Absolute Position-pressure Based Particle Finite Element Method For Free Surface Flow Problems

The traditional simulation method of fluid is mainly based on Eulerian method,one of the important reasons is that Eulerian method has the ability to deal with large deformation of fluid.However,the Eulerian method will face great challenges for the problems of flow with free surface and moving boundary.In a mesh-based Lagrangian model,the mesh moves along with the continuum,the boundary and interface can be naturally tracked and identified during the movement.However,when the deformation is large,the mesh will be extremely distorted,and the accuracy of the solution will decrease or even not converge.Therefore,the traditional Lagrangian finite element method can only deal with the flow problems with small deformation.Absolute nodal coordinate(ANC)element uses the slope coordinates to describe local direction,which allows the use of a small number of elements to represent complex shapes,so it has recently been applied to the field of fluid simulation,especially the large deformation of the free surface of a liquid-filled system.Nevertheless,the absolute nodal coordinate element based on the total Lagrangian description is still be limited by the extreme deformation of the mesh and the complex contact boundary.The particle finite element method(PFEM)is a background-mesh based method.It uses the updated Lagrangian description and discretizes the domain through the finite element mesh.The nodes of the finite element mesh can be regarded as particles used to transfer the momentum of the fluid and all physical properties.These particles can move freely and even separate from the main fluid domain.Therefore,on the basis of the absolute nodal coordinate method,this paper combines the particle finite element method and its efficient mesh update technology to describe the flow with free surface,which can not only be combined with the multi-body algorithm,but also be applicable to all kinds of complex boundaries.In addition,related improvements have been made in the algorithm to avoid the time step limitation of the traditional Lagrangian method due to mesh distortion.The main works of this paper are as follows:The two-dimensional finite element model of incompressible Newtonian fluid is established by using the absolute nodal coordinate formulation and the total Lagrangian description.The penalty method is used to deal with the weak incompressibility of the fluid,and the explicit expressions of the tangent stiffness matrix corresponding to the generalized viscous force and the penalty force are given.In order to establish a unified model of the rigid-liquid system in the global coordinate system,the absolute nodal coordinate reference node(ANCF-RN)is used to describe the movement of the rigid tank.In addition,Lagrange multipliers are introduced to impose free-slip and non-penetration constraints on the boundary.In order to ensure the stability of long-term simulation,the kinetic equations of the liquid-solid system are solved by the Bathe compound integral scheme.The large deformation capability of ANCF fluid element is verified by relevant examples.Then,the free surface displacement and the pressure at the monitoring points under different external excitations are compared with the experimental data in literature,and the convergence analysis is carried out.In addition,the practicability and limitation of using absolute nodal coordinate element to analyze fluid problems are analyzed.Combined with the idea of absolute nodal coordinates and the traditional Lagrangian particle finite element method,the absolute position-pressure based particle finite element method(AP-PFEM)is proposed.Based on the Galerkin finite element method,the equivalent integral form of Navier-Stokes equations with updated configuration are derived,and the pressure stabilization of the continuity equation is carried out by using the finite incremental calculus method(FIC)to avoid the inf-sub condition.In order to improve the accuracy of the solution,a generalized-? method with high-frequency numerical dissipation and second-order accuracy is adopted,and the dynamic equations of system are solved by the "discrete-predictor-corrector" scheme.In addition,a new "predictor-discrete-corrector" model based on streamline integral is proposed,in which explicit streamline integral is used to predict the initial configuration of nonlinear iterations.This streamline prediction model based on the current background mesh can greatly reduce the time step limitation of the traditional Lagrangian model,especially when the element inversion may occur within a time step.In addition,using the absolute position as the main variable of motion can directly update the current mesh position to satisfy the momentum conservation equation.Then,based on the streamline integral prediction,further improvement is made to consider the change of streamline in different time steps.The stability of the algorithm in complex flow and large time step is verified by the relevant examples.The limitation of the traditional particle finite element method(PFEM)with no-slip boundary is analyzed.It is found that the viscous effect of the boundary has a great influence on the whole field when the coarse mesh is used.However,due to the mesh updating characteristics of PFEM,it is difficult to impose the free-slip boundary.Therefore,the virtual contact elements generated at each time step are used to identify the real contact nodes,and the free-slip constraints are imposed by Lagrange multipliers.Then,the absolute position particle finite element method is combined with the multi-body algorithm to establish a unified Lagrangian coupling system.In order to avoid the deviation of the interface nodes on the large curvature boundary in a large time step,the corresponding adjustment method is proposed.In addition,the traditional Lagrangian method is difficult to deal with the inlet and outlet boundary,mainly because the fluid particles cannot maintain the profile shape of the inlet and outlet when they moving.With the help of the idea of virtual contact layer,the inlet and outlet and driving boundary conditions are imposed.Some numerical examples show that the free-slip boundary has good mass conservation characteristics under the condition of coarse mesh and large time step.The pressure and deformation are compared with the literature experimental results to prove the stability and accuracy of the proposed method.The mesh deformation and the solution of 3D AP-PFEM with free-slip boundary are discussed and analyzed in detail.The non-physical oscillation of pressure and velocity are eliminated by using consistent normal vector to impose the free-slip boundary.In order to avoid the excessive distortion of the contact surface mesh in the simulation,and maintain the geometric features of the solid wall at the same time,an effective contact node identification method and the contact surface mesh smoothing method are proposed,and the holes prone to appear in the contact surface are repaired.In addition,through the refinement of the free surface mesh and the adjustment of the surface flux,the fluid mass loss caused during the simulation is corrected.The particle finite element model based on the absolute position-pressure scheme and the corresponding improved algorithm proposed in this paper provide a new solution idea for the simulation of engineering liquid-filled multi-body systems.