Table Of ContentLecture Notes on Numerical Methods
in Engineering and Sciences
Genki Yagawa
Atsuya Oishi
Computational
Mechanics
with Deep
Learning
An Introduction
Lecture Notes on Numerical Methods
in Engineering and Sciences
SeriesEditor
EugenioOñate ,JordiGirona,1,EdificiC1-UPC,UniversitatPolitecnicade
Catalunya,Barcelona,Spain
EditorialBoard
CharbelFarhat,DepartmentofMechanicalEngineering,StanfordUniversity,
Stanford,CA,USA
C.A.Felippa,DepartmentofAerospaceEngineeringScience,Universityof
Colorado,CollegeofEngineering&AppliedScience,Boulder,CO,USA
AntonioHuerta,UniversitatPolitècnicadeCataluny,Barcelona,Spain
ThomasJ.R.Hughes,InstituteforComputationalEngineering,UniversityofTexas
atAustin,Austin,TX,USA
SergioIdelsohn,CIMNE-UPC,Barcelona,Spain
PierreLadevèze,EcoleNormaleSupérieuredeCachan,CachanCedex,France
WingKamLiu,Evanston,IL,USA
XavierOliver,CampusNordUPC,InternationalCenterofNumericalMethods,
Barcelona,Spain
ManolisPapadrakakis,NationalTechnicalUniversityofAthens,Athens,Greece
JacquesPériaux,CIMNE-UPC,Barcelona,Spain
BernhardSchrefler,MechanicalSciences,CISM-InternationalCentrefor
MechanicalSciences,Padua,Italy
GenkiYagawa,SchoolofEngineering,UniversityofTokyo,Tokyo,Japan
MingwuYuan,Beijing,China
FranciscoChinesta,EcoleCentraledeNantes,NantesCedex3,France
This series publishes text books on topics of general interest in the field of
computationalengineeringsciences.
Thebookswillfocusonsubjectsinwhichnumericalmethodsplayafundamental
role for solving problems in engineering and applied sciences. Advances in finite
element, finite volume, finite differences, discrete and particle methods and their
applicationstoclassicalsingledisciplinefieldsandnewmultidisciplinarydomains
areexamplesofthetopicscoveredbytheseries.
Themainintendedaudienceisthefirstyeargraduatestudent.Somebooksdefine
thecurrentstateofafieldtoahighlyspecialisedreadership;othersareaccessibleto
finalyearundergraduates,butessentiallytheemphasisisonaccessibilityandclarity.
Thebookswillbealsousefulforpractisingengineersandscientistsinterestedin
stateoftheartinformationonthetheoryandapplicationofnumericalmethods.
·
Genki Yagawa Atsuya Oishi
Computational Mechanics
with Deep Learning
An Introduction
GenkiYagawa AtsuyaOishi
ProfessorEmeritus GraduateSchoolofTechnology
UniversityofTokyoandToyoUniversity IndustrialandSocialSciences
Tokyo,Japan TokushimaUniversity
Tokushima,Japan
ISSN 1877-7341 ISSN 1877-735X (electronic)
LectureNotesonNumericalMethodsinEngineeringandSciences
ISBN 978-3-031-11846-3 ISBN 978-3-031-11847-0 (eBook)
https://doi.org/10.1007/978-3-031-11847-0
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Preface
ComputationalMechanics
It is well known that various physical, chemical, and mechanical phenomena in
natureandthebehaviorsofartificiallycreatedstructuresanddevicesaredescribed
bypartialdifferentialequations.Whilepartialdifferentialequationsarerarelysolved
by analytical methods except those under idealized and special conditions, such
numericalmethodsasthefiniteelementmethod(FEM),thefinitedifferencemethod
(FDM), and the boundary element method (BEM) can approximately solve most
partial differential equations using a grid or elements that spatially subdivide the
object.Thus,thedevelopmentofthesenumericalmethodshasbeenacentralissue
incomputationalmechanics.Inthesemethods,thefinenessofthegridorelementsis
directlyrelatedtotheaccuracyofthesolution.Therefore,manyresearchresources
havebeendevotedtosolvinglargersimultaneousequationsfaster,andtogetherwith
the remarkable advances in computers, it has become possible to solve very large
problemsthatwereconsideredtobeunsolvablesomedecadesago.
Nowadays,ithasbecomepossibletoanalyzeavarietyofcomplexphenomena,
expandingtherangeofapplicationsofcomputationalmechanics.
DeepLearning
Ontheotherhand,theadvancesofcomputershavebroughtaboutsignificantdevel-
opmentsinmachinelearning,whichaimsclassifyingandmakingdecisionsbythe
process of finding inherent rules and trends in large amounts of data based on
algorithmsratherthanhumanimpressionsandintuition.
Feedforwardneuralnetworksareoneofthemostpopularmachinelearningalgo-
rithms.Theyhavetheabilitytoapproximatearbitrarycontinuousfunctionsandhave
beenappliedtovariousfieldssincethedevelopmentoftheerrorbackpropagation
learning in 1986. Since the beginning of 21st century, they have become able to
v
vi Preface
usemanyhiddenlayers,calleddeeplearning.Theirareasofapplicationshavebeen
furtherexpandedduetotheperformanceimprovementbyusingmorehiddenlayers.
ComputationalMechanicswithDeepLearning
AlthoughthedevelopmentofcomputationalmechanicsincludingtheFEMhasmade
itpossibletoanalyzevariouscomplexphenomena,therestillremainmanyproblems
thataredifficulttodealwith.Specifically,numericalsolutionmethodssuchasthe
FEMaresolidsolutionmethodsbasedonmathematicalequations(partialdifferential
equations),sotheyareusefulwhenfindingsolutionstopartialdifferentialequations
based on given boundary and initial conditions. However, it is not the case when
estimating boundary and initial conditions from the solutions. In fact, the latter is
oftenencounteredinthedesignphaseofartifacts.
Inaddition,asdeeplearningandneuralnetworkscandiscovermappingrelations
between data without explicit mathematical formulas, it is possible to find inverse
mappingsonlybyswappingtheinputandoutput.Forthisreason,deeplearningand
neuralnetworkshavebeenacceptedinthefieldofcomputationalmechanicsasan
importantmethodtodealwiththeweakpointsofconventionalnumericalmethods
suchastheFEM.
They were mainly applied to such limited areas as the estimation of constitu-
tivelawsofnonlinearmaterialsandnon-destructiveevaluation,butwiththerecent
developmentofdeeplearning,theirapplicabilityhasbeenexpandeddramatically.In
otherwords,afusionhasstartedbetweendeeplearningandcomputationalmechanics
beyondtheconventionalframeworkofcomputationalmechanics.
Readership
Theauthors’previousbooktitledComputationalMechanicswithNeuralNetworks
publishedin2021fromSpringercoversmostoftheapplicationsofneuralnetworks
and deep learning in computational mechanics from its early days to the present
togetherwithapplications ofothermachinelearningmethods.Itsconcisedescrip-
tionsofindividualapplicationsmakeitsuitableforresearchersandengineerstoget
anoverviewofthisfield.
On the other hand, the present book, Computational Mechanics with Deep
Learning:AnIntroduction,isintendedtoselectcarefullysomerecentapplicationsof
deeplearningandtodiscusseachapplicationindetail,butinaneasy-to-understand
manner. Sample programs are included for the readers to try out in practice. This
bookisthereforeusefulnotonlyforresearchersandengineers,butalsoforawide
rangeofreaderswhoareinterestedinthisfield.
Preface vii
StructureofThisbook
The present book is written from the standpoint of integrating computational
mechanicsanddeeplearning,consistingofthreeparts:PartI(Chaps.1–3)coversthe
basics,PartII(Chaps.4–8)coversseveralapplicationsofdeeplearningtocomputa-
tionalmechanicswithdetaileddescriptionsofthefieldsofcomputationalmechanics
towhichdeeplearningisapplied,andPartIII(Chaps.9–10)describesprogramming,
wheretheprogramcodesforbothcomputationalmechanicsanddeeplearningare
discussedindetail.Theauthorshavetriedtomaketheprogramnotablackbox,but
ausefultoolforreaderstofullyunderstandandhandletheprocessing.Thecontents
ofeachchapteraresummarizedasfollows:
PartIFundamentals:
InChap.1,theimportanceofdeeplearningincomputationalmechanicsisgiven
first and then the development process of deep learning is reviewed. In addition,
variousnewmethodsusedindeeplearningareintroducedinaneasy-to-understand
manner.
Chapter2isdevotedtothemathematicalaspectsofdeeplearning.Itdiscussesthe
forwardandbackwardpropagationsoftypicalnetworkstructuresindeeplearning,
such as fully connected feedforward neural networks and convolutional neural
networks,usingmathematicalformulaswithexamples,andalsolearningacceleration
andregularizationmethods.
Chapter 3 discusses the current research trends in this field based on articles
published inseveral journals.Many ofthesearticlesarecompiled inthereference
list,whichmaybeusefulforfurtherstudy.
PartIICaseStudy:
Chapter 4 presents an application of deep learning to the elemental integration
processofthefiniteelementmethod.Itisshownthatageneral-purposenumerical
integrationmethodcanbeoptimizedforeachintegrandbydeeplearningtoobtain
betterresults.
Chapter5introducesamethodforimprovingtheaccuracyofthefiniteelement
solutions by deep learning, showing how deep learning can break the common
knowledgethatafinemeshisessentialtoobtainanaccuratesolution.
Chapter6isdevotedtoanapplicationofdeeplearningtothecontactpointsearch
process in contact analysis. It deals with contact between smooth contact surfaces
definedbyNURBSandB-splinebasisfunctions,showinghowdeeplearninghelps
toaccelerateandstabilizethecontactanalysis.
Chapter7presentsanapplicationofdeeplearningtofluiddynamics.Aconvolu-
tionalneuralnetworkisusedtopredicttheflowfield,showingitsunparalleledspeedy
calculationagainstthatofconventionalcomputationalfluiddynamics(CFD).
Chapter 8 discusses further applications of deep learning to solid and fluid
analysis.
PartIIIComputationalProcedures:
Chapter 9 describes some programs to be used for the application problems:
Sect. 9.1 programs in the field of computational mechanics, such as the element
viii Preface
stiffnessmatrixcalculationprogram,andSect.9.2thoseinthefieldofdeeplearning,
suchasthefeedforwardneuralnetwork,bothofwhicharegivenwithbackground
mathematicalformulas.
Chapter10presentsprogramsfortheapplicationofdeeplearningtotheelemental
integrationdiscussedinChap.4.WiththeseprogramsandthosepresentedinChap.9,
thereadersofthepresentbookcouldeasilytry“ComputationalMechanicswithDeep
Learning”bythemselves.
Tokyo,Japan GenkiYagawa
Tokushima,Japan AtsuyaOishi
May2022
Acknowledgements
WewouldliketoexpressourgratitudetoY.Tamura,M.Masuda,andY.Nakabayashi
for providing the data for Chap. 7. We also express our cordial thanks to all the
colleaguesandstudentswhohavecollaboratedwithusoverseveraldecadesinthe
fieldofcomputationalmechanicswithneuralnetworks/deeplearning:S.Yoshimura,
M.Oshima,H.Okuda,T.Furukawa,N.Soneda,H.Kawai,R.Shioya,T.Horie,Y.
Kanto,Y.Wada,T.Miyamura,G.W.Ye,T.Yamada,A.Yoshioka,M.Shirazaki,H.
Matsubara, T. Fujisawa, H. Hishida, Y. Mochizuki, T. Kowalczyk, A. Matsuda, C.
R.Pyo,J.S.Lee,andK.Yamada.
We are particularly grateful to Prof. E. Oñate (CIMNE/Technical Univ. of
Catalonia,Spain)forhiskindandimportantsuggestionsandencouragementsduring
thepublicationprocessofthisbook.
Tokyo,Japan GenkiYagawa
Tokushima,Japan AtsuyaOishi
ix
