Numerical modelling of increasingly complex physical systems helps the community to address important questions such as identifying environmental problems, improving technological processes, developing biomedical applications, etc. The involved mathematical models often consist of systems of coupled partial differential equations (PDEs) that are ultimately solved numerically. Modern numerical analysis is dominated by the Finite element method (FEM), as it offers a mature solution approach that includes all kinds of adaptivities, well-understood error indicators and an established toolset for isogeometric analysis, i.e. coupling with computer-aided design (CAD). However, despite the wide acceptance of FEM, the meshing of realistic 3D domains remains a problem. In fact, meshing is one of the most complex and time-consuming steps in the entire process. In response to the tedious meshing, an alternative class of meshless methods emerged in the 1970s. As the name suggests, meshless methods define the relationship between the nodes using only the internodal distances, thus freeing themselves from the shackles of using mesh.
However, despite the many romantic descriptions of meshless methods in the literature, there are still many unanswered questions in the community. Unlike FEM (and other mesh-based methods), meshless methods are far from being fully understood, leading to a vivid research in the field. Despite decades of research, there is still no consensus on the relative performance of various methods, which is clearly problem-dependent.
In our group, we focus on the synergy between computer science and meshless numerical analysis, where we strive to combine the generic programming paradigm in C++ to achieve modularity at all levels with the elegance of meshless mathematical formulation in an open-source meshless library Medusa (https://e6.ijs.si/medusa/). We launched the Medusa project in 2015 to support our research on meshless methods and to facilitate the implementation of research and applied projects (https://e6.ijs.si/ParallelAndDistributedSystems/projects/). There are several open research topics that can be tackled within the PhD research
Mathematical analysis of meshless methods:
- Studying the basic properties of meshless methods (convergence, stability, …)
- development of adaptive methods (hp-adaptivity, error indicators, solution smoothness indicators),
- analysis of the impact of stabilisation methods on the quality of the solution (hyperviscosity, adaptive upwind scheme, etc.).
Numerical solutions to different problems:
- Computational fluid dynamics
- Simulations of the thermal state of power grid elements,
- solidification simulations,
- meteorological simulations.
Computer science:
- development of parallel algorithms for meshless numerical analysis,
- optimisation of code execution with respect to the computer architecture through low-level measurements on the central processing unit (CPU).
- development of code for execution on graphics cards (GPGPU).