Extension of the Meshless Approach for the Cell Method to Three-Dimensional Numerical Integration of Discrete Conservation Laws

In this work, an extension to the three-dimensional space of the meshless approach proposed by the authors in a preceding paper is presented. The Cell Method (CM) for the numerical integration of discrete conservation laws is adopted. As in the two-dimensional case, for every node a local mesh is generated, formed by all tetrahedra whose vertices coincide with the node itself and its neighbors. Conservation equations are then written directly in discrete form on the tributary region (dual cell) made up of the polyhedron whose faces are represented by the planes passing through the circumcenters and/or the barycenters of the local mesh, the circumcenters and/or barycenters of the faces of the tetrahedra, and the midpoints of their sides. Such an approach avoids the construction of the global mesh, and is particularly efficient for non-linear problems whose iterative solution would require large CPU resources (problems with time-varying domain, or mixed Lagrangian-Eulerian formulations are examples). Moreover, the flexibility of the method allows the run-time addition or deletion of nodes (in order to increase solution accuracy, especially at critical zones of the domain) which, for a global mesh in 3D domains, would result computationally very intensive. The paper describes some algorithms (applied to Laplace equation) in order to compare their convergence order as a function of the number of neighbor nodes and the type of boundary conditions.

Keywords:

• Discrete Formulation
• Cell Method
• Meshless Approach
• FEM

Notes

^a The length of side a can be determined as a = R _m arccos( _ik · _im ), where _ik and _im represent the versor of sides ik and im, respectively. The procedure is similar for sides b and c.

^b The components of the six versors along the three rotating cartesian axes are function of the angles α, β, γ, and are given by:

^c For example a polynomial, or the functions cos (w), cos (wt), t ⁿ exp (−w), …