovery technique are proposed. Subsequently, a local extrapolation technique (BF) is developed based on the best-fit point. The recovered derivatives are obtained at nodes via extrapolation from the sampling points and subsequent averaging. Afterwards, the discretization error is assessed by comparing the finite element solution and the recovered solution. Numerical tests show that the BF method provides the most accurate error estimation in these methods. In an adaptive simulation, remeshing techniques are required to re-discretize computational domain while the old spatial discretization is not suitable for further simulation. Unstructured meshing techniques have been shown to be effective and robust in generating a new mesh to replace the old distorted mesh. However, it could have difficulties in generating local dense mesh or yield distorted elements in graded mesh due to mesh transition. In contrast, hangingnode- based hierarchical mesh refinement can easily achieve desired local dense mesh though it doesn’t help the improvement of mesh quality. Therefore, in Chapter 2, we develop a combined unstructured and hanging-node-based remeshing strategy by exploiting the advantages of unstructured meshing technique and hanging-node-based mesh refinement technique. Mesh refinement and coarsening on boundary is realized by using a boundary node placement algorithm. It is well known that a severely distorted mesh reduces the solution accuracy (Oddy et al., 1988). Mesh smoothing techniques such as Laplacian smoothing have been shown to be effective in improving geometrical mesh quality. However, when a badly shaped mesh contains invalid elements, most existing methods are not able to optimize such a mesh. In Chapter 3, an optimization based mesh smoothing scheme based on the mesh quality measure, derived from the condition number of the Jacobian matrix, is presented to optimize both invalid and valid meshes. The corresponding optimization problem is solved with the help of the steepest descent method. The method can be used together with any type of mesh refinement approach, e.g., hanging nodes. Numerical examples using the current approach demonstrate its robustness and effectiveness. In Chapter 4, each of the parameters including error estimator, mapping algorithm, remeshing technique and element type in adaptive metal forming simulations are discussed and evaluated. The simulations of four types of manufacturing processes such as extrusion, cutting, forging and rolling have been carried out to validate the proposed adaptive remeshing procedure. In the applications, bilinear quadrilateral elements seem to be more efficient and robust than linear triangular elements. In the adaptive simulation of metal cutting, numerical comparison shows that the mapping algorithm based on local extrapolation technique (BF) transfers state variables with the least numerical diffusion. Mesh coarsening included in the adaptive remeshing procedure is shown to be able to reduce computational costs without decreasing the solution accuracy. For large deformation problems with damage, the adaptive remeshing, including a damaged element elimination procedure, is shown to be efficient