The Modified Direct Method: an Approach for Smoothing Planar and Surface Meshes

The Modified Direct Method (MDM) is an iterative mesh smoothing method for smoothing planar and surface meshes, which is developed from the non-iterative smoothing method originated by Balendran [1]. When smooth planar meshes, the performance of the MDM is effectively identical to that of Laplacian smoothing, for triangular and quadrilateral meshes; however, the MDM outperforms Laplacian smoothing for tri-quad meshes. When smooth surface meshes, for trian-gular, quadrilateral and quad-dominant mixed meshes, the mean quality(MQ) of all mesh elements always increases and the mean square error (MSE) decreases during smoothing; For tri-dominant mixed mesh, the quality of triangles always descends while that of quads ascends. Test examples show that the MDM is convergent for both planar and surface triangular, quadrilateral and tri-quad meshes.Comment: 18 page

OceanMesh2D 1.0: MATLAB-based software for two-dimensional unstructured mesh generation in coastal ocean modeling

OceanMesh2D is a set of MATLAB functions with preprocessing and post-processing utilities to generate two-dimensional (2-D) unstructured meshes for coastal ocean circulation models. Mesh resolution is controlled according to a variety of feature-driven geometric and topo-bathymetric functions. Mesh generation is achieved through a force balance algorithm to locate vertices and a number of topological improvement strategies aimed at improving the worst-case triangle quality. The placement of vertices along the mesh boundary is adapted automatically according to the mesh size function, eliminating the need for contour simplification algorithms. The software expresses the mesh design and generation process via an objected-oriented framework that facilitates efficient workflows that are flexible and automatic. This paper illustrates the various capabilities of the software and demonstrates its utility in realistic applications by producing high-quality, multiscale, unstructured meshes.</p

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

A Direct Smoothing Method For Surface Meshes

Various algorithms employed in automated meshing of surfaces often produce distorted elements in the first phase. Different methods are used in the second phase to improve the quality of the mesh. Smoothing is one such method. In this method, the nodes are relocated without altering the element connectivity to improve the mesh quality. The most commonly used smoothing is called Laplacian smoothing. It is an iterative method. In this paper, a direct, method is presented for smoothing surface meshes. It is shown that a system of equations can be generated for the new nodal locations and then be solved for the constraints at once. The generation of the system of equations is very similar to assembling of global stiffness matrix from the element stiffness matrices. The element stiffness matrices are constant in local coordinate systems. In this method, final mesh is independent of the initial locations of unconstrained nodes