## Transient incompressible flow PFEM Algorithm

 PFEM nsteps #(in) deltaT #(rn) material #(in) cs #(in) pressure #(in) [mindeltat #(rn)] [maxiter #(in)] [rtolv #(rn)] [rtolp #(rn)] [alphashapecoef #(rn)] [removalratio #(rn)] [scheme #(in)] [lstype #(in)] [smtype #(in)]

Solves the transient incompressible flow using particle finite element method based on the Lagrangian formulation of Navier-Stokes equations.

Mesh nodes are represented by PFEMParticles 4.2, which can freely move and even separate from the main domain. To integrate governing equations in each solution step, a temporary mesh, built from particles, is needed. The mesh is rebuilt from scratch in each solution step to prevent large distortion of elements. Paramters cs and material assign types from cross section and material record to created elements. Thus, the problem is defined without any elements in the input file.

Mesh is generated using Delaunay triangulation and Alpha shape technique for the identification of the free surface. The parameter alphashapecoef should reflect initial distribution of PFEMParticles. Value approximately equal to 1,5-multiple of shortest distance of two neighboring particles has been found well. On the free surface the zero-pressure boundary condition is enforced. This must be defined in boundary condition record under the number defined by pressure.

Parameter scheme controls whether the equation system for the components of the auxiliary velocity is solved explicitly (0) or implicitly (1). The last is the default option.

Parameter nsteps determines number of solution steps. Parameter deltaT is time step length used for integration. To ensure numerical stability, step length is adapted upon mesh geometry and velocity of paricular nodes. To avoid to short time length a minimal size can be defined by mindeltat. Alternatively prescribing limit removalratio of the element edge length too close particles can be removed from solution.

Optional parameters rtolv and rtolp allow to specify relative norms for velocity and pressure difference of two subsequent iteration step. Default values are 1.e-8. By default maximal 50 iterations are performed, if not specified by maxiter.

Parameter lstype allows to select solver for linear system of equations. Parameter smtype allows to select sparse matrix storage scheme. Please note, that the present algorithm leads to a non-symmetrical system matrix. The scheme should be compatible with solver type. See section 5.1 for further details.

Borek Patzak
2018-01-02