Elliptic and Parabolic Problems
A Special Tribute to the Work of Haim Brezis
1st Edition
Published on 17. January 2006
IX, 470 pages
978-3-7643-7384-9 (ISBN)
Description
Haim Brezis has made significant contributions in the fields of partial differential equations and functional analysis, and this volume collects contributions by his former students and collaborators in honor of his 60th anniversary at a conference in Gaeta. It presents new developments in the theory of partial differential equations with emphasis on elliptic and parabolic problems.
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Elliptic and Parabolic Problems
A Special Tribute to the Work of Haim Brezis
Book
07/2005
Birkhäuser
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Content
One-Layer Free Boundary Problems with Two Free Boundaries.- New Numerical Solutions for the Brezis-Nirenberg Problem on $$\mathbb{S}^n $$ .- On some Boundary Value Problems for Incompressible Viscous Flows with Shear Dependent Viscosity.- Radiative Heat Transfer in Silicon Purification.- A Decay Result for a Quasilinear Parabolic System.- The Shape of Charged Drops: Symmetrybreaking Bifurcations and Numerical Results.- On the One-dimensional Parabolic Obstacle Problem with Variable Coefficients.- Hardy Potentials and Quasi-linear Elliptic Problems Having Natural Growth Terms.- Recent Advances on Similarity Solutions Arising During Free Convection.- Rellich Relations for Mixed Boundary Elliptic Problems.- Lyapunov-type Inequalities and Applications to PDE.- Gaeta 2004. Elliptic Resonant Problems with a Periodic Nonlinearity.- Harnack Inequality for p-Laplacians on Metric Fractals.- Wave Propagation in Discrete Media.- A Solution of the Heat Equation with a Continuum of Decay Rates.- Finite Volume Scheme for Semiconductor Energy-transport Model.- Asymptotic Behavior of Nonlinear Parabolic Problems with Periodic Data.- Geodesic Computations for Fast and Accurate Surface Remeshing and Parameterization.- On the Newton Body Type Problems.- Some Open Problems on Water Tank Control Systems.- Hölder Estimates for Solutions to a Singular Nonlinear Neumann Problem.- Asymptotic Analysis of the Neumann Problem for the Ukawa Equation in a Thick Multi-structure of Type 3:2:2.- On the Haïm Brezis Pioneering Contributions on the Location of Free Boundaries.- Fractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization.- Stationary and Self-similar Solutions for Coagulation and Fragmentation Equations.- Orlicz Capacities and Applications to PDEs and SobolevMappings.- Energy Forms on Non Self-similar Fractals.- Measure Data and Numerical Schemes for Elliptic Problems.- Brezis-Nirenberg Problem and Coron Problem for Polyharmonic Operators.- Local and Global Properties of Solutions of a Nonlinear Boundary Layer Equation.- Mathematical Models of Aggregation: The Role of Explicit Solutions.- Metastable Behavior of Premixed Gas Flames.- Recent Progress on Boundary Blow-up.- Maximum Principle for Bounded Solutions of the Telegraph Equation: The Case of High Dimensions.- Kolmogorov Equations in Physics and in Finance.- Harnack Inequalities and Gaussian Estimates for a Class of Hypoelliptic Operators.- How to Construct Good Measures.- Bifurcation and Asymptotics for Elliptic Problems with Singular Nonlinearity.- A Model for Hysteresis in Mechanics Using Local Minimizers of Young Measures.- The Precise Lp-theory of Elliptic Equations in the Plane.- Essential Spectrum and Noncontrollability of Membrane Shells.- The Porous Medium Equation. New Contractivity Results.- Large Solutions of Elliptic Equations with Strong Absorption.- Relaxation in Presence of Pointwise Gradient Constraints.
Radiative Heat Transfer in Silicon Purification (P.33)
A. Bermudez, R. Leira, M.C. Müniz and F. Pena
Abstract.
We present a numerical model describing the thermal behavior of a silicon purification process which takes place into a so-called casting ladle. We consider, simultaneously, the phase change in the silicon and a nonlinear non-local boundary condition arising from the Stefan-Boltzmann radiation condition at the enclosure surfaces within the ladle. We also propose a numerical approximation using a finite element method. An iterative algorithm and numerical results are presented.
1. Introduction
In many engineering applications involving high-temperature processes numerical simulation provides an insight into the radiative analysis of these complex systems and it promotes improvements of several process optimization (see [3, 5]). The motivation of this work is to compute the numerical solution of the problem addressed in [6] applied to a silicon purification process - see [1]. We consider, simultaneously, the phase change in the silicon and the non-local boundary condition arising from the Stefan-Boltzmann radiation condition at the enclosure surfaces within the ladle.
The outline of this paper is as follows. In Section 2 the physical problem is introduced. In Section 3 using the axisymmetry of the domain, we formulate the mathematical problem in a two-dimensional domain by means of cylindrical coordinates. Section 4 is devoted to introduce space and time discretization of the aforementioned problem and to present an iterative algorithm. Finally, in Section 5, several numerical results are shown.
2. The physical problem
Metallurgical grade silicon (MG-Si) is obtained from a silicon oxide in electrical submerged arc furnaces. A technique of MG-Si puri.cation is to melt it and to induce its directional solidification. This method of removing impurities is based on the fact that most impurities tend to remain in a molten region rather than re-solidify.
This purification process is taking place into a casting ladle which consists of a finite axisymmetric cylinder containing a cylindrical enclosure. After the casting ladle being electrically heated, its lid is open and molten silicon is poured into its inner cavity keeping a gap between the top of the silicon and the upper part of the inner ladle surface where several heating elements are located. The objective is now to push upwards the metal impurities by means of inducing its one-directional solidification switching on the heating elements and then keeping molten the top of the silicon ingot.
In doing so, the solid silicon grows gradually upwards into the liquid and the metallic impurities are segregated into the melt region during solidification, thus, at the end of the process most of impurities are concentrated at the top of the silicon ingot. Radiation heat transfer is considered at the inner cavity and materials of the enclosure are assumed to be opaque (see [4]), therefore radiation may be treated as a surface phenomenon.
Moreover, we assume both that the walls of the cylindrical enclosure behave as black surfaces and that the medium within the enclosure is radiatively nonparticipating so that it has no effect on the radiation transfer between inner surfaces.
A. Bermudez, R. Leira, M.C. Müniz and F. Pena
Abstract.
We present a numerical model describing the thermal behavior of a silicon purification process which takes place into a so-called casting ladle. We consider, simultaneously, the phase change in the silicon and a nonlinear non-local boundary condition arising from the Stefan-Boltzmann radiation condition at the enclosure surfaces within the ladle. We also propose a numerical approximation using a finite element method. An iterative algorithm and numerical results are presented.
1. Introduction
In many engineering applications involving high-temperature processes numerical simulation provides an insight into the radiative analysis of these complex systems and it promotes improvements of several process optimization (see [3, 5]). The motivation of this work is to compute the numerical solution of the problem addressed in [6] applied to a silicon purification process - see [1]. We consider, simultaneously, the phase change in the silicon and the non-local boundary condition arising from the Stefan-Boltzmann radiation condition at the enclosure surfaces within the ladle.
The outline of this paper is as follows. In Section 2 the physical problem is introduced. In Section 3 using the axisymmetry of the domain, we formulate the mathematical problem in a two-dimensional domain by means of cylindrical coordinates. Section 4 is devoted to introduce space and time discretization of the aforementioned problem and to present an iterative algorithm. Finally, in Section 5, several numerical results are shown.
2. The physical problem
Metallurgical grade silicon (MG-Si) is obtained from a silicon oxide in electrical submerged arc furnaces. A technique of MG-Si puri.cation is to melt it and to induce its directional solidification. This method of removing impurities is based on the fact that most impurities tend to remain in a molten region rather than re-solidify.
This purification process is taking place into a casting ladle which consists of a finite axisymmetric cylinder containing a cylindrical enclosure. After the casting ladle being electrically heated, its lid is open and molten silicon is poured into its inner cavity keeping a gap between the top of the silicon and the upper part of the inner ladle surface where several heating elements are located. The objective is now to push upwards the metal impurities by means of inducing its one-directional solidification switching on the heating elements and then keeping molten the top of the silicon ingot.
In doing so, the solid silicon grows gradually upwards into the liquid and the metallic impurities are segregated into the melt region during solidification, thus, at the end of the process most of impurities are concentrated at the top of the silicon ingot. Radiation heat transfer is considered at the inner cavity and materials of the enclosure are assumed to be opaque (see [4]), therefore radiation may be treated as a surface phenomenon.
Moreover, we assume both that the walls of the cylindrical enclosure behave as black surfaces and that the medium within the enclosure is radiatively nonparticipating so that it has no effect on the radiation transfer between inner surfaces.
