Weakly Compressible Material Point Method For Fluid-structure Interaction

Fluid-structure interaction(FSI)problems exist widely in the nature and engineering fields,and have a strong application background and needs in hydraulic civil engineering and aerospace engineering.Fluid-structure interaction problems often involve large deformations and geometric nonlinearities,such as the fracture and fusion of the liquid surface.The traditional grid-like numerical methods have difficulties and challenges in grid distortion and tracking of free interfaces.As a new meshless method,the weakly compressible material point method(WCMPM)can avoid the mesh distortion and particles can automatically track the fluid surface.However,in the case of medium and small deformation,the accuracy of the material point method is often not as good as the finite element method.The coupled finite element point method(CFEMP)uses the finite elements to simulate the solid region,while the material points to simulate the fluid region.The interaction between the two is realized by a coupling algorithm.This method not only retains the high precision and efficiency of the finite element method but also avoids the difficulty of mesh distortion,which is very suitable for solving FSI problems.The coupling algorithm in the CFEMP is implemented based on the background grid.Thus the mesh inconsistency of FEM and MPM will results in the non-physical penetration.In order to relieve the dependence on the mesh consistency in the CFEMP,an improved CFEMP(ICFEMP)method is proposed,which is based on a particle-to-surface contact algorithm.The contact detection in the ICFEMP method is no longer based on the background grid nodes.Instead,it depends on the surface information of the FE mesh.Therefore,the size of the finite element can be much larger than the background grid size and the non-physical penetration is avoided.In order to suppress the volumetric locking and pressure oscillation in the WCMPM,the v-p MPM based on the two-field manner is presented,which treats the velocity v and the pressure p as the independent variables.Compared with the Hu-Washizu variational principle,the v-p manner has only one extra variable(pressure)needed to reconstruct.Besides,if a large slope difference exists between different control volumes,a false pressure oscillation will occur at the discontinuous interface.Thus,a slope limiter is employed to further suppress pressure oscillations.In addition,the v-p manner can be easily introduced into the ICFEMP method,and the resulting v-p ICFEMP can be used to model FSI problems.For the low-speed and long-term FSI problems,the explicit time integration is inefficient because of the small time step.Using the implicit time integration is more reasonable.Thus,a contact algorithm for the implicit material point method based on the augmented Lagrange multiplier method is proposed.The contact problem belongs to the saddle point problems,which is undefined and highly ill-conditioned.By using the Uzawa algorithm,the condition number of the original problem is significantly improved and the positive definiteness of the effective stiffness matrix is preserved.In addition,the Uzawa algorithm decouples the unknown variables from multipliers,which simplifies the solution process.The implicit contact algorithm guarantees a large time step,which lay a good foundation for the further study of the FSI problem with the implicit WCMPM.