A Lagrangian finite element method for the simulation of 3D compressible flows

The numerical solution of compressible fluid flows is of paramount importance in many industrial and engineering applications. Compared to the classical fluid dynamics, the introduction of the fluid compressibility changes the formulation of the problem and consequently its computational treatment. Among the possible numerical solutions of compressible flow problems, the finite element method has always been privileged. However, the standard Eulerian approaches with fixed domain are not particularly suited to represent the strong shock waves and the significant movement of the external boundaries. On the contrary, in problems characterized by evolving surfaces, Lagrangian approaches can be very effective. The governing equations of compressible flow problems are mass, momentum and energy conservation. These equations are discretized in the spirit of the Lagrangian Particle Finite Element Method (PFEM). The strong distortions of the mesh, typical of the Lagrangian approaches, are managed with a continuous remeshing of the computational domain. The nodal unknowns are velocities, density and internal energy. To fully exploit the potential of continuous remeshing, only nodal variables are stored and consequently only linear interpolation are used. In addition, an artificial viscosity has been introduced to stabilize the formation and propagation of shock waves. Finally, explicit time integration of the governing equations enables a highly efficient solution of the discretized problem. The proposed approach has been validated against typical benchmarks of gas dynamics in the presence of strong shock waves. A very good agreement has been shown in all the tests proving the excellent accuracy and versatility of the proposed method.