Open AccessThe aim of the present research is to develop software for the Particle Finite Element Method (PFEM) and its verification on the model problem of viscous incompressible flow simulation in a square cavity. The Lagrangian description of the medium motion is used: the nodes of the finite element mesh move together with the fluid that allows to consider them as particles of the medium. Mesh cells deform when in time-stepping procedure, so it is necessary to reconstruct the mesh to provide stability of the finite element numerical procedure.
Meshing algorithm allows us to obtain the mesh, which satisfies the Delaunay criteria: it is called \the possible triangles method". This algorithm is based on the well-known Fortune method of Voronoi diagram constructing for a certain set of points in the plane. The graphical representation of the possible triangles method is shown. It is suitable to use generalization of Delaunay triangulation in order to construct meshes with polygonal cells in case of multiple nodes close to be lying on the same circle.
The viscous incompressible fluid flow is described by the Navier | Stokes equations and the mass conservation equation with certain initial and boundary conditions. A fractional steps method, which allows us to avoid non-physical oscillations of the pressure, provides the timestepping procedure. Using the finite element discretization and the Bubnov | Galerkin method allows us to carry out spatial discretization.
For form functions calculation of finite element mesh with polygonal cells, \non-Sibsonian interpolation" is used, which allows to construct and reconstruct the functions with desired properties for finite elements with arbitrary number of nodes. The model problem of the flow simulation of viscous incompressible fluid in a square cavity is considered. The diagram of the velocity distribution after the first time step is shown and the numerical convergence of solutions for different initial meshes (regular quadrilateral and triangular) is discussed.
As the conclusion, the considered algorithm is useful for solving a number of problems in the domains with complex geometry, which can be either broken into several domains or merged. A Lagrangian approach us allows to take into account the convective terms of the governing equations automatically. However, the procedure of the mesh reconstruction has large computational cost and time step have to be reduced significantly if particles are located close one to each other. So, there are some possibilities of algorithm improvement.