Partial Differential Equation System (partial + differential_equation_system)

Distribution by Scientific Domains

Selected Abstracts

Error estimation in a stochastic finite element method in electrokinetics

S. Clénet
Abstract Input data to a numerical model are not necessarily well known. Uncertainties may exist both in material properties and in the geometry of the device. They can be due, for instance, to ageing or imperfections in the manufacturing process. Input data can be modelled as random variables leading to a stochastic model. In electromagnetism, this leads to solution of a stochastic partial differential equation system. The solution can be approximated by a linear combination of basis functions rising from the tensorial product of the basis functions used to discretize the space (nodal shape function for example) and basis functions used to discretize the random dimension (a polynomial chaos expansion for example). Some methods (SSFEM, collocation) have been proposed in the literature to calculate such approximation. The issue is then how to compare the different approaches in an objective way. One solution is to use an appropriate a posteriori numerical error estimator. In this paper, we present an error estimator based on the constitutive relation error in electrokinetics, which allows the calculation of the distance between an average solution and the unknown exact solution. The method of calculation of the error is detailed in this paper from two solutions that satisfy the two equilibrium equations. In an example, we compare two different approximations (Legendre and Hermite polynomial chaos expansions) for the random dimension using the proposed error estimator. In addition, we show how to choose the appropriate order for the polynomial chaos expansion for the proposed error estimator. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Global existence for a contact problem with adhesion

Elena Bonetti
Abstract In this paper, we analyze a contact problem with irreversible adhesion between a viscoelastic body and a rigid support. On the basis of Frémond's theory, we detail the derivation of the model and of the resulting partial differential equation system. Hence, we prove the existence of global in time solutions (to a suitable variational formulation) of the related Cauchy problem by means of an approximation procedure, combined with monotonicity and compactness tools, and with a prolongation argument. In fact the approximate problem (for which we prove a local well-posedness result) models a contact phenomenon in which the occurrence of repulsive dynamics is allowed for. We also show local uniqueness of the solutions, and a continuous dependence result under some additional assumptions. Copyright © 2007 John Wiley & Sons, Ltd. [source]

The normal and cancerous living cell

Janos Ladik
Abstract We do not have a definition of the living and cancerous states; we can give only their main characteristics at the different levels of organization: cell, organ, and organism. A simple model is proposed for a normal eukaryotic cell based on Prigogine's equation of chemical kinetics with diffusion. In this model, possibly only a few hundred key biochemical reactions should be selected together with their rate and diffusion constants. To solve these coupled nonlinear partial differential equation systems, it is proposed that the model cell be subdivided into compartments and that the problem be worked out always for one compartment (finite element method). This is possible, since the most important biochemical reactions and reaction cycles occur in different parts of the cell. The solutions (concentrations) obtained in one compartment can be used as input to the other compartments (together with the components entering from the environment). As an example, the problem of 10 reactions and 3 compartments has been solved by discretizing the space coordinates and choosing time steps. The solutions obtained by solving the 10 differential equations directly and by the compartmentalization agree very well. The main obstacles to further progress lie in the right choice of reactions and compartments, as well as in the correct estimation of the rate and diffusion constants, which were measured in only a few cases. If such a model cell can be obtained, the solutions should be investigated to determine (i) for their stability (homeostasis); (ii) whether changing the input concentrations to a larger degree one would obtain a new stationary state showing the characteristics of a precancerous state; and (iii) a method of extracting those input concentrations, or functions of them, which are the most important regulatory parameters. If successful, this would provide a scientific definition of the living state in the normal and cancerous states, respectively, at least at the cell level. Finally, outline is provided showing how the model might be extended to multicellular cases, as well as the main difficulties of such a process. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source]

Crystal temperature control in the Czochralski crystal growth process

AICHE JOURNAL, Issue 1 2001
Antonios Armaou
This work proposes a control configuration and a nonlinear multivariable model-based feedback controller for the reduction of thermal gradients inside the crystal in the Czochralski crystal growth process after the crystal radius has reached its final value. Initially, a mathematical model which describes the evolution of the temperature inside the crystal in the radial and axial directions and accounts for radiative heat exchange between the crystal and its surroundings and motion of the crystal boundary is derived from first principles. This model is numericully solved using Galerkin's method and the behaviour of the crystal temperature is studied to obtain valuable insights which lead to the precise formulation of the control problem, the design of a new control configuration for the reduction of thermal gradients inside the crystal and the derivation of a simplified 1-D in a space dynamic model. Then, a model reduction procedure for partial differential equation systems with time-dependent spatial domains (Armaou and Christofides, 1999) based on a combination of Galerkin's method with approximate inertial manifolds is used to construct a fourth-order model that describes the dominant thermal dynamics of the Czochralski process. This low-order model is employed for the synthesis of a fourth-order nonlinear multivariable controller that can be readily implemented in practice. The proposed control scheme is successfully implemented on a Czochralski process used to produce a 0.7 m long silicon crystal with a radius of 0.05 m and is shown to significantly reduce the axial and radial thermal gradients inside the crystal. The robustness of the proposed controller with respect to model uncertainty is demonstrated through simulations. [source]