Mathematical Models and Methods in Applied Sciences

We consider the problem of a clamped elastic thin plate unilaterally constrained by an elastic obstacle. A discrete, finite element approach to solve this nonlinear problem is given here and its convergence properties are discussed.

We use the formalism of wave-packet propagation in passive media to characterize the spread of the electrical excitation in excitable media, namely the cardiac myocardium. We introduce equivalent concepts of group and phase velocities, attenuation coefficient and refraction index to describe the myocardial excitation wave and apply the wavelet approach to construct an analogue of the classical dispersion dependence for active media — the "equivalent dispersion dependence". Using the wavelet decomposition we develop a method for the reconstruction of the equivalent dispersion dependence for the myocardium on the basis of electrical intracardiac signals that are measured in two spatially separated points. The novel method is applied to two different sets of experimental data and to data obtained from a numerical simulation of the atrial myocardium. We show that the introduced equivalent dispersion dependence under physiological conditions is similar to the one that is obtained for resonant wave-medium interaction. The analysis of both experimental data sets clearly shows that the number of cardiac cycles with a resonant form of the equivalent dispersion dependence predominates in the normal state of the myocardium while it decreases early before the onset of atrial fibrillation. We set up the hypothesis that an increasing number of non-resonant cardiac cycles is a precursor of atrial fibrillation and thus can serve to predict fibrillation already at an early stage before its onset. The proposed conception can be applied to investigate the properties of the atrial as well as the ventricular myocardium.

In refined network analysis, a compact network model is combined with distributed models for semiconductor devices in a multiphysics approach. For linear RLC networks containing diodes as distributed devices, we construct a mathematical model that combines the differential-algebraic network equations of the circuit with elliptic boundary value problems modeling the diodes. For this mixed initial-boundary value problem of partial differential-algebraic equations a first existence result is given.

We consider two classical problems which are widely used as benchmark tests for shell numerical methods: the Scordelis–Lo roof and the pinched roof. Due to the particular load and boundary conditions applied, neither belongs to the well-known classes of purely bending or purely membrane dominated shells. Consequently the asymptotic energy norm behavior, which is useful not only because it represents the structure stiffness, but also for numerical comparison purposes, is not a priori known. In this work, using space interpolation techniques and a recently developed "intermediate" shell theory, the asymptotic energy behavior of both problems is found analytically. The results are in agreement with the numerical estimates obtained in other papers.

We study a system of nonlinear partial differential equations governing the motion of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain, with a non-Newtonian viscosity depending on the second invariant of the rate of deformation tensor. Considering the equations in a suitably decomposed form, we establish, for small and suitably regular data, existence of a unique solution using a fixed point argument in an appropriate functional setting. This model includes the classical Oldroyd-B fluid as a particular case.

In this paper we propose a way to analyze certain classes of dimension reduction models for elliptic problems in thin domains. We develop asymptotic expansions for the exact and model solutions, having the thickness as small parameter. The modeling error is then estimated by comparing the respective expansions, and the upper bounds obtained make clear the influence of the order of the model and the thickness on the convergence rates. The techniques developed here allows for estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).

We establish some existence and uniqueness results for a nonlinear elliptic equation. The problem has a diffusion matrix A(x, u) such that A(x, s)ξξ ≥ β(s)|ξ|², with β : (s₀, + ∞) ↦ ℝ a continuous, strictly positive function which goes to infinity when s is near s₀. On the other hand, . Also, the right-hand side f belongs to L¹(Ω). We make use of the concept of renormalized solutions adapted to our problem.