Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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Dimension reduction for a coupled electro-elastic saddle-point problem at finite strains
Full text linkWe study the finite deformation of a thin, elastically heterogeneous sheet subject to electrostatic coupling. The interaction between mechanics and electrostatics is formulated as a saddle-point problem involving the deformation and the electrostatic potential. Starting from a three-dimensional electro-elastic model with prestrain in the elastic energy, we rigorously derive a reduced plate model in the bending regime. To perform the dimension reduction, that is, to derive the energy of a thin object by taking a suitable limit as its thickness tends to zero, we apply Gamma-convergence-type methods to the underlying saddle-point problem. In the case of bivariate functionals, this convergence is understood in an adapted epi/hypo-convergence sense. In this concept, we demonstrate the convergence of the rescaled electro-elastic problems to an effective two-dimensional bending model coupled to electric effects. We verify that cluster points of saddle points are saddle points for the limit
Chaotic semiconductor laser systems—bandwidth enhancement predicted by numerical simulations
No full textNumerical simulations using a traveling wave model are used to predict the radio frequency (rf) bandwidth and relative total rf power of chaotic output from a semiconductor laser (SL) with a delayed optical feedback system. By tailoring the linewidth enhancement factor (LEF) and nonlinear gain compression of the SL, improvements of a factor of two in the 80%-of-the-rf-power bandwidth, five in the rf bandwidth above 40 dB, and ten in the integrated rf power of the chaos, compared to values for a commercial-like SL, are predicted to be possible. The SL parameter values to be targeted will require device designers and fabricators to explore ways to decrease relaxation oscillation damping while, preferably, still achieving a high relaxation oscillation frequency. Higher values of LEF are a feature of poorer quality SLs of the past and can be expected to be achieved by reversion to past device recipes
Well-posedness and relaxation in a simplified model for viscoelastic phase separation via Hilbertian gradient flows
Full text linkThis article is concerned with a gradient-flow approach to a Cahn--Hilliard model for viscoelastic phase separation introduced by Zhou et al. (Phys. Rev. E, 2006) in its variant with constant mobility. By means of time-incremental minimisation and generalised contractivity estimates, we establish the global well-posedness of the Cauchy problem for moderately regular initial data. For general finite-energy data we obtain the existence of gradient-flow solutions and a stability estimate of weak--strong type. We further study the asymptotic behaviour for relaxation time and bulk modulus depending on a small parameter. Depending on the scaling, we recover the Cahn--Hilliard, the mass-conserving Allen--Cahn or the viscous Cahn--Hilliard equation. A challenge in the well-posedness analysis is the failure of semiconvexity of the appropriate driving functional, which is caused by a phase-dependence of the bulk modulus
Time-asymptotic self-similarity of the damped compressible Euler equations in parabolic scaling variables
Full text linkWe study the long-time behavior of solutions to the compressible Euler equations with frictional damping in the whole space, where we prescribe direction-dependent values for the density at spatial infinity. To this end, we transform the system into parabolic scaling variables and derive a relative entropy inequality, which allows to conclude the convergence of the density towards a self-similar solution to the porous medium equation while the associated limit momentum is governed by Darcy's law. Moreover, we obtain convergence rates that explicitly depend on the flatness of the limit profile. While we focus on weak solutions in the one-dimensional case, we extend our results to energy-variational solutions in the multi-dimensional setting
Modeling Cellular Self-Organization in Strain-Stiffening Hydrogels
No full textWe derive a three-dimensional hydrogel model as a two-phase system of a fibre network and liquid solvent, where the nonlinear elastic network accounts for the strain-stiffening properties typically encountered in biological gels. We use this model to formulate free boundary value problems for a hydrogel layer that allows for swelling or contraction. We derive two-dimensional plain-strain and plain-stress approximations for thick and thin layers respectively, that are subject to external loads and serve as a minimal model for scaffolds for cell attachment and growth. For the collective evolution of the cells as they mechanically interact with the hydrogel layer, we couple it to an agent-based model that also accounts for the traction force exerted by each cell on the hydrogel sheet and other cells during migration. We develop a numerical algorithm for the coupled system and present results on the influence of strain-stiffening, layer geometry, external load and solvent in/outflux on the shape of the layers and on the cell patterns. In particular, we discuss alignment of cells and chain formation under varying conditions
Unravelling the mystery of enhanced open-circuit voltages in nanotextured perovskite solar cells
Full text linkPerovskite solar cells have reached power conversion efficiencies that rival those of established silicon photovoltaic technologies. Nanotextures in perovskite solar cells optimise light trapping and scattering, thereby improving optical absorption. In addition, nanotextures have been experimentally shown to enhance electronic performance, in particular, by increasing the open-circuit voltage -- a phenomenon that, until now, has remained not fully understood. This study investigates the underlying reasons by combining multi-dimensional optical and charge-transport simulations for a single-junction perovskite solar cell. Our results reveal that the increased open-circuit voltage is not driven by optical effects but by the textured geometry itself. For voltages near , texturing one of the absorber/transport layer interfaces increases the imbalance between electron and hole densities in the absorber, thereby reducing Shockley-Read-Hall (SRH) recombination, which is the dominant loss mechanism in this study. While idealised solar cells benefit unconditionally from increasing texture height, in realistic cells there is an optimal texture height which maximizes the power conversion efficiency. These findings provide new insights into the opto-electronic advantages of texturing and offer guidance for the design of next-generation textured perovskite-based solar cells, light emitting diodes, and photodetectors
A frame approach for equations involving the fractional Laplacian
Full text linkExceptionally elegant formulae exist for the fractional Laplacian operator applied to weighted classical orthogonal polynomials. We utilize these results to construct a solver, based on frame properties, for equations involving the fractional Laplacian of any power, s ∈ (0, 1), on an unbounded domain in one or two dimensions. The numerical method represents solutions in an expansion of weighted classical orthogonal polynomials as well as their unweighted counterparts with a specific extension to Rd, d ∈ 1, 2. We examine the frame properties of this family of functions for the solution expansion and, under standard frame conditions, derive an a priori estimate for the stationary equation. Moreover, we prove one achieves the expected order of convergence when considering an implicit Euler discretization in time for the fractional heat equation. We apply our solver to numerous examples including the fractional heat equation (utilizing up to a 6th-order Runge-Kutta time discretization), a fractional heat equation with a time-dependent exponent s(t), and a two-dimensional problem, observing spectral convergence in the spatial dimension for sufficiently smooth data
Representation formulas and far-field behavior of time-periodic incompressible viscous flow around a translating rigid body
No full textThis paper is concerned with integral representations and asymptotic expansions of solutions to the time-periodic incompressible Navier--Stokes equations for fluid flow in the exterior of a rigid body that moves with constant velocity. Using the time-periodic Oseen fundamental solution, we derive representation formulas for solutions with suitable regularity. From these formulas, the decomposition of the velocity component of the fundamental solution into steady-state and purely periodic parts and their detailed decay rate in space, we deduce complete information on the asymptotic structure of the velocity and pressure fields
On the parallelized efficient computation of high dimensional Voronoi diagrams on bounded, unbounded, spherical and periodic domains
Full text linkWe investigate a recently implemented new algorithm for the computation of a Voronoi diagram in high dimensions and generalize it to N nodes in general or non-general position using a geometric characterization of edges and vertices. The algorithm consist of local computations, is well suited for parallelization and can be applied to the Euclidean geometry or on the sphere. We provide a mathematical proof that the algorithm is exact, convergent and has computational costs of O(E NN (N))$, where E is the number of edges and NN (N) is the computational cost to calculate the nearest neighbor among N points. We also provide data from performance tests in the recently developed Julia package „;HighVoronoi.jl”; and compare it to the quickhull algorithm. It turns out that the new approach is particularly well suited for bounded domains, periodic domains and parallelization of computations
