High Performance Dirichlet Parametrization Through Triangular Be´zier Surface Interpolation for Deformation of CAE Meshes

Abstract

We present a method that extends the physics-based Dirichlet parametrization for applications concerning deformation of CAE meshes. Developed for a geometric surface feature framework called Direct Surface Manipulation, Dirichlet parametrization offers a number of operational flexibilities, such as its ability to use a single polynomial blending function to control deformation of a surface region subject to multiple user-specified displacement conditions. Dirichlet parametrization considers the domain of deformation as 2D steady-state conductive heat flow and solves for unique temperature distribution over the deformation domain using the finite element analysis (FEA) method. The result is used for evaluation of the polynomial blending function during surface deformation. The original Dirichlet parametrization, however, suffers from two limitations. First, because the 2D FEA mesh required for solving the steady-state heat transfer problem is obtained by directly projecting the affected 3D mesh onto a plane (deformation domain), both parameterization quality and performance depend on the structural characteristics of the projected 2D mesh (type of elements, node density, etc.) rather than geometrical characteristics of the deformation domain. Second, projecting a 3D mesh to create a 2D FEA mesh can be problematic when multiple areas of a 3D mesh are projected on the plane and overlap each other. Improvement techniques are presented in this paper. Instead of projecting the 3D mesh onto the plane to form the 2D FEA mesh, an auxiliary mesh is created based on geometric characteristics of the deformation domain, such as its size and boundary shape. Delaunay triangulation with an area constraint is applied in meshing the deformation region. The result is used as the 2D FEA mesh for solving the steady-state heat flow problem using the finite element method. Temperature of an affected node of the 3D mesh is obtained by interpolation in two steps. First, the node is projected onto the 2D FEA mesh, and the intersecting triangle is found. Second, the temperature at the intersection is obtained by interpolating the temperatures at the three vertices of the triangle using the cubic, triangular Be´zier interpolant. The result is equated to the temperature of the node. The use of an auxiliary mesh eliminated mesh-dependency for Dirichlet parametrization. The use of triangular cubic Be´zier interpolant results in better continuity condition of the interpolating surface between adjacent elements than linear interpolation. This allows us to employ a moderate size FEA mesh for computational efficiency. Implementation of the method is discussed and results are demonstrated.