# Numerical Treatment of Coupled Systems

### January 20th to 22nd, 1995.

->[GAMM-Homepage]

### Reference

W. Hackbusch, G. Wittum (Eds.): Numerical Treatment of Coupled Systems.
Proceedings of the 11th GAMM-Seminar Kiel, January 20 to 22, 1995.
Notes on Numerical Fluid Mechanics Vol. 51. Vieweg-Verlag, Braunschweig,

### Foreword

The GAMM Committee for ''Efficient Numerical Methods for Partial Differential Equations'' organizes seminars and workshops on subjects concerning the algorithmic treatment of partial differential equations. The topics are discretisation methods like the finite element and the boundary element method for various type of applications in structural and fluid mechanics. Particular attention is devoted to the advanced solution methods.

The series of such seminars was continued in 1995, January 20-22, with the 11th Kiel-Seminar on the special topic

Numerical Treatment of Coupled Systems

at the Christian-Albrechts-University of Kiel. The seminar was attended by 100 scientist from 9 countries. 23 lectures were given, including two survey lectures.

Different kinds of couplings are considered in this volume. The coupling of different components may occur in the physical model. On the other hand, a coupling of subsystems can be generated by the numerical solution technique. General examples of the latter kind are the domain decomposition (see p. 128) or subspace decomposition (p. 117). The local defect correction method couples different discretizations of the same problem in order to improve the results, although the basic linear system to be solved remains unchanged (p. 47). In general, the aim of the numerical coupling is to make use of (efficient) subsystem solvers (p. 1). The combination of different discretization techniques is mentioned on page 59.

Various examples of coupling different physical components are discussed: the interaction of solid and fluid (p. 96, p. 175, and p. 198), the microelectronic systems (p. 11 and p. 86), and the coupled modelling in groundwater flow (p. 106 and p. 139).

A topic where the coupling may be of physical and numerical nature is the finite element boundary element coupling (FEM-BEM). This very interesting problem is discussed in four contributions (p. 25, p. 37, p. 73, p. 151).

Kiel, May 1995,
W. Hackbusch, G. Wittum

### Contents

• ST. ARTLICH, W. MACKENS:
Newton-coupling of fixed point iterations, pp. 1
• H.-J. BUNGARTZ, ST. SCHULTE:
Coupled problems in microsystem technology, pp. 11
• C. CARSTENSEN, D. ZARRABI:
An a posteriori error estimate for the unsymmetric coupling of FEM and BEM, p. 25
• M. FEISTAUER, G.C. HSIAO, R.E. KLEINMAN:
On numerical treatment of coupled BEM and FEM for nonlinear exterior problems, pp. 37
• P.J.J. FERKET:
Coupling of a global coarse discretization and local fine discretizations, pp. 47
• B. HEINRICH:
The Fourier-finite-element method for elliptic problems in axisymmetric domains, pp. 59
• B. HEISE:
Parallel solvers for coupled FEM-BEM equations with applications to non-linear
magnetic field problems, pp. 73
• R. HIPTMAIR, R.H.W. HOPPE, B. WOHLMUTH:
Coupling problems in microelectronic device simulation, pp. 86
• F. IHLENBURG, CH. MAKRIDAKIS:
Error estimates of Galerkin FEM for a system of coupled Helmholtz equations
in one dimension, pp. 96
• H. KASPER, G. KOSAKOWSKI, T. TANIGUCHI, W. ZIELKE:
Coupled geometric modelling for the analysis of groundwater flow
and transport in fractured rock, pp. 106
• E. KATZER:
Subspace decomposition methods for solving the Euler equations, pp. 117
• A. KLAR:
Domain decomposition schemes and coupling conditions for kinetic
and hydrodynamic equations, pp. 128
• O. KOLDITZ, R. RATKE, W. ZIELKE, H.-J. DIERSCH:
Coupled physical modelling for the analysis of groundwater systems, pp. 139
• H. SCHIPPERS, J.A. WENSING:
Coupling of boundary and finite elements in aeroacoustic calculations, pp. 151
• W. SCHMID, F. WAGNER:
Numerical solution of the neutron diffusion equation - adaptive concepts
in time and space, pp. 163
• U. SPECHT, CH. DICKOPP, J. BALLMANN:
A numerical scheme for stress waves at a fluid-solid interface, pp. 175
• C. VUIK, A. SEGAL:
Solution of the coupled Navier-Stokes equations, pp. 186
• D. WEISS, G. LUBE:
Numerical simulation of temperature distribution and seam forming
in narrow gap welding, pp. 198

### Collection of Abstracts

#### ST. ARTLICH, W. MACKENS: Newton-coupling of fixed point iterations

To solve a coupled system of two PDEs it may be intended not to use the Newton-Raphson-method, for example due to the non-sparsity of the Jacobian of the entire system or because there exist solvers for subsystems.

For this type of problems we present an iterative Newton type method which requires only a finite number of iterations for the single equations in each step. The algorithm is based on a formal Block Gauss elimination of the full Newton system and the solution of the resulting Schur complement equation by a Bi-CGSTAB-iteration. No computation of the Jacobian of the whole system is necessary. In contrast to alternative approaches, the two sets of equations resulting from the discretization of the two PDEs may be of the same size.

The above method had to be developed to solve a system of PDEs describing the behaviour of a chemical reactor for the combustion of coal. This model consists of an enthalpy balance of convection-diffusion-type and two mass balances, one of them of reaction-diffusion-type, the other one an ODE. For the latter there exists an analytical solution depending on the unknowns of the two other balances.

The numerical results presented deal with this combustion model.

#### H.-J. BUNGARTZ, ST. SCHULTE:Coupled problems in microsystem technology

In microsystem technology, the numerical simulation of coupled problems is one of the principal challenges. There are three main reasons for the fact that, here, the coupling of different physical effects (structural dynamics, fluid dynamics, heat transfer, or electromagnetics, e.g.) is more frequently encountered than in the macro world: First, aspects of scaling often lead to a dominance of surface effects on volume dependent effects. Second, especially in sensors a lot of different physical phenomena are used for signal conversion, and, finally, in some microsystems different physical effects have an influence on each other.

We present a classification of the most important couplings in the microsystem world, and we give a survey on existing solution techniques with emphasis on methods based on the so-called partitioned solution. Here, there is no joint model, neither continuous nor discrete, but the coupled problem is solved by an outer iteration realizing the coupling and by arbitrary inner solution processes for each single problem. The coupling is done via changed boundary conditions, geometries, or parameters after each step of iteration. This approach seems to be advantageous, since its modularity allows the use of existing and efficient codes for each sub-problem. Therefore, only the outer iteration has to be organized with some kind of interface for the coupling. Furthermore, this technique seems to be perfectly suited for parallelization, especially for the use of (heterogeneous) workstation clusters.

Finally, some first numerical results concerning the simulation of a micro-miniaturized two-valve membrane pump are presented.

#### C. CARSTENSEN, D. ZARRABI: An a posteriori error estimate for the unsymmetric coupling of FEM and BEM

Some engineering applications are concerned with a nonlinear material behaviour in a bounded domain $\Omega$ and a linear elastic material behaviour in the surrounding unbounded exterior domain $\Omega_c$, e.g., in a tunnel problem, in crack problems or in Kirsch's problem. The paper is concerned with such interface problems where the exterior problem is rewritten via boundary integral operators. We give an overview on certain nonlinear materials in $\Omega$ where convergence of Costabel's symmetric coupling and a priori error estimates can be proved. Then we focus on a priori error estimates for the direct coupling (where no a priori estimates are available) and for the symmetric method. Numerical examples are given.

#### M. FEISTAUER, G.C. HSIAO, R.E. KLEINMAN: On numerical treatment of coupled BEM and FEM for nonlinear exterior problems

The contribution presents main results (obtained in cooperation with G. C. Hsiao and R. E. Kleinman) of the investigation of the coupled BEM and FEM applied to a nonlinear exterior boundary value problem. The problem consists of a nonlinear partial differential equation considered in an annual bounded domain and the Laplace equation outside. These equations are equipped with boundary and transmission conditions. The problem is reformulated in a weak sense and combined with an integral equation. The attention is paid to the solvability of this problem and of the corresponding discrete finite element - boundary element problem, the convergence of approximate solutions to an exact one and to a sufficiently robust solver of the discrete coupled nonlinear problem. Computational results will also be presented.

#### P.J.J. FERKET: Coupling of a global coarse discretization and local fine discretizations

In elliptic boundary value problems with strong local variations in the solution, a basic global discretization can be improved significantly by introducing a local discretization defined in a subdomain. The local defect correction iteration [1] is based on the combination of a local and a global discretization method. We present theoretical insights assessing that the approximation obtained after two iteration steps is much more accurate than the approximation obtained after only one iteration step and that often the use of more than two iterations is not efficient. Also we will explain that in many cases a suitable choice for the "overlap" parameter $d$ in the local defect correction method is $d=0$. Numerical examples confirm the theoretical results.

The limit value of the local defect correction iteration with $d=0$ can be written as exact solution of a discrete problem on the composite grid that is composed of the global coarse grid and the local fine grid. This composite grid problem can be solved by the fast adaptive composite grid method [2]. We will show that, under reasonable assumptions, the local defect correction iteration with $d=0$ yields the same results as the fast adaptive composite grid iteration.

1. Hackbusch, W.: Local defect correction method and domain decomposition techniques , Computing, Suppl 5 (1984), pp 89-113.
2. Mc Cormick, S.F.: Multilevel adaptive methods for partial differential equations , SIAM, Philadelphia, 1989.

#### B. HEINRICH: The Fourier-finite-element method for elliptic problems in axisymmetric domains

The paper deals with the Fourier--finite--element method for solving elliptic problems in three--dimensional axisymmetric domains and gives some survey of this approach to the Poisson equation and to the Lam\'e equations. The method combines the approximating Fourier method with the finite--element method and reduces the approximate solution of a problem in 3D to the solution of coupled or even decoupled systems of elliptic problems in 2D. Algorithmic aspects of this method, its parallelization and basic results of the numerical analysis of this method, particularly for the Poisson equation in domains with edges, are discussed. The results are illustrated by numerical examples.

#### B. HEISE: Parallel solvers for coupled FEM-BEM equations with applications to non-linear magnetic field problems

An efficient parallel algorithm for solving nonlinear boundary value problems arising, e.g., in magnetic field computation, is presented. It is based on both the idea of ''nested iteration'' (also called Full Multilevel Method) and the parallel domain decomposition (DD) linear solvers suited for computations on MIMD computers with local memory and message-passing principle. It makes use of the parallel data structure of these solvers, the linearization is done by Newton's method, the linear system is solved by CG with DD preconditioning.

The DD approach allows to couple Finite Element and Galerkin Boundary Element Methods in a unified variational problem. In this way, e.g., magnetic field problems in an infinite domain with Sommerfeld's radiation condition can be modelled correctly. The problem of a nonsymmetric system matrix due to Galerkin-BEM is overcome by transforming into a symmetric but indefinite matrix and applying Bramble/Pasciak's CG for indefinite systems. For preconditioning, the main ideas of recent DD research are applied.

Test computations on a Parsytec transputer based system were performed for both model problems and a real-life electric motor. The very good scale-up when using $16 \cdots 128$ processors for problems of growing size (100.000 $\cdots$ 800.000 unknowns) shows the high efficiency of the proposed algorithm.

First tests were run on the Parsytec Xplorer and GC-Power Plus systems. They have shown a reduction of calculating time by a factor of about 1/29.

#### R. HIPTMAIR, R.H.W. HOPPE, B. WOHLMUTH: Coupling problems in microelectronic device simulation

The cost-effective design of electronic microstructures requires an advanced modeling and coupled simulation of various physical effects. The classical isothermal approach leads to the basic drift-diffusion model for semiconductor device simulation. In the stationary case, it represents a coupled nonlinear system consisting of a Poisson equation for the electric potential and two continuity equations for the electron and hole flow. We discuss the pros and cons of various discretization schemes that have to take into account extreme layers in the solution caused by the nature of the coupling. We further address efficient numerical solution techniques including adaptive multilevel methods. Finally, to allow for nonisothermal and magnetic effects we consider extensions of the classical model and we discuss perspectives for their appropriate numerical treatment.

#### F. IHLENBURG, CH. MAKRIDAKIS: Error estimates of Galerkin FEM for a system of coupled Helmholtz equations in one dimension

The quality of discrete solutions to the reduced wave equation depends on the relation between the physical parameter (frequency) and the numerical parameters (stepsize, order of polynomial approximation). For a one-dimensional model problem of fluid-solid interaction we analyze continous and discrete stability (inf- sup-condition) and prove an error estimate for a Galerkin FEM (h-p-method). The estimate holds in the asymptotic range while practical computations are carried out for preasymptotic values of h. Reference to recently proven statements for the uncoupled problem and discussion of computational results for the model problem are employed to address the error behaviour in the preasymptotic range.

#### H. KASPER, G. KOSAKOWSKI, T. TANIGUCHI, W. ZIELKE: Coupled geometric modelling for the analysis of groundwater flow and transport in fractured rock

A geometric modelling method is proposed for the finite element analysis of groundwater flow in 3-dimensional rock, which is separated by an arbitrary number an location of fracture planes. The method prepares coupled meshes using hexahedral elements for the rock matrice, quadrilateral elements for the fracture planes and line elements for the boreholes. The mesh generation method is based on the Delaunay triangulation, and the efficiency of the proposed method is examined through several test problems, for examples Grimsel and Asp\'o sites. Using these numerical models the groundwater flow is analyzed. As the result it has been proved, that the modelling method is valid for flow problems. However, hexahedral elements with an improved shape are required for solving transportation problems in fractured rock. Therefore several theoretical proposals to generate meshes with improved geometric properties are included in this paper. The numerical solution shown in this investigation are obtained using the solver {\sc Rockflow}, which is developed at the Institut f\"ur Str\"omungsmechanik und elektrisches Rechnen im Bauwesen, Universit\"at Hannover.

#### E. KATZER: Subspace decomposition methods for solving the Euler equations

A robust subspace decomposition method for solving first order systems of partial differential equations is presented. The algorithm is based on a carefully constructed sequence of four subspaces. Parallel additive and sequential multiplicative subspace correction methods are applied.

The discretisation is based on a vertex centered finite volume method for a triangular grid. The discrete system is obtained by minimizing the residuals of the discrete flux equations.

We obtain robust methods with uniformly bounded asympotic error reduction rates. Convergence rates are independent of grid size and only slightly dependent of parameters of the elliptic and hyperbolic model system.

Convergence rates for the elliptic case are of order 0.1 for isotropic and 0.4 for anisotopic cases. For the isotropic elliptic case, error reduction rates are similar to results of standard twogrid methods. In the hyperbolic cases, convergence is slightly larger and of order 0.8 but still uniformly bounded for all characteristic directions. A combination of the subspace decomposition approach as preconditioner for a conjugate gradient method improves the convergence speed.

The approach opens the way to fast solutions of the Euler equations which comprise a nonlinear system of composite elliptic/hyperbolic type.

#### A. KLAR: Domain decomposition schemes and coupling conditions for kinetic and hydrodynamic equations

For numerical calculations in rarefied gas dynamics domain decomposition methods involving kinetic and aerodynamic equations have become of considerable interest. Two main problems in the coupling procedure are the statement of the correct coupling condition at the interface between the kinetic and aerodynamic domain and the development and investigation of suitable algorithms to obtain the solution of the DD - problem.

The correct choice of the coupling conditions depends on the physical situation at the interface between the two types of equations. Assuming thermal equilibrium there, the so called Marshak conditions, based on the equality of fluxes, are appropriate. If we consider instead of equilibrium situations nonequilibrium states at the interface, then the above coupling conditions will not lead to the correct results. Here the matching can be done by modelling the interface region by a transition layer. Scaling the space coordinate in the layer one obtains a kinetic linear half space problem, leading to the correct coupling conditions. The validity of the obtained coupled solution can be proven in this case. To make the approach reasonable from a numerical point of view we developed a fast numerical scheme for the half space problem, based on a Chapman Enskog type expansion procedure. In a special case this method is shown to be equivalent to the well known so called variational methods.

The algorithm to obtain finally the solution of the DD - problem is usually an iterative procedure solving in turn the kinetic and aerodynamic equations in their respective subdomains, a variant of the well known Schwarz alternating method. We consider here the linearized versions of the above equations and prove convergence of the alternating scheme.

#### O. KOLDITZ, R. RATKE, W. ZIELKE, H.-J. DIERSCH: Coupled physical modelling for the analysis of groundwater systems

Salt-water intrusion and hydrothermal convection are classic examples of density-dependent problems in groundwater hydrology. The economic importance of these phenomena has invited continuous research in the fields of aquifer contamination/remediation, waste disposal, and geothermal energy.

Especially if considering coupled nonlinear problems the numerical models have to be carefully verified. The commonly used benchmarks for testing density-dependent simulations are the Henry Problem and the Elder Problem. We calculate these benchmarks using different codes (ROCKFLOW-DM2, FEFLOW). The topic of our first study is to show the impact of different numerical schemes (explicit and implicit in time) and equation solvers (direct and iterative) on the accuracy of results and on the computation time consumed.

Based on the verified numerical models more complex problems are tackled. In this way mixed convection in geothermal fractures including thermoelastic effects due to the permanent heat minig is simulated. Furthermore, first results of modelling thermohaline processes are presented. For this purpose the flow must be coupled with both, mass and heat transfer simultaneously.

#### H. SCHIPPERS, J.A. WENSING: Coupling of boundary and finite elements in aeroacoustic calculations

A coupled computational system of finite elements and boundary elements is presented to estimate scattering and refractive effects on the acoustic pressure waves through a boundary layer surrounding an aircraft's fuselage. The noise source due to isolated propellers or propfans is assumed to be known. The fuselage is modelled by an infinitely long cylinder with a constant cross section in axial direction. The propagation effects through the boundary layer can be significant and should be included in the noise predictions. Due to the boundary layer, the incident acoustic field will be refracted.

The acoustic pressure in the boundary layer is modelled by linearizing the Euler equations about a unidirectional cylindrically sheared mean flow. The governing equation is solved in the frequency domain using piecewise linear finite elements on a triangular grid. The total pressure field is assumed to consist of a scattered field and an incident field given by a plane wave, propagating harmonically with frequency $\omega$ in time and with wavenumber $\alpha$ in axial direction. Outside the boundary layer the pressure satisfies the convective Helmholtz equation with a uniform wavenumber. Here the scattered pressure is given by a boundary integral representation formula which involves as unknown distributions the pressure and its normal derivative at the edge of the boundary layer. This formula follows from applying Green's theorems to the Helmholtz equation. The pressure at the edge is approximated by piecewise linear boundary elements, whereas the normal derivative is modelled by piecewise constant boundary elements. The boundary element grid coincides with the outer boundary of the finite element grid.

The diffraction problem is investigated for a cylinder with a cross section similar to the Fokker 50 aircraft fuselage.

#### W. SCHMID, F. WAGNER: Numerical solution of the neutron diffusion equation - adaptive concepts in time and space

An important aspect in the safety analysis of nuclear power plants is the computation of the distribution of the neutrons in the reactor. A widely used mathematical model is given by the energy-dependent diffusion equation. Integrating over a finite number of energy intervals results in the so-called multigroup diffusion equations representing a coupled system of parabolic differential equations where the coupling is due to scattering effects.

We discretize implicitly in time and by mixed finite elements on rectangular geometries in three space dimensions and present an adaptive solution concept featuring combined time step control and grid refinement techniques. We discuss in detail the proper coupling of the energy levels as well as the basic tools for realizing the adaptive solver.

#### U. SPECHT, CH. DICKOPP, J. BALLMANN: A numerical scheme for stress waves at a fluid-solid interface

The lecture deals with a numerical scheme for stress waves at the interface of an inviscid, compressible fluid and an elastic-plastic solid. The problem is to couple domains of solution of non-linear, hyperbolic PDE's with different types of waves and wave speeds.

The scheme couples a FE-solver for the Euler equations and a Godunov method for elastic-plastic wave propagation in solids proposed by Lin and Ballmann [1]. It is based on the scheme at the boundary for the solid and a bicharacteristic formulation for the fluid. Instead of integrating along the characteristic lines belonging to the speed of sound of the fluid, the integration is carried out along the characteristic directions belonging to the fastest wave speed of the solid, according to the von Schmidt wave effect. An iterative procedure applying the contact conditions to the interface leads to a convergent scheme.

As a test example demonstrating the efficiency of the numerical scheme the pressure produced by a collapsing bubble is prescribed as initial condition for the fluid.

1. X. Lin and J. Ballmann. Numerical method for elastic-plastic waves in cracked solids, part 2: Plane strain problem. Archive of Applied Mechanics, 63:283-- 295, 1993.

#### C. VUIK, A. SEGAL: Solution of the coupled Navier-Stokes equations

Discretization of the incompressible Navier-Stokes equations leads to large linear systems of equations of the following form $$\left( \begin{array}{cc} A & B\\ C & 0\\ \end{array} \right) \left( \begin{array}{c} u\\ p\\ \end{array} \right) = \left( \begin{array}{c} f_u\\ f_p\\ \end{array} \right). \label{een}$$ One of the difficulties to solve this system is the zero block at the lower right corner. Many different techniques are used to circumvent these difficulties. In most approaches the velocity and pressure equation are decoupled, e.g. pressure correction, artificial compressibility, penalty methods etc. In this paper we solve the original system (\ref{een}) using iterative methods of Krylov subspace type. In general the Krylov subspace methods for non symmetric system, GMRES, GCR, Bi-CGSTAB etc. can be applied to (\ref{een}). However in general the rate of convergence of the un-preconditioned Krylov subspace methods is very slow. This together with the fact that the spectrum of the coefficient matrix used in (\ref{een}) contains eigenvalues with positive and negative real parts motivates the search for good preconditioners. We use a preconditioner based on an incomplete LU decomposition. Due to the zeroes on the main diagonal it is possible that the ILU decomposition breaks down (zero pivot elements). To prevent such a break down special ordering techniques are used. We give results using this preconditioner combined with various Krylov subspace methods. At present, for 2D f.e.m. discretized problems direct methods are hard to beat, however for 3D f.e.m. discretized problems the preconditioned iterative methods are superior with respect to memory requirements and CPU-time.

#### D. WEISS, G. LUBE:Numerical simulation of temperature distribution and seam forming in narrow gap welding

The topic of this paper is the numerical simulation of temperature distribution and weld pool deformation during narrow gap arc welding. Knowledge of the temperature field and the seam geometry is the basis of process optimization (in particular to ascertain the influence on the fusion and solidification as well as on the change in structure.)

A strongly coupled nonlinear problem has to be considerd. The energy transport is described by a convection dominated diffusion-convection equation (Fourier-Kirchhoff model). The deformation of the pool surface is determined by minimization of the surface energy which is influenced by the arc pressure, gravitation and the surface stress. An improved model of the arc intensity distribution is introduced instead of the standard Gaussian distribution in order to study heavily deformed molten pools during the joining of plates in the range of medium sized and thick metal sheets in different positions.

The coupled non-linear system is linearized by the simple iteration method. The singularly perturbed character of the problem is taken into account by using a stabilized Galerkin finite element method of Galerkin/Least-squares type. In particular, we study the robustness of the numerical method with respect to different aspects of the model. Numerical results for several test problems will be given.

### How to contact the authors by email

Artlich, Stefan           artlich@tu-harburg.d400.de
Bungartz, Hans-Joachim    bungartz@informatik.tu-muenchen.de
Ferket, P.J.J.            peterf@win.tue.nl
Feistauer, M.             feist@karlin.mff.cuni.cz
Heise, Bodo               heise@miraculix.numa.uni-linz.ac.at
Heinrich, Bernd           heinrich@mathematik.tu-chemnitz.de
Hoppe, R.W.H.             rohop@mathematik.tu-muenchen.de
Ihlenburg, Frank          ihl@olgao.umd.edu
Katzer, Edgar             edgar.katzer@mathematik.uni-magdeburg.d400.de
Kasper, Harald            kasper@appel012.hydromech.uni-hannover.de
Klar, Axel                klar@mathematik.uni-kl.de
Kolditz, O.               kolditz@appel012.hydromech.uni-hannover.d400.de
Lube, Gert                lube@namu13.gwdg.de
Makridakis, Ch.           makr@athina.edu.uch.gr
Schippers, H.             schipiw@nlr.nl
Schulte, Stefan           stefan.schulte@zfe.siemens.de
Specht, Ulf               usp@leibniz.lufmech.rwth-aachen.de
Vuik, C.                  c.vuik@math.tudelft.nl
Wagner, Frank             wagnerf@mathematik.tu-muenchen.de
Zarrabi, Darius           zarr@ifam.uni-hannover.d400.de