A Study For The Improved B-spline Material Point Method

Numerical methods play an increasingly important role in engineering applications and scientific researches.However,the traditional mesh-based methods,such as Finite Element Method(FEM),Finite Difference Method(FDM),and Finite Volume Method(FVM),have to resort to the remesh technique for avoiding the mesh distortion when solving the problems with large deformation,material failure,perforation,interface propagation between multiphase materials,etc.,which could lead to complicated algorithm,lower accuracy,increasing computational cost,and even numerical instability.Without relying on the rigid mesh connection,the meshfree particle methods are well suited to simulating those complex problems.As one meshfree method,the Material Point Method(MPM)is a particle-grid method evolving from a Particle-in-Cell method of FLIP.In the MPM,the field variables are recorded at Lagrangian material particles and the discretized momentum equations are solved on an Eulerian background mesh.Therefore,the MPM combines the advantages of both Lagrangian and Eulerian methods but avoids their shortcomings.So far,the MPM has been widely applied to many challenging problems since its first report.However,there exists the so called "cell-crossing error" and the quadrature error in the original MPM.In this dissertation,a high-order method,B-spline Material Point Method(BSMPM),is discussed and improved as follows:(1)In the original MPM,the piece-wise linear grid basis functions are used to map the properties between the particles and the grid nodes and to calculate the internal forces at the grid nodes.Due to the discontinuous gradients of linear basis functions at the grid cell boundary,the "cell-crossing instability" would occur when the particles cross a cell boundary,resulting in the nonphysical jump in the internal forces and velocity gradient and thus reducing the solution accuracy.To solve this problem,the B-spline Material Point Method(BSMPM)has been proposed by simply using the B-spline basis functions instead of the piece-wise linear basis functions.In this dissertation,the basic idea and algorithm implementation of BSMPM are presented in detail.By means of the numerical examples,the computational accuracy,convergence,and efficiency of the BSMPM with quadratic,cubic and quartic B-spline basis functions are investigated and compared with the original MPM,and other improved MPM schemes of the Generalized Interpolation Material Point(GIMP)method,the Convected Particle Domain Interpolation(CPDI)method and Dual Domain Material Point Method(DDMP).The numerical results indicate that the BSMPM can significantly reduce the cell-crossing error resulting from the discontinuous gradient of piece-wise linear basis functions at the grid cell boundary and exhibits the higher solution accuracy and convergence than the original MPM.With the increase of B-spline function order,the accuracy and convergence of the BSMPM can be improved partly.Although the consuming time of the BSMPM is longer than the MPM,the increasing rate of consuming time is basically consistent with the MPM.In addition,the BSMPM shows better performance in term of the stress accuracy and the energy dissipation conservation as compared with the GIMP and the CPDI.(2)The quadrature error in the calculation of internal force at grid nodes,especially for large deformation problems,is one of the main sources for the solution error and is due to the fact that the material particle is used as the integration points while the weight function is the particle domain.To solve this problem,an improved BSMPM scheme,referred as Gauss Quadrature B-spline Material Point Method(gqBSMPM),is proposed by applying the Gaussian quadrature in the particle domain for the internal force calculation.Through the numerical examples,the computational accuracy,convergence,and efficiency of the gqBSMPM are investigated.It is shown that the incorporation of the Gauss quadrature into the internal force evaluation can drastically improve the accuracy and convergence of the quadratic BSMPM,especially for the case of lower particle density or smaller grid size,whereas the reduction of quadrature error is insignificant in the cubic or higher-order quartic BSMPM.Although the quadrature error in the BSMPM can be reduced by increasing the order of B-spline basis functions,the lower-order gqBSMPM exhibits higher efficiency at the comparable or higher accuracy.(3)As many scientific and engineering problems could include localized failure and large deformation,the background grid local refinement technique becomes highly desirable to improve the computational efficiency so that the fine grid mesh is employed in the area of high-stress gradients while the coarse grid mesh is used in other areas.For this purpose,a method for the local refinement of background grid in the BSMPM is proposed based on the bridge domain method and verified by representative numerical examples.In addition,the local mesh refinement methods with the Truncated Hierarchical B-spline(THB)functions and Locally Refined B-spline(LRB)basis functions are also examined.For quasi-static problems,all studied mesh refinement methods could be used for the BSMPM.However,for dynamic problems,the approaches based on the THB and LRB basis functions could result in spurious reflection of stress waves at the interface between the fine-scale mesh and coarse-scale mesh.Instead,such reflection of stress wave at the fine-coarse mesh interface is effectively reduced when the local mesh refinement based on the bridge domain method is used.