Original articles
Problems with uncertain hysteresis operators and homogenization

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Abstract

The paper deals with the reliable solution to the homogenization problem for the scalar heat equation with the uncertain hysteresis Prandtl–Ishlinskii operator. The problem is solved by the so-called worst scenario method. The contribution extends the results of paper Franců (2017), to the corresponding homogenization problem.

Introduction

Some problems in technology can be modeled by initial boundary value problems for partial differential equations with hysteresis operators. Among them let us consider the equation cut=qx+f with q=kux, where the subscripts t and x mean partial derivatives with respect to t and x. In physics, the equation can be interpreted as a heat conduction in a bar (0,) with an unknown temperature u(x,t) and a negative heat flow q(x,t). We take the negative heat flow q instead of the positive one in order to obtain the Fourier law in the form q=kux with the positive conductivity coefficient k instead of q=kux with the positive heat flow.

We replace the Fourier law q=kux by the relation q=F[ux] with a hysteresis operator F to describe behavior of a rate-independent material with memory or phase transition, see [4], Chapter IV, or [16]. The hysteresis operators are introduced in Section 2. Since we consider a heterogeneous bar, the coefficient c and the hysteresis operator F depend even on the space variable x. The equation can be interpreted also as a diffusion problem. In this way we have obtained the following nonlinear equation cut=F[ux]x+f,(x,t)(0,)×(0,T)Ω×I.completed with an appropriate initial and boundary conditions.

In case of the heterogeneous media with a fine space periodic structure the problem is approximated by a method called homogenization. For computational reasons the method replaces the strongly oscillating periodic coefficients by “equivalent” ones yielding globally the same solution. The homogenization problem is formulated in Section 4.

The problem for parabolic equation with the space-dependent hysteresis operator F and the corresponding homogenization problem was solved in [5], the homogenization problem with the scalar hysteresis operator in higher dimension was solved in [6].

Since the coefficient c and the functional dependence F in the equation were obtained by measurements, they are uncertain, i.e. their values are known to be within certain intervals. In order to obtain reliable solutions of the problem, the so-called worst scenario method was used by engineers.

The mathematical approach was formulated and analyzed by I. Babuška and I. Hlaváček in a series of papers, for survey see [10], [11]. The method properly considers all data, i.e. all material parameters from their range of uncertainty. Using a criterion-functional, which measures the badness of the situation, the worst situation that may occur is looked for, see Section 5.

Reliable solution of the homogenization problem with monotone operators was studied in [9]. Reliable solution for the heat equation with the uncertain hysteresis operator was solved in [7].

The contribution combines and extend the previous results. The initial boundary value problem for the nonlinear heat equation with the uncertain hysteresis operator is studied. The novelty of the contribution is formulation and analysis of the worst scenario method applied to the homogenization problems for the equation with hysteresis operators using the previous results.

The text is organized as follows. Section 2 contains survey of hysteresis operators and their properties, in Section 3 the initial boundary value problem with hysteresis operator is formulated and results on the existence and uniqueness of the solutions from [5] are summarized. Section 4 formulates the homogenization problem and surveys the results from [5]. Section 5 summarizes the problem of uncertain data and the worst scenario method. In Section 6 the corresponding homogenization problem is formulated, the set of admissible data is defined, the bounds for reliable homogenized operator are derived, various criterion-functionals are proposed and the existence of the reliable solution is proved. In the last section the obtained results are surveyed, reliability of the solution to the homogenized problem is discussed and several generalizations of the results are outlined.

Section snippets

Hysteresis operators

We shall deal with one-dimensional hysteresis operators. These single valued operators act on the space of absolutely continuous real functions on a time interval I=0,T, i.e. they map Sobolev space W1,1(I) into itself. They can be characterized in a simple way: the hysteresis operator T is

  • rate-independent — the output T[v] is independent of speed of the input v, i.e. T[vφ](t)=T[v](φ(t))  for any continuous increasing function φ from the interval I onto I,

  • causal — the output is independent of

Initial boundary value problem with hysteresis operator

We start with the formulation and solvability of the problem with the space-dependent mutually inverse hysteresis operators Fη and Gζ determined by the couple of distribution functions (η,ζ) satisfying (η(x,),ζ(x,))PI(α,β), where PI(α,β) is introduced in Definition 2.4. There are two possible mutually equivalent relations q=Fη[ux]  and  ux=Gζ[q]. For convergence of the coefficients in the homogenization problem the latter will appear to be better, see remark after Theorem 4.1.

Homogenization

Composite materials are modern heterogeneous materials with fine structure, we suppose that the structure is periodic with a small period. Thus, their mathematical model has periodic coefficients in the constitutive relations. For practical computational reasons this heterogeneous material is replaced by a fictitious homogeneous material having the same properties, the strongly in space oscillating coefficients are replaced by constant ones, the so-called homogenized coefficients. What are

Problems with uncertain data and the worth scenario method

Mathematical models of engineering problems contain some uncertain data, mainly material constants in constitutive relations. These data obtained by measurements are uncertain, their values are known to be within certain interval only. Calculation of the response of a loaded construction with mean value of the material properties such as strength, caused already several failures of construction in the engineering practice. To obtain reliable solutions engineers started to use the so-called

The worst scenario method for the homogenization problem

The worst scenario method will be applied in two steps. In the first one we obtain the reliable bounds for the data of the homogenized problem, in the second one the worst scenario method is applied to the homogenized solution.

Concluding remarks

The contribution studied the nonlinear heat conduction or diffusion equation in the heterogeneous one-dimensional media. The hysteresis operator of Prandtl–Ishlinskii type describes behavior of a rate-independent material with memory or phase transition. The conductivity coefficient c and the distribution function ζ of the hysteresis operators are assumed to be uncertain, i.e. known in some extent only.

The corresponding homogenization problem was formulated. The set of compact admissible data

Acknowledgment

This research was supported by Brno University of Technology, Czech Republic from the Specific Research Project No. FSI-S-20-6187.

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