Table Of ContentISNM99:
Intemational Series of Numerical Mathematics
Intemationale Schriftenreihe zur Numerischen Mathematik
Serie Intemationale d'Analyse Numerique
Vol. 99
Edited by
K.-H. HotTmann, Augsburg; H. D. Mittelmann, Tempe;
J. Todd, Pasadena
Springer Basel AG
Numerical Methods for
Free Boundary Problems
Proceedings of a Conference held at the
Department of Mathematics, University of
Jyvaskyla, Finland, July 23-27, 1990
Edited by
P. Neittaanmaki
1991
Springer Basel AG
Editor
P. Neittaanmiiki
University of Jyviiskylii
Department of Mathematics
P.O. Box 35
SF-40351 Jyviiskylii
Finland
Deutsche Bib60thek Cataloging-in-Pub6cation Data
Numerical methods for free boundary problems: proceeding of
a conference held at the Department of Mathematics,
University of Jyviiskylii, Finland, July 23-27, 1990/ ed. by
P. Neittaanmiiki.
(International series of numerical mathematics ; Vol. 99)
ISBN 978-3-0348-5717-8 ISBN 978-3-0348-5715-4 (eBook)
DOI 10.1007/978-3-0348-5715-4
NE: Neittaanmiiki, Pekka [Hrsg.]; Matematiikan Laitos <Jyviiskylii>; 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
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made for other than private use a fee is payable to »Verwertungsgesellschaft Wort«,
Munich.
© 1991 Springer Basel AG
Originally published by Birkhauser Verlag, Basel in 1991
Softcover reprint of the hardcover 1st edition 1991
Printed from the authors' camera-ready manuscripts on acid-free paper in Germany
ISBN 978-3-0348-5717-8
v
Preface
About 80 participants from 16 countries attended the Conference on Numerical Methods
for Free Boundary Problems, held at the University of Jyviiskylii, Finland, July 23-27,
1990. The main purpose of this conference was to provide up-to-date information on
important directions of research in the field of free boundary problems and their numerical
solutions.
The contributions contained in this volume cover the lectures given in the conference.
The invited lectures were given by H.W. Alt, V. Barbu, K-H. Hoffmann, H. Mittelmann
and V. Rivkind. In his lecture H.W. Alt considered a mathematical model and existence
theory for non-isothermal phase separations in binary systems. The lecture of V. Barbu
was on the approximate solvability of the inverse one phase Stefan problem. K-H. Hoff
mann gave an up-to-date survey of several directions in free boundary problems and listed
several applications, but the material of his lecture is not included in this proceedings.
H.D. Mittelmann handled the stability of thermo capillary convection in float-zone crystal
growth. V. Rivkind considered numerical methods for solving coupled Navier-Stokes and
Stefan equations.
Besides of those invited lectures mentioned above there were 37 contributed papers
presented. We shall briefly outline the topics of the contributed papers:
Stefan like problems. Modelling, existence and uniqueness.
In the contribution of I.G. Gotz a nonuniqueness theorem for a self-similar solution in a
model for phase transitions in binary mixtures is proved. Moreover, the existence of the
generalized solution and the stability criterion are studied. In a joint paper of I. Gotz and
B.B. Zaltzman the disappearance of the mushy region in the multidimensional Stefan prob
lem is discussed. N.V. Shemetov considers the Stefan problem with the time relaxation of
Cattaneo, proves the existence of the weak solution of the relaxed Cattaneo-Stefan problem
and shows that the weak solution of this problem converges to the weak solution of the
Stefan problem. In the contribution of A.M. Maximov and G.G. Tsypkin a mathematical
model of phase transition in frozen soil containing unfrozen water is proposed.
Modelling and numerical methods.
Z. Chen presents a numerical scheme to solve the continuous casting problem. The scheme
is based on the nonequilibrium phase relaxation to smooth the enthalpy and the character
istic finite element method to treat the convection term. D.A. Kurtze presents a fixed grid
finite element method for solidification. The method is intended for problems in which su
percooling (and possibly superheating) occurs, and in which the effects of nonequilibrium
interface kinetics and surface tension at the front are important. R.H. Nochetto, M. Paolini
and C. Verdi study the mesh adaptation for the two phase Stefan problem in 2D domains
and consider the efficient implementation of an adaptive FEM for parabolic free boundary
problems. A. Bossavit considers a two-phase free boundary problem in magnetostatics and
introduces special "mixed" finite elements for its numerical solution. K-H. Hoffmann and
K Liu treat a thermodynamical model of ferromagnetism and its numerical simulation.
Concerning front-tracking there were two contributions. G. Bellettini, M. Paolini and
C. Verdi consider front-tracking and variational methods to approximate interfaces with
VI
prescribed mean cUlyature and G.H. Meyer handles front-tracking of free boundaries with
curvature terms and among others presents numerical results of front-tracking for 3D Stefan
problem with Gibbs-Thomson interface condition.
A. Lapin proposes domain decomposition method to realize implicit mesh schemes for
a class of singular parabolic equations including, among others, one-phase and two-phase
Stefan problems. S. Jensen, E. Laitinen, P. Neittaanmaki and T. Seidman handle a free
boundary problem modelling the growth/dissolution of a crystal and study the question
of stability of a radially symmetric solution in two dimensions. The contribution of J.
Steinbach is devoted to the numerical solutions of variational inequalities which are used
to model the injection moulding process of a plastic melt into a mould cavity. S. Mazzullo,
M. Paolini and C. Verdi give an analysis of thermal effects during the cementation of
femoral prosthesis and present several numerical simulations to illustrate critical aspects
of this implant.
Optimal control, optimal shape design, identification, applications.
V. Arnautu considers approximation of the inverse one-phase Stefan problem in terms of
an optimal control problem.
J. Haslinger and P.D. Panagiotopoulos study the approximation of the optimal control
problem coverned by hemivariational inequalities. A.M. Khludnev considers the conver
gence of the solutions into problem of optimal control of the fourth order variational
inequality (contact problem). T. Miinnikk6 and M.M. Makela apply a nonsmooth op
timization technique (Proximal Bundle method) to solve control problems arising from
the continuous casting process. M. Trigub considers numerical solutions of optimal con
trol problems for a dynamical system described by ordinary differential equations. The
proposed approach combines analytical expansions of nonlinear functions with error evalu
ation by Krotov's formula. V.N. Phat studies the control problem of nonlinear dynamical
systems with phase constraints and gives some controllability criterion for discrete time
systems. A. Kadoya and N. Kenmochi consider problems of shape optimization in which
the driving actions are performed by geometry of the domain. V. Danek and R. Makinen
study the feasibility of finite element and mathematical programming methods for find
ing an optimal shape for a symmetric airfoil in the case of transonic fiow. A. Mysklinski
proposes a mixed variational approach for shape optimization in contact problem with
prescribed friction. X.-C. Tai and P. Neittaanmaki present a linearization method con
cerning a nonlinear distributed parameter identification problem in which one recovers
the nonlinear parameters from the identified linear parameters. Ch. Grossmann considers
the numerical methods for solving variational inequality and proposes an iteration pro
cess for solving the discrete problems (obtained by mixed finite element method) and also
investigates the selection rules for penalty parameters.
Dam, fluid fiow and free boundary problems
N. Van Luoc deals with a free boundary problem of systems for earth dams with unknown
intermediate water levels and separated variable filtration coefficients. C.A. Papadopoulos,
J.M. Sloss and J.C. Bruch Jr. apply a domain decomposition to a free surface seepage
problem. E.A. Karabut gives a semi-analytical investigation of unsteady free boundary
VII
flows and presents the solution in the form of a power series in time, the coefficients for
which have been computed. V.M. Teshukov considers long wave approximation for vortex
free boundary flows. The work of B.E. Protopopov is devoted to numerical investigation
of the process of soliton generation by a moving disturbance.
P.J. Shopov presents a numerical method for free surface hydro dynamical problems.
With the contribution of C. Cuvelier and R.M.S.M. Schulkes the authors consider numer
ical methods for capillary free boundaries governed by the Navier-Stokes equations and
in the contribution of R.M.S.M. Schulkes and C. Cuvelier eigenfrequencies of a viscous,
imcompressible fluid with a capillary free boundary are studied.
The organizers are grateful to the participants, and especially to the speakers who made
this meeting an important scientific event. We are indebted to the Administration of the
Faculty of Mathematics, to the younger members of the numerical analysis group and to
the secretatiat of the Department of Mathematics whose support was an indispensable
contribution to the success of this conference. We also acknowledge the financial support
for this meeting given by Academy of Finland, Research Council of Technology and Rolf
Nevanlinna Institute.
Jyviiskylii, April 1991 P. Neittaanmiiki
IX
CONTENTS
List of contributors ....................................... XII
Invited lectures
H. W. AU and L Pawlow
A mathematical model and an existence theory for non-isothermal phase separation
V.Barbu
The approximate solvability of inverse one-phase Stefan problem 33
H.D. Mittelmann, C.C. Law, D.F. Jankowski and G.P. Neitzel
Stability of thennocapillary convection in float-zone crystal growth 45
V. Rivkind
Numerical solution of coupled Navier-Stokes and Stefan equations 57
Contributed papers
V. Arnautu
On approximation of the inverse one-phase Stefan problem 69
G. Bellettini, M. Paolini and C. Verdi
Front-tracking and variational methods to approximate interfaces with
prescribed mean curvature ...................... . 83
A. Bossavit
Mixed elements and two-phase free-boundary problem in magnetostatics 93
z.
Chen
Numerical solutions of two-phase continuous casting problem ................ 103
C. Cuvelier and R.M.S.M. Schulkes
Numerical analysis of capillary free boundaries governed by the
Navier-Stokes equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
V. Danek and R. Makinen
Optimal design for transonic flows .............................. 129
x
Ch. Grossmann
Penalties and mixed finite element analysis for variational inequalities ......... " 137
I.G. Gotz
The nonuniqueness in a model for phase transitions in binary alloys ............. 147
I.G. Gotz and B.B. Zaltzman
On the behaviour of mushy region in a Stefan problem .................... 155
J. Haslinger and P.D. Panagiotopoulos
Optimal control of hemivariational inequalities. Approximation results ........... 165
K.-H. Hoffmann and Liu Kunkun
A thermodynamical model of ferromagnetism and its numerical simulation ......... 175
S. Jensen, E. Laitinen, P. Neittaanmiiki and T.I. Seidman
Computational stability of an initially radial solution of a growth/dissolution
problem in a nonradial implementation ............................ 191
A. Kadoya and N. Kenmochi
Shape optimization for multi-phase Stefan problem ..................... 203
E.A. Karabut
Semi-analytical investigation of unsteady free-boundary flows ................ 215
A.M. Khludnev
On the convergence of solutions in problems of optimal control of
variational inequality of the fourth-order ........................... 225
D.A. Kurtze
A fixed-grid finite element method for solidification ..................... 235
A.V. Lapin
Domain decomposition method to realize implicit mesh schemes for some
singular parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
A.M. Maximov and G.G. Tsypkin
A mathematical model of a two-phase region in thawing soil ................ 251
S. Mazzullo, M. Paolini and C. Verdi
An axisymmetric analysis of thermal effects during cementation of femoral prostheses ... 259
G.H. Meyer
Front tracking of free boundaries with curvature terms .................... 271
