Gregor Kosec (Jožef Stefan Institute Ljubljana)

Adaptive meshfree approach to solving partial differential equations

Many natural and technological phenomena are modelled through Partial Differential Equations (PDEs), which can rarely be solved analytically - either because of geometric complexity or because of the complexity of the model at hand. Instead, realistic simulations are performed numerically. There are well-developed numerical methods that can be implemented in a more or less effective numerical solution procedure and executed on modern computers to perform virtual experiments or simulate the evolution of various natural or technological phenomena.
The key element of any numerical method for the solution of partial differential equations (PDE) is discretization of the domain. In traditional numerical methods such as the finite element method (FEM), this discretization is typically performed by partitioning the domain into a mesh, i.e., a finite number of elements that entirely covers it. Despite substantial developments in the field of mesh generation, the process of meshing often remains the most time-consuming part of the whole solution procedure while the mesh quality limits the accuracy and stability of the numerical solution.
In response to the tedious meshing of realistic 3D domains, required by FEM, and the geometric limitations of FDM and FVM, a new class of mesh-free methods emerged in the 1970s. The conceptual difference between mesh-based and mesh-free methods lies in the consideration of relationships between the computational nodes. Mesh-free methods, as their name implies, fully define the relationship between nodes only by the internodal distances and thus free themselves from the shackles of using mesh. An important implication of this simplification is that mesh-free methods can operate on a set of scattered nodes. Although it is generally accepted that certain rules must be followed when generating such scattered nodes, the positioning of nodes is significantly less complex compared to meshing and can be automated regardless of the dimensionality or shape of the domain under consideration.
In this seminar we will discuss core elements of adaptive mesh-free numerical analysis, e.g. quasi-uniform and variable density node generation including consideration of problems on domains whose boundaries are represented as computer-aided design (CAD) models, high order differential operator approximation, and adaptive solution procedure. The discussion will be supported with examples ranging from simple solution of Poisson equation to the coupled system of non-linear PDEs describing thermo-fluid dynamics in irregular 3D domain.