S. Bulent Biner Programming Phase-Field Modeling Programming Phase-Field Modeling S. Bulent Biner Programming Phase-Field Modeling S.BulentBiner IdahoNationalLaboratory IdahoFalls,ID,USA Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com. ISBN978-3-319-41194-1 ISBN978-3-319-41196-5(eBook) DOI10.1007/978-3-319-41196-5 LibraryofCongressControlNumber:2016951933 #SpringerInternationalPublishingSwitzerland2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeor part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway, andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To my wife Zu€lal Preface Thisbookaimstointroducestudentsandscientiststotheprogrammingofthe phase-fieldmethod.Thecodes provided inthebookwillserveasafounda- tion and template for developing other phase-field models with more com- plexity as demanded by their underlying physics. The Matlab/Octave programming language was chosen for the codes presented in the book. This approach provides a very efficient and compact connection with the mathematical formulism and its numerical implementations; moreover, they canbeeasilyexpandedtohigherlevelprogramminglanguages(e.g.,Fortran, C/C++, and MPI). Therefore, this book provides a fast track to numerical implementation of phase-field modeling, the numerical details of which are usuallyomittedinliterature.Particularattentionwasdevotedtocomputational efficiency and clarity during development of the codes, with the latter most often being givena higherpreference. Therefore, ifitisdesired,theycan be furtherimproved,withalittleeffort,andcanbeturnedintoproductioncodes duetotheextremelywell-optimizednatureofMatlab/Octave. This book is not intended to provide an extensive survey of phase-field modelingortobeanexercisebookforMatlab/Octaveprogramming.There areplentyofreferencesandtextbooksthatcoverbothofthesesubjects.Itis hopedthatthisbookwillserveasacookbookforprogrammingofphase-field modeling. However, with this in mind, the book starts with the historical backgroundandfundamentalformulismofthephase-fieldmethod. In the following chapters, three numerical algorithms, namely finite differencing, spectral methods, and finite element methods, are developed with increasing complexity and the capability to solve the equations of the phase-fieldmethod.Foreachalgorithm,thepresentedcasestudiesstartwith a description of the model and are followed by its formulism, numerical implementation, results obtained from the solution, discussion, and the source codes. The same two case studies are repeated with each of the threealgorithms.Anynumericalalgorithm canbecharacterizedintermsof its properties, which are accuracy, flexibility to handle many different problems,robustness,andcomputationalefficiency.Veryoften,itisdifficult toachieveallofthesepropertiesinoneparticularalgorithm.Theaimofthis book is that the reader, after hands-on experience, can understand the fine details and strengths and weaknesses of each given solution’s methods sufficientlywellandcanadoptthesuitablealgorithmneededintheirmodels. vii viii Preface This book facilitates a comparative study with a collection of algorithms. Chapter7isdevotedtoamorecomplexandnewclassofphase-fieldmodels termed “Phase-Field Crystal Method.” This method bridges the traditional phase-field theory to the atomistic scale, enabling simulations on diffusive timescalesthatarenotachievablewithcurrentatomisticsimulations. The computer programs and background materials discussed in each sectionofthisbookalsoprovideaforumforundergraduatelevelmodeling– simulation courses as part of their curriculum. Even though there are no specific exercises provided, if desired, exercises can be easily devised in variety of ways (for example, by modifying either material-specific propertiesorsimulation-specificproperties). Of course, writing this book is the product of many stimulating discussionswithcolleagues,graduatestudents,andresearchassociatesover theyears,andalltheirvaluablecontributionsaregreatlyappreciated.Special thankstoMichaelLuby,senioreditorofSpringer,forhispatience,skills,and recommendations;itwasagreatpleasureworkingwithhim. IdahoFalls,ID S.BulentBiner Contents 1 AnOverviewofthePhase-FieldMethod andItsFormalisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 IntroductiontoNumericalSolutionofPartial DifferentialEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 BasicNumericalMethods. . . . . . . . . . . . . . . . . . . . . . . 9 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 PreliminariesAbouttheCodes. . . . . . . . . . . . . . . . . . . . . . . 13 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 SolvingPhase-FieldModelswithFinite DifferenceAlgorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 One-DimensionalTransientHeatConduction: ASolutionwithFiniteDifferenceAlgorithm. . . . . . . . . . 19 4.3 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 CaseStudy-I Simulationofthespinodaldecompositionofabinary alloywithexplicitEulerfinitedifferencealgorithm. . . . . 21 4.4.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 22 4.4.3 NumericalImplementation. . . . . . . . . . . . . . . . 22 4.4.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 23 4.4.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 26 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.5 CaseStudy-II Phase-fieldmodelingofgraingrowthwithfinite-difference algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 36 4.5.3 NumericalImplementation. . . . . . . . . . . . . . . . 36 ix x Contents 4.5.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 37 4.5.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 39 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 CaseStudy-III Phase-fieldmodelingofsolid-statesintering. . . . . . . . . . 51 4.6.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.6.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 51 4.6.3 NumericalImplementation. . . . . . . . . . . . . . . . 52 4.6.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 52 4.6.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 53 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.7 CaseStudy-IV Phase-fieldmodelingofdendiriticsolidification. . . . . . . 69 4.7.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.7.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 70 4.7.3 NumericalImplementation. . . . . . . . . . . . . . . . 71 4.7.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 71 4.7.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 72 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.8 CaseStudy-V Multicellularsystems,theroleofelasticmismatch andcellmotility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.8.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.8.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 82 4.8.3 NumericalImplementation. . . . . . . . . . . . . . . . 83 4.8.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 83 4.8.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 85 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5 SolvingPhase-FieldModelswithFourier SpectralMethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2 One-DimensionalTransientHeatConduction: ASolutionwithFourierSpectralAlgorithm. . . . . . . . . . 99 5.3 SourceCode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4 CaseStudy-VI Simulationofspinodaldecompositionofabinaryalloy withsemi-implicitFourierspectralalgorithm. . . . . . . . . 102 5.4.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 102 5.4.3 NumericalImplementation. . . . . . . . . . . . . . . . 103 5.4.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 103 5.4.5 SourceCode. . . . . . . . . . . . . . . . . . . . . . . . . . . 104 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.5 CaseStudy-VII Phase-fieldmodelingofgraingrowthwithsemi-implicit Fourierspectralalgorithm. . . . . . . . . . . . . . . . . . . . . . . 110 5.5.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Contents xi 5.5.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 110 5.5.3 NumericalImplementation. . . . . . . . . . . . . . . . 111 5.5.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 111 5.5.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 113 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6 CaseStudy-VIII Phase-fieldsimulationofprecipitationbehavior ofaFe-Cu-Mn-Nialloy. . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 122 5.6.3 NumericalImplementation. . . . . . . . . . . . . . . . 124 5.6.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 124 5.6.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 124 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.7 CaseStudy-IX Theroleofelasticinhomogeneitiesandappliedstress onthephase-separationbehaviorofabinaryalloy. . . . . . 137 5.7.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.7.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 138 5.7.3 SolvingtheEquationofMechanical Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.7.4 NumericalImplementation. . . . . . . . . . . . . . . . 140 5.7.5 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 140 5.7.6 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 144 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.8 CaseStudy-X Theroleoflatticedefectsonspinodaldecomposition ofaFe-Cralloy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.8.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.8.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 156 5.8.3 NumericalImplementation. . . . . . . . . . . . . . . . 157 5.8.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 157 5.8.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 161 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6 SolvingPhase-FieldEquationswithFiniteElements. . . . . . . 169 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.2 IsoparametricRepresentationandNumerical Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.3 IntroductiontoStrongandWeakFormsofFEM Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.3.1 FEMDiscretizationofWeakForm. . . . . . . . . . 172 6.3.2 DiscretizationinTimeforTransient HeatTransfer. . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.4 FEMFormulationofLinearElasticityBased onthePrinciplesofVirtualWork. . . . . . . . . . . . . . . . . . 174 6.4.1 ANumericalExampleforFEMSolution ofLinearElasticity. . . . . . . . . . . . . . . . . . . . . . 176