Finite element analysis using uniform B-spline approximation and implicit boundary method

The finite element method (FEM) is a widely used numerical technique for solution of boundary value problems arising in engineering applications. FEM is well studied and has several advantages; however the stress and strain solutions obtained by FEM are discontinuous across element boundaries and require smoothing algorithms to make the stress/strain appear smooth in the analysis domain. Automatic mesh generation is still a difficult task in FEM for complicated geometries and often requires user intervention for generating quality meshes.;Finite elements based on uniform B-spline basis functions defined on a structured grid are developed which provide at least C 1 continuous solution through out the analysis domain. Numerical techniques are presented in this work based on structured grids and implicit equations of the boundaries of analysis domain, material interface boundaries or coarse/fine grid interfaces for solution of various engineering boundary value problems. Solution structures are constructed using approximate Heaviside step functions for imposition of Dirichlet boundary conditions, treatment of material discontinuity to perform micromechanical analysis and for local refinement of grid. The use of structured grid eliminates the need for constructing a conforming mesh and results in significant savings of time in pre-processing stage of the design cycle.;Numerical examples are presented to demonstrate the performance of B-spline elements. The results are compared with analytical solutions as well as traditional finite element solutions to demonstrate the ability of B-spline elements to represent continuous stress and strain through out analysis domain. Convergence studies show that B-spline elements can provide accurate solutions for many engineering problems with fewer numbers of elements and nodes as compared to traditional FEM. Solution structure for treatment of material boundary is validated by performing a convergence analysis on a problem involving circular inclusion in a square matrix and by determining effective properties of fiber reinforced composite. Solution structure for local grid refinement is validated by analyzing classical stress concentration problems.