Let us consider a domain containing both fluid and solid subdomains. The moving fluid particles interact with the solid boundaries thereby inducing the deformation of the solid which in turn affects the flow motion and, therefore, the problem is fully coupled.
In the PFEM approach, both the fluid and the solid domains are modelled using an updated Lagrangian formulation. The finite element method (FEM) is used to solve the continuum equations in both domains. Hence a mesh discretizing these domains must be generated in order to solve the governing equations for both the fluid and solid problems in the standard FEM fashion. The nodes discretizing the fluid and solid domains can be viewed as material particles which motion is tracked during the transient solution.
The Lagrangian formulation allows us to track the motion of each single particle (a node) in either the fluid and solid domains. This is particularly useful to model the separation of water particles from the main fluid domain and to follow their subsequent motion as individual particles with an initial velocity and subject to gravity forces. The lagrangian formulation also allows to naturally track the large deformation of a solid subjected to the forces induced by the fluid.
The quality of the numerical solution depends on the refinement of the discretization chosen, as in the standard FEM. Adaptive mesh refinement techniques are used to improve the solution in zones where large motions of the fluid or the structure occur.
In summary, a typical solution with the PFEM involves the following steps.
1. Identify the external boundaries for both the fluid and solid domains. This is an essential step as some boundaries (such as the free surface in fluids) may be severely distorted during the solution process including separation and re-entering of nodes. The Alpha Shape method is used for the boundary definition (Figure 3).
2. Discretize the fluid and solid domains with a finite element mesh. The mesh generation process is based on an enhanced Delaunay discretization of the analysis domain using an initial collection of points which then become the mesh nodes. Additional nodes can be created during the mesh generation process.
3. Solve the coupled Lagrangian equations of motion for the fluid and the solid domains. Both an implicit (iterative) solution or a fully explicit solution are possible. Compute the relevant state variables in both domains at each time step: velocities, pressure and viscous stresses in the fluid and displacements, stresses and strains in the solid.
4. Move the mesh nodes to a new position in terms of the time increment size.
5. Go back to step 1 and repeat the solution process for the next time step.
Figure 4 shows a typical example of a PFEM solution in 2D. The pictures correspond to the analysis of the problem of breakage of a water column. Figure 4a shows the initial grid of four node rectangles discretizing the fluid domain and the solid walls. Boundary nodes identified with the Alpha-Shape method have been marked with a circle. Figures 4b and 4c show the mesh for the solution at two later times. An animation of this problem can be seen in the Gallery of images.