Projects, Bachelor and Master and Theses – André Massing

For å underlette egne internettsøk/undersøkelser er teksten med bakgrunn om min forskning og beskrivelsene av temakretser skrevet på engelsk, men all veiledning pågår vanligvis på norsk (alternativt på engelsk eller tysk).

My research centers around both theoretical analysis, algorithmic realization, and application of novel computational methods for partial differential equations (PDEs). In particular, I have been developing novel finite element method for the efficient solution of complex multiphysics problems arising in e.g., engineering, biology or medicine. A common theme in my research is to challenging issue arising from considering realistic complex 3D domain geometries, handling of intricate interface conditions and moving boundaries. Prime examples of such problems are the simulation of blood flow dynamics in realistic vessel geometries, fluid-structure interaction problems, acoustic shape optimization, electrical signal propagation in 3D resolved neurons and remodelling processes of cell membranes. Recently, I have written a short piece on my research which you can find at the ECMI (European Consortium for Mathematics in Industry) blog.

A general theme is the development and/or use of open source libraries, in particular, we make heavy use of the finite element libraries ngsolve (Python-based) and Gridap (Julia-based) in our research.

Below you find a list of possible themes/current research areas, each of which can give rise to multiple concrete master's thesis topics. As a computational mathematician, I care for both analysis, implementation and applications of the numerical methods we develop, and most topics can be spun towards your preferences. Just as an example: During your project, you could e.g. derive stability and a prior error estimates for a novel scheme that we design together, or implement new AI-driven integration scheme for complex geometries, or simulate the structural properties of biological tissues.

Most projects also offers the possibility to collaborate with our external collaborators from Norway (Oslo), Germany (Augsburg, Paderborn, Berlin), Sweden (Stockholm), US (San Diego, Pittsburgh).

You find my contact information on my employee page

The CutFEM is a finite element-based framework for the numerical solution of PDEs which allows for a flexible representation of complicated and evolving geometries by decoupling the domain description from the underlying discretization, while maintaining the accuracy and robustness of a standard finite element method. This approach's generality has enabled researchers worldwide to design more flexible solvers for a multitude of problems, and below we present a small selection to illustrate the versatility of the Cut Finite Element Method.

Figure: Numerical solution of complex flow problems. (Left) Simulation of blood flow passing an aneurysm. (Right) Flexible domain decomposition in fluid-structure interaction problems.

For instance, in close collaborations with our external partners, we are currently developing CutFEMs to computationally model

• mechanics of the brain, which can be modelled as a poroelastic medium
• complex fluid interface problems arising in artificial cell models
• the dynamics of glaciers and ice sheets

Surface partial differential equations (PDEs) are crucial in modeling phenomena that occur on surfaces, such as diffusion processes on cell membranes, fluid flow on curved surfaces, and wave propagation on the Earth's surface. These equations are essential in various applications, including biological systems, material science, and geophysics. One of the main challenges in solving surface PDEs numerically is accurately representing the geometry of the surface and ensuring that the discretization respects the intrinsic properties of the surface. Additionally, handling complex and evolving geometries, as well as ensuring stability and convergence of the numerical methods, are significant hurdles that researchers must overcome.

Mixed-dimensional problems, which involve coupling equations defined on domains of different topological dimensions, are crucial in accurately modeling multi-physics phenomena where interactions occur across various scales and dimensions. For instance, in biological systems, the interaction between the processes in the 3D intra- and extracellular cell domain and on the 2D cell membranes is a typical mixed-dimensional problem. These problems pose significant numerical challenges, such as ensuring the consistency and stability of the coupling conditions, accurately capturing the interface dynamics, and efficiently solving the resulting large-scale systems.

Figure: (Left) Solution of diffusion-type problems on embedded surfaces and curves. (Right). CutFEM allows for a seamless discretization of coupled diffusion-type surface-bulk problems.

We are currently developing fitted FEM and cutFEM to simulate cell membrane remodelling processes, e.g. to model process cell motility, cell division and cell growth.

Active soft matter refers to a class of materials composed of self-driven units that consume energy to generate motion and mechanical stresses. These systems are inherently out of equilibrium and exhibit complex behaviors such as spontaneous flow, pattern formation, and collective dynamics. In the context of biological systems, active soft matter is crucial for modeling processes such as cellular motility, tissue morphogenesis, the mechanics of the cytoskeleton, and collective behavior in bacterial colonies. Understanding and simulating active soft matter can provide insights into the fundamental principles governing the behavior of living organisms and lead to advancements in biomedical engineering and synthetic biology. See this review article in Nature Review Physics which can be downloaded from arxiv. In particular, the study of phase separation phenomena in active matter systems have drawn much attention in the past decade, as a key organization principle in many biological systems, see e.g this very recent review article on arxiv

Figure: Time evolution of two, initially fully phase separated species towards microphase separation. This is sterk contrast to the classical Oswald ripening phenomena. Image taken from https://arxiv.org/abs/2412.02854

While there a plethora of different model approaches for active soft matter systems, I am interested in developing numerical methods for continuum models described by PDEs. Due to the "activity" part of the material, the PDEs typically contain non-standard terms compared to their passive counterparts, such as active stresses, active fluxes, and active sources. These terms are responsible for the emergence of complex behaviors such as self-organization, pattern formation, and non-equilibrium phase transitions. Consequently, there is a great demand for developing stable and efficient numerical methods that can accurately capture the dynamics of active soft matter systems and provide insights into their emergent behaviors.

We are currently developing numerical methods for the simulation of active droplets and fluids.