Numerical Study Of The Singular Potential-Flow Hydrodynamic Problems Based On The Local Enrichment

Based on potential flow theory,achieving accurate numerical results of hydrodynamic loads of structures with sharp edges is extremely challenging,due to the singular behavior of the local-flow velocities at sharp corners.In this paper,we introduce,perhaps the first time in the literature on the marine hydrodynamics,the Extended Finite Element Method(XFEM)to the fluid-structure interaction problems involving sharp edges on structures.Compared with the conventional Finite Element Method(FEM),the Extended Finite Element Method adds local singular functions into the finite element approximation space through partition of unity to accurately capture the local singularity of the solution.Four different finite element method solver,including linear conventional FEM,quadratic conventional FEM,linear XFEM and quadratic XFEM with local enrichment by singular function at the sharp edges,are implemented and compared.In order to demonstrate the implementation of the conventional FEM and the XFEM,a one-dimensional Dirichlet-Neumann mixed boundary value problem with local properties is calculated by the linear FEM and the linear XFEM.The numerical implementation details of the FEM and the XFEM are well demonstrated by this example,and the local characteristics of solutions with local characteristics are captured well by the extended finite element method,but the local characteristics of solutions with local characteristics are difficult to be captured by the traditional FEM.The singularity of solutions affects the convergence order of numerical methods.In this paper,for the sake of exploring how conventional FEMs converge without singular solutions,linear and quadratic conventional FEM are used to solve two Dirichlet-Neumann mixed boundary value problems and Neumann-Robin mixed boundary value problems without singularity.The numerical results show that the conventional FEMs converges by order or superorder for the Dirichlet-Neumann mixed boundary value problem and Neumann-Robin mixed boundary value problem without singularity.To demonstrate the accuracy,efficiency and convergence of the XFEMs for solving the hydrodynamic problems of potential flow with singular,four FEMs are used to study the flow around a thick-less plate in a two-dimensional infinite domain.In this paper,the convergence of linear and quadratic conventional FEMs and XFEMs are analyzed by using the velocity potential in the fluid domain and the added mass of the plate.The convergence order of linear and quadratic conventional FEMs is affected by the singularity of the solution,which is obviously lower than the convergence order of theory.On the contrary,the XFEMs enriched by local singular function at sharp corners shows obvious advantages in accuracy,efficiency and convergence order.In this paper,three different local enrichment strategies of XFEMs are considered and compared.Different local enrichment strategies improve the convergence order and accuracy of numerical method differently.After numerical comparison,this paper recommends using radius enrichment to reinforce the local approximation space at the shape edges.In this paper,four FEMs are used to study the forced heave of a two-dimensional truncated cylinder on free-surface,and the linear and second-order hydrodynamic loads are considered.The results show that it is not difficult to obtain convergent numerical results for added mass and damping coefficients using traditional numerical models.However,the effect of singularity becomes extremely significant for the second-order mean wave load if the direct pressure integration is adopted along the body surface,as direct pressure integration along the body surface involving the term of velocity squared.It is extremely difficult to obtain a convergence result by using the traditional FEMs to calculate the second-order average wave load.Even if a convergence result can be obtained,the calculation burden is too heavy to be acceptable.On the contrary,using locally reinforced linear and quadratic XFEMs to calculate the second-order mean wave loads of the truncated cylinder can converge rapidly even with coarse meshes,especially for the quadratic XFEM.Unlike other methods based on domain decomposition when deal with singularities,the XFEMs,with local singular function enriched,possess a flexible framework to deal with the complex geometry,which allows use of unstructured grids without changing the numerical program.Using unstructured grid significantly reduced the mesh modeling of labor costs,moreover,using local stretched unstructured grid can significantly reduce the computational burden.Therefore,in this paper,the linear hydrodynamic loads and second-order hydrodynamic loads of two-dimensional forced heave truncated cylinder on free-surface are studied by using unstructured grids.The numerical results show that,regardless of the linear or second-order hydrodynamic loads,the labor cost and computation amount of grid modeling are significantly reduced by using unstructured grids.The present paper,though,only studies two-dimensional frequency domain,it lays a solid foundation for the follow-up work of three-dimensional frequency domain,twodimensional time domain and three-dimensional time domain.