95
Numerical Analysis and Applications
of Differential EquationsBjrn Engquist and Jesper Oppelstrup
The last two years (19981999) have been a period of consolidation and
hard work towards our goals. The NA group works with very close ties to
PDC, TDB (Scientific Computing, Uppsala University), a number of
application oriented departments at KTH, notably Aeronautics and
Mechanics, and Mathematics. External funding is supplied by Nutek, TFR,
SSF, NFR, and the EC.
The Parallel and Scientific Computing Institute, Psci, a national Nutek
center of excellence, has been a significant catalyst for the amalgamation
and profiling of research areas. It is now in the final year of its Phase II,
19972000. The Psci projects are all carried out in collaboration with
industry. They focus on present applications with short to medium time
duration. This makes the TFR Basic Program grant on Numerical and
Mathematical Study of Continuum Mechanics to the NA-group instrumen-
tal in its support for more basic research with long-term goals.
The SSF support to the NA-group comes through a new program on
Multi-Phase Flows, the National Network in Applied Mathematics, and the
National Graduate School in Scientific Computing. The National Council
for High-Performance Computers was discontinued in 1998 and its role in
the financing of PDC taken over by NFR. The EC projects cover applications
of computational electromagnetics and non-linear conservation laws.
Our traditional research areas with applications to fluid mechanics
and wave propagation are now complemented by Stochastic Differential
Equations with applications to porous media flows and financial mathema-
tics. Parallelization techniques for field problems are currently converging
to spatial domain composition and standard message passing libraries, and
Software for parallel architectures has become an important theme which is
developed in most projects, such as CFD and CEM.
The KTH International Masters Program Scientific Computing
hosted by the department since fall 98 has a lively interaction with the
research groups. The students carry out examination projects, and the
researchers are strongly involved in curriculum development and as teachers.
The NA-group members have also been active technology providers in
the EC Technology Transfer Node Network projects coordinated by PDC.
Models for flow of paper pulp, airfoil shape optimization by CFD, and high-
performance simulation tools for building industry have fit well into on-
going research in the NA-group.
www.nada.kth.se/na
http://www.nada.kth.se/na
96
The research will be presented in the following paragraphs. The Psci projects
are described in more detail in Psci Progress Reports 19951996 and
19971999
Applications to combustion, phase change problems, and semiconductor
devices are all modeled by strongly non-linear partial differential equations.
The computational tools involve grid construction, adaptation heuristics
and algorithms, and theoretical solution estimates, which are all the subject
of Nonlinear PDEs and Adaptive Methods. Computational Electromagnetics is overall the major effort with a cleargoal to produce algorithms and software for the next generation of
industrially applicable program packages for simulation of electromagnetic
wave propagation. The group is also represented in two EU Esprit projects,
Impact on inverse problems and JACO3 on standardized coupling of simu-lation programs across the Internet.
Multi-Phase Flows are studied in an applied Psci project as well as basictheoretical issues and method studies for moving boundary problems. The
SSF project investigates the properties of the commonly used models that
allow useful simulations although the standard solution estimates
demonstrate perturbations that can grow catastrophically. A new promising
Level Set method for separated flows has reduced mass conservation errors
usually associated with interface tracking schemes. The model for flocculated
suspensions has been subjected to extensive parameter studies and it is now
clear that it must be refined to reproduce known properties of shear flows.
In Computational Harmonic Analysis Wavelet bases are used foraggregating the influence of small scales on the large scales in an adaptive
spatial resolution setting. The rapidly rising interest for applications,
primarily in signal processing and image treatment, both in industry and
academia, is reflected in the increasing number of collaborative projects
undertaken by the wavelet group.
Computational Physics Our work in computational physics is inspired byapplications to materials science, and might fit equally well under the heading
of Computational Materials Science. From simulations of assemblies of
particles: atoms in lattices, and molecules in liquids, we try to deduce
macroscopically observable properties of phase change, crystal formation,
and processes such as diffusion. A few results from the study of vacancydynamics in solids, and icosahedral clusters in super-cooled mono-atomicliquids are given below. Composite materials and porous media are oftentreated by ad-hoc averaged models, but new algorithms are emerging for
detailed, microscopic models with thousands of cracks, inclusions, or pores.
The techniques are now applied to elastostatics, electrostatics, and creeping
flow. We give some spectacular examples achieved by fast and stablealgorithms for fracture mechanics.
97
Non-linear PDEs and Adaptive MethodsMichael Hanke, Gunilla Kreiss, Bjrn Sjgreen, and Anders Szepessy
The aim of our research is to mathematically analyze phenomena in fluid
dynamics and develop efficient numerical methods for fluid flow and re-
lated problems. In particular, we study non-linear waves in partial differen-
tial equations, modelling e.g. compressible flow, and construct adaptive
finite element methods and finite difference methods. To study convergence
of approximations, construct adaptive refinement criteria or to analyze it-
erative solution methods one needs to understand the stability properties of
the equations. In the case of the non-linear partial differential equations for
compressible flow this is a difficult problem since, so far, there are no gen-
eral results on well posedness.
Another topic concerns the development of efficient numerical algo-
rithms for stationary Schrdinger-Poisson systems describing charge car-
rier distributions in semiconductors with quantum-mechanical effects.
Viscous Shock Waves
The stability of viscous shocks is important from a numerical point of view
since all shock capturing schemes for hyperbolic conservation laws are based
on artificial viscosity. [Efraimsson 1998a; Efraimsson 1998b] are detailed
studies on the effect of artificial viscosity terms for discrete waves.
By restricting to particular, but typical, cases we have proved stabil-
ity of viscous rarefaction waves and detonation waves [Szepessy 1999a] in
one dimension based on energy and pointwise estimates. Using estimates
of resolvents we have studied stability of viscous shocks [Kreiss and Kreiss
1998a; Kreiss and Kreiss 1998b; Liefvendahl 1999]. See also [Szepessy
1999c] and [Kreiss et al., 1999]. We have begun to study stability of wavesof systems of conservation laws in several space dimensions. High frequency
perturbations of two dimensional viscous shock waves have been analyzed
using techniques in geometric optics [Szepessy 1999b].
The convergence rate of numerical methods for shock problems is
the subject of [Engquist and Sjgreen 1998; Efraimsson and Kreiss 1999;
The aim of our research is tomathematically analyze phen-omena in fluid dynamics...
Computational Aerodynamics is pursued in collaboration with the depart-ment of Aeronautics and Mechanics. The dual time-stepping scheme for
time-accurate simulations was applied to pressure disturbances in a jet engine
air intake. Dynamic load balancing for the NSMB compressible flow code
is being developed in the Parallel Computational Aerodynamics Psci project,
while nozzle design by CFD, and MATLAB automatic differentiation for
airfoil shape optimization are the remaining subjects studied.
98
Efraimsson, et al., 1998]. In the domain upstream of the shock, we have
proved that, under certain conditions, the convergence rate of the numerical
method is equal to the formal convergence rate. The inevitable numerical
errors local to a shock will in general propagate and pollute the solution in
the entire region downstream from the shock.
Adaptive Schemes
To be useful, a computation needs some sort of estimate of its accuracy. In
adaptive schemes information is extracted from the computed solution to
achieve a computational error below a given tolerance level, with as few
degrees of freedom as possible in local refinements. Flow problems often
exhibit shock waves or reaction fronts, making localized resolution offered
by adaptive meshes appropriate. Efficient adaptive flow calculations need a
robust mesh generator, a stable and reasonably accurate discretization
method, an adaptive refinement criterion, and an efficient solution method.
In joint work with Jonathan Goodman at the Courant Institute, New York,
and Klas Samuelsson, CTH, we have developed an adaptive finite element
program based on a new robust mesh generator. Using mesh enrichment
and successive partitions of the elements, the shape of the elements adapt to
features of the flow, for example, fitting high aspect ratio elements around
shocks and boundary layers [Goodman et al., 1998].We have begun to explore the use of a posteriori error estimates in
stochastic differential equations. Models based on stochastic differential
equations are widely used in e.g. ground water hydrology. Compared to the
deterministic case, surprisingly little is known for adaptive methods with
stochastic data. [Szepessy et al., 1999]In earlier work, we computed two dimensional detonation waves.
The grid convergence of the numerical method for this inviscid problem
was not sufficiently demonstrated in numerical tests. In an ongoing project
together with Patrik Skogqvist, we use adaptive grids to increase the reso-
lution. A new error estimator based on solving an error equation along with
the problem was inserted into the AMR code from University of Berkeley
together with our solver for the compressible Euler equations with one chemi-
cal reaction [Skogqvist, 1999]. The density distribution at two different
grid resolutions, at the same time, are shown in Figs.1 and 2. Grid conver-
gence is not obtained.
To be useful, a computationneeds some sort of estimate ofits accu-racy.
99
In recent years, there has been a considerable interest in the so called
pulse detonation engine. This engine works in cycles, and a detonation isignited once in each cycle. It is important to be able to ignite detonations in
an efficient way. We have therefore started to study the problem of comput-
ing a deflagration to detonation transition (ddt). This work is done in col-
laboration with researchers from the defence research establishment (FOA).
In a ddt, the instability of an initially plane, diffusion dominated wave gives
acceleration up to a point where the wave changes into a detonation. Here
we solve the Navier-Stokes equations for compressible fluid flow with one
chemical reaction. A new parallel solver for such problems has been devel-
oped, and run on the PDC IBM/SP2. From these computations, we con-
clude that the ddt process consists of two steps. First, the flame front wrin-
kles, which give rise to high pressure in front of the flame. Second, when
the pressure has become sufficiently high, the situation becomes very un-
stable, and a small perturbation makes the high pressure region ignite, thereby
creating severe over pressures, and finally a detonation. A study of ignition
mechanisms and sensitivity is presented in [Sjgreen et al., 1999).The computation of ddt puts high demands on accuracy and effi-
ciency of numerical algorithms. Because of increasing modes in the solu-
tion, numerical errors also increase, and the computation becomes very sen-
sitive to grid resolution. We study such numerical effects in [Sjgreen and
Tegnr, 1999]. We are currently working on the improvement of the algo-
rithms.
Numerical methods for semiconductor equations
Modern production technologies in semiconductor manufacturing can manu-
facture structures in the range of a few nanometers with almost ideal bounda-
ries. Thus, we have an electron gas which is reduced in dimension making
quantum mechanically motivated models for the density of the electrons
necessary. An appropriate model is a coupled Schrdinger-Poisson system.
The numerical treatment is challenging because the system is frequently
... we solve the Navier-Stokesequations for compressiblefluid flow with one chemicalreaction.
...the so called pulse detonationengine.
Figure 1:The density distribu-tion at two different grid reso-lutions, at the same time.
100
singularly perturbed. Moreover, the quantum mechanical effects introduce
global dependencies such that the discretized system leads to full, rather
than sparse matrices.
The aim of the project is the construction of fast solvers for this sys-
tem. Standard methods for the Schrdinger equation failed or lead to ex-
cessive computing time. Therefore, a reliable program for Sturm-Liouville
problems was developed in cooperation with N.B. Konyukhova (Russian
Academy of Sciences, Moscow) and implemented in a parallel environ-
ment using PVM. The computation time is further reduced by an efficient
implementation of a method developed in collaboration with T. Zhanlav
(University Ulaanbaatar). A further collaboration is settled with Yu. Kurskij
(Technion, Haifa, Israel).
In close cooperation with Comsol AB, the applicability of the pro-
gram package FEMLAB in the two-dimensional case was considered (see
figure 2). As a result, a new nonlinear solver for highly nonlinear problems
was developed.
References-Non-linear PDE and Adaptive Methods
[Efraimsson, 1998] Efraimsson G., Kreiss G., and Nordstrm J. (1998),
Elimination of first order errors in shock calculations, TRITA-NA-
9823.
[Efraimsson, 1998a] Efraimsson, G., (1998a), A numerical method for thewave equation, accepted by J. Num. Meth. for PDE.
[Efraimsson, 1998b] Efraimsson, G., (1998b) A 2D Analysis of the Influence of Artificial Viscosity Terms on Solutions of the Euler Equations, J. Comp. Physics, 138, 1998.
[Efraimsson and Kreiss, 1996] Efraimsson, G., and Kreiss, G., (1996), Anote on the effect of artificial viscosity on solutions of conservationlaws, J. Appl. Num. Math., 21, 1996, pp. 155173.
[Efraimsson and Kreiss, 1999] Efraimsson, G., and Kreiss, G., (1999) Aremark on numerical errors downs team of slightly viscous shocks,SIAM J. Numer. Anal. 36 (1999), no. 3, 853863
In close cooperation withCOMSOL AB ...
Figure 2: Solution of a (muchsimplified 2D) semiconductormodel using FEMLAB.
The aim of the project is theconstruction of fast solvers forthis system.
101
[Engquist and Sjgreen, 1998] Engquist, B., and Sjgreen, B., (1998), Theconvergence rate of numerical methods in the presence of shocks,SIAM J. Num. anal. 35 no 6, 24642488.
[Goodman et al., 1998] Goodman, J., Samuelsson, K., and Szepessy, A.,(1998), Anisotropic refinements algorithms for finite elements, pre-print.
[Kreiss and Kreiss, 1998a] Kreiss, G., and Kreiss, H.-O., (1998a), Nonlinearstability of steady solutions to Burgers equation, SIAM J. Numer.Anal. 35, no. 6, 23292349
[Kreiss and Kreiss, 1998b] Kreiss, G., and Kreiss, H.-O., (1998b), Stabilityof systems of viscous conservation Laws, Comm. Pure Appl. Math.51, no. 11-12, 1397--1424.
[Kreiss et al., 1999] Kreiss, G; Kreiss, H-O; Lorenz, J (1999) On stabilityof conservation laws. SIAM J. Math. Anal. 30, no. 2, 401--430
[Liefvendahl, 1999] Liefvendahl M., (1999), On stability of viscousshock waves, Licentiate Thesis, Nada, KTH, October 1999 (ReportTRITA-NA-9908)
[Persson et al., 1999] Persson I., Samuelsson K. and Szepessy A.(1999),On the convergence of multigrid methods for flow problems,Electron. Trans. Numer. Anal. 8, 4687
[Sjgreen et al., 1999] Sjgreen B., Skogqvist P., and Tegnr J. (1999),Accuracy in computation of combustible flows, To appear in Proc.Int. Symp. on Combustible Flow Dynamics, Bremen, September 1999
[Sjgreen and Tegnr, 1999] Sjgreen B., Tegnr J. (1999), Mechanismsgoverning detonation waves and their initiation Implication on thepulse detonation engine. AIAA-99-IS-134
[Skogqvist, 1999] Skogqvist, P. (1999) An adaptive finite difference methodfor the simulation of combustible flows. Licentiate Thesis, Nada,KTH, May 1999 (Report TRITA-NA-9905)
[Szepessy, 1999a] Szepessy A.(1999a), Dynamics and Stability of a WeakDetonation Wave, Comm. Math. Phys. 202 (1999), no. 3, 547569.
[Szepessy, 1999b] Szepessy A.(1999b), High frequency asymptotics for 2Dviscous shocks, To appear in Indiana Univ. Math. J.
[Szepessy et al., 1999] Szepessy A., Tempone R., and Zouraris G.E.(1999),Adaptive weak approximation of stochastic differential equations,preprint.
[Szepessy, 1999c] Szepessy A.(1999c), Lectures on stability of nonlinear
waves in viscous media and numerics, in "Analysis of systems ofconservation laws". Ed. H.Freisthler, Chapman & Hall/CRCMonographs and Surveys in Pure and Applied Mathematics,
1999.
102
Computational ElectromagneticsBjrn Engquist, Bo Strand, Erik Abenius, Ulf Andersson, GunnarLedfelt, Lars Eriksson, ke Rydell, Fredrik Bergholm, Stefan Hagdahl,Anders Nilsson, Anders hlund, Andreas Atle, Christer Johansson,Lennart Hellstrm, Sandy Sefi, Jesper Oppelstrup, Lars Lovius, Perster, and Eva Prt-Enander
The last few years have seen a rapid increase in academic and industrial
interest in Computational ElectroMagnetics (CEM). CEM is one of the major
research programs conducted by Psci and the C2M2 activities are closely
coordinated with the industrial projects at Psci. More detailed descriptions
of Psci projects are available in the Psci progress report. The most impor-
tant project is the General Electromagnetic Solvers (GEMS) project which
is a collaboration between Psci, the Swedish Defence Research Establish-
ment (FOA), Ericsson Microwave Systems (EMW), Saab Ericsson Space
(SES) and Ericsson Saab Avionics (ESB). During the last few years a strong
collaboration has been established between Psci and ESB. Together the NA
group and ESB are involved in several Esprit projects, the ongoing IM-
PACT and JACO3, and the recently finished EMCP2. The GEMS project is
funded by the National Program for Aeronautics Research (NFFP), Nutek
and KTH.
GEMS
In the General Electromagnetic Solvers (GEMS) project general purpose
time domain (TD) and frequency domain (FD) hybrid codes are being de-
veloped as well as waveguide mode solvers. The objective is that the soft-
ware suite on the international level should be state of the art and form a
platform for future development by Swedish industry and academia. Appli-
cations are antenna analysis, scattering and Electromagnetic Compatibility
(EMC) computations and are shown in Figure 1. Parallel with the code
development there is a supporting research program to back up the devel-
opments as well as doing research on potentially interesting fields. The
total project workload is 408 man months with a Psci part of 282 man months.
The most important project isthe General ElectroMagneticSolvers (GEMS) project...
The objective is that thesoftware suite on the internatio-nal level should be state of theart and form a platform forfuture development by Swedishindustry and academia.
103
The TD code is a multi-block and out of core solver based on Finite Differ-
ences (FDTD) on structured grids, explicit Finite Volumes (FVTD) and
implicit Finite Elements (FETD) on unstructured grids. Figure 2 illustrates
a simulation of a SAAB 2000 aircraft that has been hit by lightning.
Figure 1: Applications in theGEMS project.
Figure 2 : Surface currents onthe Saab 2000 after a lightningstrike
104
This calculation used over one billion grid cells and was performed on the
IBM SP at PDC. For more details see http://www.nada.kth.se/~ledfelt/CEM/
SC98/sc98.html and [Andersson and Ledfelt, 1999].The FD hybrid code is based on Method of Moments (MM), Physical Op-
tics (PO) and Geometrical Theory of Diffraction (GTD). Since conven-
tional MM has a computational complexity that grows with the sixth power
of frequency, it is hybridized with PO and GTD for medium and high fre-
quency problems.
A large supporting research program runs in parallel with the code
development. The research is focused on topics that should support the code
development in the short term as well as on topics that could increase the
accuracy, performance and functionality of the codes in the future. For the
TD codes stability and accuracy for hybridisation techniques [Ledfelt etal., 1999], high order FDTD methods, subcell models, parallelisation andfrequency dispersive materials in FETD are studied. Research for the FD
codes covers hybridisation techniques between MM, PO, and GTD, and
iterative methods for MM including fast multipole techniques.
IMPACT
Understanding the response of a structure to the emission of a radiated sig-
nal has many industrial applications in the fields of electromagnetism, acous-
tics and elastodynamics. The objective of IMPACT is to address these needs
by developing a parallel implementation of an inverse scattering method
applicable within different industrial domains. The method will be used to
determine the unknown physical properties of a given object.
By numerical solution of a specific set (Maxwell, Hemholtz, etc.) of
partial differential equations corresponding to the physical properties un-
der study, the response of a given structure to an illuminating wave can be
found. There are well-known algorithms for the solution of such direct
problems. The inverse method consists in searching the origin of a known
response. IMPACT implements a time domain inverse scattering method
which uses deterministic optimisation algorithms. Project participants are
Aerospatiale Matra (F), Ericsson SAAB Avionics (S), Esaote (I), KTH (S),
CRS4 (I) and QSW (UK).
JACO3
The optimal design of a new complex, large and expensive product like an
aeroplane, a satellite or a car requires the study of different components
interacting with each other and of the corresponding global behaviour. Nu-
merical methods and simulation together with HPCN are established key]
technologies for improving the design process.
Understanding the response ofa structure to the emission of a radiated signal...
JACO3=JAVA and CORBAbased COllaborativeEnvironment for COupledSimulations
A large supporting researchprogram runs in parallel withthe code development.
105
Component simulation however does not allow for full optimisation
of the system under design. Improved collaboration among a number of
experts and easy and flexible coupling of simulation codes wll reduce the
total solution time with its impact on the design cycle and catch design
flaws desugns earlier.
JACO3 addresses these needs by developing a CORBA based high
performance distributed computing environment for coupling simulation
codes. More precisely, JACO3 will allow engineering departments, their
partners, their subcontractors and their suppliers, especially SMEs
to develop and use distributed coupled simulation applications withexisting in-house and commercial code
to access and to optimise the use of computing resources distributed over the Internet
to exploit the results in collaborationIt also allows code providers to plug in their simulation codes without ex-tensive modification and promotes an approach based on standards (CORBA
and JAVA) to protect already made and future investments, provide oppor-
tunities for other contributors to join, and significantly improve the way
complex systems are designed through simulation.
References
[Andersson and Ledfelt, 1999] Andersson, U. and Ledfelt, G. (1999).
Large Scale FD-TDA Billion Cells, in 15th Annual Review of
Progress in Applied Computational Electromagnetics, Vol. 1, p.
572577.
[Ledfelt et al., 1999] Ledfelt, G., Edelvik, F., Eriksson, L. and Andersson,U. (1999) Hybrid Time Domain Solver for the Maxwell Equations,in RadioVetenskap och Kommunikation 99, p. 7074.
[Andersson, 1998] Andersson,U. (1998), Parallelization of a 3D FD-TDcode for the Maxwell equations using MPI. B. Kgstrm et al.,editors, Applied Parallel Computing, PARA98, Lecture Notes in
Computer Science, No. 1541, pages 1219, June 1998.
[Andersson, 1998] Andersson,U. (1998) Parallelization of a 3D FD-TDcode for the Maxwell equations using MPI. In G. Kristensson,editor, EMB 98Electromagnetic Computations for analysis and
design of complex systems, pages 94-101. SNRV, November 1998.
[Andersson et al., 1999] Andersson, U., Engquist, B., Ledfelt, G., and
Runborg, O. (1999), A contribution to wavelet-based subgridmodeling. Applied and Computational Harmonic Analysis, 7:151164, 1999.
106
[Edelvik, 1998] Edelvik, F. (1998), A survey of finite volume time domainmethods for solving Maxwells equations. Technical report,Department of Scientific Computing, Uppsala Universitet, 1998.
[Edelvik, 1999] Edelvik, F. (1999), Analysis of a finite volume solver forMaxwells equations, In Vilsemeier R., editor, Finite Volumes forComplex Applications II, pages 141148, Paris, France, July 1999.
IVG, University of Duisburg, Hermes.
[Edlund, 1999] Edlund. J. (1999), Solving the electric field integralequation using a block LDLT method, UPTEC f99 007, UppsalaUniversity, February 1999.
[Karlsson, 1999] Karlsson, M. (1999), Hgre ordningens FDTD-metod frlsning av maxwells ekvationer. Masters thesis, Uppsala University,January 1999 (in Swedish).
[Ledfelt, 1998] Ledfelt, G. (1998), A thin wire sub cell model for arbitrarilyoriented wires for the FD-TD method, In G. Kristensson, editor,EMB 98 - Electromagnetic Computations for analysis and design of
complex systems, pages 148155. SNRV, November, 1998.
[Ledfelt et al., 1999] Ledfelt ,G., Edelvik, F., Eriksson, L., and Andersson,U., (1999), Hybrid time domain solver for the Maxwell equations, InJan Olov Gustafsson, editor, RadioVetenskap och Kommunikation
99, pp. 7074, 371 79 Karlskrona, Sweden, June 1999. SNRV,
Nutek, Hgskolan i Karlskrona/Ronneby.
[Nilsson, 1999] Nilsson, M. (1999), Solving the electric field integralequation using a sparse approximate inverse preconditionediterative method, UPTEC f99 004, Uppsala University, February1999.
[Strand, 1998] Strand, B. (1998), Numerical studies of hyperbolic IBVPwith high-order finite difference operators satisfying a summationby parts rule, Appl. Numer. Math., 26(4):497521, 1998.
[Strand, 1999] Strand, B. (1999), Simulations of acoustic wave phenomenausing high-order finite difference approximations, SIAM Journal onScientific Computing, 20(5):15851604, 1999.
[Syrowattchenko, 1999] Syrowattchenko, O. (1999), Creeping waves onNURBS, TRITA-NA-9970, KTH, November 1999.
107
Multi-phase flows
Bjrn Engquist, Reynir Gudmundsson, Katarina Gustavsson, Heinz-Otto Kreiss, Jesper Oppelstrup, Anna-Karin Tornberg, and JacobYstrm
Multi-phase flow denotes the simultaneous motion of two or more phases
(gas, liquid or solid substances) and is probably the most common type of
natural and engineering flow. Blood circulation, clouds moving in the at-
mosphere, boiling liquids, sewage water, pneumatical transport of parti-
cles, and mixing of particles with a liquid are obvious examples..
The study of multi-phase flows is one of the fastest growing areas in
fluid mechanics. Modern computers have made it possible, at least in rea-
sonable first attempts, to attack both basic microscopic models and realistic
application problems. However, the mathematical knowledge of the equa-
tions and the numerical methods devised to solve them are on a rather primi-
tive level and the opportunities for applied mathematics are far from ex-
hausted.
General classifications of two-phase flow problems are [Ishii, 1975]
based on the constituents: gas-solid, gas-liquid, solid-liquid, and two im-miscible liquids as well as on the topology: separated and dispersed flows.In separated flow the phases are separated by a distinct interphase, e.g. a
liquid jet in a gas or gas film in a liquid. In dispersed flow the phases are
mixed on a macroscopic level, e.g. gas bubbles in a liquid or solid particles
in gas or liquid. The research at Nada concerns separated and dispersed
flows.
Separated flows
The main challenge is the representation of the interphase. Material proper-
ties, e.g. the density and the viscosity, typically are discontinuous at the
interphase, and important properties like the surface tension are directly
dependent on its shape which can be extremely complex with e.g. break-up
and merging.
The level-set method introduced by [Osher and Sethian, 1988] intro-
duces an extra continuous function, designed to be an approximate signed
distance to the interphase which is represented simply the zero level set of
this function.
We analyze errors associated with approximating integrals of dis-
continuous functions. Two approaches have been used, first a special
quadrature for the computational cells affected by the discontinuity, and
second, a smoothing of the discontinuity in a transition region. The regular-
The stydy of multi-phase flowsis relatively new and one of thefastest growing areas in fluidmechanics.
The main challenge is the rep-resentation of the interphase.
108
ity of the smooth approximation is shown to be critical for the latter ap-
proach. A 2D finite element method for two immiscible liquids has been
developed using these techniques, see [Tornberg, 1998] and Figure 1. This
method is also compared to an earlier front-tracking method described in
[Tornberg et. al., 1997]. Second order accuracy for the spatial discretizationand good agreement between the methods were concluded.
Figure 1: The Level-Set method used for a 2D simulation with two immisci-ble fluids A(blue) and B(green). At t=0, two circular bubbles of fluid A withradius r=0.4 and r=0.5 are positioned at (0,1) and (0,2) respectively with aflat surface at y=3. Top left: t=0.05, top right: t=0.1, bottom left:t=0.325and bottom right: t=0.3625.
Dispersed flows
The Lagrangian formulation is based on following the individual particles
in the dispersed phase, and in the Eulerian formulation both phases are
considered continuous, averaged over a finite volume. The first has the po-
tential of being very detailed since particle-particle and particle-fluid inter-
actions can be modeled in a completely controlled way. The drawback is
109
that the number of particles one can treat limits the applicability to very
small problems. The general Eulerian two-fluid approach is to formulate
integral balances for mass, momentum and energy for a control volume
containing both phases. Averaging in space, time or ensemble gives an ef-
fective formulation but additional unknowns appear and the system of equa-
tions obtained by the averaging has to be closed. Empirical closures have
been developed for several cases
The two-fluid Eulerian model is the focus for two projects at Nada.
The first dates back to 1993, and aims at developing simulation tools for
de-watering of suspensions. The goal is to understand the one-dimensional
effect of gravity induced separation by buoyancy forces in combination
with shearing flow of the two-phase fluid. Simulation tools have been de-
veloped for a simplified model for slow flows, see [Gustavsson, 1999] and
Figure 2.
The second project, started 1999, is a part of the SSF program in
multi-phase flow and aims at understanding the mathematical structure of
general two-fluid models and thereby to design efficient and robust nu-
merical methods. It has been known for a long time that the commonly used
models, in the absence of viscous-type dissipation, possess linear instabili-
ties and that the equations are not of standard type. However, numerical
calculations for these problems indicate that the problems have bounded
and reasonably well-behaved solutions.
Figure 2: Simulation of the de-watering of a particle-liquid two-phase fluid.The stress-strain relation for the particle phase is shear-thinning'. Topleft; mean volume fraction for wall speed 0 (blue), 0.1 (green), and 1 cm/s(red). Color plots of the volume fraction of particles and arrows for pseudo-streamlines of the particle phase velocity at t=250, top right: speed 0, bot-tom left: 0.1 and bottom right:1 cm/s.
The two-fluid Eulerian model isthe focus for two projects atNada.
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References-Multiphase Flows
[Gustavsson, 1999] Gustavsson, K. (1999) Simulation of ConsolidationProcesses by Eulerian Two-Fluid Models. Licentiate thesis.Technical Report TRITA-NA-9907, Department of Numerical
Analysis and Computing Science, Royal Institute of Technology,
Sweden.
[Ishii, 1975] Ishii, M. (1975) Thermo-Fluid Dynamic Theory of Two-PhaseFlow, Eyrolles.
[Osher and Sethian, 1988] Osher, S and Sethian, J.A. (1988) FrontsPropagating with Curvature Dependent Speed: Algorithms Basedon Hamilton-Jacobi Formulations, J. Comp. Phys. 79:12-49
[Tornberg et al., 1997] Tornberg,A-K., Metcalfe,R.W., Scott,R. and
Bagheri,B. (1997) A Fluid Particle Motion Simulation Method.Computational Science for the 21st Century 312-321. John Wiley
and Sons, New York.
[Tornberg, 1998] Tornberg,A-K. (1998) Finite Element Based Level-SetMethod for Multiphase Flow Simulations. Licenciate thesis.Technical Report TRITA-NA-9817, Department of Numerical
Analysis and Computing Science, Royal Institute of Technology,
Sweden.
[Kreiss and Ystrm, 1998] Kreiss, H.-O., and Ystrm,J. (1998), ANumerical Study of the Solution to the 3D Incompressible Navier-Stokes Equations, CAM Report 98-24, Dept. Math., UCLA, 1998
[Kreiss and Ystrm, 1998] Kreiss, H.-O., and Ystrm,J. (1998), ANumerical Study of the Solution to the 3D Incompressible Navier-Stokes Equations with Forcing Function, CAM Report 98-31, Dept.Math., UCLA, 1998
[Gustavsson and Oppelstrup, 1999] Gustavsson,K., and Oppelstrup,J.
(1999), A Numerical Study of the Consolidation Process of aFlocculated Suspension using a Two Fluid Model, Proc ofENUMATH, World Scientific, 1999, to appear
[Gustavsson and Oppelstrup, 1999] Gustavsson,K., and Oppelstrup,J.
(1998), Consolidation of Concentrated Suspensions - NumerialSimulations Using a Two-Phase Fluid Model, to be published inAMIF volume of Computing and Visualization in Science.
111
Computational Harmonic AnalysisJan-Olof Strmberg
The wavelet theory has been developed during the last decades emerging
from harmonic analysis and signal analysis. The interest of this field has
grown rapidly and many applications have been found, both in signal process-
ing and in numerical analysis.
To stimulate the application of these new mathematical methods, the
Swedish Strategical Research Foundation through the National Network in
Applied Mathematics (NTM) decided to fund a chair in Computational Har-
monic Analysis at KTH. The new professor was installed in 1998 in joint
position between Nada and the Department of Mathematics.
In applications to Image processing, data compression, and integraloperators we have had well established contacts with Yale UniversityDepartments of Mathematics and Computer Science since before 1998. Fast
wavelet processing and coding techniques have been developed for image
and data compression, for manipulating and enhancing the images by
denoising, sharpening (deconvolution), etc. Our methods and codes are also
used for medical image processing and for compression of seismic data by
PEGASUS and Amoco , viz.
Singular integral operators are used for solving PDEs, see e.g. the
section on Computational Electromagnetics. One graduate student is sup-
ported by NFR (Norskt Forskningsrd) in studying alternative ways of
representing and compressing these operators with wavelets.
The group is in the phase of establishing a number of new activities.We are starting a project in collaboration with Dept. of Clinical Neuro-
science, KI about confocal problems in 3D imaging. The iterative statisticalmethods that are used can be stabilized by using wavelet denoising tech-
niques. The other image manipulating methods mentioned above may also
be useful in this context.
The first step, an examination project, has been taken in a project
with Dept. Neurophysology, KI, on analyzing EEG measurements. The ba-sic problem is to detect transients in a collection of parallel measurements
for localization of defective brain activities.
We are interested in supporting other research groups when they want
to use wavelet methods for analyzing data from enginering experiments.Through the more advanced time-frequency analysis that the wavelet meth-
ods offer one can often extract more information than through the tradi-
tional spectral analysis. We have been contacted by several research groups,
and we have also held a wavelet seminar where they have presented their
problems.
The wavelet theory has beendeveloped during the last deca-des emerging from harmonicanalysis and signal analysis.
112
As an example, in an application to vibration analysis we will use
wavelets to analyze signals from bearings. This is a project together with
Niden Teknik, Lule, and the Mathematics Center, Lule Institute of
Technology supported by NTM. Niden Teknik manufactures equipment
for monitoring paper machines. Several paper mills participate in the project
which started October 1999.
References
[ Averbuch et al., 1998] Averbuch, A., Meyer, F.G., and Strmberg, J.-O.Fast adaptive wavelet packet image compresson. Submitted to IEEETrans.Image Compression, April 1998.
[Meyer et al., 1997] Meyer F.G., Averbuch A., Strmberg, J.-O, and CoifmanR.R., Motion Compensation of Wavelet Coefficients with WaveletPacket Based Motion Residual Coding. Technical report, October1997.
[Meyer et al., 1998] Meyer F.G., Averbuch A., Strmberg, J-O., and CoifmanR.R. Multi-layered image compression. Wavelet Applications inSignal and Imaging Processing VI, 1998.
[Strmberg, 1997] Strmberg. J.-O, Wavelets and application to medicalimages. Proceedings SSAB'97, Swedish Symposium on Image Analysis in Stockholm in March 1997, pp. 110113.
Computational PhysicsMikhail Dzugutov, Sergei Simdyankin, Fredrik Zetterling, FredrikHedman, Johan Helsing
Our work in computational physics is inspired by applications to materials
science, and might fit equally well under the heading of Computational
Materials Science. The interaction of processes with widely different scales
makes the subject difficult and fascinating.. On the smallest scale, we study
assemblies of particles: atoms in lattices, and molecules in liquids, and try
to deduce macroscopically observable properties of phase change, crystal
formation, and processes such as solvation of salts in water. A few results
from the study of vacancy dynamics in solids, and icosahedral clusters insuper-cooled mono-atomic liquids are given below.
The next scale is most relevant for composite materials and porous
media. Traditionally simulated by (more or less) ad-hoc averaged models,
new algorithms are emerging which allow computer simulation of detailed,
microscopic models with thousands of cracks, inclusions, or pores. The
techniques have evolved in the last decade from applications in celestial
mechanics and are now applied to elastostatics, electrostatics, and creeping
flow.
113
The most important tool for atom as well as crack scales is an effi-
cient code for treatment of N-body interactions. For the particle simulationsproper, we use in-house codes, and for the composite materials models we
are continuously exchanging algorithmic ideas and codes with the groups
at Yale and Courant Institute. We give some spectacular examples achieved
by our fast and stable algorithms for fracture mechanics.
Molecular dynamics simulations of condensed matter
The main tool for simulations in materials science is an efficient molecular
dynamics (MD) code. The group is developing a program which runs well
on RISC-type architectures, both on a single processor and in parallel, us-
ing MPI. The most time-consuming part of any MD program is the compu-
tation of the force exerted on a particle by all the other particles. It is enough
to take into account only the neighbors within interaction range, but the
time required to select the interacting neighbors is proportional to the square
of the number of particles, N. This is because all pairs of atoms must beexamined.
In our algorithm, the particles are sorted so that geometrical neighbors
are also close in memory. This locality of reference results in more infre-
quent cache misses, and the time becomes essentially proportional to N.The resulting re-use of data in the cache is frequent enough that execution
speed scales linearly with clock frequency. The data allocation scheme also
makes the program easily parallelizable for shared and distributed memory
machines.
The parallel versions are used for large-scale projects, but for system
sizes of some twenty thousand particles we use one SMP node in the PDC
IBM / SP2 interactively. Interactivity is particularly useful for systems with
slow dynamics; Regular analysis of the results, suitably visualized, enables
constructive interaction with the simulation and we have found that to sig-
nificantly reduce overall run times.
Vacancy dynamics in solids
The formation of macroscopic size voids is a problem in metallic structures
exposed to radiation, such as in space and nuclear applications. Void forma-
tion also occurs in semiconductor materials as a result of the electronic
wind.
Atomistic simulations of formation and evolution of vacancies give
unique information which cannot be accessed by real experiments. In the
present study, the kinetics of a vacancy network in a solid is investigated by
a molecular dynamics simulation of a crystal. Simulations with fcc lattices
and a LennardJones pair potential showed a tendency for large-scale den-
114
sity fluctuations, and eventually the formation of a large void region. This
separation of the system into two phases with different densities invariably
occurred for different temperatures and total number of vacancies, whereas
other potentials investigated did not form voids. Thus the experiments sup-
port the belief that void-forming vacancy dynamics is related to the energetics
of inter-atomic interactions rather than to the particular structure of the sys-
tem. Figure. 1 shows the void formed in one of the experiments.
Icosahedral clusters in super-cooled monoatomic liquids
Metallic glasses possess many desirable properties that cannot be obtained
in crystalline alloys. Icosahedral local order is a prominent feature of glass
forming alloys because it enables the formation of a solid amorphous phase
if the cooling rate is high. However, with slower cooling the systems crys-
tallize into the periodic Frank-Kasper structure, and avoiding this is the
most important problem in the metallurgy of metallic glasses. A particle
simulation approach is crucial for understanding of this process.
We have explored glass formation in a one-component system, and
have found that Frank-Kasper phase crystallization can be avoided if the
forces are generated by the proper pair potential. A tendency for clustering
of the local icosahedra was observed, and the growth of the clusters pro-
ceeds in a low-dimensional way. The simulation, illustrated in fig. 1, is
evidence that glass formation in such systems can be viewed as percolation
of a global cluster composed of icosahedrally ordered units.
Figure 1: Left, a large scale ag-gregation of vacancies in the fcccrystal. Right, a cluster oficosahedra close to the perco-lation threshold.
115
Fast and stable algorithms for fracture mechanics
Experimental studies of reinforced composites have shown that matrix mi-
cro-cracking and fiber-matrix de-bonding can result in complicated crack
patterns which eventually lead to macroscopic fractures. An efficient and
realistic simulation model for the process would open up new possibilities
for optimization of composite materials.
Properly chosen boundary integral equation formulations when cou-
pled with the fast multi-pole scheme can solve problems with thousands of
cracks with speed and accuracy not obtainable by any competing technique
such as the finite element method. The reason is that the algorithms are
stable also under extreme refinement, and that computational complexity
grows only superlinearly with the number of discretization points.
The first task in the solution of a problem is to devise a second kind
Fredholm equation for the case at hand. This is the key element of our
method, but may not be easy. The classical equations are seldom if ever
the most suitable for numerical work ! The discretized problem inherits the
stability properties of the second kind equation, and standard iterative
schemes like GMRES can be employed. It becomes fast by computing the
necessary matrix-vector multiplications by the multi-pole method.
Here follows a gallery of geometries and solutions, taken from work
with L.Greengard, Courant Institute, A.Jonsson, Solid Mechanics, KTH,
and G.Peters, Mathematics, KTH.
116
Figure 1 : Left, a unit cell of an elastic membrane with eight ellipticalholes and some VERY close encounters. Right: Convergence of the effec-tive elastic modulus with increased number of discretization points. Notethe log-scale for the error!
Figure 2: Large-scale computing on complicated domains. Left, the elec-trostatic field computed in a unit cell with 3,502 overlapping disks at areafraction 0.675. Right, the plane stress field in a unit cell with 10,000 ran-domly oriented cracks. The estimated relative errors in the solutions aresmaller than 0.5 per mille.
117
ReferencesFast and stable algorithms for fracture mechanics
[Greengard, L. and Helsing, 1998] Greengard, L. and Helsing, J. (1998)
On the numerical evaluation of elastostatic fields in locallyisotropic two-dimensional composites, J. Mech. Phys. Solids 4614411462.
[Helsing, J. 1998] Helsing, J. (1998) A high-order accurate algorithm forelectrostatics of overlapping disks, J. Stat. Phys. 90, 14611473.
[Helsing, J. 1999] Helsing, J. (1999) On the numerical evaluation ofstress intensity factors for an interface crack of a general shape,Int. J. Num. Meth. Engn 44 729741.
[Helsing, J. 1999] Helsing, J. (1999) Fast and accurate numericalsolution to an elastostatic problem involving ten thousandrandomly oriented cracks, Int. J. Fracture (in the press).
[Helsing, J., 1999] Helsing, J. (1999) Stress intensity factors for a crackin front of an inclusion, Engn. Fracture Mech. 64 245253.
[Helsing, J., 1999] Helsing, J. (1999) Corner singularities for ellipticproblems: special basis functions versus brute force, Int. J.Num. Meth. Engn (in press).
[Helsing, J., 1999] Helsing J., and Jonsson A. (1999) Elastostatics forplates with holes, Int. J. Solids Structures (submitted).
[Helsing, J., and Peters, G., 1999] Helsing, J., and Peters, G. (1999)
Integral equation methods and numerical solutions of crack andinclusion problems in planar elastostatics, SIAM J. Appl. Math.59, 965982.
[Helsing, J., and Peters, G., 1999] Helsing, J., and G. Peters (1999) Anefficient numerical algorithm for a crack partly in frictionless contact, SIAM J. Appl. Math. (accepted after revision)
118
Computational AerodynamicsArthur Rizzi, Joakim Mller, Christian Wauquiez, and AndersYtterstrm
The research is focused on the efficient use of high-performance computers
for aerodynamic applications. Parallel Computational Aerodynamics is an
enabling technology for development projects where shorter design cycles
are ever more important. Parallel CFD is now used also in process industry,
as exemplified by the nozzle shape design in [Mller].
As another example, the large computer resources required in automatedoptimal design can be cost-effectively supplied by parallel computing.
Dynamical load balancing by block splitting, merging, and re-distribution
is now under development and will improve the adaptivity of the solution
process both to the simulated process, and to the computer network used
for the calculation.
The dual time-stepping algorithm for time-accurate simulations
developed for the NSMB compressible flow code was applied in a Psci
project to the Hammershock pressure wave simulation . The time step
could be increased by four orders of magnitude (estimated) and made the
calculation at all feasible, [Ytterstrm and Axelsson].
Airfoil shape optimization by CFD requires derivatives of the
aerodynamic forceslift, moment, and drag with respect to design variables.
We have investigated automatic differentiation which is now supported by
the MATLAB programming environment. An experimental code using po-
tential flow coupled to integral laminar and turbulent boundary layer models
was written in MATLAB. After automatic differentiation to yield first and
second derivatives it was applied to a number of optimization exercises,
[Wauquiez].
References
[Ytterstrm and Axelsson] Hammershock Calculations in the Air Intake ofJAS39 Gripen, using Dual Timestepping, Proc. 17th Appl. Aerodyn.Conf., June 1999, Norfolk, VA., AIAA 993113.
[Mller] CFD Analysis in Nozzle Design for Gas Atomization,Sv. Mekanikdagarna, 1999.
[Wauquiez] Shape Optimization of Low Speed Airfoils using MATLAB andAutomatic Differentiation, Lic. Thesis, TRITA-NA report 00-07, Feb.2000.