Computational Methods in Engineering

 Grid dispersion is one of the criteria of validating the finite element method (FEM) in simulating acoustic or elastic wave propagation. The difficulty usually arisen when using this method for simulation of wave propagation problems, roots in the discontinuous field which causes the magnitude and the direction of the wave speed vector, to vary from one element to the adjacent one. To solve this problem and improve the response accuracy, two approaches are usually suggested: changing the integration method and changing shape functions. The Finite Element iso-geometric analysis (IGA) is used in this research. In the IGA, the B-spline or non-uniform rational B-spline (NURBS) functions are used which improve the response accuracy, especially in one-dimensional structural dynamics problems. At the boundary of two adjacent elements, the degree of continuity of the shape functions used in IGA can be higher than zero. In this research, for the first time, a two dimensional grid dispersion analysis has been used for wave propagation in plane strain problems using B-spline FEM is presented. Results indicate that, for the same degree of freedom, the grid dispersion of B-spline FEM is about half of the grid dispersion of the classic FEM.

This study shows how to create different types of crack and discontinuities by using isogeometric analysis approach (IGA) and extended finite element method (XFEM). In this contribution, two unique features of isogeometric analysis approach are utilized to create discontinuous zones. Discontinuities consist of crack and cohesive zone. In isogeometric analysis method NURBS is used to approximate both geometry and primary variable. NURBS can create quadratic shapes exactly. Also, stress intensity factors are calculated in the vicinity of the crack tips for two dimensional problems and are compared with corresponding analytical and numerical counterparts. Extended finite element method is the other numerical method which is used in this work. The enrichment procedure is utilized in extended finite element method to create discontinuities. The well-known path independent J-integral approach is used in order to calculate the stress intensity factors. Also, in mixed mode situation, the interaction integral (M-integral) is considered to calculate the stress intensity factors. Results show that isogeometric analysis method has desirable accuracy as it uses lower degree of freedoms and consequently lower computational efforts than extended finite element method. In addition, creating the internal cohesive zone as one of the most important issues in computational fracture mechanics is feasible due to the special features of isogeometric analysis. The present study demonstrates the capability of isogeometric analysis parametric space to control the inter-element continuity and create the cohesive zone.