Automatic generation of node spacing function

Abstract A node spacing funnction defines the sizes of mesh elements over the entire domain. This page describes a way to generate smoothly varying node space functions from input data of (1) two-dimensional domain and (2) element sizes specified on some locations of the domain. This method generates the node spacing functions by minimizing the strain energy of a thin-plate-bending model. Background A node spacing function must satisfy the four requirements listed below. Being a function generated over any shape of two-dimensional domain. Being a function satisfying the element sizes given on some locations of domain. Being a function varying smoothly. If the node spacing function does not vary smoothly, ill-shaped elements may be generated. Being a function generated by less input data of specifying element sizes. Conventionally, a node spacing function is represented by (1) a single equation over the entire domain, or (2) a set of constant values, each for a subdomain obtained by subdividing the entire domain. The conventional methods, however, do not satisfy the requirements listed above. Automatic generation of node spacing function This method automatically generates smoothly varying node spacing functions from input data of (1) two-dimensional domain and (2) element sizes specified on some locations of the domain. The above figure shows the input data. The given domain is represented by the yellow lines and the element sizes are represented by the length of the green lines. The above figure shows a node spacing function obtained by this method. This method consists of three steps: Generation of an orthogonal lattice over the given domain, Mapping the given element sizes to the lattice points, Generation of a node spacing function by minimizing the strain energy. Details of each step are described below. 1. Generation of an orthogonal lattice over the domain An orthogonal lattice is generated to cover the entire domain depicted in the above figure. 2. Mapping element sizes to the lattice points Given element sizes at some points of the domain are mapped to the lattice points surrouding the points. The mapping is weighted so that a larger value is asssigned to the lattice point which is closer to the element-size-given point. 3. Generation of a node spacing function by minimizing the strain energy The generated orthogonal lattice is assumed to be a thin plate. The state of minimizing the strain energy of the thin plate is calculated by using the thin-plate-bending FEM theory of structural dynamics, where the values mapped to the lattice points are employed as the boundary conditions of FEM. The node spacing function generated by this method varies smoothly while satisfying the given element sizes. The above figure shows the triangular mesh generated by using the node spacing function generated by this method. This figure shows that the sizes of elements are varying smoothly. An example of adaptive subdivision using the node spacing function The generation method of node spacing function requires the two input data: region, and sampling element size values. Calculating sampling element size by referring the solution of initial analysis, adaptive meshing can be automatically done. The above figures are the image shaded by radiosity method, and the mesh used for radiosity method. Radiosity method calculates intensity for each element. Therefore, smaller elements are preffered at shadows or highlights. The above figures are the image shaded by radiosity method after adaptive meshing, and the mesh used for radiosity method. Sampling element size is calculated by referring the difference of intensity each node. Shape shadows are represented, because smaller elements are generated around shadows. However, the number of elements is reduced, because larger elements are generated where the distribution of intensity is flat. To the top page of meshing project

Last modified 30 June 1998