Lecture Notes in Applied and Computational Mechanics Volume 12 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. Dr.-Ing. Peter Wriggers Springer-Verlag Berlin Heidelberg GmbH ONLINE LIBRARY http://www.springer.de/engine/ Analysis and Simulation of Multifield Problems Wolfgang Wendland Messoud Efendiev (Eds.) , Springer Professor WOLFGANG WENDLAND Professor MESSOUD EFENDIEV Universitat Stuttgart Lehrstuhl fur Angewandte Mathematik Institut fur Angewandte Analysis und Numerische Simulation pfaffenwaldring 57 D-70569 Stuttgart Germany e-mail: [email protected] e-mail: [email protected] With 131 Figures Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche BibJiothek Die Deutsche Bibliothek lists this publication in the Deutsche Na tionalbibliografiej detailed bibliographic data is available in the Internet at <http://dnb.dd ISBN 978-3-642-05633-8 ISBN 978-3-540-36527-3 (eBook) DOI 10.1007/978-3-540-36527-3 This work is subject to copyright. 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Cover design: design & production GmbH, Heidelberg Printed on acid-free paper 62/3020Rw -5 4 3 2 1 0 Preface The analysis and simulation of multifield problems became recently one of the most actual and vivid areas of research. Of course, it is clear that complex technical and physical phenomena modelled by the interaction of different fields require a corresponding complex mathematical modelling and numeri cal simulation. Although the corresponding subproblems often are quite well understood separately, however, their interaction and coupling creates not only new difficulties but also a completely new level and quality of the inter acting coupled field problems. In order to tackle multifield problems, the Ger man Research Foundation DFG started in 1995 the Collaborative Research Center on Multifield Problems (Sonderforschungsbereich Mehrfeldprobleme, SFB 404) at the University of Stuttgart where engineers from applied me chanics and mathematicians work together in multidisciplinary interaction on multifield problems from continuum mechanics. Whereas during the first years the models of multifield interaction involved mostly the coupling of different fields within their common space-time regions of definition as well as transmission problems connecting different field models, the focus moved recently more to the interactions between different spacial, time and state scales. The International Conference on Multifield Problems (April 8 - 10, 2002) at the University of Stuttgart was organized by the members of the Collaborative Research Center. (The program of the conference can be found in the appendix.) The organizers of the conference are very happy that three of the keynote speakers are willing to publish their contributions in this volume. (The lectu res of the other five keynote speakers have already been published elsewhere.) In addition, we have selected 38 further contributions to our conference and this whole collection will present a rather colourful picture of corresponding ongoing research. Keynote speakers An inherent difficulty in the treatment of multifield problems is the fact that the subproblems usually are well understood and even corresponding robust and reliable software is available, whereas the coupling of these subproblems in general cannot be done in a naive manner. Hermann Matthies and Jan Steindorf present a fundamental approach to this problem by viewing the strong coupling of these subproblems as a block GauE-Seidel procedure. Since without particular care such a procedure will not converge in general, they give a modification in terms of a block Newton formulation which then allows to use the available subproblem solvers appropriately. As an example fluid structure coupling is treated. Many of current important applications of multifield problems involve multiphase flows either in porous media or in dispersed particle fluid mixtu res. Donald Drew shows how to develop a rigorous mechanical and mathe- VI Preface matical model for two-phase flows of particle or bubble suspensions involving averaging based on the conservations of mass and momentum. He shows how the additional constitutive laws can be obtained by an appropriate analysis of the microscale behaviour of ensembles and obtains the dispersion tensor, an appropriate diffusion model and the Reynolds stress tensor which allow to define the constitutive laws needed for closing the system of equations. Mary Wheeler, Shuyu Sun, Owen Eslinger and Beatrice Riviere present the complex model and corresponding numerical simulation of reactive flows through porous media. These multifield problems are very complicated since they involve not only multi phase flows but also chemical reactions through time, space and phase regions where the corresponding physical and time scales by drastically different magnitudes whereas the physical properties of the porous materials admit strongly variable permeabilities. The nume rical method used is based on discontinuous Galerkin approximation which needs to be locally conservative and to avoid high numerical diffusion. This contribution shows that it is nowadays possible to simulate the transport of radionucleids, chemicals and biological species in order to control water pollution in large geographical areas. Part I - Material Modelling and Ml1ltiscale Problems is devoted to the mathematical and mechanical modelling of composite ma terials and materials showing nonlinear behaviour. Here we find essentially three different types of models: Variational models based on nonconvex energy functionals describing phase transitions as in the contribution by Marcel Arndt, the abstract approach to rate-independent processes based on two energy functionals as presented by Messoud Efendiev and the modelling of elastoplasticity without and with hardening via energy minimization as in the article by Klaus Hackl, Alexander Mielke and Dirk Mittenhuber. Tomas Roubicek presents the functional analysis with measure-valued solutions to relaxed convexification of the nonconvex energy functional associated with laminated composites. The second type of problems consists of two-scale mo dels (which are related to the first type). The corresponding contributions are due to Claus Bayreuther and Cristian Miehe who treat the homogeniza tion by employing appropriate transfer operators to a multigrid formulation to couple heterogeneous linear elastic microstructures with the macroscopic nonlinear behaviour. Romana Piat and Eckart Schnack use an experimentally determined stochastic distributional function describing the geometrical pro perties of mesoscopic ellipsoids and cracks within pyrolytic carbon composites in order to obtain macroscopic material laws by homogenization. The third type of material modelling involves interior variables. Klaus Hackl, Alexander Mielke and Dirk Mittenhuber present a strain-based mathematical model of materials without and with hardening where the energy functional does not need to be smooth in the state space, and they are able to model elastopla sticity and the flow rules for single crystal deformation with switching slip systems. Sergey Lurie, Petr Belov and Dmitrij Volkov-Bogorodsky use a mul- Preface VII tiscale variational formulation and an extended Cosserat model for two-scale formulations of materials with a periodic distribution of inclusions having simple geometry. Slav Dimitrov and Eckart Schnack present a two-scale da mage model based on additional interior variables in order to simulate the stress-based evolution of materials containing a big number of cracks. Part II - Numerical Methods In this section, the contributions deal with flow problems, elliptic and free boundary value problems and with inverse problems. Raimund Burger and Kenneth Karlsen present a new version of the Enquist-Osher finite diffe rence method which now can be applied to nonstandard conservation laws with discontinuous flux functions. Cristian Co clici , Ivan Sofronov and Wolf gang Wendland present a nonlocal nonreflecting boundary condition based on the discretization of the Dirichlet to Neumann map subject to a farfield gi ven by the solution to the linearized compressible Euler equations. Christian Wieners defines a complete family of p2-pl Taylor-Hood finite elements on tetrahedrons, pyramids, prisms and hexahedrons which now can be used wit hin three-dimensional saddlepoint discretizations involving adaptively refined 3D-meshes. He shows that for his elements the inf-sup stability conditions are satisfied. Andrzej Nowakowski, Wanwilai Kraipech and Tom Dyakow sky present numerical experiments for the three-dimensional simulation of Navier-Stokes flows in hydrocyclones and show that the p2-pO Taylor-Hood elements will provide robust approximations whereas the p2-pl elements fail. Iuliu Sorin Pop shows how transient Darcy flows can be approximated by the use of a lumped upwind box scheme applied to an Euler implicit time di scretization of the Krichhoff transform and regularized Darcy flow. He shows asymptotic convergence of the method and presents corresponding numerical simulations. Elliptic problems are treated by Gunther Of and Olaf Stein bach who use a multipole method for solving stabilized boundary element Galerkin equations with the hypersingular boundary integral operator. The corresponding discrete systems are solved with an appropriately preconditio ned conjugate gradient iteration. This method can be used to solve Neumann problems which will be one of the basic blocks in 3D domain decomposition. Victor Ryaben'kii is using finite difference potentials which can be seen as to be a discrete version of potential methods. He applies this approach to the solution of the coupling of different elliptic equations which again is nee ded in domain decomposition methods. A highly nonlinear inverse problem is treated by Bernd Scholz and Wolfgang Ehlers who use a least-squares fit of the solution to the micropolar model of a Cosserat medium with experi mental biaxial tests in order to identify the physical constants characterizing the Cosserat medium. Tom-Alexander Langhoff and Eckart Schnack simulate the growth of the solid pyrolytic carbon layer with a free boundary in pores by solving corresponding reaction diffusion equations. Part III - Solid Mechanics In this section one finds contributions to contact and crack problems and VIII Preface to the asymptotic singularity behaviour of stress fields in the vicinity of corners and edges. One contribution is devoted to the material behaviour of fine grained structures on the microscale. Jifi Jarusek and Christo f Eck give a survey on friction laws under which the incremental and quasista tic formulations of contact problems with generalized Coulomb friction as well as the corresponding transient formulation with viscoelastic material behaviour allow a mathematically rigorous existence proof. Gwideon Szefer and D. Kedzior show results for contact with materials that have negative Poissons ratio where the corresponding stress field at the contact zone is significantly different from fields observed in traditional materials. Bin Hu, Werner Schiehlen and Peter Eberhard show that the simulation of dynamic elastic contact problems needs three different time scales so that the ela stodynamic boundary approach for the wave propagation can be combined with flexible multiple body systems for structural vibrations and with the ri gid multibody systems for the rigid body motion. Victor Kovtunenko shows how the quasistatic modelling of crack propagation including nonpenetra tion within the crack can be formulated in terms of variational inequalities based on the Griffith criterion which allow the simulation of the growing crack geometry. Ch. Zhang presents transient dynamic crack analysis in pie zoelectric materials by employing a time-domain boundary integral equation approach where the time integration is based on Lubich's convolution qua drature. Atanas Dimitrov and Eckart Schnack use Kondratiev's singularity analysis for three-dimensional structures and present a complete algorithm to compute stress concentrations and stress intensity distributions at corners and edges including those that appear at surface penetrating cracks. Dorothee Knees is able to characterize the corresponding stress intensity distributions for linear elastic structures composed of polyhedral sub domains occupied by different linear elastic materials, by some geometric ordering so that global bounds for the corrresponding stress intensity factors can be found which are basic for the preconditioning of corresponding finite element computati ons. Alain Miranville and Messoud Efendiev extend the Cahn-Hilliard theory to fine grained structures and obtain new results for corresponding material properties. Part IV - Porous Media and Fluid - Structure Interaction In this section we collect the contributions to multiphase flows in porous me dia and polydisperse particle flows as well as the coupling of vibrations and wave phenomena. Mal:gorzata Peszynska is using a heterogeneous domain de composition in order to couple reactive transport models of two-phase and three-phase flows with applications to crude oil exploration. Hartmut Jakobs, Andreas Bielinski, Holger Class and Rainer Helmig analyze the influence of interface conditions on the performance of numerical algorithms for the simu lation of multiphase flows through porous media with heterogeneities. Wolf gang Ehlers, Ayhan Acartiirk and Bernd Markert present a finite element simulation of an electrochemical multiphase formulation of electrolytes and Preface IX fluids in porous media modelling the behaviour of swelling tissues. Martin Schanz and Dobromil Pryl use a time-dependent boundary integral equation method to compare two different models of wave propagation phenomena in poro-elastic media. Stefan Berres, Raimund Burger and Elmer Tory inve stigate the deterministic macroscale mathematical modelling of polydisperse sedimentation by using the theory of mixtures; whereas Elmer Tory, R.A. Ford and M. Bargiel show how to use steady state parameters of a Markov model for the stochastic n particle simulation of the same kind of polydisperse fluid-particle mixtures. Dieter DinkIer, Bjorn Hubner and Elmar Walhorn use a discontinuous Galerkin method to simulate the fluid-structure interaction between buildings and strong airflows. Matthias Fischer and Lothar Gaul use a finite element - boundary element coupling procedure to simulate the cou pling between a vibrating structure with an exterior acoustic field by using mortar elements on the coupling surfaces. Harmen Schippers, Guus Vos and Hans van Tongeren investigate the influence of the structure vibrations gene rated by aerodynamic loads on the vibrations of antenna appertures and the corresponding perturbations of the electromagnetic fields. Thomas Schwartz kopff and Claus-Dieter Munz use a nonoverlapping domain decomposition with different physical models, namely coupling of the nonlinear Euler equati ons with the linearized Euler equations for modelling acoustic noise generated by flows with discontinuities as shocks. Decan Ivanovic and Viktor Saljinikov use perturbation analysis to model flows round porous airfoils where injection or ejection influences the separation of the flow and will therefore be useful for the control of laminar-turbulent transition. Bosko Rasuo investigates the wall corrections needed to modify wind tunnel experiments appropriately due to the perforated tunnel walls by using two-dimensional stationary flow mo dels and corresponding perturbation analysis. Yuli Lifshitz and David Degani show how a nonsymmetric three-dimensional flow about a conical obstacle can be modelled by an appropriate decomposition into two simplified two dimensional flows which then are coupled. This allows a significant reduction of the computational effort for the corresponding flow simulations. The authors and editors express their sincere gratitude to the German Research Foundation which supported this activity significantly and the con ference participants from the East. Special thanks are due to Jan Jung who did the technical work in preparing the final form of this book and to Mr. Richardson and his team for all the organizational work in connection with our conference. We also thank the Springer-Verlag for its patience and excel lent outfit of this volume. Stuttgart, January 2003 Messoud Efendiev and Wolfgang Wendland