A fast computational framework for the linear peridynamic model

The non-locality property of peridynamic poses significant computational challenges, especially for problems with discontinuous solutions (e.g., cracks). An efficient matrix-structure-based fast (MSBF) algorithm is proposed for the meshfree method (MF) to solve the linearized peridynamic models, including bond-based and ordinary state-based peridynamic models. Taking full advantage of the intrinsic Toeplitz structure of the block-diagonal stiffness matrix, the fast Fourier transform (FFT) algorithm is utilized to significantly reduce the number of floating point operations (FLOPS) count to achieve a substantial speed up at fine grid resolutions than the classical matrix–vector multiplication (MVMP) algorithm. Both methods are applied to multi-dimensional static and dynamic problems, including general boundary conditions and unsteady crack propagation, with good results. In addition, the proposed algorithm can be applied to handle problems on arbitrary domains by volume penalization technique. The MF-MSBF algorithm demonstrates its substantial computational efficiency with significantly reduced CPU times and smaller memory footprint over the MF-MVMP algorithm in multi-dimensional peridynamic simulations with fine girds.