# Master's Theses offered by Douglas R. Q. Pacheco

## Background

I am a computational mechanist working in the **intersection between Applied Mathematics and Computational Engineering**, with experience in developing **finite element methods** (FEMs) to simulate complex physical phenomena. I am particularly (but not exclusively) interested in **Computational Fluid Dynamics**, for different application fields such as industrial, biomedical and aerospace engineering. In the past years I have been working towards efficient discretisation methods for *non-Newtonian flow* problems, as often found in hemodynamics (blood flow), industrial processes (e.g., polymer extrusion), among many other scenarios.

I am very open to new ideas and discussions, so if you have some topic you are really interested in — and provided that finite element methods play a central role —, you are more than welcome to approach me with your own ideas! Other than that, you will find below a list of some project proposals I believe to be promising and for which some new, relevant developments can be made within the duration of a master's thesis.

Students working with me will join a team of applied mathematicians, physicists and engineers, and will be co-supervised by the leader group, Prof. André Massing. The general requirements for being able to take on such projects are:

- Knowledge on and interest in finite element methods
- Experience with coding, regardless of the language (e.g., Matlab, Python, C++)

## Efficient discretisation methods for non-Newtonian fluids with pressure-dependent viscosity

The Navier–Stokes momentum equation for a Newtonian fluid with viscosity \(\mu\) and density \(\rho\) is commonly written as
\[ \rho\partial_t\mathbf{u} + (\rho\nabla\mathbf{u})\mathbf{u} - \nabla\cdot(2\mu\nabla^s\mathbf{u}) + \nabla p = \mathbf{\rho g}\, ,\]
where \(\mathbf{g}\) is gravity and the *two unknowns* are the pressure field \(p\) and the flow velocity \(\mathbf{u}\), often assumed to be incompressible: \[\nabla\cdot\mathbf{u} = 0\, .\]
While a Newtonian fluid (e.g., air, water) has a constant \(\mu\), the behaviour of certain fluid-like materials can be modelled through a *nonlinear viscosity*. A very interesting and challenging application is the **modelling of granular motion**, where the solid material is so thin that it basically "flows". Examples are sand, snow and grains, which can behave almost like fluids under certain conditions (think of an avalanche or a landslide).

This similarity motivates **using the Navier–Stokes equations to model dense granular flows**, but that requires a more complex description of the viscosity. In particular, the more *compacted* a granular medium is, the more "viscous" it tends to be. In other words, one can consider an *apparent viscosity* depending at least on the pressure, but usually on the velocity field as well: \(\mu = \mu(p,\nabla\mathbf{u}) \). This modelling approach usually produces very good results, but it increases the mathematical complexity of the problem. More specifically, considering a pressure-dependent viscosity introduces a **pressure nonlinearity** in the Navier–Stokes equations, which is something not at all well studied or developed. Other than possibly compromising the well-posedness of the problem, it renders most classical numerical solvers useless. Efficient numerical algorithms are therefore scarce for these problems, which means a development opportunity.

The idea with this project is to **investigate efficient discretisation ideas for the incompressible Navier–Stokes equations with pressure-dependent viscosity**. This is especially relevant for time-dependent problems, where the pressure nonlinearity can be handled through semi-implicit stepping.

## Ultra-weak finite element methods for problems with low-regularity data

The standard model problem for elliptic equations is the Poisson problem
\[-\Delta u = f \ \ \text{in} \ \ \Omega\, ,\\
\quad \ \, u = g \ \ \text{on} \ \ \partial\Omega\, , \]
where \(f\) and \(g\) are data. In a domain \(\Omega\subset \mathbb{R}^d\) with a smooth boundary \(\partial\Omega\), standard variational formulations for this problem require \( u\in H^1(\Omega) \), \( g\in H^{1/2}(\partial\Omega) \) and \(f\in H^{-1}(\Omega)\). However, there are various **practical applications where some of those regularity requirements cannot be met** (e.g., problems with discontinuous boundary conditions).

Instead of considering the standard weak formulation
\[ \int_{\Omega} \nabla u \cdot \nabla v \, \mathrm{d}\Omega = \int_{\Omega} vf\, \mathrm{d}\Omega\, , \]
one can use smooth test functions \(v\in H^2(\Omega)\) and apply integration by parts to derive the **ultra-weak problem** of finding \(u\in L^2(\Omega) \) such that
\[ -\int_{\Omega} u \, \Delta v \, \mathrm{d}\Omega = \int_{\Omega} vf\, \mathrm{d}\Omega \, - \int_{\partial\Omega} g\frac{\partial v}{\partial n}\, \mathrm{d}\Gamma \]
is satisfied for all \( v\in H^2(\Omega)\cap H^1_0(\Omega) \). This allows, in particular, discontinuous solutions and boundary conditions, and very non-smooth data \(f\in H^{-2}(\Omega)\) — but that comes at a price. Shifting the regularity requirements to the test functions brings complications both in the theory (analysis) and practice (discretisation). Especially in finite element methods, constructing \(H^2\)–conforming test spaces is a challenge.

The idea with this project is to **investigate different finite element pairs** (of test and trial functions) and **study their stability** numerically and/or theoretically. One practical application is in **reconstructing arterial pressure** from MRI blood-flow measurements, which is typically done using a Poisson equation with very low-regularity right-hand side. We will also investigate other possible applications that can benefit from the ultra-weak framework.