Barbora Benešová

Global optimization numerical strategies for rate-independent processes

This paper presents an approach to numerical solution of problems posed in the framework of quasi-static rate-independent processes. As soon as a problem allows for an energetic formulation there are known methods of its time discretization by time incremental minimization problems, which demand for global optimization of a non-convex functional. Moreover the two-sided energy inequality, a necessary condition for optimization, can be formulated. Here we present an algorithm for finding solutions of rate-independent processes that verifies this condition and uses the strategy of backtracking if it is violated. We present the selectivity of the mentioned necessary condition in general and give numerical examples of the efficiency of such an algorithm, but also of situations that are beyond its limits. For those we propose a second strategy relying on wisely chosen combinations of spatial discretizations.

A microscopically motivated constitutive model for shape memory alloys: Formulation, analysis and computations

We present a three-dimensional constitutive model for NiTi polycrystalline shape memory alloys exhibiting transformations between three solid phases (austenite, R-phase, martensite). The ‘full modelling sequence’ comprised of formulation of modelling assumptions, construction of the model, mathematical analysis and numerical implementation and validation is presented. Namely, by formulating micromechanics-inspired modelling assumptions we concentrate on describing the dissipation mechanism: a refined form of this description makes our model especially useful for complex loading paths. We then embed the model into the so-called energetic framework (extended to our case) while taking advantage of describing the dissipation mechanism through the so-called dissipation distance. We prove the existence of energetic solutions to our model by a backward Euler scheme. This is then implemented into finite element software, and numerical simulations compared with experiments are also presented.

Characterization of gradient young measures generated by homeomorphisms in the plane

We characterize Young measures generated by gradients of bi-Lipschitz orientationpreserving maps in the plane. This question is motivated by variational problems in nonlinear elasticity where the orientation preservation and injectivity of the admissible deformations are key requirements. These results enable us to derive new weak∗ lower semicontinuity results for integral functionals depending on gradients. As an application, we show the existence of a minimizer for an integral functional with nonpolyconvex energy density among bi-Lipschitz homeomorphisms. Mathematics Subject Classification. 49J45, 35B05. Received May 13, 2014. Revised November 14, 2014. Published online January 28, 2016.

An Implicit Midpoint Spectral Approximation of Nonlocal Cahn--Hilliard Equations

The paper is concerned with the convergence analysis of a numerical method for nonlocal Cahn--Hilliard equations. The temporal discretization is based on the implicit midpoint rule and a Fourier spectral discretization is used with respect to the spatial variables. The sequence of numerical approximations in shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme are derived. The uniform bounds on the sequence of numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial discretization parameter needs to be assumed.

Young measures supported on invertible matrices

Motivated by variational problems in non-linear elasticity, we explicitly characterize the set of Young measures generated by gradients of a uniformly bounded sequence in where the inverted gradients are also bounded in . This extends the original results due to the studies Kinderlehrer and Pedregal. Besides, we completely describe Young measures generated by a sequence of matrix-valued mappings , such that is bounded, too, and the generating sequence satisfies the constraint .

Weak Lower Semicontinuity of Integral Functionals and Applications

Minimization is a reoccurring theme in many mathematical disciplines ranging from pure to applied ones. Of particular importance is the minimization of integral functionals that is studied within the calculus of variations. Proofs of the existence of minimizers usually rely on a fine property of the involved functional called weak lower semicontinuity. While early studies of lower semicontinuity go back to the beginning of the 20th century the milestones of the modern theory were set by C.B. Morrey Jr. [176] in 1952 and N.G. Meyers [169] in 1965. We recapitulate the development on this topic from then on. Special attention is paid to signed integrands and to applications in continuum mechanics of solids. In particular, we review the concept of polyconvexity and special properties of (sub)determinants with respect to weak lower semicontinuity. Besides, we emphasize some recent progress in lower semicontinuity of functionals along sequences satisfying differential and algebraic constraints which have applications in elasticity to ensure injectivity and orientation-preservation of deformations. Finally, we outline generalization of these results to more general first-order partial differential operators and make some suggestions for further reading.Minimization is a recurring theme in many mathematical disciplines ranging from pure to applied. Of particular importance is the minimization of integral functionals, which is studied within the calculus of variations. Proofs of the existence of minimizers usually rely on a fine property of the functional called weak lower semicontinuity. While early studies of lower semicontinuity go back to the beginning of the 20th century, the milestones of the modern theory were established by C. B. Morrey, Jr. [Pacific J. Math., 2 (1952), pp. 25--53] in 1952 and N. G. Meyers [Trans. Amer. Math. Soc., 119 (1965), pp. 125--149] in 1965. We recapitulate the development of this topic from these papers onwards. Special attention is paid to signed integrands and to applications in continuum mechanics of solids. In particular, we review the concept of polyconvexity and special properties of (sub-)determinants with respect to weak lower semicontinuity. In addition, we emphasize some recent progress in lower semicontinuity o...

Two-dimensional electronic spectroscopy can fully characterize the population transfer in molecular systems

Excitation energy transfer in complex systems often proceeds through series of intermediate states. One of the goals of time-resolved spectroscopy is to identify the spectral signatures of all of them in the acquired experimental data and to characterize the energy transfer scheme between them. It is well known that in the case of transient absorption spectra such decomposition is ambiguous even if many simplifying considerations are taken. In contrast to transient absorption, absorptive 2D spectra intuitively resemble population transfer matrices. Therefore, it seems possible to decompose the 2D spectra unambiguously. Here we show that all necessary information is encoded in the combination of absorptive 2D and linear absorption spectra. We set up a simple model describing a broad class of absorptive 2D spectra and prove analytically that they can be inverted uniquely towards physical parameters fully determining the species-associated spectra of individual constituents together with all connecting intrinsic rate constants. Due to the matrix formulation of the model, it is suitable for fast computer calculation necessary to efficiently perform the inversion numerically by fitting the combination of experimental 2D and absorption spectra. Moreover, the model allows for decomposition of the 2D spectrum into its stimulated emission, ground-state bleach, and excited-state absorption components almost unambiguously. The numerical procedure is illustrated exemplarily.

Gradient Young Measures Generated by Quasi-Conformal Maps in the Plane

In this contribution, we completely and explicitly characterize Young measures generated by gradients of quasi-conformal maps in the plane. By doing so, we generalize the results of Astala and Faraco [Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), pp. 1045--1056], who provided a similar result for quasi-regular maps and Benesova and Kružik [ESAIM Control Optim. Calc. Var., (2015)] who characterized Young measures generated by gradients of bi-Lipschitz maps. Our results are motivated by nonlinear elasticity where injectivity of the functions in the generating sequence is essential in order to assure noninterpenetration of matter.

Thermodynamically consistent mesoscopic model of the ferro/paramagnetic transition

A continuum evolutionary model for micromagnetics is presented that, beside the standard magnetic balance laws, includes thermomagnetic coupling. To allow conceptually efficient computer implementation, inspired by relaxation method of static minimization problems, our model is mesoscopic in the sense that possible fine spatial oscillations of the magnetization are modeled by means of Young measures. Existence of weak solutions is proved by backward Euler time discretization.

Micro-to-Meso Scale Limit for Shape-Memory-Alloy Models with Thermal Coupling

Modeling of shape-memory alloys represents a multiscale problem due to the occurrence of a martensite/austenite phase transformation and a microstructure in the deformation gradient typical for a martensitic phase. Inspired by relaxation in a static situation, a limit passage between two modeling scales, called micro- and mesoscales, is performed for the corresponding evolution variants while considering activated phase transformation and even thermodynamically consistent thermal coupling. The mesoscopic model captures possible fine spatial oscillations of the deformation gradient by means of gradient Young measures. In particular, the mesoscopic model is justified as a limit from the microscopic scale and existence of its solutions is proved.