Free Boundary Value Problems: Proceedings of a Conference held at the Mathematisches Forschungsinstitut, Oberwolfach, July 9–15, 1989

283 Pages·1990·6.584 MB·English

by  Hans Wilhelm Alt, Irena Pawlow (auth.), K.-H. Hoffmann, J. Sprekels (eds.)

D
ISNM95:
International Series of Numerical Mathematics
Internationale Schriftenreihe zur Numerischen Mathematik
Serie internationale numerique
d~nalyse
Vol. 95
Edited by
K.-H. Hoffmann, Augsburg; H. D. Mittelmann, Tempe;
J. Todd, Pasadena
Birkhauser Verlag
Basel· Boston· Berlin
Free Boundary
value Problems
Proceedings of a Conference held at the
Mathematisches Forschungsinstitut, Oberwoifach,
July 9-15,1989
Edited by
K.-H. Hoffmann
J. Sprekels
1990 Birkhiuser Verlag
Basel . Boston . Berlin
Editors
K.-H. Hoffmann J. Sprekels
Institut flir Mathematik Fachbereich 10 - Bauwesen
Universitiit Augsburg Universitiit-GH Essen
Universitiitsstrasse 8 Postfach 103 764
D-8900 Augsburg D-4300 Essen
Deutsche Bibliothek Cataloguing-in-Publication Data
Free boundary value problems: proceedings of a conference
held at the Mathematisches Forschungsinstitut, Oberwolfach,
July 9-15,1989/ ed. by K.-H. Hoffmann; J. Sprekels. - Basel
Boston ; Berlin : Birkhiiuser, 1990
(International series of numerical mathematics ; Vol. 95)
ISBN-13: 978-3-7643-2474-2 e-ISBN-13: 978-3-0348-7301-7
DOl: 10.1007/978-3-0348-7301-7
NE: Hoffmann, Karl-Heinz [Hrsg.]; Mathematisches Forschungsinstitut
<Oberwolfach >; GT
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically those of translation, reprinting, re-use of
illustrations, broadcasting, reproduction by photocopying machine or similar means,
and storage in data banks. Under § 54 of the German Copyright Law where copies are
made for other than private use a fee is payable to >Nerwertungsgesellschaft Wort«, Munich.
© 1990 Birkhiiuser Verlag Basel
Softcover reprint of the hardcover 1st edition 1990
ISBN-13: 978-3-7643-2474-2
v
PREFACE
This monograph contains a collection of 16 papers that were presented at the confer
ence "Free Boundary Problems: Numerical 7reatment and Optimal Control", held at the
Mathematisches Forschungsinstitut Oberwolfach, West Germany, July 9-15, 1989.
It was the aim of the organizers of the meeting to bring together experts from different
areas in the broad field of free boundary problems, where a certain emphasis was given to
the numerical treatment and optimal control of free boundary problems. However, during
the conference also a number papers leading to important new theoretical insights were
presented. The strong connection between theory and applications finds its reflection in
this monograph which contains papers of high theoretical and numerical interest, as well
as applications to important practical problems. Many of the contributions are concerned
with phase transition phenomena, a field which was of particular importance during the
meeting; topics like spinodal decomposition, shape memory alloys, crystal growth and
flow through porous media are addressed. Another field of major interest during the con
ference was fluid flow; also this field is addressed in this volume.
The volume opens with a contribution by H.W.Alt and I.Pawlow. In their paper the
problem of spinodal decomposition is treated in the non-isothermal situation. For the first
time the existence of a weak solution to the corresponding system of evolution equations
could be proved. The results of some numerical experiments are also reported.
In the following paper, M.Bornert and I.Miiller introduce a term reflecting interfacial
energies into the free energy in order to explain hysteresis phenomena in phase transitions.
The consequences of this theory for the temperature-dependence of the size of hysteresis
loops in pseudoelastic materials is discussed in detail by the authors.
The continuous casting problem is addressed in the contribution of J.N.Dewynne, S.D.
Howison and J.R.Ockendon. Their main emphasis lies in the numerical solution of a
model problem.
In their contribution, C.M.Elliott and S.Zheng give an analysis of the global behaviour
of the phase-field equations which constitute a model for solidification processes with
interfaces of finite thickness. In particular, the asymptotic stability of the solutions is
investigated in this work.
Near shore coastal processes, such as erosion and sediment transport, are strongly influ
enced by the actions of shallow water waves. In their contribution, R.B.Guenther and
J.A.Crow give a model for the damping of such water waves and test it numerically.
The paper of J.Haslinger and P.Neittaanmiiki brings an extension of former results on
an elliptic identification problem to some problems in linear elasticity which play an
important role in the stability analysis of constructions. It is shown that a curve can be
found in the solid along which the tangential components of the stress tensor attain a
maximal value.
B.Kawohl considers in his contribution the question how two materials with different shear
moduli should be mixed in a given bar so as to maximize its torsional rigidity. For this
optimal design problem, regularity and uniqueness results are proved for cross-sections of
general shape. Some numerical results support the theoretical findings.
VI
In his paper, N .Kenmochi provides a new uniqueness proof for nonlinear two-phase Stefan
problems which also applies to some classes of nonlinear boundary conditions; for instance,
unilateral boundary conditions of the Signorini type are included in the theory.
The contribution of N.Kenmochi and M.Kubo is concerned with flows in partially sat
urated porous media. The existence of a unique periodic solution is proved, and the
solution is shown to be asymptotically stable (in a suitable sense).
The numerical solution of free boundary problems with globally defined free boundary
conditions is the topic of the paper contributed by G.H.Meyer. As prototype a Stefan
Signorini ablation problem is considered. The numerical algorithm consists of a front
tracking method, based on the method of lines.
In the following paper, H.D.Mittelmann presents a numerical technique for the compu
tation of energy-stability bounds for the thermocapillary convection in a model of the
float-zone crystal growth of high-quality semiconductors in microgravity environments.
The numerical results are compared with model experiments.
A finite element method for the solution of two-phase Stefan problems is proposed in
the contribution of R.H. Nochetto, M.Paolini and C.Verdi. The mesh modification is
adaptive, based on both heuristic considerations and a detailed analysis of the local error
behaviour. Several numerical examples are presented.
The following paper of P.D.Panagiotopoulos deals with the optimal control of processes
governed by hemivariational inequalities. The existence of optimal controls is shown, and
a derivation of the first order necessary conditions of optimality is provided.
The contribution of J.F .Rodrigues is devoted to the study of a two-dimensional stationary
model problem for the continuous casting process. He takes advantage of the Lipschitz
continuity of the free boundary to control the process via a direct observation of the free
boundary curve.
Incompressible inviscid flow problems with moving boundaries are studied in the paper by
J.C.W.Rogers, W.G.Szymczak, A.E.Berger and J.M.Solomon. They present a numerical
method based on a fixed domain formulation using conservation laws and a unilateral
constraint to describe the energy loss in the collision of fluid particles.
The final contribution to this monograph by T.Roubicek brings a discussion of a finite
element technique for the numerical solution of Stefan problems in heterogeneous media,
including an error analysis.
The editors wish to express their gratitude to all contributors to this monograph. We also
thank the director and the staff of the Mathematisches Forschungsinstitut Oberwolfach
whose professional performance helped much to create the pleasant atmosphere in which
the conference took place.
Augsburg and Essen, 1990
Karl-Heinz Hoffmann, Jurgen Sprekels
VII
TABLE OF CONTENTS
ALT, H.W.; PAWL OW , I.: Dynamics of non-isothermal phase separation. . ..... 1
BORNERT, M.; MULLER, I.: Temperature dependence of hysteresis in pseudo-
elasticity. . ....................................... .-............................ 27
DEWYNNE, J.N.; HOWISON, S.D.; OCKENDON, J.R.: The numerical solution
of a continuous casting problem. . ............................................. 36
ELLIOTT, C.M.; ZHENG, S.: Global existence and stability of solutions to the
phase field equations. . ........................................................ 46
GUENTHER, R.B.; CROW, J.A.: Damping of shallow water waves ........... 59
HASLINGER, J.; NEITTAANMAKI, P.: On one identification problem in linear
elasticity. . .................................................................... 66
KAWOHL, B.: Regularity, uniqueness and numerical experiments for a relaxed
optimal design problem. . ..................................................... 85
KENMOCHI, N.: A new proof of the uniqueness of solutions to two-phase Stefan
problems for nonlinear parabolic equations. . ................................. 101
KENMOCHI, N.; KUBO, M.: Periodic stability of flow in partially saturated porous
media. ...................................................................... 127
MEYER, G.H.: Numerical solution of diffusion problems with non-local free bound-
ary conditions. . ............................................................. 153
MITTELMANN, H.D.: Computing stability bounds for thermo capillary convection
in a crystal-growth free boundary problem. . ................................. 165
NOCHETTO, R.H.; PAOLINI, M.; VERDI, C.: Selfadaptive mesh modification for
parabolic FBPs: theory and computation. . .................................. 181
PANAGIOTOPOULOS, P.D.: Optimal control of systems governed by hemivaria-
tional inequalities. Necessary conditions ...................................... 207
RODRIGUES, J.F.: On a steady-state two-phase Stefan problem with extraction.
............................................................................. . 229
ROGERS, J.C.W.; SZYMCZAK, W.G.; BERGER, A.E.; SOLOMON, J.M.: Nu-
merical solution of hydrodynamic free boundary problems. . .................. 241
ROUBICEK, T.: A finite-element approximation of Stefan problems in heteroge-
neous media ................................................................. 267
International Series of 1
Numerical Mathematics, Vol. 95
© 1990 Birkhauser Verlag Basel
DYNAMICS OF NON-ISOTHERMAL PHASE SEPARATION
Hans Wilhelm Alt1 & Irena Pawlow2
ABSTRACT
A mathematical model of non-isothermal phase separation in binary systems is presen
ted. The model, constructed within the Landau-Ginzburg theory of phase transitions,
has the form of a coupled system of evolutionary nonlinear equations that describe mass
diffusion and heat conduction in a quenched system. Existence of weak solutions to the
model is discussed. Numerical results are presented in the case of one space dimension.
1. Introduction
By thermal treatment, a binary system can be thermodynamically destabilized what
may lead to separation of phases. In particular, by cooling, a stable spatially homo
geneous structure can be transformed into an unstable mixture of both components.
Consequently, it undergoes phase transition to a new equilibrium state involving non
homogeneous spatially modulated structure. Pattern formation resulting from phase
separation is observed in metallic alloys, glasses and other amorphous materials, poly
mers and liquid mixtures (cf., [16] ).
A phenomenological theory of isothermal phase separation has been proposed by Cahn
and Hilliard [5 - 8]. Stochastic thermal fluctuations for the Cahn-Hilliard equation have
been treated by Cook [9] and Langer [19].
Mathematical properties of the isothermal Cahn-Hilliard model were recently studied
by Novick-Cohen & Segel [24], von Wahl [26], Elliott & Zheng [14] and Zheng [27]. A
survey of various aspects and generalizations of the Cahn-Hilliard model was given by
Elliott [12]. Besides, the qualitative behaviour of solutions was extensively studied by
numerical simulation, see Elliott [12] and the references cited therein, Elliott & French
[13], Miyazaki et al. [21,22], and Swanger et al. [25] .
The Cahn-Hilliard model does not reflect all physically relevant developments. It assu
mes elimination of all non-isothermal effects and keeps in view only the mass diffusion.
In real processes, changes in concentration are coupled to changes in temperature, in
particular for systems with rapid diffusion time scale (cf., Cahn [6] ). The effect oftem
perature dependent coefficients on the solution of the Cahn-Hilliard equation has been
studied by Huston et al. [18]. External thermal activation can be used to controlling
the phase separation kinetics.
1 Supported by SFB 256, University of Bonn
2 Partially supported by the Research Program RP.1.02 of the Ministry of Education, Warsaw, and by
the SFB 256, University of Bonn
2
In this paper we propose a mathematical model of phase separation that describes
coupled phenomena of mass diffusion and heat conduction in binary systems subject
to thermal activation. The model is constructed within the Landau-Ginzburg theory
of phase transitions and, simultaneously, is based on non-equilibrium thermodynamics
considerations. Conceptually, the model is close to ideas developed by Alt, Hoffmann,
Niezgodka & Sprekels [1], Niezgodka & Sprekels [23] for shape memory alloys and by
Luckhaus & Visintin [20] for phase transitions in multi-component systems.
Phenomenological foundations of the model and the corresponding constitutive relations
are introduced in Section 2. The mathematical model is specified in Section 3. Results
on the existence of solutions are discussed in Section 4. In Section 5 results of numerical
experiments performed in the case of one space dimension are reported.
A more extensive discussion of numerical simulation results for the model is given in
[3]. The existence proof is contained in [4].
There is a correspondence between the process of phase separation in quenched binary
systems and the process of mushy zones formation (with modulated phase structure)
in a pure material by rapid internal heating of a solid (cf., Fife & Gill [15] ). In this
context we refer to the phase-field models for solid-liquid phase transitions in a pure
material and their connection to the Cahn-Hilliard model (cf., Elliott [12], Fife & Gill
[15] and the references therein). The phase-field approach seems to offer an alternative
to the modelling of phase separation processes.
2. Non-isothermal phase separation
2.1. Phenomenology
We consider a binary system with components A and B, occupying a spatial domain
n. Let 9 denote the Kelvin temperature, CA and CB be the local concentrations of
components A and B, scaled so that
CA + CB = 1
=
For simplicity we shall write C CA •
The molar free energy f of the system exhibits a different qualitative behaviour as
a function of C in various ranges of temperature. At high temperatures f is convex,
whereas below a critical temperature ge it assumes a characteristic non-convex double
well form (see Fig. 1a). In the corresponding phase diagram in Fig. 1b the locus of
concentration values Cel, Ce2, defined by the supporting tangent to f , determines the
coexistence curve in the (9, c) - plane, usually referred to as the binodal. The region
above this curve corresponds to a stable single phase while the inner region contains
the states which are thermodynamically unstable. The inflection points CSI' CS2 of f
determine a curve, referred to as the spinodal. This curve separates regions of metastable
and unstable states in the phase diagram.
Suppose that the system initially is in an isothermal equilibrium at temperature 9
0
higher than ge, and has spatially homogeneous composition with the mean value Cm.