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- ItemA molecular dynamics view of hysteresis and functional fatigue in martensitic transformations(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2010) Kastner, Oliver; Ackland, Graeme J.; Eggeler, Gunther; Weiss, WolfShape memory alloys (SMA) exhibit a number of features which are not easily explained by equilibrium thermodynamics, including hysteresis in the phase transformation and ?reverse? shape memory in the high symmetry phase. Processing can change these features: repeated cycling can ?train? the reverse shape memory effect, while changing the amount of hysteresis and other functional properties. These effects are likely to be due to creation of persistent localised defects, which are impossible to study using non-atomistic methods. Here we present a molecular dynamics simulation study of this behaviour. To ensure the largest possible system size, we use a two dimensional binary Lennard-Jones model, which represents a reliable qualitative model system for martensite/austenite transformations. The evolution of the defect structure and its excess energy is investigated in simulations of cyclic transformation/ reverse transformation processes with 160,000 atoms. The simulations show that the transformation proceeds by non-diffusive nucleation and growth processes and produces distinct microstructure. Upon transformation, lattice defects are generated which affect subsequent transformations and vary the potential energy landscape of the sample. Some of the defects persist through the transformation, providing nucleation centres for subsequent cycles. Such defects may provide a memory of previous structures, and thereby may be the basis of a reversible shape memory effect.
- ItemFast scatterometric measurement of periodic surface structures plasma-etching processes(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2018) Klesse, Wolfgang Matthias; Rathsfeld, Andreas; GroÃŸ, Claudine; Malguth, Enno; Skibitzki, Oliver; Zealouk, LahbibTo satisfy the continuous demand of ever smaller feature sizes, plasma etching technologies in microelectronics processing enable the fabrication of device structures with dimensions in the nanometer range. In a typical plasma etching system a plasma phase of a selected etching gas is activated, thereby generating highly energetic and reactive gas species which ultimately etch the substrate surface. Such dry etching processes are highly complex and require careful adjustment of many process parameters to meet the high technology requirements on the structure geometry. In this context, real-time access of the structures dimensions during the actual plasma process would be of great benefit by providing full dimension control and film integrity in real-time. In this paper, we evaluate the feasibility of reconstructing the etched dimensions with nanometer precision from reflectivity spectra of the etched surface, which are measured in real-time throughout the entire etch process. We develop and test a novel and fast reconstruction algorithm, using experimental reflection spectra taken about every second during the etch process of a periodic 2D model structure etched into a silicon substrate. Unfortunately, the numerical simulation of the reflectivity by Maxwell solvers is time consuming since it requires separate time-harmonic computations for each wavelength of the spectrum. To reduce the computing time, we propose that a library of spectra should be generated before the etching process. Each spectrum should correspond to a vector of geometry parameters s.t. the vector components scan the possible range of parameter values for the geometrical dimensions. We demonstrate that by replacing the numerically simulated spectra in the reconstruction algorithm by spectra interpolated from the library, it is possible to compute the geometry parameters in times less than a second. Finally, to also reduce memory size and computing time for the library, we reduce the scanning of the parameter values to a sparse grid.
- ItemHÃ¶lder-estimates for non-autonomous parabolic problems with rough data(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2015) Meinlschmidt, Hannes; Rehberg, JoachimIn this paper we establish HÃ¶lder estimates for solutions to non-autonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al., which also serves as the starting point for our investigations.
- ItemWeak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2019) Lasarzik, Robert; Rocca, Elisabetta; Schimperna, GiulioIn this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.
- ItemTwo-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Reichelt, Sina; Thomas, MaritaWe derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.
- ItemDynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2022) Jahnel, Benedikt; KÃ¶ppl, JonasWe consider irreversible translation-invariant interacting particle systems on the d-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs w.r.t. the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure w.r.t. the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems.
- ItemInverse elastic scattering from rigid scatterers with a single incoming wave : this paper is dedicated to the memory of Armin Lechleiter(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2019) Elschner, Johannes; Hu, GuanghuiThe @first part of this paper is concerned with uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern corresponding to a single elastic plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern.
- ItemConstrained evolution for a quasilinear parabolic equation(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JÃ¼rgenIn the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the well-posedness and some regularity results for the CauchyNeumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set K of L2 (Omega). Then, we consider convex sets of obstacle or double-obstacle type, and we can act on the factor of the feedback control in order to be able to reach the convex set within a finite time, by proving rigorously this property.
- ItemGlobal existence result for phase transformations with heat transfer in shape memory alloys : dedicated to 75th birthday of K. GrÃ¶ger(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2011) Paoli, Laetitia; Petrov, Adrien; GrÃ¶ger, K.We consider three-dimensional models for rate-independent processes describing materials undergoing phase transformations with heat transfer. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. Under appropriate regularity assumptions on the initial data, we prove the existence a global solution for this thermodynamically consistent system, by using a fixed-point argument combined with global energy estimates.
- ItemOptimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures(Berlin : WeierstraÃŸ-Institut fÃ¼r Angewandte Analysis und Stochastik, 2016) Liero, Matthias; Mielke, Alexander; SavarÃ©, GiuseppeWe develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.