Table Of ContentMultifield Problems
ONLINE LIBRARY
Engineering
http://www.springer.de/engine/
Springer-Verlag Berlin Heidelberg GmbH
A. -M. Sandig W. Schiehlen
W.L. Wendland (Eds.)
Multifield Problems
State of the Art
With 93 Figures
Springer
Professor Dr. Professor Dr. Dr. h.c.
Anna-Margarete Sandig Wolfgang L. Wendland
Mathematisches Institut A Mathematisches Institut A
Universitat Stuttgart Universitat Stuttgart
Pfaffenwaldring 57 Pfaffenwaldring 57
70569 Stuttgart 70569 Stuttgart
Germany Germany
E-mail: E-mail:
[email protected] [email protected]
Professor Dr. Dr. h.c.
Werner Schiehlen
Institut B fiir Mathematik
Universitat Stuttgart
Pfaffenwaldring 9
70569 Stuttgart
Germany
E-mail:
[email protected]
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Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Multifield problems: state of the art / S.-M. Sandig ... (ed.).
-Berlin; Heidelberg; New York; Barcelona; Hong
Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000
ISBN 978-3-642-08693-9 ISBN 978-3-662-04015-7 (eBook)
DOI 10.1007/978-3-662-04015-7
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Preface
This book on the state of the art in "Multifield Problems" consists of selected
articles based on a conference on this topic at the University of Stuttgart in
1999. The first two articles are contributions to the general modelling of
multifield problems. S.S. Antman presents the important role of viscoelastic
dissipation in the mathematical modelling of bifurcation analysis of nonlinear
elasticity for large deformations. G.A. Maugin presents the basic theoretical
foundations for the combination of three scales - the microscopic lattice of
crystals, the mesoscopic thermomechanical model and the macroscopic con
tinuum mechanics model for describing the propagation of phase transition
fronts in terms of the Landau-Ginzburg theory and the modelling of nonlinear
waves.
The other contributions are associated with five main areas of multifield
modelling such as two and multiphase flows, the mechanics of materials in
terms of multiscaling, the interaction of solids and fluids,efficient solution
methods of the discrete equations including adaptivity, and the modelling of
contact and fracture.
Within Chapter 1 on two- and multi phase flows we present three articles
on the modelling of the sedimentation of particles: First, a measure-valued
stochastic model which is able to justify macroscopic constitutive hindered
settling models, and second, computer simulations of the sedimentation of
particles in comparison with experiments, which also provide constitutive
laws for the phenomenological theory of sedimentation-consolidation. This
phenomenological theory, which assumes that the material behaviour of the
mixture is modelled by constitutive laws for the hindered settling function
and the effective solid stress, is topic of the third contribution on sedimen
tatin theory. The model arising from this theory is an initial-boundary value
problem of parabolic-hyperbolic type, where a free surface in the solution
separates the parabolic and hyperbolic regions. This free surface separates
the hindered settling zone, where the equation is hyperbolic, from the com
pression zone, where it is parabolic. Another field of applications consists of
multiphase flows through porous media. In this volume, we present a model
which is based on Lattice-Boltzmann simulations and includes hysteresis ef
fects. Here gas particle and gas-water flows are analyzed and simulated. The
fifth contribution is on front tracking for the accurate simulation of interfaces
between different fluids; a very interesting example shows the simulation of
rising bubbles in fluids.
Chapter 2 is devoted to the mechanics of materials where homogenization
and multiscales play a dominant role in the corresponding modelling, dis
cretization and algorithmic implementation of complex inelastic response of
solid materials. The mathematical foundation of these models is currently an
VI Preface
intensive research field. Microstructures form if oscillating sequences infimize
a given non-convex energy functional. Different forms of convexification allow
macroscopic approximations and the microscopic quantities can be removed
from relaxed models. Corresponding finite element approximations including
a priori and a posteriori error bounds allow numerical computations which
simulate, e.g., phase transitions in elastic solids. A very important role in
the transition from micro to macroscales play homogenization techniques.
Here the mathematical analysis as well as numerical homogenization strate
gies allow to model and to simulate numerically the behaviour of strongly
heterogeneous elastic materials. Multigrid discretizations with local finite el
ements and patchwise homogenization provide efficient numerical simulation
techniques for these rather complex materials. For the transition from the
micro to macroscales one needs a reliable analysis of the mixture of the be
haviour in multiphase models for alloys in order to get reliable models on the
macroscopic scale. The two-scale micro-macro approach allows also to model
the behaviour of materials with finite crystal plasticity of the grains for large
strain problems.
Solid-fluid interaction is considered in Chapter 3. Here models for the in
teraction of fluid flow and mechanical deformation mechanics are of central
interest, and we present two fundamentally different approaches: The classi
cal coupling of fluid flow with moving, perhaps elastic and vibrating bodies;
and on the other hand, modelling of interface conditions for the complex ma
terial behaviour of two-phase continua with fluid-solid interaction. For the
first type of problems we present the modelling and numerical computation
of the vibrational response of ships to the fluid forces generated by the ship's
movement in the fluid. Here coupling of finite and boundary elements and
the application of model adaption, homogenization and mesh refinement al
lows the numerical simulation of complete ship vibrations. Also coupling of
existing computational fluid dynamics methods based on the Navier-Stokes
equations and finite element structural dynamics can be used to compute e.g.
the behaviour of a pendulum in a channel flow. Fluid-structure interaction
like fluids sloshing in a tank, wind induced vibrations of a bridge and vibrat
ing off-shore structures are rather complex and therefore complexity has to
be taken into account when modelling the different fields as well as their in
teraction. Here a finite element Lagrangian-Eulerian formulation (ALE) leads
to a three field model with moving mesh and a robust iterative substructur
ing scheme. For the second type of problems we present the derivation of
interface conditions for the sliding between deformable porous media involv
ing viscosity of the interstitial fluid. In a boundary layer, the viscous drag
competes with the resistive drag between solid and fluid; a set of boundary
conditions like solid to solid frictional fraction interaction is derived. Local
concentration of plastic strains in frictional geomaterials yields the develop
ment of sheerbands which is simulated with a time and space adaptive finite
element method. The complex material behaviour of viscoelastic two-phase
Preface VII
models with fluid-solid interaction and intrinsic skeleton viscoelasticity leads
to a macroscopic theory of porous media with an appropriate viscoelastic
constitutive law. Here finite elements can be used for the simulation of flow
dependent consolidation processes.
Chapter 4 is devoted to efficient solvers and adaptivity. Due to the com
plexity of multifield problems, the construction of effective solution algo
rithms for the discretized equations is of extraordinary importance. Here we
present efficient finite and boundary element methods. For multifield prob
lems, domain decomposition methods provide a natural tool for solving com
plex problems and also for the application of hierarchical finite element anal
ysis. For geometrical domain decomposition involving non-matching grids on
the interfaces between the individual subdomains, the mortar methods based
on an additional Lagrangian field on the interface boundaries provide an
efficient method for relaxed interface linkage of adjacent subdomains. Here
such methods are applied to the design of high power electrical devices as,
e.g., in high-speed train engines. The corresponding electrothermal mechan
ical coupling with Joule selfheating is modelled by a combination of finite
elements, mortars on the interfaces and curl-conforming edge-elements for
the Maxwell equations. The corresponding mixed saddlepoint equations then
are solved with an efficient preconditioned iterative solver. A posteriori error
estimators drive the adapt ion of the grids. Geometric domain decomposition
also underlies the finite element tearing and interconnecting method for solv
ing elasticity problems for compressible elastic materials; and a very efficient
preconditioned conjugate gradient method is obtained by using approximate
Neumann sub domain solvers and Dirichlet preconditioners. Parallel finite el
ement multigrid methods can be used to perform efficient elastoplastic sim
ulations where the radial return algorithm is applied to viscoplasticity with
nonlinear isotropic and kinematic hardening. Boundary element methods can
very efficiently be used for nonlocal equations such as in the modelling of ra
diation heat transfer applied to the modelling of industrial exhaust pipes and
catalytic converters which is needed in the automobile industry. Here a degen
erate kernel approximation of the boundary integral operators in combination
with adaptive compression reduces the originally fully populated matrices to a
sparse representation in terms of low-rank matrices which allow an extremely
fast matrix times vector multiplication. In combination with preconditioned
conjugate gradient methods, boundary integral equations in this form can be
solved with almost optimal efficiency. For the boundary integral equations,
efficient and reliable error estimators are still an important task. Based on
an appropriate approximation of the classical Neumann series for solving a
Fredholm integration of the second kind for the error approximately, one ob
tains efficient residual-based error estimators that drive adaptive boundary
element methods.
In Chapter 5 we consider contact and fracture. A permanent challenge
in structural mechanics is the design of engineering multistructures in the
VIII Preface
presence of stress singularities, and in contact and fracture mechanics. For
dynamic problems involving contact and Coulomb friction, sensitivity and
optimal control is formulated in terms of the adjoint dynamical system. The
dynamic evolution of contact problems for viscoelastic bodies without and
with heat production from friction can be formulated in terms of an evo
lution based on variational inequalities. Here new existence and regularity
results are obtained. The longitudinal wave propagation in conical rods can
be formulated as a wave equation on a one-dimensional interval. D' Alembert 's
solution formula yields with the reflection at the boundaries recurrence rela
tions and and explicit solution algorithm which can be used with computer
algebra for computing the wave's propagation. The acoustic radiation of noise
generated by the tire-road interaction can numerically be simulated by us
ing e sequential finite element-boundary element formulation based on an
Eulerian-Lagrangian model. Quasistatic contact problems with elastic plates
can be treated by an augmented Lagrangian method. For these non-convex
problems existence results and an iterative solution algorithm is presented.
For mildy nonlinear elastic deformation of composites, where stress singu
larities develop at crossing points of interfaces between different materials,
a regularity analysis based on the Mellin technique yields explicit asymp
totic characterizations of the stress fields which in this case behave similar to
corresponding solutions of linearized problems.
As these examples show, multifield problems need careful investigations
with respect to the efficiency of the individual models and solution algorithms.
The successful performance requires a close multi-disciplinary interaction of
engineers from applied mechanics with mathematicians who work on the anal
ysis and numerical solution of partial differential equations; hence, scientists
from different disciplines have created this work on multifield problems.
The authors and editors express their sincere gratitude to the German
Research Foundation DFG which supported this activity. Special thanks are
due to Dr. Werner Kolbe who did all the technical work in preparing this final
form. We also thank the Springer-Verlag for its patience and the opportunity
to publish this book.
Stuttgart, March 2000 Wolfgang Wendland
Table of Contents
The Many Roles of Viscosity in Solid Mechanics
Stuart S. Antman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Introduction................................................. 1
2. A two-dimensional shearing problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3. Self-sustained oscillations of a viscoelastic layer . . . . . . . . . . . . . . . . . . 4
4. Longitudinal motions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5. Longitudinal motions of incompressible rods. . . . . . . . . . . . . . . . . . . . . 6
6. The motion of rods in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7. Shocks and viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
References ..................................................... 10
Multiscale Approach to a Basic Problem of Materials
Mechanics (Propagation of Phase-Transition Fronts)
Gerard A. Maugin .............................................. 11
1. Introduction................................................. 11
2. Microscopic condensed-matter-physics approach: solitonics ... . . . .. 12
3. Macroscopic engineering approach: singular surface and thermody-
namic criterion ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
4. Mesoscopic applied-mathematics approach: structured front. . . . . .. 16
5. Theoretical-physics approach: quasi-particle and transient motion.. 16
6. Numerics: from finite-differences to continuous cellular automata. .. 17
6.1 Finite-Difference Method ................................. 17
6.2 Finite-Element Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18
6.3 Finite-Volume Method and cellular automaton. . . . . . . . . . . . . .. 18
7. Conclusion.................................................. 19
References ..................................................... 21
