Multifield Problems: State of the Art PDF
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Multifield 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] Library of Congress Cataloging-in-Publication Data applied for. 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 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 2000 Originally published by Springer-Verlag Berlin Heidelberg New York in 2000. Softcover reprint ofthe hardcover I st edition 2000 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro tective laws and regulations and therefore free for general use. Typesetting: Camera-ready copy by the authors/editors Cover design: de'blik, Berlin Printed on acid-free paper SPIN 10768587 62/3144/tr 5 4 3 2 1 0 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
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