Numerical Simulation of Fluid-Structure Interaction Between Acoustic and Elastic Waves

144 Pages·2011·7.45 MB·English

by  Sanna Mönkölä

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JYVÄSKYLÄ STUDIES IN COMPUTING 133
Sanna Mönkölä
Numerical Simulation of
Fluid-Structure Interaction Between
Acoustic and Elastic Waves
Esitetään Jyväskylän yliopiston informaatioteknologian tiedekunnan suostumuksella
julkisesti tarkastettavaksi yliopiston Agora-rakennuksen Auditoriossa 2
lokakuun 29. päivänä 2011 kello 12.
Academic dissertation to be publicly discussed, by permission of
the Faculty of Information Technology of the University of Jyväskylä,
in the building Agora, Auditorium 2, on October 29, 2011 at 12 o'clock noon.
UNIVERSITY OF JYVÄSKYLÄ
JYVÄSKYLÄ 2011
Numerical Simulation of
Fluid-Structure Interaction Between
Acoustic and Elastic Waves
JYVÄSKYLÄ STUDIES IN COMPUTING 133
Sanna Mönkölä
Numerical Simulation of
Fluid-Structure Interaction Between
Acoustic and Elastic Waves
UNIVERSITY OF JYVÄSKYLÄ
JYVÄSKYLÄ 2011
Editors
Timo Männikkö
Department of Mathematical Information Technology, University of Jyväskylä
Pekka Olsbo, Ville Korkiakangas
Publishing Unit, University Library of Jyväskylä

Cover: Simulation results for a cross-section of an ABS plastic tube surrounded
by water. The displacements are presented in the solid domain as streamlines
(in blue), whereas the acoustic field in the fluid domain is described as a surface
plot of pressure waves (in purple). The velocity vectors in the solid domain and
the contour lines of the pressure variable in the fluid domain are projected to
the 2D plane.
URN:ISBN:978(cid:16)951(cid:16)39(cid:16)4439(cid:16)1
ISBN 978-951-39-4439-1 (PDF)
ISBN 978-951-39-4438-4 (nid.)
ISSN 1456-5390
Copyright © 2 0 1 1 , by University of Jyväskylä
Jyväskylä University Printing House, Jyväskylä 2011
ABSTRACT
Mönkölä,Sanna
NumericalSimulationofFluid-StructureInteractionBetweenAcousticandElasticWaves
Jyväskylä: UniversityofJyväskylä,2011,136p.
(JyväskyläStudiesinComputing
ISSN1456-5390;133)
ISBN978-951-39-4438-4(nid.)
ISBN978-951-39-4439-1(PDF)
Finnishsummary
Diss.
This study considers developing numerical solution techniques for the computer sim-
ulations of the fluid-structure interaction. The focus is especially on the efficiency of
the iterative methods based on exact controllability and spectral element methods. In
particular, the thesis concentrates on the coupling between two linear wave equations:
thescalar-valuedequationconcerningthepropagationofacousticwavesandthevector-
valuedequationmodelingthepropagationofwavesinanelasticmedium. Thesefunda-
mentalequationsoccurinanumberofphysicalapplications, suchasacoustics, medical
ultrasonics,andgeophysics.
We consider both transient and time-harmonic problems. Traditionally, the com-
plex-valued time-harmonic equations and low-order finite elements are used for solving
thetime-harmonicproblems. Thisleadstolarge-scaleindefinitesystems,forwhichitis
challengingtodevelopefficientiterativesolutionmethods. Takingaccountofthesediffi-
culties, weturntotime-dependentequations. Itisknownthattime-dependentequations
can be simulated with respect to time until a time-harmonic solution is reached, but the
approach suffers from poor convergence. Thus, we accelerate the convergence rate by
employingtheexactcontrollabilitymethod. Theproblemisformulatedasaleast-squares
optimizationproblem,whichissolvedwiththeconjugategradient(CG)algorithm. Com-
putationofthegradientofthefunctionalisdonedirectlyforthediscretizedproblem. A
graph-basedmultigridmethodisusedforpreconditioningtheCGalgorithm.
Theaccuracyofthemethodisimprovedbyutilizinghigher-orderspectralelements
forspatialdiscretization. Thedegreesoffreedomassociatedwiththebasisfunctionsare
situatedattheGauss–Lobattoquadraturepointsoftheelements,andtheGauss–Lobatto
quadratureruleisused. Thisleadstohighaccuracyanddiagonalmassmatrices,thuspro-
vidingcomputationalefficiencywhencombinedwithanexplicittime-steppingscheme.
Thesoftwareimplementationofthemethodsisdonesidebysidewiththemethod
development. Problemsrelatedtothefluid-structureinteractionbetweenelasticmaterials
andacousticwavesareanalyzedandsolvedbycomputersimulations,whichareefficient
toolsinthetestingandoptimizingofmodelparameters. Forinstance,inplanningunder-
waterstructures,thedesignprocesscanbeimprovedandthedevelopmentcycleshortened
withcomputeraidedmodeling.
Keywords:exact controllability, spectral element method, coupled problem, numerical
simulation,fluid-structureinteraction,acoustic,elastic,waveequation
Author SannaMönkölä
DepartmentofMathematicalInformationTechnology
UniversityofJyväskylä
Finland
Supervisors ProfessorTuomoRossi
DepartmentofMathematicalInformationTechnology
UniversityofJyväskylä
Finland
AdjunctprofessorErkkiHeikkola
NumerolaOy
Finland
AdjunctprofessorTimoMännikkö
DepartmentofMathematicalInformationTechnology
UniversityofJyväskylä
Finland
Reviewers ProfessorGaryCohen
INRIAResearchCentre,Rocquencourt
France
AdjunctprofessorTomiHuttunen
DepartmentofAppliedPhysics
UniversityofEasternFinland,Kuopio
Finland
Opponent ProfessorSeppoPohjolainen
DepartmentofMathematics
TampereUniversityofTechnology
Finland
ACKNOWLEDGEMENTS
This thesis was carried out at the Department of Mathematical Information Technology
ofUniversityofJyväskylä. Thewholeprocesshasbeenspicingupmylifewithchalleng-
ing but enjoyable moments, and I wish that this is rather the beginning than the end of
my research work. The doctoral study was financially supported by Finnish Foundation
for Technology, Jyväskylä Graduate School in Computing and Mathematical Sciences,
University of Jyväskylä, National Graduate School in Engineering Mechanics, Alfred
KordelinFoundation,FinnishCulturalFoundation,EllenandArtturiNyyssönenFounda-
tion, Finnish Concordia Fund, and European Community on Computational Methods in
Applied Sciences. I gratefully acknowledge the importance of this support that enabled
thefull-timeresearchandinternationalcontacts. Thefinishingstageofthisresearchwas
partly carried out in the Digital Product Process technology program supported by the
FinnishFundingAgencyforTechnologyandInnovation(TEKES).
I am grateful to my supervisors, Prof. Tuomo Rossi, Dr. Erkki Heikkola and Dr.
Timo Männikkö for their help and useful advice especially at the beginning of my re-
search. I would like to express my gratitude to Prof. Raino A. E. Mäkinen, Dr. Jari
Toivanen, andProf. TimoTiihonenforreadingthemanuscriptandcommentingonit. I
alsothankDr. DirkPaulyforhisguidanceintothescatteringtheoryinexteriordomains
and Prof. Enrique Zuazua for the fruitful discussions considering mathematical control
theory during the 19th Jyväskylä Summer School. I would like to thank the reviewers,
Prof. GaryCohenandDr. TomiHuttunen,fortheirvaluablecomments. Iappreciatetheir
reviews,whichdefinitelyimprovemywork. ThepeopleatNumerolaLtd. areacknowl-
edgedforencouragingmetopursuedoctoralstudiesandgivingmetheopportunitytouse
the Numerrin simulation tool to confirm that my own implementation was functioning
correctly. I am thankful to Dr. Janne Martikainen and Dr. Anssi Pennanen for pro-
viding the multigrid solver that was combined to my code for solving the linear system
involvedatthepreconditioningstage. Ithankthecomputerguys,especiallyHarriTuomi
andTapaniTarvainen,forkindhelpandforkeepingthecomputersrunningandpromptly
repairing the damages. I am also grateful to Tuula Blåfield for the English checking of
this manuscript. I would also like to thank my colleagues for their inspiring company
andfriendship. Ithankmyfriendsforsharingdelightfulmomentsandbeingthereforme
althoughlatelyIhavenothadmuchtimetospendwiththem.
Thelast,andsurelythemost,Iwouldliketothankmyfamilyforalltheirloveand
encouragement. Iamgratefultomyparentsforprovidingsparetimeactivitiesthathave
beenanexcellentcounterbalancetoofficeworkandgivenmesuchanenterprisingwork-
ing attitude. Without that attitude I would probably have given up at least after my sick
leave. Mybrotherisacknowledgedforhissympatheticunderstandingandprovidingthe
computerfacilitiesatthebeginningofmyuniversitystudies. Thehelpfrommyparents
andmybrotherinthecurrentrenovationprojectisalsogratefullyacknowledgedsinceit
broughtmetheopportunityforbetterconcentratingonfinishingthisthesis. Finally,Itake
theopportunitytothankSamiforhislovingsupportbothatworkandinprivatelife.
Jyväskylä,October2011
SannaMönkölä
LISTOFFIGURES
FIGURE1 The physical phenomenon is presented as a mathematical model
which is simulated by using the computer implementation of nu-
mericalsolutionmethods.......................................................... 11
FIGURE2 Shipforunderwaterresearchwithasonarsubmarine.Image:xedos4
/FreeDigitalPhotos.net............................................................. 14
FIGURE3 S(cid:2)cattering(cid:3)by the obstacle Θ in the exterior domain G = R2 \
Θ∪Γ ............................................................................. 32
0f
FIGURE4 ObstacleΘ,domainΩ ,andthetwopartsoftheboundary∂Ω =
f f
Γ ∪Γ ofthedomainΩ ....................................................... 35
0f ef f
FIGURE5 ThedomainΩisdividedintothesolidpartΩ andthefluidpartΩ . 37
s f
FIGURE6 Spectralelementoforderr =1.................................................. 46
FIGURE7 Spectralelementoforderr =4.................................................. 46
FIGURE8 Anexampleoftheglobalnumberingofnodesandelementsinstruc-
tureandfluiddomainsdividedintoquadrilateralelements................ 47
FIGURE9 One-dimensional Lagrangian interpolants of degree r. Each of the
r+1 functions has the value one at one GL quadrature point and
zeroattheotherGLquadraturepoints. ........................................ 50
FIGURE10 Geometryofthedomainandthecoarsestmeshused....................... 57
∞
FIGURE11 Maximum error computed as L -norms versus number of degrees
offreedom. Thenumberoftimestepsisfixedtobe100................... 58
FIGURE12 ThedomainΩisdividedintothesolidpartΩ andthefluidpartΩ .. 60
s f
FIGURE13 CPU time (in seconds) and memory (in kilobytes) consumed for
solving a time-dependent fluid-structure interaction problem with
differentformulations. Thenumberoftimestepsisfixedtobe400,
andsquare-elementmesheswithmeshstepsizeh = 0.1areusedin
bothmedia............................................................................. 61
∞
FIGURE14 Maximum errors, computed as L -norms, with respect to the ele-
mentorder. Thenumberoftimestepsisfixedtobe400,andsquare-
elementmesheswithmeshstepsizeh =0.1areusedinbothmedia. .. 63
FIGURE15 ComparisonbetweenmaximumerrorswiththeCDandtheRKtime
discretizations. Themeshstepsizeisfixedtobeh = 1/20,andthe
timesteprefinementgivesaseriesofnumericalresultswithvarious
lengthsofthetimestepforeachelementorder................................ 67
FIGURE16 AccuracywithrespecttoCPUtimeconsumptions(inseconds)with
theCDandtheRKtimediscretizations.Themeshstepsizeisfixedto
beh =1/20,andthetimesteprefinementgivesaseriesofnumerical
resultswithvariouslengthsofthetimestepforeachelementorder..... 69
FIGURE17 AccuracywithrespecttoCPUtimeconsumptions(inseconds)with
theRKandtheABtimediscretizations.Themeshstepsizeisfixedto
beh =1/20,andthetimesteprefinementgivesaseriesofnumerical
resultswithvariouslengthsofthetimestepforeachelementorder..... 71
FIGURE18 CPU time (in seconds) for computing the gradient by the adjoint
equation technique and by the central finite difference approxima-
tionfortheelementordersr = 1,2,3withtheRKtimediscretiza-
tionschemeandangularfrequencyω =4π.Thenumberoftimesteps
ischosentoeliminatethetemporalerror,andsquare-elementmeshes
withmeshstepsizeh =1/20areusedinbothmedia...................... 78
FIGURE19 Comparisonbetweenthecomputationaleffortofpreconditionedand
unpreconditionedCGalgorithm. CPUtimeinsecondsispresented
withrespecttothenumberofdegreesoffreedom........................... 88
FIGURE20 ProportionofCPUtime(inpercent)requiredbytheGBMGcycles
andcomputingstate(FWD)andadjointstate(BWD)inoneCGit-
eration.................................................................................. 89
FIGURE21 Geometricalshapesoftheobstacles............................................. 91
FIGURE22 Scatteringbyaconvexobstaclewithr =3andh =1/28................ 92
FIGURE23 Scatteringbyanon-convexsemi-openobstaclewithr = 3andh =
1/28. ................................................................................... 93
FIGURE24 Scatteringbyasystemoftwonon-convexsemi-openobstacleswith
r =3andh =1/28................................................................ 93
FIGURE25 Comparison between the convergence histories of the relative eu-
clideannormoftheresidualwithrespecttothenumberofiterations
when the functional (157) is minimized by the preconditioned CG
algorithmwiththenon-symmetricandthesymmetricformulation..... 98
FIGURE26 Values of the functional (157) with respect to the number of itera-
tionsinthecaseofnon-symmetricandsymmetricformulations......... 99
FIGURE27 Errorsbetweenthesolutionofthecontrolalgorithmandtheanalyt-
∞
icalsolutionmeasuredintheL norm.........................................100
FIGURE28 TheelasticcircleΩ surroundedbytheacousticmediaΩ . ............101
s f
FIGURE29 Displacementamplitudesolutionsinthesoliddomain.....................102
FIGURE30 Velocitypotentialsolutionsinthefluiddomain..............................103
FIGURE31 ErrorsbetweenthecontrolalgorithmandtheFourierseriessolution
∞
measuredintheL norm..........................................................104
FIGURE32 Maximum errors obtained in the case of the CD and the RK time-
steppingwithfourdifferentstoppingcriteriaε...............................105
FIGURE33 ErrorswithrespecttoCPUtime(inseconds)withangularfrequen-
ciesω = {2π,4π,8π,16π}suchthatωh =rπ/10......................106