Approximation of phase-field models with meshfree methods

234 Pages·2014·5.2 MB·English

by  Christian Peco Regales

APPROXIMATION OF
PHASE-FIELD MODELS
WITH MESHFREE METHODS:
EXPLORING BIOMEMBRANE DYNAMICS
Christian Peco Regales
Doctoral Thesis
Advisor: Marino Arroyo
Barcelona, September 2014
Departament de Matemàtica Aplicada III
Programa de Doctorat en Enginyeria Civil

To my parents
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A good hockey player plays where the puck is. A great hockey player plays where the puck
is going to be.
Wayne Gretzky
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Abstract
Approximation of phase-field models with meshfree methods: exploring
biomembrane dynamics
Christian Peco Regales
Biomembranes are the fundamental separation structure in animal cells, and are
alsousedinengineeredbioinspiredsystems. Theirsimulationischallenging,particu-
larlywhenlargeshapechangesanddynamicsareinvolved,ormicrometersystemsare
considered,rulingoutatomisticorcoarse-grainedmolecularmodeling. Themaingoal
of this thesis is to develop a computational framework to understand the dynamics
of biomembranes embedded in a viscous fluid using phase-field models. Phase-field
modelsintroduceascalarcontinuousfieldtodefineadiffusemovinginterface,whose
physics is encoded in partial differential equations governing it. These models can
dealwithdramaticshapeandtopologicaltransformationsandareamenabletomulti-
physicscoupling. However,theypresentsignificantnumericalchallenges,suchasthe
high-order character of the equations, the resolution of sharp and moving fronts,
or the efficient time-integration. We address all these issues through a combina-
tion of meshfree spacial discretization using local maximum-entropy basis functions,
and a Lagrangian variational formulation of the coupled elasticity-hydrodynamics.
The smooth meshfree approach provides accurate approximations of the phase-field
and can easily deal with local adaptivity, the Lagrangian approach naturally extend
adaptivity to dynamics, and the variational formulation enables nonlinearly-stable
robustvariationaltimeintegration. Thenumericalimplementationofthesemethods
inahigh-performancecomputingframeworkhasmotivatedthedevelopmentofanew
computer code, which integrates state-of-the-art parallel libraries and incorporates
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important technical contributions to overcome bottlenecks that arise in meshfree
methodsforlarge-scaleproblems. Theresultingcodeisflexibleandhasbeenapplied
to other scientific problems in a number of collaborations dealing with flexoelectric-
ity, metal forming, creeping flows, or fracture in materials with strongly anisotropic
surface energy.
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Acknowledgments
I would like to express my special appreciation and thanks to my advisor, Prof.
Marino Arroyo. First, for his guidance, support, enthusiasm and fearless attitude
towards research and work. I have deeply enjoyed the vast majority of the projects
wehaveembracedandalsofelthisoptimism,helpandpatienceatthetoughandun-
certainmomentsthatinevitablyappeartogivevalueandmeaningtoanysignificant
task. Second, Iwouldliketothankhimforhispricelessexamplenotonlywithinthe
professional field but also within the human side. His balanced and flexible way of
dealing with delicate issues regarding the group and the individual has undoubtedly
improvedmyperceptionofleadership,managingandeducation. Iamtrulyindebted
andthankfultothefacultyofLaC`aNandDepartamentdeMatem`aticaAplicadaIII.
In particular, Professors Antonio Huerta, Antonio Rodr´ıguez, Sonia Fern´andez and
JoseMun˜oz. ImetthemattheCivilEngineeringdegreeandallofthemareresponsi-
ble for the seeds that made me love and focus my attention in this new way of doing
science that is Computational Mechanics. Without their motivation and support,
this journey would not have been possible. I would also like to thank the reviewers
of the thesis and the members of the committee for their useful comments and ad-
vice. I am very grateful to the people at LaC`aN for providing such an enjoyable and
stimulatingworkingenvironment. Averyspecialthankstomycolleaguesandfriends
at Omega lab. I’ll never forget the guidance, kindness and sense of humor of Adri´an
Rosolen, the neverending but wise and valuable advices of Daniel Mill´an, the deep
conversationswithSusantaGhoshaboutthe”trueflavourofresearch”,therigorous-
nessofMohammadRahimiinsidethelabandhishappyviewofthelifeoutside,that
wonderful trip to Porto and the “Old Friends” bottle I shared with Amir Abdollahi,
the passion for cleanness, order and beer of Behrooz Hashemian, the enjoyable con-
versationsaboutCataloniaindependencewithAlejandroTorresandeverythingthat
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he uses to bring to the lab when he goes home, the example and kindness of Juan
Vanegas, the wonderful coffee and beach time with Aditya, the James Bond smile of
BinLiwhenhedoesnotunderstandsomething, thebasketballmatchesofKuanand
last but not least, that crazy man known as Dimitri Kaurin. My special thanks also
toDavidModesto,AnaTamayoandCristinaDiaz-Cereceda,amongothercolleagues
of the “far, far side of the Campus”. We spent perhaps too little but truly enjoyable
and genuine time together. I’m going to miss my lunch time with my dear friend
Gonzalo and our hot chocolates in the break of the french course.
I have so much to be grateful for my friends, specially Xavi and David, and my
dear loved ones, my parents, sisters, little brother and brother in law (”Aut viam
inveniam aut faciam!”), who have been a source of unconditional inspiration and
encouragement. During this time I had the privilege of walking through heaven and
hell, and it was in the darkness where I felt you brighter and closer. Life is like
a collective sport, no real success is achieved just by oneself. Thank you for being
my team now and always, everyone of you is a Ph.D. to me. Following with the
sport, let me bewilder yet and again of how much happiness and inspiration could a
pair of blades bring into my existence. Ice and hockey are already part of me, and
so my dear friends Avi (gracias por encontrarme, profeta del hockey!), Judit and
Cristina (hockey girls!), Ram´on, Max, Artur, Octavi, Oriol, Ruth (la campeona),
Sebastian (the doctor), Sebasti´an (el maestro), Anna, Xavi, Carles and Cristina (el
tr´ıomaravilladelstaff),Marc(Ididitmywaaaayyy!!!!!),Petit(locurasobrelapista)
and Jarek (crack!).
AndfinallyIthanktoGod. Orchaos, orchanceorluckorfateorwhateverword
that is able to describe an astonishing chain of circumstances that ends up with you
finding a treasure. In my case this treasure has a name and it is Mar´ıaPaz. Thank
you for such many moments that worth a lifetime. “Le vent se l`eve... Il faut tenter
de vivre!”.
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Contents
Abstract v
Acknowledgments vii
Contents ix
1 Introduction and overview 1
2 Approximation of meshfree phase-field models 9
2.1 Model complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Meshfree methods and the Local Maximum Entropy Approximants . . 12
3 Phase-field modeling of biomembranes 17
3.1 An introduction to biomembranes. . . . . . . . . . . . . . . . . . . . . 17
3.2 Vesicle modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Vesicle statics: equilibrium shapes . . . . . . . . . . . . . . . . . . . . 25
3.4 Vesicle dynamics : an adaptive Lagrangian approach . . . . . . . . . . 32
3.4.1 Lagrangian phase-field model formulation . . . . . . . . . . . . 33
3.4.2 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Complex biological processes : influence of kinetics and adhesion in
vesicle shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.2 Modeling adhesion . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Kinetics and morphogenesis . . . . . . . . . . . . . . . . . . . . . . . . 55
4 High Performance Computing 59
4.1 Supercomputing: towards an efficient parallel sparse LME environment 59
4.1.1 Neighborhood coarsening algorithm . . . . . . . . . . . . . . . 61
4.1.2 Compressed meshfree basis functions storage . . . . . . . . . . 64
4.1.3 Meshfree parallel sparse matrices in PETSc . . . . . . . . . . . 67
4.2 A brief code overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
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5 Other applications 75
5.1 Stabilization of Stokes equations with LME approximants . . . . . . . 75
5.2 A stabilized formulation for viscoplastic flow in metal forming . . . . . 81
5.3 Computational evaluation of the flexoelectric effect in dielectric solids 84
5.4 Fracture in brittle materials of anisotropic surface energy . . . . . . . 89
6 Concluding remarks and future directions 93
6.1 Conclusions and future directions . . . . . . . . . . . . . . . . . . . . . 93
6.2 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References and list of figures 97
Appendix A
An adaptive meshfree method for phase-field models of biomem-
branes. Part I: approximation with maximum-entropy approxi-
mants. 119
Appendix B
An adaptive meshfree method for phase-field models of biomem-
branes. Part II: a Lagrangian approach for membranes in viscous
fluids. 149
Appendix C
Efficient implementation of meshfree Galerkin methods for large-
scale problems with an emphasis on maximum entropy approxi-
mants. 177
Appendix D
Meshfree Parallel Algorithms 211
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