Table Of Content

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 ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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. 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 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 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 6.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.6 CaseStudy-XI Simulationofspinodaldecomposition ofabinaryalloywithfiniteelementmethod. . . . . . . . . . 196 6.6.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.6.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 197 6.6.3 FEMImplementation. . . . . . . . . . . . . . . . . . . . 197 6.6.4 NumericalImplementation. . . . . . . . . . . . . . . . 197 6.6.5 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 199 6.6.6 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 199 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.7 CaseStudy-XII Phase-fieldmodelingofgraingrowthwithfinite elementmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.7.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.7.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 219 6.7.3 FEMFormulation. . . . . . . . . . . . . . . . . . . . . . . 219 6.7.4 NumericalImplementation. . . . . . . . . . . . . . . . 219 6.7.5 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 220 6.7.6 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 222 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 6.8 CaseStudy-XIII Phase-fieldmodelingofinstabilitiesinlayered thinfilms. .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . 237 6.8.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 238 6.8.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 238 6.8.3 FEMFormulation. . . . . . . . . . . . . . . . . . . . . . . 238 6.8.4 NumericalImplementation. . . . . . . . . . . . . . . . 239 6.8.5 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 240 6.8.6 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 243 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 6.9 CaseStudy-XIV Phase-fieldmodelingofmulti-variantmartensitic transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 6.9.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 273 6.9.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 273 6.9.3 FEMFormulation. . . . . . . . . . . . . . . . . . . . . . . 275 6.9.4 NumericalImplementation. . . . . . . . . . . . . . . . 276 6.9.5 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 277 6.9.6 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 279 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 6.10 CaseStudy-XV Phase-fieldmodelingofbrittlefracture. . . . . . . . . . . . . . 303 6.10.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 303 6.10.2 Phase-FieldModel. . . . . . . . . . . . . . . . . . . . . . 303 6.10.3 FEMFormulation. . . . . . . . . . . . . . . . . . . . . . . 305 6.10.4 NumericalImplementation. . . . . . . . . . . . . . . . 306 6.10.5 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 307 6.10.6 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 310 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 7 Phase-FieldCrystalModelingofMaterialBehavior. . . . . . . 337 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 7.2 CaseStudy-XVI NumericalImplementationsofPhase-Field CrystalMethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 7.2.1 Phase-FieldCrystalModel. . . . . . . . . . . . . . . . 340 7.2.2 NumericalImplementation. . . . . . . . . . . . . . . . 340 7.2.3 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 341 7.2.4 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 341 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 7.3 CaseStudy-XVII Phase-FieldCrystalModelingofGrainGrowth. . . . . . . . 355 7.3.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 355 7.3.2 Phase-FieldCrystalModel. . . . . . . . . . . . . . . . 355 7.3.3 NumericalImplementation. . . . . . . . . . . . . . . . 355 7.3.4 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 356 7.3.5 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 356 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 7.4 CaseStudy-XVIII Phase-FieldCrystalModelingofDeformation BehaviorofaBicrystal. . . . . . . . . . . . . . . . . . . . . . . . . 361 7.4.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . 361 7.4.2 NumericalImplementation. . . . . . . . . . . . . . . . 362 7.4.3 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . 362 7.4.4 SourceCodes. . . . . . . . . . . . . . . . . . . . . . . . . . 364 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 8 ConcludingRemarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 AppendixA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 AppendixB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 AppendixC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 AppendixD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 List of Programs and Functions Manyfunctionsarerepeatedlyusedinmanyprogramsgiveninthetext.This listgivestheirfirstappearancesinthetext. Chapter/ Chapter/ Case Page Case Page Program/FunctionName Study Number Program/FunctionName Study Number Chapter4/CaseStudy-V Chapter4 fd_cell_dyna_v1.m 85 heat_1d_fd.m 19 fd_cell_dyna_v2.m 90 micro_poly_cell.m 93 Chapter4/CaseStudy-I free_energ_v1.m 96 fd_ch_v1.m 26 free_energ_v2.m 96 fd_ch_v2.m 28 Chapter5 micro_ch_pre.m 31 heat_1d_fft.m 101 calculate_energ.m 31 Chapter5/CaseStudy-VI free_energ_ch_v1.m 32 fft_ch_v1.m 104 free_energ_ch_v2.m 32 fft_ch_v2.m 107 write_vtk_grid_values.m 32 prepare_fft.m 109 laplacian.m 34 Chapter5/CaseStudy-VII Chapter4/CaseStudy-II fft_ca_v1.m 113 fd_ca_v1.m 39 fft_ca_v2.m 117 fd_ca_v2.m 43 free_energ_fft_ca_v1.m 120 init_grain_micro.m 46 free_energ_fft_ca_v2.m 120 free_energ_fd_ca_v1.m 49 Chapter5/CaseStudy-VIII free_energ_fd_ca_v2.m 50 fft_FeCuNiMn_v1.m 125 Chapter4/CaseStudy-III fft_FeCuNiMn_v2.m 128 fd_sint_v1.m 53 init_FeCuNiMn_micro.m 131 fd_sint_v2.m 61 Fe_Cu_Ni_Mn_free_energy.m 132 micro_sint_pre.m 64 Chapter5/CaseStudy-IX free_energ_sint_v1.m 66 fft_raft_v1.m 144 free_energ_sint_v2.m 67 fft_raft_v2.m 147 Chapter4/CaseStudy-IV green_tensor.m 149 fd_den_v1.m 72 solve_elasticity_v1.m 150 fd_den_v2.m 77 solve_elasticity_v2.m 153 nucleus.m 79 Chapter5/CaseStudy-X gradient_mat.m 80 fft_FeCr_v1.m 161 vect2matx.m 80 fft_FeCr_v2.m 164 (continued) (continued)

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