As the automotive industry increasingly shifts toward electrification, reducing vehicle drag becomes crucial for enhancing battery range and meeting consumer expectations. Additionally, recent European regulations require tire and car manufacturers to provide reliable drag data. A significant factor influencing vehicle drag is the highly turbulent wake generated by rotating tires. Accurate correlation between wind tunnel experiments and numerical simulations is essential, yet discrepancies often arise due to the dynamic behavior and technical challenges in measuring tire deformation, leading to inconsistencies between tire deformations observed in wind tunnels and those modelled in simulations.

This Master’s Thesis investigates the aerodynamic impact of realistic tire deformation parameters—specifically, bulge and contact patch deformations—using Computational Fluid Dynamics (CFD) simulations conducted with PowerFLOW. The deformation parameter tests are carried out in a standalone tire setup, which is validated from previous work by Dassault Systèmes and experimental results. To further assess the impact of these deformations on vehicle drag, the tests were repeated using the DrivAer model, an IP free model provided by the Technical University of Munich. Given the complexity of the flow features in a tire wake, the analysis of the results consisted of two different approaches: a statistical approach based on Principal Component Analysis (PCA), and a vortex behaviour and wake development analysis, including a vortex tracking algorithm based on the Gamma 2-Criterion and single-link hierarchical clustering.

The results identify the most influential deformation parameters based on their impact on the drag coefficient and the wake behaviour changes. Two primary wake development behaviours were identified: wake contraction and wake expansion. An analysis of the transient flow showed how these two trends resulted from changes in the unsteady behaviour introduced by certain deformation parameters. The relevance of these wake development changes was portrayed by the impact of certain deformations on the full vehicle drag as a result of
complex wake interactions.

One of the novel methodologies in computational physics research is to use mimetic discretisation techniques. Among these, the mimetic spectral element method holds special promise as it not only has the benefits of mimetic methods but also the additional benefit of higher-order discretisations using higher polynomial degrees. These methods are aided by the development of algebraic dual polynomials, resulting in a sparser system for better computational efficiency. This combination was used to develop a formulation that would result in topological relations for equilibrium of forces as well as the symmetry of the stress tensor for linear elasticity as well as the first steps for Stokes flows in an orthogonal domain. As a result, this study was extended to look at how a modified formulation would behave for an unsteady linear elastic solid, with the intention to extend this method to Fluid-Structure Interaction cases. However, the choice of both primal and nodal basis functions makes it impossible to undertake this challenge, demanding a rethink in strategy towards looking at linear elastic solids when the physical domain is not orthogonal. With the use of bundle-valued forms to represent physical quantities, a new hybrid formulation is developed where the equivalent of the physical problem is computed on a reference domain, which is orthogonal and thus can utilise the spectral bases defined before. The physical problem is defined on a skewed domain, where partial transformation of components results in a formulation that can conserve linear momentum point-wise, but not conservation of angular momentum, although angular momentum does converge on refinement of polynomial degree and mesh parameters. A change in bases with partial transformation aiming to make angular momentum conservation topological is not fruitful, although the value of the error decreases in the process. The final attempt is through full transformation, which results in a formulation with an inherent error in the formulation, owing to erroneous assumptions.

Interatrial shunting is a proposed technique to reduce elevated left heart and pulmonary pressures in heart failure patients. Clinical trials show promising results in relief of symptoms and improvements in quality of life, but little is still known about the working principles of interatrial shunting and important questions remain regarding the optimal diameter. The current research investigates the patient-specific optimal diameter of an interatrial shunt through computational fluid dynamics simulations. An idealized two-dimensional model of the left and right atria, four pulmonary veins, the two vena cavae and the two atrioventricular valves is used to study the intra- and interatrial flow fields in absence and presence of an interatrial shunt through steady-state and transient simulations. For the transient simulations, the inlets and outlets are coupled through a Windkessel including four reservoirs, representing the two ventricles and the pulmonary and systemic circulations. This coupling represents the closed-loop behavior of the circulatory system and ensures realistic inlet and outlet pressures throughout the cardiac cycle. Furthermore, wall movement is applied in the transient simulation to model the atrial deformation during the systolic and diastolic phases of the heart cycle. The introduction of an interatrial shunt significantly influences the atrial flow field and a shunt flow from the left atrium to the right atrium is observed in all the simulations throughout the cardiac cycle, increasing the ratio of pulmonary to systemic blood flow (Qp:Qs). This reduces pulmonary, left atrial and left ventricular pressures while slightly increasing systemic, right atrial and right ventricular pressures. The patient-specific optimal shunt diameter is studied by applying shunts with diameters ranging between 0 and 20 mm to three patients of varying left ventricular stiffness. From this, it is found that the pressure reductions of the smallest (< 6 mm) as well as the largest shunts (> 12 mm) are little sensitive to shunt diameter, whereas medium-sized shunts are the most sensitive to the diameter. It is concluded that the optimal shunt diameter is patient-specific and is defined as the smallest diameter that manages to reduce a patient’s peak pulmonary pressure to below 15 mmHg, as long as its Qp:Qs ratio does not exceed 1.5. Otherwise, the optimal shunt is the one with the largest diameter that does not exceed Qp:Qs = 1.5.

The mimetic spectral element method is a relatively young method in numerical solutions of partial differential equations and actions that describe physical systems. Its advantage is that it takes the geometrical structure of the problem into account which guarantees consistency of the numerical scheme and the conservation of relevant quantities. This prohibits the solution to diverge into non-physical states since the discretization mimics the physical behaviour of the problem. With relation to physics the mimetic method makes use of the stationary action principle which contains all information about the system and reduces the continuous variables to discrete ones that become ready for computation. The stationary action principle requires a formulation of the Lagrangian such that it implies Newton’s laws of motion or equivalent physical laws. However these Lagrangians are only easily formulated for monogenic systems. This work investigates the class of (discrete) non-monogenic physical systems. The goal is to present a general method to numerically mimic non-conservative systems such that energy (loss) is exact. This is done by modifying a newly introduced systematic method that transforms the non-conservative problem to a system with a doubled degrees of freedom that makes it a conservative or monogenic one. In this modification process new degrees of freedom are introduced which are allowed to have arbitrary variations at the initial and final states of the new action. The system can than be mimicked with the spectral element method where algebraic dual polynomials are used as both approximating polynomials and as test functions that serve as arbitrary variations. Furthermore a gauge transformation is introduced by the modified action such that it contains the initial and final states of the system to make the solution unique.
The introduced method is generally applicable to Lagrangian mechanics of discrete systems with holonomic constraints. The method is applied to the damped harmonic oscillator (DHO) as a test case. The computational results are clearly accurate and converging. However they deviate with one order from the h-refinement spectral element theory of Gauss-Lobatto-Legendre interpolation basis polynomials. Theoretical analysis of the equations of the mimicked (single and doubled) DHO show that energy (loss) is exact. Nevertheless, the computations show an oscillation in energy that remains closely around the conserved value. This oscillation in energy in time was observed in earlier works that applied the mimetic spectral element method to conservative systems. The reason behind this behavior of energy remains a topic for further research. Furthermore, in a follow up research the method can be extended to the Lagrangian field theory and tested with simplified non-conservative fluid problems to verify it for fields.

The mimetic spectral element method (MSEM) is a structure-preserving discretization scheme based on the Galerkin Method, which strongly constrains the topology relations by discretizing and reconstructing variables in specific function spaces in order to preserve certain critical structures of the PDE in the numerical solution. In studying the 2D incompressible Navier-Stokes equations, the conservation law of mass, energy, vorticity, and enstrophy (or helicity for 3D cases) are expected to be preserved. According to the de Rham complex, the mimetic spectral element method uses differential forms rather than vector or scalar fields to present physical variables and discretize differential forms on specified function spaces. It has two significant advantages. Firstly, the topological relations between discretized variables depend only on the grid's topology structure, which means no numerical errors are introduced into the discretized conservation equations. Secondly, the variables are reconstructed with spectral functions, which can be of arbitrary high order.

Based on the MSEM, a more efficient hybrid dual mimetic spectral element method (hdMSEM) was proposed. In the hybrid mimetic spectral element method (hMSEM), a set of trace function spaces and trace variables are introduced at the interface between subdomains, applying a Lagrange multiplier to strongly couple the variables of bordered subdomains so that domain decomposition is feasible, and the solver can run in parallel efficiently. In addition, designing a proper set of dual grid and dual function spaces for trace variables can further increase the sparsity of the matrix, thus saving computational resources. However, a singularity problem arises in the structure-preserving simulation of incompressible flows with the primal hdMSEM when Lagrange multipliers are applied to couple variables of vorticity at the edge where more than two subdomains meet.

This thesis proposes the hybrid dual mimetic spectral element method with a novel dual grid, which can avoid the singularity and simultaneously keep the matrix of the discrete system symmetrical and the mathematical definition of the matrix equations rigorous. The basic idea is to introduce a dummy degree of freedom at the edge where singularity arises and design a curvilinear dual grid for trace variables to couple the degree of freedom of vorticity and the dummy degree of freedom to eliminate singularity. Besides, this thesis studies the implementations of several kinds of boundary conditions with the novel dual grid and the corresponding grid topology near boundaries. Then, we extend the hdMSEM with the novel dual grid to solve steady and unsteady 2D incompressible Navier-Stokes equations. In numerical experiments, the accuracy and structure-preserving capability are verified numerically with several benchmark cases.

The need of computational power for engineering applications has been ever increasing and with classical computers approaching their physical limits, new ways of improvement have to be investigated. One of the promising solutions is quantum computing. Most engineering problems require solving a system of linear equations of higher dimensions and the Variational Quantum Linear Solver (VQLS) algorithm seems like a promising near term solution. This algorithm evaluates a cost function on a quantum machine and uses a classical optimizer to minimize this cost function. The minimum of this cost function corresponds to the solution of the linear system. This work aims at finding what the practical limitations are when solving the Poisson equation by means of VQLS. Results showed that problems small in size can be solved, however for larger problems obtaining the solution becomes effectively impossible due to barren plateaus, which are flat spots in the cost function landscape.

Advection is at the heart of fluid dynamics and is responsible for many interesting phenomena. Unfortunately, it is also the source of the non-linearity of fluid dynamics. As such, its numerical treatment is challenging and often suboptimal. One way to more effectively deal with advection is by using a Lagrangian formulation instead of the conventional Eulerian view.

This work aims to show that the Lagrangian formulation, and its underlying geometric and physical character, are fundamental in overcoming the challenges posed by (non-)linear advection. The Mimetic Spectral Element Method ensures that this geometric and physical character is kept when the equations are discretised. The advection term can then be dealt with exactly. The method is put to the test by solving the inviscid Burgers and isentropic Euler equations in one spatial dimension. The results confirm the exactness of the advection term and good overall accuracy.

Structure-preserving or mimetic discretisations are a class of advanced discretisation techniques derived by employing concepts from differential geometry. Such techniques can attain specific conservation properties at the discrete level such as conservation of mass, kinetic energy, etc when applied to conservation laws. However, like traditional techniques, they are not entirely robust in specific multiscale cases such as under-resolved simulations which are often encountered in industrial applications. This ties into the concept of Large Eddy Simulations (LES) of which an enticing approach is the Variational Multiscale (VMS) method. Both Mimetic methods and VMS approaches have been extensively studied independently, however, their combination offers the potential of achieving a favorable robust solver capable of handling complicated industrial problems. Moreover, the extension of the Hybridised variant of the Mimetic methods towards advection-dominated problems is an interesting avenue yet to be explored. Therefore, this thesis focuses on the extension of the Hybridised Mimetic method and its combination with the VMS theory. The said combination is tested on simple linear equations, namely the advection-diffusion equation in both steady and unsteady cases. Additionally, the Hybridised Mimetic method is studied in more complex test cases such as the Burgers' equation and the incompressible Euler/Navier-Stokes equations.

Mimetic formulations, also known as structure-preserving methods, are numerical schemes that preserve fundamental properties of the continuous differential operators at a discrete level. Additionally, they are well-known for satisfying constraints such as conservation of mass or momentum.

In the present work, a Mimetic Spectral Element Method based on quadrilaterals is explored. As an introduction, the framework is first implemented and tested on the classical Poisson equation, the Hartmann Flow system and several eigenvalue problems for the Laplacian operator. Solutions are attained by direct/mixed formulations and the extension to multi-element approaches is dealt with using either gathering or connectivity matrices. Different boundary conditions and various geometries are utilized.

Afterwards, the Maxwell Eigenvalue problem for the electric field E with general material properties is tackled in an attempt to generate spurious-free solutions by incorporating the condition ∇·D = 0 into the discrete system. The formulation is further scrutinized on geometries with Betti number b₁ > 0 as to verify if the proposed scheme captures the physical zero eigenvalues.

In the end, a mixed formulation for the eigenproblem is proposed in which the curl-curl operator is separated. The approximation of the electrostatic field energy is then computed with this formulation and compared to the solution obtained with a direct method allowing to create an upper and a lower bound for this variable.

This thesis poses a new geometric formulation for compressible Euler flows. A partial decomposition of this model into Roe variables is applied; this turns mass density, momentum and kinetic energy into product quantities of the Roe variables. Lie derivative advection operators of weak forms constructed with this decomposed model naturally follow to be self-adjoint, which results in skew-symmetric discrete advection operators in any number of dimensions. Under certain conditions these conserve products of the Roe variables, leading to a discrete model formulation with advection operators that simultaneously conserve mass, momentum, kinetic energy, internal energy and total energy in compressible Euler flows. While this idea is not new the novelty of this work lies in its extension to mimetic finite element methods and its application to discontinuous compressible Euler flows.The regular geometric Euler model and its Roe variable decomposition have been discretized through mimetic isogeometric analysis. At the core of mimetic discretization methods lies the idea of retaining the De Rham sequence of differential form spaces when projecting these to finite-dimensional approximations and when constructing discrete operators. B-spline differential form spaces have been defined such that the exterior derivative maps in a topologically exact and metric-free way, while the interior product has been discretized in a weak way in order to retain its map between appropriate spaces in the De Rham sequence. Only primal grids are used; the Hodge star operator is discretized through the definition of an L2 inner product to resolve weak forms. Cartan’s homotopy formula allows for a consistent discretization of the Lie derivative through compositions of the interior product and exterior derivative.Several tests were carried out to determine the efficacy and behavior of this regular geometric Euler model, its Roe variable decomposition and the resulting discrete advection operators. Testing one-dimensional linear advection and Burgers’ equation on periodic domains shows that discretizations of the self-adjoint advection operators are consistent with conservative formulations. Sod’s shock tube is used as one-dimensional discontinuous compressible flow test. Both the regular Euler model and its Roe variable decomposition display strong oscillatory tendencies, yet solution convergence is obtained without any issues. While application of a simple moving average-filter removes the worst oscillations more sophisticated methods are necessary for obtaining solutions that are free of unphysical oscillations. The Roe variable decomposition negatively affects the accuracy of shock speed predictions. The presence of strong oscillations likely affects the numerical conservation errors of both methods. Compared to finite volume package Clawpack and the nodal Discontinuous Galerkin (DG) method of Hesthaven & Warburton both models have less numerical diffusion on coarse meshes with the regular model outperforming the Roe variable decomposition. Convergence of momentum conservation error is slow and the errors are large compared to the reference methods. For two-dimensional periodic vortices both the regular geometric Euler model and its Roe variable decomposition outperform both reference methods for stationary and moving vortices. Clawpack displays diffusive behavior, resulting in large L2 errors and high amounts of numerical diffusion. While the L2 errors of the DG method are comparable to those of the two models developed in this work for all basis function orders considered, the resulting DG discretization requires significantly more degrees of freedom to attain this. To obtain similar levels of numerical diffusion the DG method needs up to ten times as many degrees of freedom as the methods presented in the current research.

This research investigates the possibility of solving one dimensional Poisson's equation on quantum computers using the Variational Quantum Linear Solver (VQLS) as a simplified test case for fluid dynamics applications. In this work, Poisson's equation is discretized with the finite element method and the resulting matrix is decomposed as a linear combination of unitaries to be cast in VQLS. When using Pauli Gates as a basis, the matrix is inefficiently decomposed with an exponential number of terms in the number of qubits. On the other hand, using gates with higher entanglement allowed for an efficient decomposition but requires high qubit interconnectivity. Numerical experiments were carried out on a quantum simulator: iterations to success were larger than the best classical counterpart and scaled exponentially for increasing qubits numbers. Across all the experiments performed, scalability was an issue mainly because of the vanishing of the gradient in the cost function

Several investigations have been undertaken to study the velocity and temperature fields associated with the thermal mixing of fluids, and resulting thermal striping in a T-junction. The T-junction thermal mixing and fatigue phenomenon is a major area of study for the purposes of safety, maintenance and operational life in the nuclear industry, in which fluid mixing occurs in cooling circuits for the nuclear reactor. The existing body of work on T-junctions mainly comprises of experimental references performed at high values of Reynolds numbers. However, these available experimental databases are not
sufficient to describe the involved physics in adequate detail, and, due to experimental limitations, accurate data on velocity and temperature fluctuations in regions close to the wall are not accessible. Computational Fluid Dynamics (CFD) can play an important role in predicting such complex flow features. However, predicting complex thermal fatigue phenomena is a challenge for the available momentum and heat flux turbulence models, which also require extensive validation.
It was realised that a comprehensive Direct Numerical Simulation (DNS) of a T-junction was required as a benchmark for validation purposes, but also to better understand the underlying physical phenomena of thermal mixing in the fluid and thermal fatigue in the solid walls. The aim of the thesis is to thus design such a reference DNS experiment of a thermal fatigue scenario calibrated using Reynolds-Averaged Navier-Stokes (RANS) simulations. The feasibility of scaling down the Reynolds number from experimental cases to a computationally-feasible range is investigated. The junction corner shape is also modified to a slightly rounded corner, ensuring that the underlying fundamental physical phenomena of turbulence and thermal mixing flow features were preserved. The pipe lengths of the model were calibrated to ensure there would be no interference of the upstream developing region on the thermal mixing at the junction, and the outlet boundary conditions. A sample proof-of-concept under-resolved DNS (UDNS) case, with high- and low-Prandtl number passive temperature scalars, with iso-temperature, iso-flux and mixed (Robin) wall boundary conditions, is simulated and presented. This proof-of-concept simulation contributes to the finalization of the fully-resolved DNS in computational grid size selection, transient characteristics, computational costs, and additionally, the implementation of the Robin boundary condition in the fully-resolved DNS.

Numerical modelling of high frequency waves is a complex and challenging area. Although the underlying equation seems simple, −∆φ − k2φ = f, the numerical challenges are not. This time harmonic wave equation is known as the Helmholtz equation. The main challenge to be studied is the pollution error, which is the difference between the actual and numerical wavenumber. Due to this error, the solution of most numerical methods deteriorates rapidly when increasing the wavenumber. So far, either problem specific solutions have been found or solutions which require knowledge of the solution itself. In this thesis a numerical method is developed to approach this challenge. A mimetic discretization of the Helmholtz equation using C1-continuous Hermite interpolating polynomials is developed to model the Helmholtz equation in 2 dimensions. Mimetic theory is briefly introduced after which the Hermite polynomials are defined and analysed for their interpolating properties. A least-squares variational problem is defined and discretized using 49 basis function at its lowest order. The model is verified using a single sinusoidal wave after which
plane wave problems are solved and analysed for their wavenumber-dependence. A diffraction and interference problem is set-up and compared to analytical solutions. The proposed method shows no significant advantages over the use of C0-continuous Lagrange polynomials in terms of effectiveness. Both methods show deterioration at the same wave numbers and equal polynomial order. The Hermite polynomials require less unknowns to reach the same accuracy. In short, a p-accurate solution is found using Hermite polynomials
at the cost of a p − 1 model using Lagrange polynomials. This benefit in efficiency does not outweigh the additional effort and boundary information necessary to construct the problem. However, the variety of coefficients offer an opportunity for further research to find relations to the numerical wavenumber in the search of a wavenumber conserving discretization.

The present research aims at establishing a numerical technique that allows for simple discretization of curved domains. The method of implementation features use of covariant exterior derivatives that are used alongside structure preserving mixed mimetic spectral methods, that is primal and algebraic dual polynomials. A single element implementation is used to demonstrate the applicability of the method over two diffeomorphisms, a horizontal shear with non-linear skew and a modified polar coordinate transformation, for 0-forms and 1-forms. Finally, an extension of this framework towards continuum mechanics is discussed with a discussion over a co-vector valued stress and elasticity formulation.

Physical systems in the continuous domain are often solved using computer-aided software because of their complexity. Preserving the physical quantities from the continuous domain in the discrete domain is therefore of utmost importance. There is however a broad range of techniques that can accomplish the translation between the continuous and discrete domain, e.g. finite difference, volume and element techniques, fourth order Runge-Kutta or Störmer-Verlet to name a few. Accompanied with the aforementioned come strengths and weaknesses but have the common thread to try maintaining the physical behaviour of the continuous system closely. The mimetic spectral element technique is used to develop an energy-conserving spectral element scheme through a Lagrangian formulation. This new formulation of the mimetic spectral element technique allows for solving time-dependent problems and the simple harmonic oscillator serves as the sample problem in this thesis. The solution has been derived from Lagrangian mechanics in a variety of ways. Discrete Lagrangian formulations have been investigated at rst and their respective equations of motions have been tested against the exact solution of the simple harmonic oscillator. This method achieved marching in time and no damping of the solution, yet energy was only bounded and not exactly conserved. The mimetic spectral element formulation of the Lagrangian formulation showed diculties when using variational analysis, i.e. boundary treatment in the future. Arbitrary domain mapping was among the possible solutions, but this formulation was found to be unreliable and unsuccessful. It was found that a more robust formulation, i.e. the spectral marching method, was most suitable. Throughout this thesis the focus was put on the conservation of energy using a Lagrangian formulation. Except the spectral scheme using arbitrary domain mapping, all schemes kept the energy of the system bounded, but no energy was conserved up to machine precision. Using arbitrary domain mapping, the energy seemed to grow over time.