arXiv subjects

Martin Kružík

Publications and source records attributed to Martin Kružík.

66 records · Page 4Linked to original sources

arXiv2016

Computational modeling of magnetic hysteresis with thermal effects

We study computational behavior of a mesoscopic model describing temperature/external magnetic field-driven evolution of magnetization. Due to nonconvex anisotropy energy describing magnetic properties of a body, magnetization can develop fast spatial oscillations creating complicated microstructures. These microstructures are encoded in Young measures, their first moments then identify macroscopic magnetization. Our model assumes that changes of magnetization can contribute to dissipation and, consequently, to variations of the body temperature affecting the length of magnetization vectors. In the ferromagnetic state, minima of the anisotropic energy density depend on temperature and they tend to zero as we approach the so-called Curie temperature. This brings the specimen to a paramagnetic state. Such a thermo-magnetic model is fully discretized and tested on two-dimensional examples. Computational results qualitatively agree with experimental observations. The own MATLAB code used in our simulations is available for download.

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arXiv2016

Generalized $\mathbf{W^{1,1}}$-Young measures and relaxation of problems with linear growth

We completely characterize generalized Young measures generated by sequences of gradients of maps from $W^{1,1}(Ω;\R^M)$ where $Ω\subset\R^N$. This extends and completes previous analysis by Kristensen and Rindler where concentrations of the sequence of gradients at the boundary of $Ω$ were excluded. We apply our results to relaxation of non-quasiconvex variational problems with linear growth at infinity. We also link our characterization to Souček spaces \cite{soucek}, an extension of $W^{1,1}(Ω;\R^M)$ where gradients are considered as measures on $\barΩ$.

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arXiv2015

A sharp interface evolutionary model for shape memory alloys

We show the existence of an energetic solution to a quasistatic evolutionary model of shape memory alloys. Elastic behavior of each material phase/variant is described by polyconvex energy density. Additionally, to every phase boundary, there is an interface-polyconvex energy assigned, introduced by M. Šilhavý. The model considers internal variables describing the evolving spatial arrangement of the material phases and a deformation mapping with its first-order gradients. It allows for injectivity and orientation-preservation of deformations. Moreover, the resulting material microstructures have finite length scales.

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arXiv2015

Characterization of gradient Young measures generated by homeomorphisms in the plane

We characterize Young measures generated by gradients of bi-Lipschitz orientation-preserving 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.

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arXiv2015

$\mathcal{A}$-quasiconvexity and weak lower semicontinuity of integral functionals

We state necessary and sufficient conditions for weak lower semicontinuity of $u\mapsto\int_Ωh(x,u(x))\,d x$ where $|h(x,s)|\le C(1+|s|^p)$ is continuous and possesses a recession function, and $u\in L^p(Ω;\mathbb{R}^m)$, $p>1$, lives in the kernel of a constant-rank first-order differential operator $\mathcal{A}$ which admits an extension property. Our newly defined notion coincides for $\mathcal{A}=\operatorname{curl}$ with quasiconvexity at the boundary due to J.M. Ball and J. Marsden. Moreover, we give an equivalent condition for weak lower semicontinuity of the above functional along sequences weakly converging in $L^p(Ω;\mathbb{R}^m)$ and approaching the kernel of $\mathcal{A}$ even if $\mathcal{A}$ does not have the extension property.

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arXiv2014

Semi-definite relaxations for optimal control problems with oscillation and concentration effects

Converging hierarchies of finite-dimensional semi-definite relaxations have been proposed for state-constrained optimal control problems featuring oscillation phe-nomena, by relaxing controls as Young measures. These semi-definite relaxations were later on extended to optimal control problems depending linearly on the con-trol input and typically featuring concentration phenomena, interpreting the control as a measure of time with a discrete singular component modeling discontinuities or jumps of the state trajectories. In this contribution, we use measures intro-duced originally by DiPerna and Majda in the partial differential equations litera-ture to model simultaneously, and in a unified framework, possible oscillation and concentration effects of the optimal control policy. We show that hierarchies of semi-definite relaxations can also be constructed to deal numerically with noncon-vex optimal control problems with polynomial vector field and semialgebraic state constraints.

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arXiv2014

Quasistatic evolution of magnetoelastic thin films via dimension reduction

A rate-independent model for the quasistatic evolution of a magnetoelastic thin film is advanced and analyzed. Starting from the three-dimensional setting, we present an evolutionary $Γ$-convergence argument in order to pass to the limit in one of the material dimensions. By taking into account both conservative and dissipative actions, a nonlinear evolution system of rate-independent type is obtained. The existence of so-called {\it energetic solutions} to such system is proved via approximation.

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arXiv2014

Boundary effects and weak$^*$ lower semicontinuity for signed integral functionals on $\mathrm{BV}$

We characterize lower semicontinuity of integral functionals with respect to weak$^*$ convergence in $\mathrm{BV}$, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the boundary. In this case, both the integrand and the shape of the boundary play a key role. This is made precise in our newly found condition -- quasi-sublinear growth from below at points of the boundary -- which compensates for possible concentration effects generated by the sequence. Our work extends some recent results by J. Kristensen and F. Rindler (Arch. Rat. Mech. Anal. 197 (2010), 539--598 and Calc. Var. 37 (2010), 29--62).

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arXiv2013

Existence results for incompressible magnetoelasticity

We investigate a variational theory for magnetoelastic solids under the incompressibility constraint. The state of the system is described by deformation and magnetization. While the former is classically related to the reference configuration, magnetization is defined in the deformed configuration instead. We discuss the existence of energy minimizers without relying on higher-order deformation gradient terms. Then, by introducing a suitable positively $1$-homogeneous dissipation, a quasistatic evolution model is proposed and analyzed within the frame of energetic solvability.

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arXiv2013

Young measures supported on invertible matrices

Motivated by variational problems in nonlinear elasticity depending on the deformation gradient and its inverse, we completely and explicitly describe Young measures generated by matrix-valued mappings $\{Y_k\}_{k\in\N} \subset L^p(Ø;\R^{n\times n})$, $Ø\subset\R^n$, such that $\{Y_k^{-1}\}_{k\in\N} \subset L^p(Ø;\R^{n\times n})$ is bounded, too. Moreover, the constraint $\det Y_k>0$ can be easily included and is reflected in a condition on the support of the measure. This condition typically occurs in problems of nonlinear-elasticity theory for hyperelastic materials if $Y:=\nabla y$ for $y\in W^{1,p}(Ø;\R^n)$. Then we fully characterize the set of Young measures generated by gradients of a uniformly bounded sequence in $W^{1,\infty}(Ø;\R^n)$ where the inverted gradients are also bounded in $L^\infty(Ø;\R^{n\times n})$. This extends the original results due to D. Kinderlehrer and P. Pedregal.

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arXiv2013

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 thermo-magnetic 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.

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arXiv2011

Oscillations and concentrations up to the boundary

Oscillations and concentrations in sequences of gradients $\{\nabla u_k\}$, bounded in $L^p(Ω;\R^{M\times N})$ if $p>1$ and $Ω\subset\R^n$ is a bounded domain with the extension property in $W^{1,p}$, and their interaction with local integral functionals can be described by a generalization of Young measures due to DiPerna and Majda. We characterize such DiPerna-Majda measures, thereby extending a result by Kałamajska and Kruž\'ık (2008), where the full characterization was possible only for sequences subject to a fixed Dirichlet boundary condition. As an application we state a relaxation result for noncoercive multiple-integral functionals.

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