Finite and infinite block Petrov–Galerkin method for cracks in functionally graded materials

Abstract

• A novel finite and infinite block Petrov–Galerkin method is applied for crack problems in functionally graded materials . • The problem domain is discretized by a few blocks in the computation which saves tedious mesh generation . • The displacements and stresses are continuous at interfaces of neighboring blocks. • Due to the usage of Petrov–Galerkin formulation, the BPGM is a weak-form method. • Numerical results show that the proposed approach is accurate and efficient. This paper presents a novel weak-form block Petrov–Galerkin method (BPGM) for linear elastic and crack problems in functionally graded materials with bounded and unbounded problem domains. The main idea of this approach is to combine the meshless local Petrov–Galerkin method with block method. Once the problem domain is discretized into several sub-regions, named blocks, which can be mapped into normalized square domains. The weak-form Petrov–Galerkin method and polynomial series of interpolations are employed in each block. The computational efficiency is rigorously examined against the strong-form finite block method, the finite element technique and meshless approaches. Numerical results demonstrate that the BPGM possesses the following important properties: (1) only a few blocks are required for calculating problems in unbounded regions which saves tedious work of meshing; (2) the displacements and stresses are continuous at the interfaces of neighboring blocks; (3) due to the use of weak formulation, the continuity requirements of the approximation functions are reduced and numerical results are stable; (4) because of using Lagrange polynomial interpolation, highly accurate solutions can be obtained with a small amount of nodes in each block.