21,338 research outputs found

    Geometric Properties of the 2-D Peskin Problem

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    The 2-D Peskin problem describes a 1-D closed elastic string immersed and moving in a 2-D Stokes flow that is induced by its own elastic force. The geometric shape of the string and its internal stretching configuration evolve in a coupled way, and they combined govern the dynamics of the system. In this paper, we show that certain geometric quantities of the moving string satisfy extremum principles and decay estimates. As a result, we can prove that the 2-D Peskin problem admits a unique global solution when the initial data satisfies a medium-size geometric condition on the string shape, while no assumption on the size of stretching is needed

    Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients

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    In this article, we build on previous work to present an optimization algorithm for nonlinearly constrained multi-objective optimization problems. The algorithm combines a surrogate-assisted derivative-free trust-region approach with the filter method known from single-objective optimization. Instead of the true objective and constraint functions, so-called fully linear models are employed, and we show how to deal with the gradient inexactness in the composite step setting, adapted from single-objective optimization as well. Under standard assumptions, we prove convergence of a subset of iterates to a quasi-stationary point and if constraint qualifications hold, then the limit point is also a KKT-point of the multi-objective problem

    Optimal Control of the Landau-de Gennes Model of Nematic Liquid Crystals

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    We present an analysis and numerical study of an optimal control problem for the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a crucial component in modern technology. They exhibit long range orientational order in their nematic phase, which is represented by a tensor-valued (spatial) order parameter Q=Q(x)Q = Q(x). Equilibrium LC states correspond to QQ functions that (locally) minimize an LdG energy functional. Thus, we consider an L2L^2-gradient flow of the LdG energy that allows for finding local minimizers and leads to a semi-linear parabolic PDE, for which we develop an optimal control framework. We then derive several a priori estimates for the forward problem, including continuity in space-time, that allow us to prove existence of optimal boundary and external ``force'' controls and to derive optimality conditions through the use of an adjoint equation. Next, we present a simple finite element scheme for the LdG model and a straightforward optimization algorithm. We illustrate optimization of LC states through numerical experiments in two and three dimensions that seek to place LC defects (where Q(x)=0Q(x) = 0) in desired locations, which is desirable in applications.Comment: 26 pages, 9 figure

    Soliton Gas: Theory, Numerics and Experiments

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    The concept of soliton gas was introduced in 1971 by V. Zakharov as an infinite collection of weakly interacting solitons in the framework of Korteweg-de Vries (KdV) equation. In this theoretical construction of a diluted soliton gas, solitons with random parameters are almost non-overlapping. More recently, the concept has been extended to dense gases in which solitons strongly and continuously interact. The notion of soliton gas is inherently associated with integrable wave systems described by nonlinear partial differential equations like the KdV equation or the one-dimensional nonlinear Schr\"odinger equation that can be solved using the inverse scattering transform. Over the last few years, the field of soliton gases has received a rapidly growing interest from both the theoretical and experimental points of view. In particular, it has been realized that the soliton gas dynamics underlies some fundamental nonlinear wave phenomena such as spontaneous modulation instability and the formation of rogue waves. The recently discovered deep connections of soliton gas theory with generalized hydrodynamics have broadened the field and opened new fundamental questions related to the soliton gas statistics and thermodynamics. We review the main recent theoretical and experimental results in the field of soliton gas. The key conceptual tools of the field, such as the inverse scattering transform, the thermodynamic limit of finite-gap potentials and the Generalized Gibbs Ensembles are introduced and various open questions and future challenges are discussed.Comment: 35 pages, 8 figure

    Chiral active fluids: Odd viscosity, active turbulence, and directed flows of hydrodynamic microrotors

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    While the number of publications on rotating active matter has rapidly increased in recent years, studies on purely hydrodynamically interacting rotors on the microscale are still rare, especially from the perspective of particle based hydrodynamic simulations. The work presented here targets to fill this gap. By means of high-performance computer simulations, performed in a highly parallelised fashion on graphics processing units, the dynamics of ensembles of up to 70,000 rotating colloids immersed in an explicit mesoscopic solvent consisting out of up to 30 million fluid particles, are investigated. Some of the results presented in this thesis have been worked out in collaboration with experimentalists, such that the theoretical considerations developed in this thesis are supported by experiments, and vice versa. The studied system, modelled in order to resemble the essential physics of the experimentally realisable system, consists out of rotating magnetic colloidal particles, i.e., (micro-)rotors, rotating in sync to an externally applied magnetic field, where the rotors solely interact via hydrodynamic and steric interactions. Overall, the agreement between simulations and experiments is very good, proving that hydrodynamic interactions play a key role in this and related systems. While already an isolated rotating colloid is driven out of equilibrium, only collections of two or more rotors have experimentally shown to be able to convert the rotational energy input into translational dynamics in an orbital rotating fashion. The rotating colloids inject circular flows into the fluid, such that detailed balance is broken, and it is not a priori known whether equilibrium properties of colloids can be extended to isolated rotating colloids. A joint theoretical and experimental analysis of isolated, pairs, and small groups of hydrodynamically interacting rotors is given in chapter 2. While the translational dynamics of isolated rotors effectively resemble the dynamics of non-rotating colloids, the orbital rotation of pairs of rotors can be described with leading order hydrodynamics and a two-dimensional analogy of Faxén’s law is derived. In chapter 3, a homogeneously distributed ensemble of rotors (bulk) as a realisation of a chiral active fluid is studied and it is explicitly shown computationally and experimentally that it carries odd viscosity. The mutual orbital translation of rotors and an increase of the effective solvent viscosity with rotor density lead to a non-monotonous behaviour of the average translational velocity. Meanwhile, the rotor suspension bears a finite osmotic compressibility resulting from the long-ranged nature of hydrody- namic interactions such that rotational and odd stresses are transmitted through the solvent also at small and intermediate rotor densities. Consequently, density inhomogeneities predicted for chiral active fluids with odd viscosity can be found and allow for an explicit measurement of odd viscosity in simulations and experiments. At intermediate densities, the collective dynamics shows the emergence of multi-scale vortices and chaotic motion which is identified as active turbulence with a self-similar power-law decay in the energy spectrum, showing that the injected energy on the rotor scale is transported to larger scales, similar to the inverse energy cascade of clas- sical two-dimensional turbulence. While either odd viscosity or active turbulence have been reported in chiral active matter previously, the system studied here shows that the emergence of both simultaneously is possible resulting from the osmotic compressibility and hydrodynamic mediation of odd and active stresses. The collective dynamics of colloids rotating out of phase, i.e., where a constant torque instead of a constant angular velocity is applied, is shown to be qualitatively very similar. However, at smaller densities, local density inhomogeneities imply position dependent angular velocities of the rotors resulting from inter-rotor friction. While the friction of a quasi-2D layer of active colloids with the substrate is often not easily modifiable in experiments, the incorporation of substrate friction into the simulation models typically implies a considerable increase in computational effort. In chapter 4, a very efficient way of incorporating the friction with a substrate into a two-dimensional multiparticle collision dynamics solvent is introduced, allowing for an explicit investigation of the influences of substrate on active dynamics. For the rotor fluid, it is explicitly shown that the influence of the substrate friction results in a cutoff of the hydrodynamic interaction length, such that the maximum size of the formed vortices is controlled by the substrate friction, also resulting in a cutoff in the energy spectrum, because energy is taken out of the system at the respective length. These findings are in agreement with the experiments. Since active particles in confinement are known to organise in states of collective dynamics, ensembles of rotationally actuated colloids are studied in circular confinement and in the presence of periodic obstacle lattices in chapters 5 and 6, respectively. The results show that the chaotic active turbulent transport of rotors in suspension can be enhanced and guided resulting from edge flows generated at the boundaries, as has recently been reported for a related chiral active system. The consequent collective rotor dynamics can be regarded as a superposition of active turbulent and imposed flows, leading to on average stationary flows. In contrast to the bulk dynamics, the imposed flows inject additional energy into the system on the long length scales, and the same scaling behaviour of the energy spectrum as in bulk is only obtained if the energy injection scales, due to the mutual generation of rotor translational dynamics throughout the system and the edge flows, are well separated. The combination of edge flow and entropic layering at the boundaries leads to oscillating hydrodynamic stresses and consequently to an oscillating vorticity profile. In the presence of odd viscosity, this consequently leads to non-trivial steady-state density modulations at the boundary, resulting from a balance of osmotic pressure and odd stresses. Relevant for the efficient dispersion and mixing of inert particles on the mesoscale by means of active turbulent mixing powered by rotors, a study of the dynamics of a binary mixture consisting out of rotors and passive particles is presented in chapter 7. Because the rotors are not self-propelled, but the translational dynamics is induced by the surrounding rotors, the passive particles, which do not inject further energy into the system, are transported according to the same mechanism as the rotors. The collective dynamics thus resembles the pure rotor bulk dynamics at the respective density of only rotors. However, since no odd stresses act between the passive particles, only mutual rotor interactions lead to odd stresses leading to the accumulation of rotors in the regions of positive vorticity. This density increase is associated with a pressure increase, which balances the odd stresses acting on the rotors. However, the passive particles are only subject to the accumulation induced pressure increase such that these particles are transported into the areas of low rotor concentration, i.e., the regions of negative vorticity. Under conditions of sustained vortex flow, this results in segregation of both particle types. Since local symmetry breaking can convert injected rotational into translational energy, microswimmers can be constructed out of rotor materials when a suitable breaking of symmetry is kept in the vicinity of a rotor. One hypothetical realisation, i.e., a coupled rotor pair consisting out of two rotors of opposite angular velocity and of fixed distance, termed a birotor, are studied in chapter 8. The birotor pumps the fluid into one direction and consequently translates into the opposite direction, and creates a flow field reminiscent of a source doublet, or sliplet flow field. Fixed in space the birotor might be an interesting realisation of a microfluidic pump. The trans- lational dynamics of a birotor can be mapped onto the active Brownian particle model for single swimmers. However, due to the hydrodynamic interactions among the rotors, the birotor ensemble dynamics do not show the emergence of stable motility induced clustering. The reason for this is the flow created by birotor in small aggregates which effectively pushes further arriving birotors away from small aggregates, which eventually are all dispersed by thermal fluctuations

    Rational-approximation-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots

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    We introduce several spatially adaptive model order reduction approaches tailored to non-coercive elliptic boundary value problems, specifically, parametric-in-frequency Helmholtz problems. The offline information is computed by means of adaptive finite elements, so that each snapshot lives in a different discrete space that resolves the local singularities of the analytical solution and is adjusted to the considered frequency value. A rational surrogate is then assembled adopting either a least squares or an interpolatory approach, yielding a function-valued version of the standard rational interpolation method (V-SRI) and the minimal rational interpolation method (MRI). In the context of building an approximation for linear or quadratic functionals of the Helmholtz solution, we perform several numerical experiments to compare the proposed methodologies. Our simulations show that, for interior resonant problems (whose singularities are encoded by poles on the V-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the V-SRI method seems to be the best performing one

    A macroscopic constitutive relation for isotropic-genesis, polydomain liquid crystal elastomers

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    Liquid crystal elastomers (LCEs) are rubber-like solids that incorporate nematic mesogens (stiff rod-like molecules) as a part of their polymer chains. In recent years, isotropic-genesis, polydomain liquid crystal elastomers (I-PLCEs) has been a topic of both scientific and technological interest due to their intriguing properties such as soft behavior, ability to dissipate energy and stimuli response, as well as the ease with which they can be synthesized. We present a macroscopic or engineering scale constitutive model of the behavior of I-PLCEs. The model implicitly accounts for the complex evolution of the domain patterns and is able to faithfully capture the experimentally observed complex response to multi-axial loading. We describe a multiscale framework that motivates the model, explore various aspects of the model, validate it against experiments, and finally verify and demonstrate a numerical implementation

    Stability of space-time isogeometric methods for wave propagation problems

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    This thesis aims at investigating the first steps toward an unconditionally stable space-time isogeometric method, based on splines of maximal regularity, for the linear acoustic wave equation. The unconditional stability of space-time discretizations for wave propagation problems is a topic of significant interest, by virtue of the advantages of space-time methods compared with more standard time-stepping techniques. In the case of continuous finite element methods, several stabilizations have been proposed. Inspired by one of these works, we address the stability issue by studying the isogeometric method for an ordinary differential equation closely related to the wave equation. As a result, we provide a stabilized isogeometric method whose effectiveness is supported by numerical tests. Motivated by these results, we conclude by suggesting an extension of this stabilization tool to the space-time isogeometric formulation of the acoustic wave equation.Comment: Masters thesi

    Effects of wind turbine rotor positioning on tornado induced loads

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    This study investigates the loads induced by a large-scale tornado simulation on a horizontal axis wind turbine (HAWT) to assess the influence of three-dimensional flows with respect to the HAWT position. The loads were analyzed under two rotor operational conditions, idling and parked, at five radial distances. Subsequently, experimental validation of the numerical code HIW-TUR was conducted by evaluating the induced moments for various yaw and pitch angles. The experimental results demonstrated that the bending moment was the most important in terms of magnitude and variation with respect to the HAWT position. Furthermore, The HIW-TUR code accurately identified the magnitude and HAWT configuration that leads to the maximum mean moments induced by the tornado. It was proved that by varying the yaw angle of the rotor plane and blade orientation to parallel to the tornado tangential component, the overall loads could be reduced to the minimum values

    A stochastic population model of cholera disease

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    A cholera population model with stochastic transmission and stochasticity on the environmental reservoir of the cholera bacteria is presented. It is shown that solutions are well-behaved. In comparison with the underlying deterministic model, the stochastic perturbation is shown to enhance stability of the disease-free equilibrium. The main extinction theorem is formulated in terms of an invariant which is a modification of the basic reproduction number of the underlying deterministic model. As an application, the model is calibrated as for a certain province of Nigeria. In particular, a recent outbreak (2019) in Nigeria is analysed and featured through simulations. Simulations include making forward projections in the form of confidence intervals. Also, the extinction theorem is illustrated through simulations
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