University of Stuttgart

About: University of Stuttgart is a education organization based out in Stuttgart, Germany. It is known for research contribution in the topics: Laser & Finite element method. The organization has 27715 authors who have published 56370 publications receiving 1363382 citations. The organization is also known as: Universität Stuttgart.

Continuous chaos—four prototype equations

Abstract: If oscillation is the typical behavior of 2-dimensional dynamical systems (Euclidean and on manifolds), then chaos, in the same way, characterizes 3-dimensional continuous systems. First a method t o obtain chaos in degenerate (relaxation type) dynamical systems in two variables is outlined whereby five basic flow patterns emerge. Second, following a piecewise linear degenerate equation, four prototypically simple quadratic differential equations in three variables that realize nondegenerate analogs of those five flows are presented. Finally a possible equation for an even higher type of qualitative behavior beyond chaos is proposed.

Description of a Turbulent Cascade by a Fokker-Planck Equation

Abstract: Fully developed turbulence is still regarded to be one of the main unsolved problems of classical physics. Great efforts have been made towards an understanding of small scale turbulent velocity fluctuations, which are assumed to be stationary, homogeneous, and isotropic in a statistical sense [1]. For large Reynolds numbers these fluctuations are supposed to exhibit universal behavior on scales smaller than the integral one. The elucidation of these properties apparently has to be based on applications of the tools of statistical mechanics. The quantity of main interest is the longitudinal velocity fluctuations yi on different length scales Li, yi › usx 1 Liy2, y, zd 2 usx 2 Liy2, y, zd , (1) where usx, y, zd is the x component of the velocity field at space point x, y, z. Based on the idea of an energy cascade, as a fundamental process governing the turbulence, we know from the pioneering works of the 1940s, cf. [1], that the velocity fluctuations are of the order yi ,s e L i d 1 y 3 . edenotes the energy dissipation (transfer) rate. However, it is commonly believed that intermittent fluctuations of the energy dissipation rate alters the scaling behavior. Intermittency effects show up in the changing shape of the probability density functions (pdf ) PLi syid as a function of Li and consequently lead for the scaling of the moments ksyid n l, L z n i to nonlinear n dependence of the scaling indices zn.

On rate-independent hysteresis models

Abstract: This paper deals with a general approach to the modeling of rate–independent processes which may display hysteretic behavior. Such processes play an important role in many applications like plasticity and phase transformations in elastic solids, electromagnetism, dry friction on surfaces, or in pinning problems in superconductivity, cf. [Vis94, BrS96]. The evolution equations which govern those processes constitute the limit problems if the influence of inertia and relaxation times vanishes, i.e. the system rests unless the external loading is varied. Only the stick–slip dynamics is present in the Cauchy problem, this means that the evolution equations are necessarily non–autonomous. Although the solutions often exhibit quite singular behavior, the reduced framework offers great advantages. Firstly the amount of modelling can be reduced to its absolute minimum. More importantly, our approach is only based on energy principles. This allows us to treat the Cauchy problem by mainly using variational techniques. This robustness is necessary in order to study problems which come from continuum mechanics like plasticity, cf. [Mie00, CHM01, Mie01]. There the potential energy is invariant under the group of rigid body rotations SO(d) where d ∈ {1, 2, 3} is the dimension. This invariance implies that convexity can almost never be expected and more advanced lower semicontinuity results (like polyconvexity) are required to assure the existence of solutions for a time discretized version of the problem. An example which illustrates this remark is a problem from phase transformations in solids, see [MTL00]. Although none of the classical methods from Section 7 can be applied, we are able to prove the existence of solutions by establishing weak lower semicontinuity of certain critical quantities. Here we present an abstract framework which is based on two energy functionals, namely the potential energy I(t, z) and the dissipation ∆(ż). Here z ∈ X, X a separable, reflexive Banach space with dual X, is the variable describing the process, and ż is the time derivative. The central feature of rate–independence means that a solution z : [0, T ] → X remains a solution if the time is rescaled. This leads to a dissipation functional ∆ : X → [0,∞) which is homogeneous of degree 1, i.e., ∆(αv) = α∆(v) for α ≥ 0 and v ∈ X. Special cases of this situation are well studied in the theory of variational inequalities