Mesh generation for implicit geometries

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 119-126).We present new techniques for generation of unstructured meshes for geometries specified by implicit functions. An initial mesh is iteratively improved by solving for a force equilibrium in the element edges, and the boundary nodes are projected using the implicit geometry definition. Our algorithm generalizes to any dimension and it typically produces meshes of very high quality. We show a simplified version of the method in just one page of MATLAB code, and we describe how to improve and extend our implementation. Prior to generating the mesh we compute a mesh size function to specify the desired size of the elements. We have developed algorithms for automatic generation of size functions, adapted to the curvature and the feature size of the geometry. We propose a new method for limiting the gradients in the size function by solving a non-linear partial differential equation. We show that the solution to our gradient limiting equation is optimal for convex geometries, and we discuss efficient methods to solve it numerically. The iterative nature of the algorithm makes it particularly useful for moving meshes, and we show how to combine it with the level set method for applications in fluid dynamics, shape optimization, and structural deformations. It is also appropriate for numerical adaptation, where the previous mesh is used to represent the size function and as the initial mesh for the refinements. Finally, we show how to generate meshes for regions in images by using implicit representations.by Per-Olof Persson.Ph.D

On the mesh nonsingularity of the moving mesh PDE method

The moving mesh PDE (MMPDE) method for variational mesh generation and adaptation is studied theoretically at the discrete level, in particular the nonsingularity of the obtained meshes. Meshing functionals are discretized geometrically and the MMPDE is formulated as a modified gradient system of the corresponding discrete functionals for the location of mesh vertices. It is shown that if the meshing functional satisfies a coercivity condition, then the mesh of the semi-discrete MMPDE is nonsingular for all time if it is nonsingular initially. Moreover, the altitudes and volumes of its elements are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. Furthermore, the value of the discrete meshing functional is convergent as time increases, which can be used as a stopping criterion in computation. Finally, the mesh trajectory has limiting meshes which are critical points of the discrete functional. The convergence of the mesh trajectory can be guaranteed when a stronger condition is placed on the meshing functional. Two meshing functionals based on alignment and equidistribution are known to satisfy the coercivity condition. The results also hold for fully discrete systems of the MMPDE provided that the time step is sufficiently small and a numerical scheme preserving the property of monotonically decreasing energy is used for the temporal discretization of the semi-discrete MMPDE. Numerical examples are presented.Comment: Revised and improved version of the WIAS preprin

Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations

We introduce a new method, the Local Monge Parametrizations (LMP) method, to approximate tensor fields on general surfaces given by a collection of local parametrizations, e.g.~as in finite element or NURBS surface representations. Our goal is to use this method to solve numerically tensor-valued partial differential equations (PDE) on surfaces. Previous methods use scalar potentials to numerically describe vector fields on surfaces, at the expense of requiring higher-order derivatives of the approximated fields and limited to simply connected surfaces, or represent tangential tensor fields as tensor fields in 3D subjected to constraints, thus increasing the essential number of degrees of freedom. In contrast, the LMP method uses an optimal number of degrees of freedom to represent a tensor, is general with regards to the topology of the surface, and does not increase the order of the PDEs governing the tensor fields. The main idea is to construct maps between the element parametrizations and a local Monge parametrization around each node. We test the LMP method by approximating in a least-squares sense different vector and tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply the LMP method to two physical models on surfaces, involving a tension-driven flow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMP method thus solves the long-standing problem of the interpolation of tensors on general surfaces with an optimal number of degrees of freedom.Comment: 16 pages, 6 figure

Size preserving mesh generation in adaptivity processes

It is well known that the variations of the element size have to be controlled in order to generate a high-quality mesh. Hence, several techniques have been developed to limit the gradient of the element size. Although these methods allow generating high-quality meshes, the obtained discretizations do not always reproduce the prescribed size function. Specifically, small elements may not be generated in a region where small element size is prescribed. This is critical for many practical simulations, where small elements are needed to reduce the error of the numerical simulation. To solve this issue, we present the novel size-preserving technique to control the mesh size function prescribed at the vertices of a background mesh. The result is a new size function that ensures a high-quality mesh with all the elements smaller or equal to the prescribed element size. That is, we ensure that the new mesh handles at least one element of the correct size at each local minima of the size function. In addition, the gradient of the size function is limited to obtain a high-quality mesh. Two direct applications are presented. First, we show that we can reduce the number of iterations to converge an adaptive process, since we do not need additional iterations to generate a valid mesh. Second, the size-preserving approach allows to generate quadri- lateral meshes that correctly preserves the prescribed element size.Peer ReviewedPostprint (published version