Geometric Shape Preservation in Adaptive Refinement of Finite Element Meshes

This thesis presents a vertex repositioning based boundary smoothing technique so that the finite element mesh, as refined, converges to the geometric shape of the domain. Adaptive mesh refinements based on a posteriori error estimates is already available within FEniCS, a free software for automated solution of differential equations. However, as the refinement process runs independently of the CAD (Computer-Aided Design) data, once the initial mesh is created the geometric shape is fixed and remains unchanged. The proposed technique removes this limitation enabling the mesh to be reshaped during the mesh refinement process according to the geometry. In the proposed technique, upon mesh refinements, the boundary vertices are repositioned to the orthogonal projection point on to the surface geometry. Newton iteration based iterative closest point algorithms have been used to search points projection on to the geometry represented as NURBS (Nonuniform Rational B-splines). For the new boundary vertices, old neighbor-nodes’ projection data have been used for obtaining an initial guess for the Newton’s iteration. Projection parameters for the boundary vertices of an initial coarse mesh is gathered using exhaustive search for points projection on the whole geometry. Once the mesh boundary vertices are repositioned, positions of the internal vertices are adjusted using mesh smoothing. Experimental results show that the method is sufficiently accurate and efficient for both 2D and 3D problems for some simple test cases. Inverted cells, which could invalidate the mesh, are not developed. Cell based mesh quality analysis and time performance analysis have been presented. Experimentally, it has been shown that using the proposed technique it is sufficient to start with a coarse mesh all over the domain and adaptively refine the mesh and the coarse mesh, as refined, captures the curved geometry more accurately. Since the proposed technique does not change the mesh topology and the finite element solver or mesh elements are not required to be changed, the developed tools can be plugged in to the finite element solver without much involvements. Although the tools are developed to become integrated parts of FEniCS, the parts representing geometric data manipulation can be used with other applications as well. Implementations in this thesis are limited for single NURBS patch geometry only and recommendations are given for future improvements to incorporate multiple patch NURBS geometry and trimmed NURBS.