Study On Geometric Iterative Method In Solid Modeling

The hexahedral mesh is usually preferred to the tetrahedral mesh in finite element methods for numerical simulation and the quality of hexahedral mesh is very important. In finite element analysis, a valid hex mesh requires that the scaled Jacobian value at each mesh vertex is larger than 0. However, the hex mesh produced by lots of prevailing hex mesh generation methods cannot be guaranteed to be a valid hex mesh. In this paper, We develop a constrained volume iterative fitting (CVIF) algorithm to fill a given triangular mesh model with an all-hex volume mesh. Starting from an initial all-hex mesh model, which is generated by voxelizing the given triangular mesh model, CVIF algorithm fits the boundary mesh of the initial all-hex mesh to the given triangular mesh model by iteratively adjusting the boundary mesh vertices. In each iteration, the movements of the boundary mesh vertices are diffused to the inner all-hex mesh vertices. After the iteration stops, an all-hex volume mesh that fills the given triangular mesh model can be generated. In the CVIF algorithm, the movement of each all-hex mesh vertex is constrained to ensure that the scaled Jacobian value at each mesh vertex is positive after each iteration. Therefore, the all-hex mesh generated by the CVIF algorithm is guaranteed to be a valid all-hex mesh.On the other hand, with the advent of isogeometric analysis, the modeling of spline solids became an important topic. In this paper, we present a discrete volume parameterization method for tetrahedral (tet) mesh models and an iterative fitting algorithm with a B-spline solid. The discrete volume parameterization method maps the vertices of a tet mesh into a parameter domain by solving a system of linear equations. Each equation is explicitly constructed for an inner vertex in terms of the geometric information adjacent to the inner vertex. Moreover, we show the validity of the parameterization system of linear equations thus constructed. Next, because the number of tet mesh vertices is usually very large, we develop an iterative algorithm for fitting a tet mesh with a B-spline solid. The iterative algorithm exploits the geometric information of the control hexahedral (hex) mesh and the local support property of the spline function, so the total amount of computation in each iteration is unchanged when the number of control hex mesh vertices of the B-spline solid is increased. Therefore, the iterative fitting algorithm performs very well in incremental fitting of a tet mesh with a large number of vertices. Finally, four experimental examples presented in this paper show the efficiency and effectiveness of the developed algorithms.