2023

A peculiarity of nonlinear hyperbolic problems is that they must be interpreted as limits of second-order equations with vanishing viscosity. Despite not explicitly being present in the hyperbolic case, diffusion is needed, e. g., at discontinuities or to avoid the occurrence of nonphysical states. In the case of gas dynamics, for instance, dissipation corresponds to the production of thermodynamic entropy. To solve hyperbolic problems numerically, one needs to adapt these ideas to the discrete setting. Standard high-order methods, however, do not incorporate the appropriate amounts of artificial viscosity because these need to be chosen adaptively based on the solution. Among the high-resolution schemes capable of doing so are the recently proposed monolithic convex limiting (MCL) techniques [1] to be discussed in this talk. They offer a way to enforce physical admissibility, entropy stability, and discrete maximum principles for conservation laws. These methods can also be generalized to systems of balance laws in a well-balanced manner [2]. In addition to second-order finite element methods, extensions to high-order discontinuous Galerkin (DG) schemes shall also be presented [3]. Numerical examples for the so-called KPP problem, the nonconservative shallow water system, and the compressible Euler equations will be shown. An overview of MCL and other property-preserving methods can be found in our recently published book [4].

We combine a pressure correction scheme with interior penalty discontinuous Galerkin (dG) discretisation to solve the time-dependent Navier–Stokes equations. We prove unconditional energy stability and a priori error estimates for the velocity. With duality arguments, optimal L2 error rates are obtained. Convergence of the discrete pressure is also established.  Further, we propose a splitting scheme,  integrating the pressure correction approach, for the Cahn–Hilliard–Navier–Stokes system  The numerical analysis of dG combined with this scheme is discussed. Namely, we show well--posedness, stability, and error estimates. Numerical results with manufactured solutions display our theoretical findings, and a spinodal decomposition example portrays the robustness of our approach.

Your brain has its own waterscape: whether you are reading, thinking or sleeping, fluid flows through or around the brain tissue, clearing waste in the process. These biophysical processes are crucial for the well-being and function of the brain. In spite of their importance we understand them but little, and mathematical and computational modelling could play a crucial role in gaining new insight. In this talk, I will give an overview of mathematical, mechanical and numerical approaches to understand mechanisms underlying pulsatility, fluid flow and solute transport in the human brain. Topics include fluid-structure interactions, generalized poroelasticity, mixed finite element discretizations and preconditioning, uncertainty quantification, and optimal control.

Fluid efflux from the brain plays an important role in solute waste clearance. Current experimental approaches provide little spatial information or data collection is limited due to short duration or low frequency of sampling. One approach shows tracer efflux to be independent of molecular size, indicating bulk flow, yet also decelerating like simple membrane diffusion. In an apparent contradiction to this report, other studies point to tracer efflux acceleration following infusions. In this talk, I will share a stylized advection-diffusion model for clearance of waste, which reconciles the apparent contradiction, and discuss methods to validate it with novel MRI data. Being stylized, it is also simple enough to permit a dimensional analysis which indicates that clearance of waste from the brain is governed by three dimensionless quantities including a potential bottle-neck for clearance due to transport across the surface membranes.

Cavitation is a ubiquitous and sometimes destructive, phenomenon. For instance, cavitation bubbles may interrupt water flow in plants or severely damage the surfaces of machines such as pumps and propellers. The so-called tribonucleation of vapor bubbles has been proposed to be responsible for the cracking sound produced by the manipulation of human synovial joints. To study cavitation up close we have developed an experimental setup where a sphere in water abruptly leaves a flat surface starting from a separation of only 10 nm.

Upon upward movement of the spherical surface, a cavitation bubble forms and develops branched fingers through the Saffmann-Taylor instability. Simultaneously, negative liquid pressures in the range of ∼10atm are observed. These large tension values occasionally lead to secondary nucleation events. The bubble sizes satisfy a predicted Familiy-Vicsek scaling law where the bubble area is proportional to the inverse bubble lifetime. The fact that creeping flow cavitation bubbles are more short lived the larger they are separate them from bubbles that are governed by inertial dynamics.

Self-assembly is the spontaneous generation of order in systems driven by thermal agitation and interactions. At the molecular level, self-assembly plays an important role in the formation of giant com- plex macromolecules, being quite relevant for living systems. At the mesoscopic level, capillary driven self-assembly has been proposed for building structures in the gap between classical bottom-up and top-down fabrication methods, i.e. at the scales in between 10 micrometers and 1 millimeter. Although the method was proposed 20 years ago, only regular or simple structures were achieved so far. Using both experimental and statistical physics ideas, we demonstrate how to exploit subtle capillary interactions to create elaborate complex structures, as well as functional micromachines. On top of that, we show how such mescoscopic systems can be the analogues of many different physical systems such as folding molecules, molecular locks and keys, and crystal formation.

Innovations in fluid mechanics are leading to better food since ancient history, while creativity in cooking inspires applied and fundamental science. In this talk, I will discuss how recent advances in hydrodynamics are changing food science, and how the surprising phenomena that arise in the kitchen lead to discoveries and technologies across the disciplines, including rheology and soft matter. Central topics include cocktails and champagne (multiphase flows), whipped cream (complex fluids) and pancake making (viscous flows). For every topic, I will present the state-of-the-art knowledge, the open problems, and likely directions for future research.

Publications:
Mathijssen, A. J., Lisicki, M., Prakash, V. N., & Mossige, E. J. (2023). Culinary fluid mechanics and other currents in food science. Reviews of Modern Physics, 95(2), 025004.

Fuller, G. G., Lisicki, M., Mathijssen, A. J., Mossige, E. J., Pasquino, R., Prakash, V. N., & Ramos, L. (2022). Kitchen flows: Making science more accessible, affordable, and curiosity driven. Physics of Fluids, 34(11).

Abstract: TBA

We have developed a pump-less recirculation Organ-on-Chip (rOoC) platform that generates a directional gravity-driven flow. This platform can be adapted to various flow conditions and enables the study of endothelial lining, blood vessel sprouting, circulation of immune cells, pathogens or other particles, and incorporation of 3D cell models like organoids. Additionally, we have developed a computational model to predict shear stress and mass transport within the rOoC, allowing for customization of the platform for various use-cases.

The rOoC platform is very versatile and can be used to model for instance drug-induced liver-injury (DILI) that mimics the complex interaction between resident human stem cell-derived liver organoids (3D-HLO) and circulating immune cells. Moreover, we show the functional crosstalk between 3D-HLOs and human pancreatic islets to model the onset of type-2 diabetes.

This seminar will consist of two separate presentations, each about 15-minute long.

1) Magnetic Quincke Rollers with tunable single particle dynamics and collective states
2) Electrically controllable ferrofluids

I will discuss some of our recent results on active chiral and nematic membranes. The chiral stresses we consider give rise to a novel form of odd elasticity. To outline this phenomenology I will give explicit calculations outlining spontaneous flow transitions and shape instabilities. I will discuss the relevance of these results in developmental biology and their relation to active nematics, in particular how certain limits of active nematic membranes can reduce to a theory of an isotropic membrane with an active stress defined by the deviatoric part of the shape operator.

The human brain has no lymphatic vessels, so how does the brain clear metabolic waste? In 2012, Iliff et al. proposed a theory about waste clearance of the brain, called the "glymphatic" theory. The theory suggest that the waste clearances is bio-mechanical, and that impaired clearance may be the cause of some neurodegenerative diseases and disorders. The inaccessibility of the human brain have been a hurdle in the research, as experiments on rat brains do not translate to the human brain. Researchers at Oslo university hospital Rikshospitalet have shown clearance using tracers visible in magnetic resonance images (MRI). However, the MRI only provide snapshots of different states in time, therefore computational modeling is needed to fill in the gaps. In this presentation, we will look at computational modeling with the MRI to infer material parameters in the brain.

Stalagmites grow on the floor of caves by precipitation of calcium ions found in the residual water film covering the top of the stalagmite, which is progressively drained away. Drops dripping from stalactites ensure the renewal of these ions.
Previous models of stalagmite growth assumed that drops fall on a straight vertical line from stalactites. Through high-speed imaging during field experiments in caves, we however observe that the impact point position of the drops is scattered. Using a Langevin-like equation to describe the fall of drops in response to gravity and aerodynamic forces, we then propose a prediction of the impact point dispersal. We show that measured stalagmite widths are correlated to the impact point dispersal of the drops.
In a second time, we focus on the mixing of calcium ions between the drop and the film during impact. The drop produces a crown when impacting the film, accompanied by a large amount of secondary droplet ejections. This is at the very heart of the film thickness variability post-impact. We record high-speed imaging of drop impacts on films of controlled thickness in a lab environment and assess the mixing between the drop and the film. We deduce how much liquid coming from the initial drop remains in the film.

The Kolmogorov N-width describes the best possible error one can achieve by elements of an N-dimensional linear space. Its decay has extensively been studied in Approximation Theory and for the solution of Partial Differential Equations (PDEs). Particular interest has occurred within Model Order Reduction (MOR) of parameterized PDEs e.g. by the Reduced Basis Method (RBM). While it is known that the N-width decays exponentially fast (and thus admits efficient MOR) for certain problems, there are examples of the linear transport and the wave equation, where the decay rate deteriorates to N^-1/2. On the other hand, it is widely accepted that a smooth parameter dependence admits a fast decay of the N-width. However, a detailed analysis of the influence of properties of the data (such as regularity or slope) on the rate of the N-width seems to lack. In this work, we use techniques from Fourier Analysis to derive exact representations of the N-width in terms of initial and boundary conditions of the linear transport equation modeled by some function g for half-wave symmetric data. For arbitrary functions g, we derive bounds and prove that these bounds are sharp. In particular, we prove that the N-width decays as c_r N^(-r-1/2) for functions in the Sobolev space, g ∈ H^r. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds and give additional quantitative insigh.

We discuss discretizations and solvers for a class of numerical methods for convection diffusion equations in arbitrary spatial dimensions. Targeted applications include the Nernst-Plank equations for transport of species in a charged media. We illustrate how such exponentially fitted methods are derived. A main step in proving error estimates is showing unisolvence for the quasi-polynomial spaces of differential forms defined as weighted spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in any spatial dimension is the same as the set of such functionals used for the polynomial spaces. We are able to prove our results without the use of Stokes' Theorem, which is the standard tool in showing the unisolvence of functionals in polynomial spaces of differential forms.
This is joint work with Shuonan Wu (Beijing University)

Constructing fast solution schemes often involves deciding which errors are acceptable and which approximations can be made for the sake of computational efficiency. Herein, we consider a mixed formulation of Darcy flow in porous media and take the perspective that the physical law of mass conservation is significantly more important than the constitutive relationship, i.e. Darcy's law. From this point of view, we propose an inexact solution technique that nevertheless guarantees local mass conservation. The method is based on first solving the mass balance equation and then computing a solenoidal correction using the curl of a potential field. We extend the method to flows in fractured porous media and present numerical experiments that indicate the efficiency of the scheme.

For millennia, origami and kirigami artists have used folds and cuts to create beautiful shapes from a simple sheet of paper. I will describe our recent scientific attempts to catch up with these remarkably imaginative arts phrased as inverse problems in physical geometry that aim to control the shape and rigidity of a thin surface. Using discrete operations that vary the number, size, orientation and coordination of folds and cuts, I will show how to create piecewise isometric kirigami and origami tessellations and control their local and global morphology and mechanical response, mixing experimental, computational and theoretical approaches.

After a brief introduction to the main physical characteristics of tsunami events, the recently developed Iterative Filtering technique is presented and applied to the decomposition of tsunami signals from pressure and tide gauges. It is shown how these signals are successfully decomposed into components of different physical origins. Then, the time-frequency representation of these time series is obtained by using the IMFogram algorithm, which computes instantaneous amplitudes and frequencies for the previously obtained components. Finally, possible applications to tsunami science are discussed, such as possible applications to real time detection in early warning context.

OceanSun’s floating solar island consists of a hydro elastic membrane attached to a flexible torus, providing a more cost-efficient alternative with natural cooling of the panels leading to increased efficiency. The current research focuses on the seakeeping characteristics of OceanSun’s FSPV concept specifically. Wave induced loads are of particular interest, as the feasibility of offshore installation strongly depends on environmental loads. Important responses of the membrane based FSPV are identified by the development of a global model based on linear potential flow theory, and linearly pre-tensioned membrane motions. Based on theory formulated by Grøn (2022), a modal analysis is used to describe the vertical displacement of the membrane-floater system. A numerical implementation of the theory in WAMIT is compared to experimental results from model-scaled tests.

Finding the optimal shape is a vivid research area and has a wide range of applications, e.g., in fluid mechanics and acoustics. Moreover, there is also a close link to image registration and image segmentation. In this talk, we consider shape optimization tasks as optimal control problems that are constrained by partial differential equations. From this perspective, state-of-the-art methods can be motivated by the choice of the metric on the set of admissible shapes. Moreover, a new approach for density based topology optimization is presented in the setting of Stokes flow. It is based on classical topology optimization and phase field approaches, and introduces a different way to relax the underlying infinite-dimensional mixed integer problem. We give a theoretically founded choice of the relaxed problems and present numerical results. Moreover, in order to show the potential of the new approach, we do a comparison to a classical approach. (joint work with Michael Ulbrich and Franziska Neumann)

I will go through my PhD work at DTU. It is about the development of a fully-nonlinear finite difference based potential flow solver which imposes all of the fluid boundaries via an immersed boundary method. The convergence and stability of this approach is first established for various linear and nonlinear wave propagation problems. When it comes to the wave-body interaction problem, cautious attention is paid to the intersection point between free surface and body surface, and a scheme which meets the accuracy and stability requirements best is picked from several proposals. With the scheme introduced in this paper, piston type wave maker and forced heaving cylinder cases with high oscillation frequency have been simulated successfully.