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Pennsylvania State University
The Graduate School
College of Engineering
COUPLED BIOPHYSICAL SIMULATIONS OF CELL-CELL
INTERACTION AND DRUG DELIVERY MICROROBOTS
A Dissertation in
Bioengineering
by
Byron J. Gaskin
c© 2018 Byron J. Gaskin
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2018
The dissertation of Byron J. Gaskin was reviewed and approved∗ by the following:
Robert F. Kunz
Professor of Mechanical Engineering
Bioengineering Intercollege Graduate Degree Program Faculty
Dissertation Advisor, Chair of Committee
Robert L. Campbell
Associate Research Professor, Applied Research Laboratory
Cheng Dong
Professor of Bioengineering
Head of the Department of Bioengineering
William O. Hancock
Professor of Bioengineering
Professor-in-charge of Bioengineering Graduate Programs
Sean M. McIntyre
Assistant Research Professor, Applied Research Laboratory
Special Member
∗Signatures are on file in the Graduate School.
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Abstract
A computational tool has been developed to model flowing cellular systems and hasbeen applied to direct numerical simulation of microvascular flows with a visiontowards personalized medicine. This tool couples computational fluid dynamics(CFD), computational structural mechanics (CSM), six degree-of-freedom (6DOF)motion, and surface biochemistry (SB), in the context of interface-resolved cellgeometry, to provide a detailed model of the heterogeneous blood flow microenvi-ronment. This tool can be used to study drug-mediated cellular interactions in thevasculature and design magnetically-actuated drug delivery microrobots (DDMRs)with targeting capabilities. The research hypothesis of this dissertation was thatapplying direct numerical simulation to study drug-mediated cellular interactionsand DDMR dynamics can lead to protocols for patient-specific drug treatments,including use of DDMRs.
The goal of this dissertation work was to validate the individual components ofthis tool and explore the capabilities and limitations of the tool being developed.The specific aims of this research were to develop a new coupled interface-resolvedfluid-structure-biochemistry interaction (FSBI) numerical scheme for low Reynoldsnumber vascular flows and perform direct numerical simulations of cell-cell inter-actions and DDMRs undergoing magnetic actuation.
The first specific aim is crucial as the fluid-solid interface must be handledwith care when solving fluid-structure interaction (FSI) problems. This is due tothe inconsistency that arises from solving for velocity in the fluid subdomain anddisplacement in the solid subdomain. Several coupling procedures have been devel-oped, implemented, and evaluated to determine an appropriate coupling strategyfor handling relevant problems of interacting bodies in vascular flow. The FSIformulations have been implemented into a robust, production-ready flow solvercapable of modeling interactions between cells.
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For the second specific aim, complete FSBI problems with multiple cell typesand DDMR designs are presented to show the capabilities of the developed scheme.The input parameters for these problems include initial cell locations, biochemi-cal reaction constants, solver time step, blood cell structural properties, DDMRgeometry, and flow shear rate.
This work provides key innovations over current state-of-the-art, namely anapproach to solve the full flow-structure-biochemistry system by modifying readilyavailable vascular flow solvers, demonstrated ability of finite-volume discretizationof hyperelastic constitutive models for large motion and large strain systems, anddesign and evaluation capabilities for magnetically actuated DDMRs in microvas-cular flow.
Collectively, the developed tool can be used in future work to use patient-specific biomarker data to develop personalized drug treatments protocols, designmagnetically-actuated drug delivery microrobots for optimal tissue targeting, andeducate a reduced-order-model with decreased runtime for greater feasibility ofclinical deployment.
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Table of Contents
List of Figures ix
List of Symbols xiii
Acknowledgments xxi
Chapter 1Introduction 11.1 Background and Significance . . . . . . . . . . . . . . . . . . . . . . 11.2 Innovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Cell Computational Fluid Dynamics Simulations . . . . . . . 61.3.2 Structural Mechanics Modeling . . . . . . . . . . . . . . . . 81.3.3 Fluid Structure Interaction . . . . . . . . . . . . . . . . . . . 101.3.4 Adhesion Biochemistry . . . . . . . . . . . . . . . . . . . . . 111.3.5 Drug Delivery Microrobots . . . . . . . . . . . . . . . . . . . 13
1.4 Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter 2Theoretical Formulation 172.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Cauchy Momentum Equation . . . . . . . . . . . . . . . . . 172.1.2 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.3 Structural Mechanics . . . . . . . . . . . . . . . . . . . . . . 19
2.1.3.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . 202.1.3.2 Nonlinear Elasticity . . . . . . . . . . . . . . . . . 212.1.3.3 General Updated Lagrangian Formulation . . . . . 26
v
2.1.3.4 Slow-Process Updated Lagrangian Formulation . . 292.1.3.5 Slow-Process Updated Lagrangian Boundary Con-
ditions . . . . . . . . . . . . . . . . . . . . . . . . . 312.1.4 Surface Biochemistry . . . . . . . . . . . . . . . . . . . . . . 322.1.5 Rigid Body Motion . . . . . . . . . . . . . . . . . . . . . . . 332.1.6 DDMR Magnetic Actuation . . . . . . . . . . . . . . . . . . 35
2.2 Coupling of Dynamic Systems . . . . . . . . . . . . . . . . . . . . . 372.2.1 Hydrodynamics / Fluid-Structure Interaction . . . . . . . . 37
2.2.1.1 Boundary Conditions on Rigid Bodies . . . . . . . 372.2.1.2 Boundary Conditions on Hyperelastic Bodies . . . 38
2.2.2 Biochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.2.1 Boundary Conditions on Rigid Bodies . . . . . . . 382.2.2.2 Boundary Conditions on Hyperelastic Bodies . . . 39
2.2.3 Modeling of Surface Roughness . . . . . . . . . . . . . . . . 392.2.4 Unified Governing Equation . . . . . . . . . . . . . . . . . . 40
Chapter 3Computational Implementation 413.1 Finite Volume Discretization . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Spatial interpolation of a scalar φ . . . . . . . . . . . . . . . 423.1.2 Explicit spatial gradient of a scalar φ . . . . . . . . . . . . . 433.1.3 Implicit spatial gradient of a scalar φ . . . . . . . . . . . . . 433.1.4 Implicit temporal gradient of a scalar φ . . . . . . . . . . . . 44
3.2 Semi-implicit Six-Degrees-of-Freedom Coupling . . . . . . . . . . . 443.3 Structural Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.1 Fixed-shape Leukocyte Rolling . . . . . . . . . . . . . . . . 463.3.2 Finite-Volume Structural Mechanics Formulation . . . . . . 503.3.3 Finite Volume Linear Elastostatics . . . . . . . . . . . . . . 50
3.3.3.1 Discrete Governing Equations . . . . . . . . . . . . 503.3.3.2 Implementation of Boundary Conditions . . . . . . 51
3.3.4 Finite Volume Nonlinear Elastodynamics . . . . . . . . . . . 513.3.4.1 Discrete Governing Equations . . . . . . . . . . . . 513.3.4.2 Accommodation of Inertial Contributions . . . . . 523.3.4.3 Lagged Correction of Non-linear Terms . . . . . . . 533.3.4.4 Quantification of Constitutive Model Selection . . . 53
3.4 Proximity-based adaptive timestepping . . . . . . . . . . . . . . . . 553.5 Fourier Stability Analysis of Computational Implementations . . . . 56
3.5.1 Analysis of 1D Laplace Equations . . . . . . . . . . . . . . . 573.5.1.1 Jacobi Method . . . . . . . . . . . . . . . . . . . . 583.5.1.2 Relaxed Jacobi Method . . . . . . . . . . . . . . . 60
vi
3.5.1.3 Gauss-Seidel method . . . . . . . . . . . . . . . . . 613.5.1.4 Relaxed Gauss-Seidel Method . . . . . . . . . . . . 61
3.5.2 Analysis of 3D Steady Linear Elasticity Equations . . . . . . 653.5.3 Fourier Stability of Analysis of Multi-Step Solution Procedures 74
3.6 Miscellaneous High Performance Computing Improvements . . . . . 753.6.1 Linear Solver Performance Optimization . . . . . . . . . . . 75
3.6.1.1 Empirically-based Performance Optimization . . . 753.6.1.2 Fourier-based Performance Optimization . . . . . . 77
Chapter 4Verification, Validation, and Results 824.1 Fixed Shape Leukocyte Rolling . . . . . . . . . . . . . . . . . . . . 824.2 Finite-Volume Structural Mechanics . . . . . . . . . . . . . . . . . . 86
4.2.1 3D Linear Elastostatics . . . . . . . . . . . . . . . . . . . . . 864.2.2 Stretching of Prismatic Beam under Self-Weight . . . . . . . 864.2.3 Flexure of Prismatic Beam Due to End Loading . . . . . . . 894.2.4 Saint-Venant Kirchhoff Hyperelasticity . . . . . . . . . . . . 94
4.2.4.1 Hyperelastic Sphere undergoing Rigid Body Rota-tion . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Single Body Simulations . . . . . . . . . . . . . . . . . . . . . . . . 954.3.1 Impulsively Started Rigid TC in Uniform Flow . . . . . . . . 954.3.2 Rigid TC in Linear Shear Flow . . . . . . . . . . . . . . . . 984.3.3 Rigid Helical Microswimmer . . . . . . . . . . . . . . . . . . 103
4.3.3.1 Constant Torque . . . . . . . . . . . . . . . . . . . 1074.3.3.2 Constant Angular Velocity . . . . . . . . . . . . . . 110
4.3.4 Magnetically Actuated Hyperelastic Microbead in CouetteFlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4 Multiple Body Simulations . . . . . . . . . . . . . . . . . . . . . . . 1144.4.1 Free-Flowing Rigid TC and Wall-Adhered Hyperelastic PMN
in Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . 1144.4.2 Rigid Two-cell Aggregate Formation Simulations . . . . . . 118
Chapter 5Conclusions and Future Work 1215.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Appendix A124
A.1 SI Scaling of Microvascular System . . . . . . . . . . . . . . . . . . 124
vii
Appendix B127
B.1 Gradient Reconstruction using Least Square Optimization . . . . . 127B.1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127B.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 131
B.1.2.1 Dirichlet BC . . . . . . . . . . . . . . . . . . . . . 131B.1.2.2 Extrapolation BC . . . . . . . . . . . . . . . . . . 131
Bibliography 132
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List of Figures
1.1 Simulation of heterogeneous biological cell flow. Tumor cells andpolymorphonuclear leukocytes are near the endothelial wall. Redblood cells are near the flow centerline. . . . . . . . . . . . . . . . . 2
1.2 Flowchart of CellCFD-PSU computational tool. . . . . . . . . . . . 41.3 Pictorial of ligand and receptor distribution of interest in the Melanoma-
PMN-Endothelium system. . . . . . . . . . . . . . . . . . . . . . . . 121.4 Types of magnetic microrobots and their actuation methods. Re-
produced from [88] . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1 Microscopy image of PMN attached to endothelial wall in shearflow. The flow profile around the cell is obtained using particleimage velocimetry (PIV) [53] . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Simplified representation of deformation gradient. . . . . . . . . . . 222.3 Simplified representation of deformation gradient decomposition us-
ing updated Lagrangian approach. Total Lagrangian considers ini-tial and deformed configurations only. Updated Lagrangian alsoconsiders a time-dependent intermediary configuration. . . . . . . . 27
2.4 Methods of magnetic actuation. Reproduced from [88] . . . . . . . 352.5 Cell surfaces contain many complex structures which are modeled
using a sub-grid fictitious repulsion force. [28,103] . . . . . . . . . . 39
3.1 Arbitrarily shaped polyhedra with shared face. . . . . . . . . . . . 423.2 Analytic transformation from a sphere to deformed PMN. . . . . . 483.3 Flowchart of fixed-shape PMN rolling algorithm . . . . . . . . . . . 483.4 Spectral radii of relaxed Gauss-Seidel applied to 1D Laplace’s equa-
tion as function of wavenumber . . . . . . . . . . . . . . . . . . . . 633.5 Maximum spectral radius of relaxed Gauss-Seidel applied to 1D
Laplace’s equation as function of relaxation factor. Spectral radiiare obtained by sampling ρk at wavenumber π/64. . . . . . . . . . . 64
3.6 Spectral Radius of relaxed Jacobi method applied to linear elasticityas function of φy and φz when φx ≈ 0.0 . . . . . . . . . . . . . . . . 71
ix
3.7 Spectral Radius of relaxed Jacobi method applied to linear elasticityas function of φy and φz when φx = π . . . . . . . . . . . . . . . . . 72
3.8 Spectral radius of relaxed Jacobi method applied to linear elasticityas function of wavenumber when φx = φy = φz for various relax-ation factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.9 Sample residual profile showing solution root mean square (RMS)error as a function of solver iteration number . . . . . . . . . . . . . 76
3.10 Estimated performance increase of two-step relaxed Jacobi methodapplied to 3D linear elasticity equations. Two-step methods showthe potential of 78% increase in performance over the single steprelaxed Jacobi method. . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1 Trajectory of point on the surface of rolling PMN. . . . . . . . . . . 834.2 X coordinate of surface point vs time during PMN rolling. . . . . . 844.3 Y coordinate of surface point vs time during PMN rolling. . . . . . 854.4 Pictorial representation of prismatic beam stretching under self-
weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5 Deformation of bar due to self weight. . . . . . . . . . . . . . . . . . 884.6 Grid convergence plot for stretching of beam due to self weight. . . 894.7 Pictorial representation of prismatic beam experiencing flexure due
to end loading T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.8 Deformation of bar due to flexure traction loading at bar end. . . . 924.9 Grid convergence plot for flexure of beam due to end loading. . . . 934.10 Surface mesh of sphere. . . . . . . . . . . . . . . . . . . . . . . . . . 944.11 Pictorial representation of computational domain and boundary
conditions for rigid TC in uniform flow. . . . . . . . . . . . . . . . . 964.12 Angled view of TC and slice of flow field along the centerline of
the computational domain at t = 10µs. Slice is colored by velocitymagnitude and flow is in the +x direction. Gradients in the velocityfield indicate the TC has not yet reached Uinf . . . . . . . . . . . . 97
4.13 Analytic and computed velocity profiles of rigid TC in uniform flowas function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.14 Pictorial representation of computational domain and boundaryconditions for rigid TC in linear shear flow. . . . . . . . . . . . . . . 99
4.15 Angled view of TC and slice of flow field along the centerline ofthe computational domain at t = 1ms. Slice is colored by velocitymagnitude and flow is in the +x direction. Gradients in the velocityfield are due to the prescribed flow shear rate. . . . . . . . . . . . . 100
x
4.16 Trajectory of TC centroid in linear shear flow. Dashed vertical linesrepresents the end of the modeled computational domain. Cyclicconditions are used to increase the effective flow length. . . . . . . . 101
4.17 Velocity profile of TC centroid in linear shear flow. Simulation endsas TC collides into the top wall of the computational domain. Walleffects appear in the profile as signal noise towards the end of thesimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.18 Schematic of helical microswimmer with labeled geometric entities.Helical geometries are defined by helix angle (θ), helix radius (R),filament radius (r), helix pitch (λ), and number of turns (n) . . . . 104
4.19 Mesh of the helical microswimmer used in the simulations per-formed in this work. Geometric properties of this helix are foundin Table 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.20 Mesh of the helical microswimmer and slice of the meshed fluiddomain used in the simulations performed in this work. Pointson the interface of the fluid and helix domains match exactly asconformal meshing is used at the interface. . . . . . . . . . . . . . . 106
4.21 Helix in flow and slice of flow field along the centerline of the compu-tational domain at t = 97µs. Slice is colored by the z-componentsof velocity. Red indicates flow coming out of the page and blueindicated flow going into the page. . . . . . . . . . . . . . . . . . . 108
4.22 Comparison of computed axial velocity profile and predictions basedon the Abbott model for a helical microswimmer undergoing con-stant torque. These models are in agreement with a slight differencein the profile slope. This difference is likely caused by the idealizedviscous drag approximation in the Abbott model. . . . . . . . . . . 109
4.23 Comparison of computed normalized axial velocity profile for a he-lical microswimmer undergoing constant angular velocity. Velocityis normalized by the Abbott model prediction. The steady-statevelocity asymptotes to approximately 0.86. . . . . . . . . . . . . . . 111
4.24 Pictorial representation of computational domain for wall-adjacenthyperelastic microbead in Couette flow. . . . . . . . . . . . . . . . . 112
4.25 Side view of the hyperelastic microbead and slice of flow field alongthe centerline of the computational domain. The microbead hasreached its equilibrium shape. Flow is in the +x direction . . . . . . 113
4.26 Comparison of initial and final shape of wall-adjacent magnetic mi-crobead in linear shear flow. The final shape has a flat bottomsurface and has evolved to an equilibrium shape best suited for thisflow region. Flow is from left to right. . . . . . . . . . . . . . . . . . 114
xi
4.27 Pictorial representation of computational domain for free-flowingrigid TC and wall-adhered hyperelastic PMN in Couette flow. . . . 115
4.28 Rigid TC and wall-adhered hyperelastic PMN at various times dur-ing the simulation. The bodies are colored by magnitude of dis-placement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.29 Rigid TC, wall-adhered hyperelastic PMN, and slice of flow fieldalong centerline of computational domain at t = 9, 320µs. Theslice is colored by magnitude of flow velocity. . . . . . . . . . . . . . 117
4.30 TC velocity and number of bonds as a function of time. Initialdrop in velocity indicates collision between TC and PMN. TC-PMNaggregate forms when TC velocity reaches zero. . . . . . . . . . . . 119
B.1 Uniform rectilinear computational compact stencil. . . . . . . . . . 128
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List of Symbols
Abbreviations
2PK Second Piola-Kirchhoff
6DOF Six degree-of-freedom
ALE Arbitrary Lagrangian-Eulerian
CFD Computational Fluid Dynamics
CSM Computational Structural Mechanics
DDMR Drug Delivery Microrobot
FBM Flowing Blood Models
FEM Finite Element Method
FSA Fourier Stability Analysis
FSBI Fluid-Structure-Biochemistry Interaction
FSI Fluid-Structure Interaction
FVM Finite Volume Method
GS Gauss-Seidel Method
ICAM Intercellular Adhesion Molecule
J Jacobi Method
LE Linear Elasticity
Pa Pascals
xiii
PDE Partial Differential Equation
PMN Polymorphonuclear Leukocytes
RBC Red Blood Cell
RBF Radial Basis Function
Re Reynolds Number
RGS Relaxed Gauss-Seidel Method
RJ Relaxed Jacobi Method
RMS Root-mean Squared
SPUL Slow-process updated Lagrangian
SVK Saint Venant-Kirchhoff
TC Tumor Cell
UL Updated Lagrangian
Greek Characters
α Wavenumber per unit length (Fourier stability analysis)Constant speedup (solver performance optimization)
αa Constant for material domain a
βa Maximum spectral radius for solver configuration a
γ Flow shear rate
γn Lighthill drag approximation on thin rigid helical bodies
γp Lighthill drag approximation on thin rigid helical bodies
δij ij-th component of Kronecker delta
ε Constant (radial basis function)
εnx Solution roundoff error of scalar φ at location x and iteration n assuminginfinite mathematical precision
xiv
εnx Fourier transform of solution roundoff error of scalar φ at location x anditeration n assuming infinite mathematical precision
εijk ijk-th component of Levi-Civita symbol (i.e., permutation symbol)
θ Wavenumber (Fourier stability analysis)Helix angle (DDMR)
θi i-th component of angular displacement vector
λ Lame Constant (solid domain)Equilibrium distance (biochemistry)Helix pitch (DDMR)
µ Fluid viscosity (fluid domain)Shear modulus (solid domain)
µ Material magnetic moment per atom
ν Poisson’s ratio
ρ DensitySpectral radius (Fourier stability analysis)
ρk Spectral radius at wave mode k (Fourier stability analysis)
ρak Spectral radius at wave mode k for method a (Fourier stability analysis)
σij ij-th component of Cauchy stress tensor
τij ij-th component of shear tensor
φ Arbitrary scalar
φx Arbitrary scalar at location x
φnx Arbitrary scalar at location x and iteration n assuming infinite mathemat-ical precision
φnx Arbitrary scalar at location x and iteration n assuming finite mathematicalprecision
χ Deformation vector from reference to deformed configurations
Ψ Hyperelastic strain energy function
xv
ω Relaxation factorAngular velocity
ωimp Relaxation factor of semi-implicit 6DOF motion solver
Roman Characters
A Arbitrary constant
At Arbitrary constant at time t
Ax Prescribed shape constant (PMN shape transform)Stability Constant (Fourier stability analysis)
Ay Prescribed shape constant (PMN shape transform)Stability Constant (Fourier stability analysis)
Az Stability Constant (Fourier stability analysis)
A Arbitrary second-order tensor
b Nonlinear spring constant
bi i-th component of body force vector
B Arbitrary constant
Bt Arbitrary constant at time t
Bi i-th component of magnetic field vector
Bx Stability Constant (Fourier stability analysis)
By Prescribed shape constant (PMN shape transform)Stability Constant (Fourier stability analysis)
Bz Stability Constant (Fourier stability analysis)
c Constant
Ca Body configuration a
Cx Stability Constant (Fourier stability analysis)
Cy Stability Constant (Fourier stability analysis)
xvi
Cz Stability Constant (Fourier stability analysis)
C Right Cauchy-Green deformation tensor
d Distance between points
dc Distance between centroids of two cells
Dx Stability Constant (Fourier stability analysis)
Dy Stability Constant (Fourier stability analysis)
Dz Stability Constant (Fourier stability analysis)
ej j-th component of unit normal along bond line of action
E Modulus of elasticity
E Green-Lagrangian strain tensor
E′′ij ij-th component of Green-Lagrangian strain tensor from meshed to de-
formed (unknown) configurations
f(x) Arbitrary function of x
f cost Optimization routine cost function
f ref Reference value of optimization routine cost function
f bondj j-th bond formed
fnk Finite-difference coefficient for k-th order derivative at the n-th grid point
Fi i-th component of force vector
f′(x) First spatial derivative of arbitrary function of x
F′ij ij-th component of deformation gradient from referenced to meshed con-
figurations
F′′ij ij-th component of deformation gradient from meshed to deformed (un-
known) configurations
Fij ij-th component of deformation gradient
F Deformation gradient
xvii
g Gravity
Gk Amplification factor of wave mode k
h Grid spacing
H′′ij ij-th component of displacement gradient from meshed to deformed (un-
known) configurations
Ii i-th tensor invariant
I Identity tensor
J Determinant of deformation gradient
k Wave mode
kb Boltzmann constant
kon Affinity of a molecule to form a bond
k0on Affinity of a molecule to form a bond under equilibrium conditions
koff Affinity of a molecule to break a bond
k0off Affinity of a molecule to break a bond under equilibrium conditions
lp(a, b) lp norm between vectors a and b
L Length of domain
m MassMeter (scale)
msat Saturation magnetization
Mi i-th component of magnetization vector
n Number of turns in helix (DDMR)
nL Number of molecules per surface area
ni i-th component of outward unit normal vector
nfi i-th component of fluid domain outward unit normal vector
nsi i-th component of solid domain outward unit normal vector
xviii
~n Unit normal vector of shared face of polyhedra
N Number of points in domain (Fourier stability analysis)
NA Avogadro’s number
p Fluid pressureProbability (biochemistry)Arbitrary point (adaptive timestepping)
P First Piola-Kirchhoff stress tensor
Qaij ij-th component of explicit surface stress tensor for material a
r RadiusHelix filament radius (DDMR)
ri i-th component of radius
rn Error residual at n-th iteration of linear solver
~r12 Vector from the centroid of polyhedron 1 to the centroid of polyhedron 2
δr1 Vector from the centroid of polyhedron 1 to shared polyhedra face
R Helix radius (DDMR)
s Spring constantSecond (scale)Estimated speedup (solver performance optimization)
S Surface
sts Transition spring constant
Si i-th component of surface outward area vector
S′ij ij-th component of second Piola-Kirchhoff stress tensor relating refer-
enced and meshed configurations
S′′ij ij-th component of second Piola-Kirchhoff stress tensor relating meshed
and deformed (unknown) configurations
Sij ij-th component of second Piola-Kirchhoff stress tensor
~S Surface area vector pointing from polyhedron 1 to polyhedron 2
xix
S Second Piola-Kirchhoff stress tensor
t Time
∆t Timestep
∆t∗ Estimated time scale (adaptive timestepping)
T Temperature
Ti i-th component of traction vector
u Linear velocity (DDMR)
ui i-th component of velocity (fluid domain)i-th component of displacement (solid domain)
uai i-th component of displacement on configuration a
v Deformed volume (solid domain)
vai i-th component of velocity of point P a (adaptive timestepping)
va′i i-th component of velocity of point P a relative to point P b (adaptive
timestepping)
V Volume (fluid domain)Reference volume (solid domain)
wi Weight associated with i-th radial basis function
xi i-th spatial componenti-th component of linear displacement vector (rigid body motion) i-thapproximation of value x
xai i-th spatial component in configuration a
x Point on deformed configuration
∆x Increment in x direction
X Point on reference configuration
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Acknowledgments
I would like to thank Dr. Kunz for having provided me with such a great oppor-tunity. My Penn State journey began with a summer internship in Dr. Kunz’sresearch group and has led to immense growth not only as a person but also as aresearcher.
I would like to thank my committee members, Dr. Cheng Dong, Dr. WilliamHancock, Dr. Robert Campbell, and Dr. Sean McIntyre for their helpful commentsand insights throughout this process. They have each had a great impact on howI view and approach my research.
I would like to thank the Applied Research Laboratory and everyone at theGarfield Thomas Water Tunnel. The Laboratory’s Eric Walker Assistantship pro-vided most of my funding throughout graduate school, for which I am extremelygrateful.
I would like to thank the Penn State Office of Graduate Educational EquityPrograms (OGEEP) and all the college multicultural offices at Penn State. Theseoffices play a huge role in the lives of underrepresented minority students at PennState by ensuring we have the resources needed to excel.
I would like to thank the Alfred P. Sloan Foundation’s Minority Ph.D. Pro-gram for providing me with financial support and great professional developmentopportunities.
I would like to thank all the friends I have made during my time at Penn State.They have all made this a wonderful experience.
Finally, I would like to thank my family for always supporting my dreams.They have always held me to my word and pushed me to achieve all that I set outto accomplish.
xxi
“I have no special talent. I am only passionately curious.”
— Albert Einstein
xxii
Chapter 1Introduction
In medicine, the treatment options used for a specific condition can have sig-
nificant impacts on a patient’s health outcomes and ongoing quality-of-life with
each option having its own set of advantages and disadvantages. A data-driven
simulation-based tool that accounts for patient-specific parameters can help in
creating personalized treatment strategies. In the past several decades, there has
been significant interest in patient-specific medical treatment strategies [1–3] with
approaches including (but not limited to) genomics-based profiling [4–7], clinical
endpoint based profiling (using information such as patient’s history of health and
disease) [8–10], and biomarker based profiling [11–13]; this study takes a biomarker-
based approach to personalized medicine.
1.1 Background and Significance
The scope of this research is focused on conditions that involve the interactions
of cells, drugs, and drug delivery microrobots (DDMRs) in microvascular flow.
Developing a better understanding of cellular interaction in heterogeneous flow
environments, like the system depicted in Figure 1.1, is critical to developing new
medical treatments as these interactions may cause complex changes in the overall
system dynamics. Any change to the cellular microenvironment may alter variables
such as flow properties, body geometries, or biochemical reaction rates which, in
2
turn, affect the rates of cellular interaction.
Figure 1.1: Simulation of heterogeneous biological cell flow. Tumor cells and poly-morphonuclear leukocytes are near the endothelial wall. Red blood cells are nearthe flow centerline.
Understanding the coupling between the cellular flow microenvironment and
rates of cellular interactions is also useful in the design and evaluation of DDMRs.
Many DDMR designs have been proposed in the literature but there is a lack
of computational platforms available to evaluate and compare arbitrary designs.
Some of the proposed designs include bio-inspired synthetic microstructures, re-
programmed bacteria [14–17], and microbiorobots (i.e., aggregates of synthetic mi-
crostructures and flagellated bacteria) [16], with each design being quite different
from the others. Several groups have developed preliminary models to determine
flow characteristics of DDMRs. However, these models are usually design specific,
often approximate blood as being a monophasic Newtonian fluid without account-
ing for the presence of discrete cells, and neglect cellular interactions that occur
in the cellular flow microenvironment. There is also work to be done on optimiz-
ing magnetic actuation fields for specific DDMR designs. The computational tool
developed through this research can be used to evaluate the preliminary DDMR
3
models presented in the literature and develop models from novel DDMR designs.
Another major thrust of this research is the development of interface resolved
fluid-structure interaction (FSI) capabilities to better capture the influences of the
cell membrane, cell molecular scale structure, and DDMR design on the overall
system dynamics. This work focused on developing FSI formulations and test
cases for physiologically-relevant flows and structures. For initial feasibility stud-
ies, these FSI models were developed for Newtonian fluids and linear elastic solids.
There has been some promising work done on the feasibility of applying finite vol-
ume formulations to linear elasticity equations [18–24] with some work focusing
specifically on FSI applications [25–27] that helped in this effort. The FSI mod-
els were subsequently enriched with more sophisticated treatments of structural
mechanics.
A general workflow for the computational tool is shown in Figure 1.2. There
have been many changes to the workflow compared to earlier versions of this tool
[28, 29]. One large area of modification is in the “Solve coupled CFD / CSM /
Biochem / 6DOF Problem” block. Within this block, numerics of each physically-
and biologically-relevant subsystem have been implemented and accommodated in
a way that efficiently solves the complete biological cell flow problem; namely, this
work focused on the coupling of the CFD and CSM through the FSI and 6DOF
formulations.
Solving the CSM problem required additional modification to the meshing
strategy being used. The computational tool implemented here recomputes the
fluid domain mesh at every timestep and retains the the solid body meshes through-
out the simulation. Retention of the solid body meshes allows for easier accumu-
lation of deformation over time. The computed displacement field within the solid
body is then used to update the solid mesh at the end of each timestep. This mesh-
ing strategy allows for the accomodation of large deformation structural mechanics
models which typically rely on stress history in the body.
4
Figure 1.2: Flowchart of CellCFD-PSU computational tool.
1.2 Innovation
The tool developed here resolves the physics and biology necessary to study the
effects of interactions among micrometer scale bodies in vascular flow. One impor-
tant feature of this tool is interface-resolved modeling of arbitrarily-shaped bodies
through the use of adaptive conformal meshing; this gives the tool an advantage
not only in analyzing novel DDMR designs but also in studying diseases that cause
irregular cell morphologies (e.g., sickle-cell anemia [30]).
In the analysis of DDMRs, the tool’s capablility of modeling microrobots that
are fully synthetic (e.g., microfabricated structures), fully biological (e.g., repro-
grammed bacteria), or synthetic-biological aggregates (e.g., microfabricated struc-
tures with monolayer coating on bacterial swarmer cells) is demonstrated.
In the development of an interface-resolving FSI formulation for low Reynolds
number flow, this research has explored the feasibility of implementing these FSI
5
formulations into existing, production-ready vascular flow solvers. This research
has also explored techniques to increase computational efficiency of these imple-
mented FSI formulations.
Key innovations over current state-of-the-art:
• Developed an approach to solve the full flow-structure-biochemistry system
by modifying readily available vascular flow solvers.
• Demonstrated the ability of finite-volume discretization of hyperelastic con-
stitutive models for large motion and large strain systems.
• Designed and evaluated capabilities for magnetically actuated DDMRs in
microvascular flow.
1.3 Literature Review
In the last several decades, there have been many attempts at modeling flowing
blood systems. Computational modeling of flowing blood systems provide valuable
insight into a variety of disease states and mechanisms. These models can advance
the understanding of disease by providing data typically unavailable through clin-
ical or experimental approaches. Example of conditions that have benefited from
the use of flowing blood models (FBMs) include sickle cell anemia [30–37], throm-
bus formation, growth, and dislodging [38–42], malaria [33, 43], hemolysis [44],
and cancer metastasis [28,45,46]. These models can additionally lead to potential
improvements in drug and therapy development for these diseases.
There are, however, significant scientific challenges associated with the devel-
opment of FBMs. Complex internal cell structures, wide-ranging fluid regimes,
and rich biochemistry must all be coupled to achieve a physically correct model.
The coupling of these nonlinear systems often leads to large computational re-
quirements that can make large scale simulation intractable. Each of these physics
6
may have effects acting across various space and time scales leading to issues of
multiscale resolution.
This work began as an effort to model cancer metastasis [28,45,46] and has since
evolved to a more general FBM which incorporates the key innovations previously
stated. The implementation of a finite volume interface-resolved large deformation
fluid-structure-biochemistry (FSBI) coupling is particularly noteworthy. Literature
reviews for each component of this FBM are presented next.
1.3.1 Cell Computational Fluid Dynamics Simulations
Literature covering cell fluid mechanics often assumes blood to be a non-Newtonian
fluid [47–49]. The non-Newtonian assumption is valid for capturing the aggre-
gate behavior of blood plasma along with all of the blood constituents (e.g., cells,
platelets, dissolved proteins). These blood constituents are explicitly modeled in
this work and their effects are captured by their corresponding models. As such,
the flow model will only be responsible for capturing the behavior of the Newtonian
blood plasma [28,29,45,47–54].
While Newtonian models are more straightforward than non-Newtonian mod-
els, there is still significant computational cost associated with solving the full
Navier-Stokes equations which contributes to that of the overall FBM. There are
a number of methods available to solve the Navier-Stokes equations for blood flow
systems including particle methods, Lattice-Boltzmann methods, interface cap-
turing methods, hydrodynamic approximation functions, and interface resolved
methods.
Particle methods are popular in FBMs due to their ability to handle systems
with various fluids or materials of differing physical properties [55–57]. This class
of method is Lagrangian in nature. Discrete particles are tracked in time as they
interact with each other and the domain boundaries. Particle differential opera-
tors are applied to the equations governing conservation of mass, momentum, and
energy producing the necessary particle interaction terms. These operators can be
derived to account only for a given particle and its neighbors within a specified
7
radius [58]. An advantage to using this type of method is that material interfaces
need not be resolved. Setting the correct material properties for each particle dur-
ing case initialization allows for interface dynamics to be captured through particle
interactions. A primary disadvantage of these methods is their inability to handle
fine, complex structures.
Lattice-Boltzmann methods are similar to particle methods but solve the dis-
crete Boltzmann equations instead of the Navier-Stokes equations [59]. Many of
the advantages of using particle methods still apply to this class of methods. An
additional advantage of using Lattice-Boltzmann methods is the ability to run
on massively parallel computing systems by design. Lattice-Boltzmann methods,
unlike particle methods, uses a fixed lattice (i.e., mesh) to create a number of
volumes within the domain. These volumes are filled with a finite number of parti-
cles that evolve in time similarly to the aforementioned particle methods. Each of
these volumes can be solved in parallel with the use of coupling terms that share
information across volume faces.
Interface capturing methods are another widely used approach to model mul-
tiphase flows where all interfacial scales are fully resolved. This class of methods
uses an Eulerian approach, where the Navier-Stokes equations are solved on a fixed
mesh. Effects of interfaces and membranes within the domain are incorporated into
the flow solution through the use of source terms representing the sub-grid scale
dynamics (e.g., surface tension). Examples of interface capturing methods are
volume-of-fluid (VOF) [60] and immersed boundary (IB) [61–69].
Hydrodynamic approximation functions are an approach to fluid dynamics
modeling that captures flow effects for a desired flow regime. Using these functions
allows for quick computation of the approximate hydrodynamic force applied to
the blood cells in the flow. This approach has been used extensively [38,44,70–72]
due to its low computational cost despite its inaccuracies as the desired flow moves
away from the flow regime for which the function was developed.
While interface-resolved methods are the most accurate approach to modeling
fluid dynamics in cell-resolving FBMs, they are also the most computationally
8
expensive. Conformal meshing techniques are used to explicitly resolve an interface
through time. The appropriate interface condition can be enforced at the correct
locations in this class of methods. This class of method is well suited for complex
geometries as comformal meshing will be used to resolve all of the desired interfaces;
although, care must be taken to ensure to balance between mesh resolution and
computational cost.
In this work, an interface-conformal finite volume approach will be used [29,45,
52, 73, 74]. This approach uses a discretization of the integral form of the Stokes
equations shown in Equation 2.7. The equations are integrated over each volume
in the domain and transformed to face integrals through the divergence theorem as
described in Section 3.1. This approach has also been implemented into a number
of commercially available CFD packages (e.g., STAR-CCM+ and OpenFOAM).
The procedures and formulations described in the work can be implemented into
a code using the interface-resolved finite volume approach.
1.3.2 Structural Mechanics Modeling
A single fluid dynamics model can be used in this work without regard for the
number of blood cells or blood cell types in the flow domain. A similar approach
cannot be used with the modeling of structural mechanics as each blood cell type
has unique structural properties due to varying molecular and internal structures.
For example, red blood cells (RBCs) are observed to be highly deformable [68,75]
and must be modeled with care, while cancerous tumor cells (TCs) experience
small deformations and can be modelled as rigid bodies [46, 51, 53, 76]. As such,
any approach used for structural mechanics modeling must be able to accommodate
a range of blood cell constitutive models.
Modeling of RBCs is particularly challanging due to their high rates of deforma-
tion. An overview of RBC structural mechanics and modeling has been presented
by Fruend [75]. RBC deformations are important as they play a large role in TC
and polymorphonuclear leukocyte (PMN) margination (i.e., migration to the vessel
wall during blood flow).
9
This work focuses on modeling the near-wall region and assumes the cell mar-
gination has already taken place. This assumption allows for priority to be placed
on capturing the structural deformations of PMNs and TCs near vessel walls.
PMNs are drastically less deformable than RBCs [75]. Previous work on PMN
modeling was able to obtain great understanding of the adhesion biochemistry
mechanisms, even when treating the PMNs as rigid bodies [29, 52]. However,
wall-adherent PMNs experience large deformations due to high shear conditions
with elongations of nearly 150% at wall shear rates of 800s−1 [77]. As such, the
structural mechanics of PMNs must be adequetly captured for modeling cellular
interactions in the near-wall region. One approach to model PMNs is using a pre-
stressed membrane with a viscoelastic Maxwell fluid interior [28,45]. This approach
works well for small deformations but requires unphysically large interior viscosities
to capture large deformations; a phenomenon likely due to the nearly rigid PMN
nucleus. An improvement to this model is a 3-layer system which places a rigid
core within the interior Maxwell fluid [28,45].
This work sought to couple FSBI systems through unified governing equations
and identical discretization operators everywhere in the computational domain.
Many CFD solvers use the finite volume method (FVM) to discretize and solve
fluid systems of interest. This work used structural mechanics models discretized
using FVM to develop FSBI formulations that could be implemented into existing
FVM flow solvers.
Early attempts to develop FVM structural mechanics models used linear elas-
ticity to compute stresses and deformations due to specified loadings on prismatic
beams [18, 23–27]. A substantial amount of work was then done to accomodate
multi-material interfaces [22] using continuity of traction and displacement at the
interface. Much of this work had been done in the context of linear elasticity and
was then extended to large deformation hyperelastic models [78]. The Saint-Venant
Kirchoff hyperelasticity model was discretized for FVM-based solvers using both
total Lagrangian and updated Lagrangian approaches. The difference between the
total and updated Lagrangian approaches are anecdotally shown in Figure 2.3. The
10
updated Lagrangian approach is of interest as it allows the structural mechanics
computations to be performed on a known geometry that can change in time. The
reference geometry in this approach is always set to the meshed deformed body
geometry used in the solver rather than an arbitrary stress-free configuration. De-
tailed description of the total and updated Lagrangian approaches, albeit in the
context of FEM, was presented by Sussman and Bathe [79].
In this work, an updated Lagrangian approach to hyperelasticity is derived
using FVM and coupled to the fluid system using continuity of traction and no-
slip conditions at the fluid-structure interface.
1.3.3 Fluid Structure Interaction
Interface-conformal methods allow for the most general description of multicom-
ponent systems by not only meshing each of the system components but also the
interfaces of each component. This description can be naturally extended to FSI
problems by meshing the flow domain, solid interior, and fluid-solid interface with
a single mesh. This class of FSI methods require mesh motion techniques and are
sometimes referred to as Arbitrary Lagrangian-Eulerian (ALE) methods [80, 81].
The advantage of using ALE methods is the ability to model interface geometry
and solve governing equations in each material exactly as needed. In the context
of cellular interaction simulation, ALE methods also allow for the modeling of
molecular biochemistry at the cell surface in a way that accounts for distribution
and individual interactions between adhesion molecules.
These methods solve the appropriate governing equations in material with nec-
essary boundary conditions at the interface. Practically, this approach could be
implemented in a number of ways.
The first, and most conceptually simple, is to use two seperate solvers (one
for the fluid and one for the solid) each using the subset of the conformal mesh
corresponding to its material domain [45]. Information from either solver can be
passed to the other for use at the interface boundaries. This is an iterative process
that continues until the two solvers have reached a consistent solution; a process
11
that may be time-consuming and computationally expensive.
An alternative approach is to use a single solver to solve all relevant governing
equations in the domain. This approach reduces communication costs as all infor-
mation transfer can be done using shared memory communication and provides a
straightforward framework in which to model highly complicated multi-material
systems. This approach was used in this work and led to the development of a
Finite Volume FSBI solver.
1.3.4 Adhesion Biochemistry
Protein expression in the vasculature causes activation of ligands and receptors
on the surface of circulating cells. Compatible ligand-receptor pairs may form
biochemical bonds creating adhesion phenomena among the cells and allowing
the formation of cellular aggregates. In the context of PMN-TC-Endothelium
adhesion, the surface molecules of interest would be ICAM-1 (intercellular adhesion
molecule) on the surfaces of the TC and endothelial cells, β-2 integrins on the PMN
surface, and E- and P-selectins on the endothelial cell surfaces [46,50,51,76,77]. A
pictorial representation of the surface molecule distribution can be seen in Figure
1.3.
This system allows for PMN-TC interaction through bonds formed between
ICAM-1 and β-2 while PMN-Endothelial interactions use strong bonds formed
between ICAM-1 and β-2 and weak bonds formed by selectin molecules. TC-
Endothelium adhesion cannot happen directly but rather must be mediated by
PMN adhesion. Simulation of such a system requires adequate modeling of the
relevant adhesion kinetics.
The adhesion kinetics model used in this work uses a probabilistic approach to
bond formation and breakage. The model used is based upon the early work of
Bell, Dembo, and Hammer [70,71,82,83]. Springs are used to model the adhesion
molecules where the bond formation and breakage rates are based on the molecule
separation distance and several empirically obtained coefficients. A uniform dis-
tribution is prescribed for each of the adhesion molecules on the surface of every
12
Figure 1.3: Pictorial of ligand and receptor distribution of interest in theMelanoma-PMN-Endothelium system.
cell in the simulation. Cell surface based models of biochemical adhesion were first
introduced by Hammer [70,71,83].
The work by Hammer and coworkers introduced surface receptor distributions
on idealized PMNs based on estimated geometric parameters and empirically com-
puted surface densities. The general approach introduced by Hammer is used in
this work with some increased complexity. This work does not comformally rep-
resent the microvilli making it necessary to modify the model by appropriately
adjusting the prescribed receptor surface distribution and incorporating a ficti-
tious repulsion force to ensure the cells do not get non-physically close [52, 54].
The adhesion kinetics model must also be consistent with both the cell surface
discretization and the local probabilistic nature of the kinetics as bonds can form
or break at any time during the simulation.
13
1.3.5 Drug Delivery Microrobots
Drug delivery microrobots (DDMRs) are an exciting technological advancement
showing great promise for the future of personalized medicine. Progress has been
made in developing biologically-inspired microscale systems capable of targeted
drug delivery. These microsystems either reproduce desired traits of microorgan-
isms or reprogram microorganisms to perform desired actions [84].
All modern DDMR research begins with the understanding of low Reynolds
number fluid mechanics. The highly viscous nature of these flows require non-
reciprocating motion for propulsion. Unlike free flowing cells, DDMRs must be
capable of more than simply travel along with the fluid flow. Biological motors such
as flagella and cilia have been used as inspiration for creating efficient propulsion
for microrobots. Flagella are typically helical and rotate to achieve propulsion.
Cilia produce locomotion through a non-reciprocal swimming motion consisting of
a powerstroke and recovery stroke.
Design of biologically-inspired motors must consider actuation of motor com-
ponents. Size limitations of these microsystems limit the amount of feasible ac-
tuation methods. A promising mode of actuation is through use of magnetic
fields. Magnetic acuation allows for the development non-invasive targeting ca-
pabities that do not require fuel storage within the microsystem [16]. Figure 1.4
shows various proposed magnetically actuated DDMR design types. Two designs
of magnetically actuated microsystems frequently discussed in the literature are
magnetic microbeads [16,85–88] and flagella-inspired magnetic helical microswim-
mers [16, 85, 86, 88, 89]. Both of these system types are explored in this work and
simulated using the developed computational tool.
14
Figure 1.4: Types of magnetic microrobots and their actuation methods. Repro-duced from [88]
Individual DDMR designs have been well-characterized for cases when the bod-
ies are rigid. Introduction of deformable microsystems presents an additional chal-
lenge to modeling of these systems due to the added fluid-structure interaction
complexity. Attempts have been made to model deformable DDMRs, specifically
elastic-tailed helical microswimmers [90], showing the need for more robust mod-
eling capabilities.
A large obstacle in the clinical use of DDMRs is the localized control of multi-
ple microrobots using a global inpul signal [91]. Clinical applications of targeting
DDMRs may require many microrobots approaching multiple target sites simul-
taneously. Using a single magnetic source to provide localized instruction to mi-
crorobots can be accomplished through the use of complex magnetic fields and
active control algorithms. The computational tool developed in this work provides
a framework in which to develop and enhance active controller systems for DDMR
swarms beyond the traditional experimental approach.
15
1.4 Research Goals
The goal of this research is to develop a computational tool to study drug-mediated
cellular interactions in the vasculature and to design drug delivery microrobots
(DDMRs) with targeting capabilities. Cellular interactions play an important role
in disease management in nearly all medical interventions [92–96]. The research
hypothesis is that applying direct numerical simulation to study drug-mediated cel-
lular interactions and DDMR dynamics can lead to protocols for patient-specific
drug treatments, including use of DDMRs. Development of this tool is an evolution
of previous work to determine the probability of circulating tumor cell vascular ex-
travasation [28,29,45,52,54,73]. This tool advances the current state of technology
by discretely resolving the interfaces of all bodies of interest to better capture the
relevant physics and biochemistry.
The specific aims of this research were to develop a new coupled interface-
resolved fluid-surface-biochemistry interaction (FSBI) numerical scheme for low
Reynolds number vascular flows and perform direct numerical simulations of cell-
cell interactions and DDMRs undergoing magnetic actuation.
The first specific aim is crucial as the fluid-solid interface must be handled
with care when solving FSI problems. This is due to the inconsistency that arises
from solving for velocity in the fluid subdomain and displacement in the solid
subdomain. Several coupling procedures have been developed, implemented, and
evaluated to determine an optimal coupling strategy for handling relevant problems
of interacting bodies in vascular flow. The FSI formulations have been implemented
into a robust, production-ready flow solver [74] capable of modeling interactions
between cells [29, 52,54].
For the second specific aim, complete FSBI problems with multiple cell types
and DDMR designs are presented to show the capabilities of the developed scheme.
The input parameters for these problems include initial cell locations, biochemi-
cal reaction constants, solver time step, blood cell structural properties, DDMR
geometry, and flow shear rate.
Collectively, the developed tool can be used in future work to:
16
• Use patient-specific biomarker data to develop personalized drug treatments
protocols.
• Design magnetically-actuated drug delivery microrobots for optimal tissue
targeting.
• Educate a reduced-order-model with decreased runtime for greater feasibility
of clinical deployment.
Chapter 2Theoretical Formulation
2.1 Governing Equations
2.1.1 Cauchy Momentum Equation
An underlying approximation throughout this work is the use of a continuum de-
scription of the fluid and solid materials. At the heart of the continuum description
is Cauchy’s Theorem which states the existence of a spatial tensor σij such that
Ti = σijnj (2.1)
where ni is the i-th component of unit normal vector at a given spatial location, Ti
is the i-th component of the traction applied at that spatial location. The spatial
tensor σij is referred to as the Cauchy stress tensor.
From this theorem it is possible to derive a description of the local balance of
linear momentum in the form
ρDuiDt
=∂σij∂xj
+ bi (2.2)
where ρ is the material density, ui is the i-th component of material velocity,
and bi is the i-th component of the body force experienced by the material.
Furthermore, it is possible to show that
18
σij = σji (2.3)
to satisfy the balance of angular momentum. A complete derivation of the Cauchy
momentum equation can be found in Chapter 19 of [97]. Starting with Equation
2.2 allows for both fluid and solid models to be obtained by substituting the model
for σij that best describes each material.
2.1.2 Fluid Dynamics
The flow system is being modeled as a highly viscous, incompressible Newtonian
fluid. The Cauchy stress, σij, for such a material can be expressed as:
σfluidij = −pδij + µ
(∂ui∂xj
+∂uj∂xi
). (2.4)
Due to the low Reynolds number (Re � 1), the flow is in the Stoke’s regime
and with negligible momentum. As such, the flow field at any time is the solution
of an elliptic boundary value problem with no time derivative term. Therefore,
it is appropriate that the flow be governed by the steady Stokes and continuity
equations [29]:
µ∂2ui∂xj∂xj
=∂p
∂xi− bi, (2.5)
∂ui∂xi
= 0. (2.6)
In Equations 2.5 and 2.6, µ is the fluid molecular viscosity, ui is the i-th com-
ponent of the velocity vector, xi is the i-th component of the spatial coordinate
vector, bi is the i-th component of the body force vector (per unit volume), and p
is the fluid pressure. Performing a volume integral on Equation 2.5 and applying
the divergence theorem gives:
∫S
µ∂ui∂xj
dSj =
∫S
pfacedSi −∫V
bidV. (2.7)
19
In finite-volume fluid dynamics solvers, Equation 2.7 is solved on every control
volume in the domain. The flow solver used in this tool is valid for this case as
show in Refs [28,74].
2.1.3 Structural Mechanics
As early as the 1950s, scientists have known that biological cells are not perfectly
rigid and have sought to develop models for cellular mechanical properties [98].
Figure 2.1: Microscopy image of PMN attached to endothelial wall in shear flow.The flow profile around the cell is obtained using particle image velocimetry (PIV)[53]
In this work, the first attempt at capturing the effects of structural mechan-
ics was fixed-shape cellular rolling. This approach is described in more detail in
Section 3.3.1. Previous studies have shown that polymorphonuclear leukocytes
(PMNs) adjacent to the vascular wall can form a weak adhesion and roll along the
wall surface [28, 45, 51, 53, 76]. During this rolling, a consistent shear-dependent
shape was observed across a range of experiments; an example of one such shape
can be seen in Figure 2.1.
The structural mechanics can be modeled using steady linear elasticity coupled
with the inter-cellular biochemistry in an effort to add more physical richness to
the computational tool. This linear elasticity approach requires those governing
equations be expressed in a finite-volume formulation for straightforward imple-
mentation into an existing fluid dynamics solver.
20
2.1.3.1 Linear Elasticity
For a compressible linear elastic (Hookean) material, the stress tensor is given by
σij = λ∂uk∂xk
δij + µ
(∂ui∂xj
+∂uj∂xi
), (2.8)
where λ and µ are the Lame constants and ui is the i-th component of the dis-
placement vector. In the case of elastostatics, the governing equation is
∂σij∂xj
+ bi = 0. (2.9)
Substituting σij into Equation 2.9, performing a volume integral, applying the
divergence theorem, and rearranging gives
∫S
µ∂ui∂xj
dSj = −(∫
S
λ∂uk∂xk
dSi +
∫S
µ∂uj∂xi
dSj +
∫V
bidV
). (2.10)
A quick observation shows that Equations 2.7 and 2.10 are of similar form. This
is one indication that it may be possible to add linear elastic capabilities into an
existing finite volume fluid solver with minimal modification. The linear elasticity
formulation will be validated in a modified fluid solver using canonical beam cases.
Implementation of linear elasticity is important in developing a general struc-
tural mechanics approach, although quick analysis shows that linear elasticity itself
is ill-suited for microvascular flow problems. Bodies in microvascular flow environ-
ments may experience large force loadings and large rotations. Large force loadings
become problematic since the resulting deformations may violate the small defor-
mation assumption used to derive the governing equations of linear elastic bodies.
Issues also arise when bodies undergoing large rotations are described using linear
elasticity. Rigid body rotation should not affect the stress field of a body, as it
does not affect the strain field. However, linear elasticity may predict non-zero
stresses resulting from rigid body rotations. For example, consider a body un-
dergoing rotation θ. The displacements associated with this motion are expressed
as,
21
u
v
w
=
(cos θ − 1) − sin θ 0
sin θ (cos θ − 1) 0
0 0 0
x− cx
y − cy
z − cz
. (2.11)
Using this expression of the body displacements, the spatial gradients of the dis-
placement field is expressed as,
∂u
∂x= (cos θ − 1), (2.12)
∂u
∂y= − sin θ, (2.13)
∂v
∂x= sin θ, (2.14)
∂v
∂y= (cos θ − 1), (2.15)
∂u
∂z=∂v
∂z=∂w
∂x=∂w
∂y=∂w
∂z= 0. (2.16)
Subsituting these values into Equation 2.8 produce non-zero stress values when
θ 6= 0. Nonetheless, solving linear elastic problems using a computational fluid
dynamics solver serves as a proof-of-concept that fluid and solid systems can be
solved simultaneously using a single unified solver.
Nonlinear elasticity is discussed in the following section and introduces concepts
necessary to capture the richness of loading application and removal on a moving
deformable body.
2.1.3.2 Nonlinear Elasticity
To explore the concepts involved in much of nonlinear elasticity, it is first necessary
to delve into the notion of body configurations and the resulting definitions. Con-
22
figurations are critical in describing a class of nonlinear elastic materials known as
Green, or hyperelastic, materials.
Figure 2.2: Simplified representation of deformation gradient.
Consider a point X on an body denoted as the reference, or undeformed, con-
figuration. Now consider that reference body is moved through space to become
a deformed body at time t. Following point X through the body motion, that
material particle will be at point x(X, t) on the deformed body; this motion can
be described using a deformation vector, χ(X, t), which maps any point X in the
reference body to its position in the deformed body. This relationship is formally
expressed as
x = χ(X, t). (2.17)
Moving one step forward, it is possible to use the descriptions of point mapping
to explore how curves and surfaces deform during body motion. Consider a vector
23
dX at point X in the reference body. Observing the vector dX through the body
motion, the vector will transform into dx at point x in the deformed body. The
deformation of this vector can be described using the deformation gradient tensor,
F(X, t); this relationship is expressed as
dx = F(X, t)dX, (2.18)
where F(X, t) is defined as
F(X, t) =∂χ(X, t)
∂X=∂x
∂X. (2.19)
Another important transformation to explore is how volume elements change
during body motion. Consider two volume elements dV and dv defined in the ref-
erence and deformed configurations, respectively. The relationship between these
volume elements is
dv = JdV, (2.20)
J(X, t) := detF(X, t) > 0, (2.21)
where J must be greater than zero due to the physical constraint of material
impenetrability. More in-depth coverage of these preliminary definitions can be
found in references [97,99,100].
Armed with the definitions of χ(X, t), F(X, t), and J it is possible to present
the hyperelastic material theory which is suitable for large-strain deformations.
Hyperelastic materials are defined by a strain-energy function, Ψ = Ψ(F), that
is solely a function of the deformation gradient, F. Taking the derivative of Ψ with
respect to F produces the first Piola-Kirchhoff stress tensor, P.
P =∂Ψ(F)
∂F(2.22)
It is important to note that Ψ = Ψ(F) = Ψ(C) = Ψ(E), that is, the strain energy
24
depends on the strain and not rotation. Here, C and E are the right Cauchy-Green
deformation tensor and Green-Lagrange strain tensor, respectively. Both of these
tensors are solely functions of F, expressed as:
C = FTF, (2.23)
E =1
2(FTF− I). (2.24)
As shown in Chapter 6 of [99], P can also be expressed as:
P =∂Ψ(F)
∂F= 2F
∂Ψ(C)
∂C= F
∂Ψ(E)
∂E. (2.25)
For isotropic hyperelastic materials, those whose strain energy function does
not change due to rigid-body motion, the strain energy function can be expressed
in terms of the tensor invariants:
Ψ = Ψ[I1(F), I2(F), I3(F)],
= Ψ[I1(C), I2(C), I3(C)],
= Ψ[I1(E), I2(E), I3(E)].
(2.26)
For an arbitrary second-order tensor over a three-dimensional vector space, A, the
three tensor invariants and derivatives of these invariants can be calculated from
the components of A expressed with respect to an orthonormal basis as:
I1(A) = tr(A) = A11 + A22 + A33, (2.27)
I2(A) =1
2[tr(A)2 + tr(A2)]
= A11A22 + A22A33 + A11A33 − A12A21 − A23A32 − A13A31,
(2.28)
I3(A) = det(A), (2.29)
25
∂I1
∂A= I, (2.30)
∂I2
∂A= I1I−AT , (2.31)
∂I3
∂A= I3A
−T . (2.32)
Having a strain-energy function as a function of the tensor invariants allows for
the derivative of the strain energy function to be written as
∂Ψ(C)
∂C=∂Ψ
∂I1
∂I1
∂C+∂Ψ
∂I2
∂I2
∂C+∂Ψ
∂I3
∂I3
∂C. (2.33)
Knowing C is symmetric, this derivative can be rewritten as
∂Ψ(C)
∂C=∂Ψ
∂I1
I +∂Ψ
∂I2
[I1(C)I−C] +∂Ψ
∂I3
[I3(C)C−1]. (2.34)
Once obtained, the first Piola-Kirchhoff tensor, P, can be related to other stress
tensors as
σ = J−1PFT , (2.35)
S = F−1P, (2.36)
where σ is the Cauchy stress tensor and S is the second Piola-Kirchhoff stress
tensor.
A model known as the Saint-Venant Kirchhoff (SVK) model will be used for
much of the nonlinear elasticity in this work. The SVK model is characterized by
the strain-energy function
Ψ(E)SV K =λ
2tr(E)2 + µtr(E2), (2.37)
where λ is the first Lame parameter of the hyperelastic material and µ is the shear
26
modulus of the hyperelastic material. This strain-energy function allows us to
obtain P as
PSV K = F∂Ψ(E)SV K
∂E,
= F[λtr(E)I + 2µE],
= λtr(E)F + 2µFE.
(2.38)
This leads to a Cauchy stress tensor of the form
σSV K =λtr(E)
JFFT +
2µ
JFEFT , (2.39)
which can be expressed solely in term of F as:
σSV K =λ[tr(FFT )− 3]
2JFFT +
µ
JFFTFFT − µ
JFFT (2.40)
Equation 2.40 shows that the Cauchy stress tensor, σ, can be expressed solely
as a function of the deformation gradient, F, for an isotropic hyperelastic mate-
rial. The logical next step is to present a description of the deformation gradient
consistent with the finite volume computational discretization. An appropriate
description of the deformation gradient can be obtained using the Updated La-
grangian approach.
2.1.3.3 General Updated Lagrangian Formulation
Using an Updated Lagrangian (UL) kinematic description of an elastic body, three
configurations are used to describe the body’s motion. The three configurations
used are the base configuration (CB) defined as the body at t = 0 (where stress is
known and often assumed to be zero), the reference configuration (Ct) defined as
the geometry of the body stored in the solver at time t, and a current configuration
(C∗) defined as the future body being solved for in the solver. At each instant,
Ct is the converged solution from the previous timestep. At t = 0, Ct and CB are
identical.
Proceeding in index notation, deformation gradients are defined as
27
Figure 2.3: Simplified representation of deformation gradient decomposition usingupdated Lagrangian approach. Total Lagrangian considers initial and deformedconfigurations only. Updated Lagrangian also considers a time-dependent inter-mediary configuration.
dxti = F′
ijdxBj , (2.41)
dx∗i = F′′
ijdxtj, (2.42)
where xai is the i-th spatial location of an arbitrary point in configuration Ca.
Expressing the deformation of the C∗ in terms of CB gives
dx∗i = F′′
ijdxtj = F
′′
ikF′
kjdxBj = Fijdx
Bj , (2.43)
Fij = F′′
ikF′
kj, (2.44)
28
where Fij is the ij-th component of the deformation gradient from CB to C∗
through Ct. Fij are the terms to be used in the constitutive model chosen for a
given material.
Computing Fij requires computing both F′ij and F
′′ij then taking the product of
the two tensors. For convenience, these terms are calculaed in the solver using the
available gradient solvers. It is important to note that spatial gradient operators
in the solver are with respect to xt. Using the built-in gradient operator, it is
straightforward to compute (F′ij)−1 as
(F′
ij)−1 =
∂xBi∂xtj
. (2.45)
The first step in computing (F′ij)−1 is to ensure the topology of the discretized
body is identical at every time t; doing this eliminates the need to perform mesh-
mesh interpolation. Next, the spatial locations at every quadrature point in CB
must be saved (in this case, the spatial locations of the volume centroids are saved).
These spatial locations are loaded into solver and stored for their corresponding
mesh element. The gradient of the spatial locations gives (F′ij)−1. Lastly, (F
′ij)−1
is inverted to give F′ij. Physical constraints require that det[F
′] > 0, ensuring F
′
is always invertible. Moreover, F′ij is a mxm square matrix when working in m
dimensional space. In this case, F′ij is represented as square, invertible, and 3x3.
Some additional care is taken when computing F′′ij to allow for future implicit
implementations of constitutive models. Ideally F′′ij should be a function of the
state variable since it is dependent on future values. Fortunately, it is possible to
express F′′ij as a function of displacement, u∗i .
uti = xti − xBi , (2.46)
u∗i = x∗i − xti, (2.47)
29
H′′
ij =∂u∗i∂xtj
=∂(x∗i − xti)
∂xtj=∂x∗i∂xtj− ∂xti∂xtj
= F′′
ij − δij, (2.48)
giving
F′′
ij =∂u∗i∂xtj
+ δij. (2.49)
This formulation of F′′ij allows for hyperelastic constitutive models to be ex-
pressed as a function of the state variable u∗i . Using this expression to compute
Fij gives
Fij = F′′
ikF′
kj =
(∂u∗i∂xtk
+ δik
)F′
kj = F′
kj
∂u∗i∂xtk
+ F′
ij (2.50)
where F′ij is constant during each timestep. This expression of Fij is then substi-
tuted into the following definition of the second Piola-Kirchhoff stress tensor for
the SVK model to obtain the stress in terms of the unknown displacement
SSV Kij = µ (FkiFkj − δij) +λ
2(FkmFkm − 3) δij (2.51)
2.1.3.4 Slow-Process Updated Lagrangian Formulation
The general updated Lagrangian formulation can be simplified in cases when ei-
ther deformation happens slowly or there is adequate temporal resolution. This
simplification of the general updated Lagrangian formulation will be referred to as
the slow-process updated Lagrangian formulation (SPUL).
The balance of linear momentum for a hyperelastic material with respect second
Piola-Kirchhoff stress is:
ρ∂2ui∂t∂t
=∂(FikSkj)
∂xj+ bi. (2.52)
The change in stress and deformation is assumed to be small when the defor-
mation process occurs slowly. As such, each of these variables can be expressed as
their value at a given point in time plus some small change.
30
ρ∂2(ui + δui)
∂t∂t=∂[(Fik + δFik)(Skj + δSkj)]
∂xj+ bi, (2.53)
where δSkj is the second Piola-Kirchhoff stress computed using the values of the
displacement increment, δui, instead of the displacement itself. δFij is defined as
δFij = F′′
ij =∂δu∗i∂xtj
+ δij. (2.54)
Subtracting Equation 2.52 from Equation 2.53 gives
ρ∂2δui∂t∂t
=∂(δFikSkj + FikδSkj + δFikδSkj)
∂xj+ bi. (2.55)
The slow-process assumption allows for the approximation that (Fij ≈ δij)
as the deformation change is captured in the δFij term. This further simplifies
Equation 2.55 to
ρ∂2δui∂t∂t
=∂δSij∂xj
+∂[δFik(Skj + δSkj)]
∂xj+ bi. (2.56)
Equation 2.56 is identical to the updated Lagrangian governing equations ob-
tained by Cardiff [78]. The Second Piola-Kirchhoff (2PK) Stress Tensor, Sij, and
its increment, δSkj, are defined for a SVK material as
Sij = µ(F′
kiF′
kj − δij)
+λ
2
(F′
kmF′
km − 3)δij
= µ
(∂uti∂xBj
+∂utj∂xBi
+∂utk∂xBi
∂utk∂xBj
)+λ
2
(∂utm∂xBm
+∂utm∂xBm
+∂utk∂xBm
∂utk∂xBm
)δij,
(2.57)
δSij = µ(F′′
kiF′′
kj − δij)
+λ
2
(F′′
kmF′′
km − 3)δij
= µ
(∂u∗i∂xtj
+∂u∗j∂xti
+∂u∗k∂xti
∂u∗k∂xtj
)+λ
2
(∂u∗m∂xtm
+∂u∗m∂xtm
+∂u∗k∂xtm
∂u∗k∂xtm
)δij.
(2.58)
These equations show the gradients of the reference stress are taken in a co-
31
ordinate system different from that of the increment stress. This need not be the
case. All gradients may be taken with respect to the xti coordinates. It is bene-
ficial to have all gradients taken with respect to the xti coordinates, as this is the
geometry represented in the flow solver, allowing gradients to be computed using
simple interpolation and the divergence theorem. Obtaining Sij using gradients
with respect to xti is done by first computing
(F′
ij)−1 =
∂xBi∂xtj
, (2.59)
at each spatial point of interest. (F′ij)−1 is the gradient of the initial geometry with
respect to the geometry at time t. The deformation gradient yields a 3x3 square
matrix when operating in three-dimensional space. As such, the values of F′ij can
be obtained by inverting a 3x3 matrix at each spatial point of interest.
The slow-process updated Lagrangian formulation was used for modeling all
hyperelastic materials in this work.
2.1.3.5 Slow-Process Updated Lagrangian Boundary Conditions
Dirichlet boundary conditions are straightforward to implement, as displacement is
specified on the boundary. The values of [δSij+δFik(Skj+δSkj)] are computed using
the prescribed displacements for the slow-process updated Lagrangian formulation.
In practice, it is also necessary to compute the displacement gradients at the
boundary to fully implement the Dirichlet boundary condition. This can be done
in a number of ways including least-square approximation methods or distance-
weighted averaging.
Traction (Neumann) boundary conditions tend to be of more importance in
FSI problems. Care must be taken when applying traction boundary conditions
at the surface of hyperelastic bodies as the effective traction is a function of the
applied traction and the internal body stresses. The traction at the surface of a
hyperelastic body using an updated Lagrangian formulation can be expressed as
32
T effectivei = T surfacei − Sijnsj . (2.60)
The traction boundary condition is introduced into the governing equations at
the appropriate faces as
T effectivei = [δSij + δFik(Skj + δSkj)]nfj , (2.61)
where T effectivei can be used at the faces without the need to compute [δSij +
δFik(Skj + δSkj)]nfj .
2.1.4 Surface Biochemistry
The surface biochemistry formulation used is nearly identical to the work of Behr
and Gaskin [29,52].
The second law of thermodynamics and equilibrium conditions are coupled to
govern the bond formation and breakage in the domain. Modifications were then
made to the biochemistry formulation to allow for localized modeling of individual
bonds.
Localized bond formation and breakage probabilities are calculated for each
compatible molecule pair in the system. The association rate, kon, and dissociation
rate, koff , are calculated as,
kon = k0onnLALexp
(−sts(d− λ)2
2kbT
), (2.62)
koff = k0offexp
((s− sts)(d− λ)2
2kbT
), (2.63)
where k0on is the association rate at equilibrium, k0
off is the dissociation rate at
equilibrium, AL is the surface area of the discrete mesh face, nL is the molecule
surface density, s is the bond spring constant, sts is the bond spring constant
during the transition state, d is the separation distance of the two molecules, λ is
the equilibrium spring length, T is the local temperature, and kb is Boltzmann’s
33
constant.
The value of kon is then corrected to allow for localized bond modeling,
kon,g = [∑faces
(nLAL)]k0onexp
(−sts(dc − λ)2
2kbT
), (2.64)
kon,ave =
∑faces
kon
number of faces, (2.65)
kon,corr = konkon,gkon,ave
, (2.66)
where dc is the distance between the centroids of the two cells.
For each molecule pair, the probability of bond formation is calculated as,
P = 1− exp(−kon,corr∆t), (2.67)
where ∆t is the elapsed simulation time since the previous calculation. A random
number is then generated and a bond is formed if the calculated probability is
greater than the random number. Similarly, bond breakage is calculated as,
P = 1− exp(−koff∆t). (2.68)
The force due to an individual bond is computed as
f bondj = s(d− λ)ej, (2.69)
where ej is a unit normal at the bond site along the line of action of the bond.
2.1.5 Rigid Body Motion
It is necessary to capture the trajectory of rigid bodies to fully understand the
cellular interactions happening during the cancer extravasation process. A common
approach is to describe the trajectories as an extension of Newton’s second law of
motion,
34
d2xidt2
=Fim, (2.70)
ICijd2θjdt2
= Ti, (2.71)
where xi is the i-th component of the linear displacement vector, θi is the i-th
component of the vector of Euler angles, Fi is the i-th component of the sum of
all forces applied at the centroid, Ti is the i-th component of the torque applied
about the centroid, m is the mass of the body, and ICij is the moment of inertia
corresponding to the ij-th axis.
To accommodate bodies of arbitrary geometry, the moment of inertia is calcu-
lated as,
ICij =
∫V
ρrirjdV, (2.72)
for each axis where ri is the i-th component of the radius vector to a point in the
body from the axis passing through the body centroid C, and ρ is the mass density
at each point ri.
Past work [29, 45, 52, 73] used an explicit Euler six degree-of-freedom (6DOF)
solver to compute trajectories of rigid bodies despite the strict limitations of such
an approach [29]. Once the right-hand sides of Equations 2.70 and 2.71 were
computed, each of the equations were marched forward in time using the explicit
Euler method.
Building upon the computational tool developed in [29], the rigid body 6DOF
solver was incorporated into the flow solver and semi-implicitly coupled to the
flow field solution procedure; an approach described in greater detail in Section
3.2. This implicit coupling of the 6DOF solver relaxed the stability restrictions of
the computational system and allows for much better runtime performance.
Another approach explored in this work is to include the contributions of inertia
into the momentum conservation equations and remove the 6DOF solver entirely.
Such an approach can be formulated to allow for body deformability. The Up-
35
dated Lagrangian technique described in Section 2.1.3.2 not only computed the
deformation in a hyperelastic material but also the rigid body contributions of
the material’s trajectory. The computational implementation of this approach is
described in detail in Section 3.3.4.2.
2.1.6 DDMR Magnetic Actuation
Many DDMRs are controlled via magnetic actuation. This conveniently allows for
the use of MRI machines to simultaneously actuate and image the device [17, 87].
Moving forward, there is much work to be done in optimization of the applied
magnetic fields for desired targeting. Figure 1.4 shows several proposed DDMR
designs, and the optimal method of magnetic actuation for each design. When
looking solely at propulsion, torque-driven helical microswimmer DDMRs are most
efficiently driven by rotating magnetic fields, while spherical DDMRs are best
driven by field gradients [88]. However, these analyses do not account for the
desired interactions of the DDMR with the surrounding microenvironment.
Figure 2.4: Methods of magnetic actuation. Reproduced from [88]
Previous work has shown that fluid shear rate affects rates of cellular adhesion
in microvascular flow [50]; where relative translational and rotational velocities of
the bodies of interest, due to flow shear, play important roles in cellular interaction.
Extending these findings to DDMR applications, it is necessary to design DDMR
36
geometry and actuation with an eye towards both efficient propulsion and desired
biochemical interactions.
The force and torque on an arbitrary body for a given magnetic field Bi are
expressed, respectively, as
Fmi = VMj
∂Bi
∂xj, (2.73)
Tmi = V εijkMjBk, (2.74)
where Mi is the i-th component of body magnetization, V is the volume of the
magnetic object, and εijk is the Levi-Civita symbol (commonly referred to as the
permutation symbol) [88]. Equations 2.73 and 2.74 can be used to determine
the components of motion due to the magnetic actuation. Therefore, DDMR
designs can be actuated in the computational tool using prescribed magnetic fields
to explore the effect of actuation on DDMR targeting capabilities (e.g., efficient
propulsion and desired biochemical interactions).
This work assumes DDMRs are ferromagnetic permanent magnets at satura-
tion magnetization. Ferromagnetic materials are made up of atoms with large
dipole moments due primarily to electron spins [101]. These dipoles in permanent
magnets align themselves due to interactions with one another in the absence of
an externally applied field. In the presence of an external field, these dipoles align
based on the strength of the applied field until all dipoles are in alignment; at which
point, the magnet has reached saturation. Saturation magnetization is defined as
the maximum magnetic dipole per unit volume [101,102].
For this class of magnets, saturation magnetization can be computed as
msat = µatom
volume= µ
mass/volume
mass/atom,
= µρatom
mass= µρ
(nucleon
mass
)(atom
nucleons
),
= µρNA
(atom
nucleons
),
(2.75)
37
where µ is the material’s magnetic moment per atom, ρ is material density, NA is
Avogadro’s number, and the last term is the reciprocal of the number of nucleons
per atom. For a pure element, the last term is simply the reciprocal of the element’s
atomic number.
At saturation, magnetization, Mi, of the body is defined as
Mi = msatBi. (2.76)
Using this description alone, it may seem that (Tmi = 0) for any possible configu-
ration of the magnetic field. However, (Tmi = 0) only holds for external magnetic
fields that are constant in time. Instantaneous changes in Bi can produce torque-
inducing misalignment of the material’s magnetic dipoles.
2.2 Coupling of Dynamic Systems
2.2.1 Hydrodynamics / Fluid-Structure Interaction
2.2.1.1 Boundary Conditions on Rigid Bodies
Given a flow field around a rigid body, the resultant forces and torques of a New-
tonian fluid acting on a rigid body are
F fluidi =
∫A
(µ∂ui∂xm
nfm + µ∂um∂xi
nfm − pnfi
)dA, (2.77)
T fluidi =
∫A
εijkrj
(µ∂uk∂xm
nfm + µ∂um∂xk
nfm − pnfk
)dA, (2.78)
where p is the fluid pressure at the surface, rj is the j-th index of the vector
pointing from the surface location to the centroid of the rigid body, and εijk is the
Levi-Civita symbol (commonly referred to as the permutation symbol).
38
2.2.1.2 Boundary Conditions on Hyperelastic Bodies
Implementation of FSI boundary conditions on hyperelastic bodies follows the Neu-
mann boundary condition approach described in Section 2.1.3.5. The traction at
the surface of a hyperelastic body in flow using an updated Lagrangian formulation
can be expressed as
T fluidi = σfluidij nfj = −pnfi + µ
(∂ui∂xj
+∂uj∂xi
)nfj , (2.79)
T FSIi = T fluidi − Sijnsj ,
= T fluidi + Sijnfj ,
=
[µ
(∂ui∂xj
+∂uj∂xi
)+ Sij
]nfj − pn
fi ,
(2.80)
The traction boundary condition is introduced into the governing equations at
the appropriate faces as
T FSIi = [δSij + δFik(Skj + δSkj)]nfj , (2.81)
where T FSIi can be used at the faces without the need to compute [δSij+δFik(Skj+
δSkj)]nfj .
2.2.2 Biochemistry
2.2.2.1 Boundary Conditions on Rigid Bodies
Forces applied at the surface of rigid bodies are transformed into equivalent forces
and torques applied at the centroid of the rigid body. The resultant force and
torque at the centroid of the rigid body due to biochemistry is
F bondsi =
∑bond
f bondi , (2.82)
39
T bondsi =∑bond
(εijkrjfbondk ). (2.83)
2.2.2.2 Boundary Conditions on Hyperelastic Bodies
Forces applied at the surface of hyperelastic bodies are added to the T surface term
in Equation 2.60.
For example, the value of T surface for an FSI face with a biochemical bond
would be
T surfacei = T fluidi + f bondi . (2.84)
This value can then be used to determine T effectivei at the boundary surface.
2.2.3 Modeling of Surface Roughness
Sub-grid scale modeling is used in this work to capture the effects of microstruc-
tures on the surface of cells. Figure 2.5 shows an image of these microstructures
which can be fairly complex. All cells must maintain a minimum separation dis-
tance to account for the presence of the surface microstructures. The specified
minimum separation distance is enforced using a fictitious localized repulsion force.
Figure 2.5: Cell surfaces contain many complex structures which are modeled usinga sub-grid fictitious repulsion force. [28, 103]
The repulsion force used in this work is
fi,rep = [b/d3]ei, (2.85)
40
where b is a constant repulsion parameter and d is the separation distance between
two points on adjacent cells. The force due to repulsion can be applied to bodies
similarly to the treatment of the effects due to biochemistry.
2.2.4 Unified Governing Equation
Re-arranging of the equations governing conservation of momentum in the fluid
and solid materials gives a single unified equation to represent all material in the
domain. The unified governing equation validates the concept of simultaneously
solving the entire computational domain in an FSI problem and is expressed as
αk∫V
ρ∂2ui∂t∂t
dV −∫S
µ∂ui∂xj
dSj =
∫V
bidV +
∫S
QkijdSj, (2.86)
αstokes = αLinElas = 0, (2.87)
αSV K = 1, (2.88)
Qstokesij = −pfaceδij, (2.89)
QLinElasij = µ
∂uj∂xi
+ λ∂uk∂xk
δij, (2.90)
QSV Kij = µ
(∂uj∂xi
+∂uk∂xi
∂uk∂xj
)+λ
2
(∂um∂xm
+∂um∂xm
+∂uk∂xm
∂uk∂xm
)δij+F
′
ik
(S′
kj + S′′
kj
),
(2.91)
where αk is a constant determined by material k. Care must be taken in the
physical meaning of state variable ui; the state variable represents flow velocity
in the fluid domain and material displacement in the solid domain. The coupling
conditions described above ensure state variable on either side of an FSI interface
are appropriately treated.
Chapter 3Computational Implementation
3.1 Finite Volume Discretization
As shown in Chapter 2, the governing equations for the fluid, linear elastic, and
hyperelastic systems all have the same form. The similarity of the equations allows
for the development of a unified solver to simultaneously compute the solutions in
all of the material regions. This work uses a finite volume discretization approach
to solve the fluid and structural domains of the system.
Figure 3.1 shows two adjacent arbitrarily shaped polyhedra with a shared face.
The vector ~r12 points from the centroid of polyhedron 1 to the centroid of poly-
hedron 2. The distance from the centroid of polyhedron 1 to the centroid of the
shared face is denoted as δr1. The area of the shared face is denoted by the vector
~S with the area vector pointing in the direction from polyhedron 1 to polyhedron
2. The unit normal vector to the shared face, ~n, is obtained by normalizing ~S.
All variable data is stored at the centroid of the polyhedron. By convention,
all terms on the left side of the equation will be treated implicitly and all terms of
the right side of the equation will be treated explicitly using lagged values. There
are several discrete operators of importance in this work:
• Spatial interpolation of a scalar φ,
• Explicit spatial gradient of a scalar φ,
42
Figure 3.1: Arbitrarily shaped polyhedra with shared face.
• Implicit spatial gradient of a scalar φ,
• Implicit temporal gradient of a scalar φ.
Each operator discretization will be individually addressed. More detailed infor-
mation about the discretization used in the baseline solver can be found in [74].
3.1.1 Spatial interpolation of a scalar φ
Spatial interpolation of a scalar φ is crucial as information is stored at the poly-
hedron centroid while the governing equations require values at the faces of the
polyhedra. The value φface is approximated as
φface = (1− k)φ1 + kφ2, (3.1)
43
k ≡ δr1
δr1 + δr2, (3.2)
where φ1 is the value of φ at the centroid of polyhedron 1. This operator returns
an exact linear interpolation if vectors ~r12 and ~S are parallel.
3.1.2 Explicit spatial gradient of a scalar φ
Gradients of φ can be computed at the centroid of an arbitrary polyhedron using
the divergence theorem
∂φ
∂xi=
1
V
faces∑f
(Sfi φ
f), (3.3)
where V is the volume of the polyhedron, Sfi is the i-th component of the outward
area vector of face f , and φf is the value of φ at face f . The value of this gradient
can then be interpolated to any face using the interpolation operator introduced
above.
3.1.3 Implicit spatial gradient of a scalar φ
The gradient of scalar φ can be arranged to allow for implicit treatment. The
gradient on the shared face shown in Figure 3.1 can be expressed as
∂φ
∂xi
∣∣∣∣face =∂φ
∂xi
∣∣∣∣face −[(
∂φ
∂xj
)face r12j
|r12|
]r12i
|r12|+
[(∂φ
∂xj
)face r12j
|r12|
]r12i
|r12|, (3.4)
where is |r12| the magnitude of ~r12. The third term in Equation 3.4 represents
the contributions of the spatial gradient parallel to ~r12 while the first two terms
represent the orthogonal contributions.
The parallel contributions can be discretely expressed as
44
[(∂φ
∂xj
)face r12j
|r12|
]r12i
|r12|=
(φ2 − φ1
|r12||r12|
)r12i , (3.5)
while the orthogonal contributions are computed and treated explicitly.
3.1.4 Implicit temporal gradient of a scalar φ
The inertial term in many systems can be expressed as the temporal gradient of a
scalar φ
∂2φ
∂t∂t. (3.6)
Applying a finite difference approach, the second time derivative of a scalar φ
can be discretely expressed as the summation,
∂2φ
∂t∂t≈
np∑n=1
(fn2 φn) , (3.7)
where np is the number of grid points in the temporal stencil, fn2 is the finite
difference coefficient for second derivative at the n-th grid point, and φn is the
value of φ at the n-th grid point. Since the term is being discretized in time, the
grid point n corresponds to an instance in time. The coefficients fn in Equation 3.7
can be computed to arbitrary orders of accuracy using Fornberg’s method [104].
3.2 Semi-implicit Six-Degrees-of-Freedom Cou-
pling
For the rigid body and linear elastostatic structural models, a semi-implicit six
degree-of-freedom (6DOF) solver was used to relax the restrictions of the explicit
off-board 6DOF solver.
The implementation of the semi-implicit 6DOF solver involved modifications to
the explicit implementation described in [29]. The governing equations for 6DOF
45
motion, as described in Section 2.1.5, can be represented as a first order ordinary
differential equation of the form
dA
dt= B. (3.8)
Beginning with the explicit approach, a first-order differential equation can be
marched forward in time as
At+∆t = At + ∆t
(dA
dt
)t= At +Bt∆t, (3.9)
where the updated value At+∆t depends solely on values known from previous
points in time. This approach is typically the simplest time marching strategy to
implement, but comes with a restrictive stability limitation. Previous work found
the maximum stable timestep for this type of flow system to be around 30µs [29].
Such a limit forces the timestep value to be much smaller than needed to adequately
resolve the system dynamics. In fact, it would take nearly 300 timesteps for an
8µm radius spherical body in a typical microvascular flow to travel the length of
one radius using a timestep of 30µs. The strict stability limit imposed by the
explicit time marching led to the exploration of implicit marching schemes that
could be implemented in a manner consistent with the FSBI solver.
The semi-implicit 6DOF coupling strategy implemented was motivated by the
iterative solution strategy used in the underlying FSBI solver. The underlying
FSBI uses the SIMPLEC algorithm [74] where the state variable being solved is
predicted then corrected iteratively until the solution converges. Each corrector
step in the predictor-corrector loop provides a potential solution to that attempts
to enforce the problems constraints. The idea for the semi-implicit solver is to use
the FSBI corrector solution to compute a potential solution for the 6DOF motion
that is coupled back into the fluid system through the no-slip condition at the
body surface. A semi-implicit discretization of Equation 3.9 can be written as
At+∆t,∗ = At +Bt+∆t,∗∆t, (3.10)
46
where the updated value At+∆t,∗ is an estimated value that depends on not only
A from the previous timestep but also the most recent potential solution of B.
However, due to the very nature of the iterative procedure, precautions must be
taken to prevent any solution error from growing uncontrollably with each iteration.
The relaxation factor, ωimp, motivated by the under-relaxation approach often
used in SIMPLEC is used to help with stability of the routine. This relaxation
factor is prescribed as a value between zero and one and implemented as
At+∆t,∗ = ωimp(At +Bt+∆t,∗∆t
)+ (1− ωimp)At+∆t,∗∗, (3.11)
where At+∆t,∗∗ is the value of At+∆t,∗ computed at the previous solver iteration.
The relaxed semi-implicit 6DOF motion solver implementation has greatly im-
proved the stability limitations. In practice, the solver remains stable even with
timestep values on the order of 1000µs.
3.3 Structural Mechanics
Rigid body motion solver performance has been greatly improved by use of the
implicit iterative discretization compared to the explicit discretization. Advances
have also been made in the structural modeling of deformable bodies in microvas-
cular flow environments.
In this work, three approaches were explored in the context of structural mod-
eling for coupling to fluid systems with biochemical interactions: 1) fixed-shape
leukocyte rolling discussed in Section 3.3.1, 2) Linear elastic solid modeling dis-
cussed in Section 3.3.3, and 3) Non-linear hyperelastic solid modeling discussed in
Section 3.3.4.
3.3.1 Fixed-shape Leukocyte Rolling
In the near-wall region, deformable leukocytes undergo complex flow-structure-
wall-biochemistry interactions. These interactions lead to PMN rolling along the
47
endothelial wall with a nominal shape and rolling velocity. Experimental observa-
tions have found the rolling shapes and velocities to be functions of flow parameters
and local biochemical expression [53]. An approximation was sought to model the
complex flow-structure-wall-biochemistry interactions of leukocytes in the near-
wall region in the manner that was computationally efficient and retained the
leukocyte surface mesh topology, to allow for modeling of long-term biochemical
interaction. A viable approximation for the leukocyte near-wall interactions is the
fixed-shape leukocyte rolling strategy.
Analytic transform equations were created to transform a sphere to match
the nominal rolling shape from experimental observations. Beginning with a unit
sphere centered at origin, the nominal rolling shape is obtained using the following
equations:
xnew = xold + Axcos
(πxold
2r
), (3.12)
ynew = yold −max(0, yold)(Ay)−min(0, yold)
[Bycos
(πxold
8r
)], (3.13)
znew = zold, (3.14)
where xold is the x-coordinate of an arbitrary point belonging to the undeformed
sphere, r is the radius of the undeformed sphere, and Ax, Ay, and By are pre-
scribed shape constants chosen based on the fluid shear rate and local biochemical
expression. An example of a shape obtained through this transform is shown in
Figure 3.2.
The obtained leukocyte shape is sent to the FSBI solver along with the pre-
scribed rolling velocities. The velocity at the surface of the leukocyte is computed
using the prescribed values of leukocyte translational and rotational velocities.
A crucial step in the leukocyte rolling strategy is transforming the deformed
shape back into a sphere to rotate the surface mesh. While the sphere-to-leukocyte
48
Figure 3.2: Analytic transformation from a sphere to deformed PMN.
Figure 3.3: Flowchart of fixed-shape PMN rolling algorithm
transform described by Equations 3.12, 3.13, and 3.14 does not have an analytic
inverse, root-finding methods can be used to approximate the inverse transform.
The Newton-Raphson method is well-suited to approximating the roots of a
49
real-valued function by using an iterative approach to find a value x that returns
the root of a given function f(x) [105]. The value of x is approximated by
xn+1 = xn −f(xn)
f ′(xn), (3.15)
where xn is the n-th approximation of the value x. This method requires as initial
guess of x, the function f(x), and the derivatives of the function f(x).
Fortunately, the derivatives of the sphere-to-leukocyte analytic transform are
easily obtained by taking the derivatives of Equations 3.12, 3.13, and 3.14. These
derivatives are found to be:
∂xnew
∂xold= 1− πAx
8sin
(πxold
8
), (3.16)
∂ynew
∂xold= 0 ∀ yold ≥ 0, (3.17)
∂ynew
∂yold= 1− Ay ∀ yold ≥ 0, (3.18)
∂ynew
∂xold=
[πByy
old
32
]cos
(πxold
32
)∀ yold < 0, (3.19)
∂ynew
∂yold= 1−Bycos
(πxold
32
)∀ yold < 0, (3.20)
∂znew
∂zold= 1, (3.21)
∂xnew
∂yold=∂xnew
∂zold=∂ynew
∂zold=∂znew
∂xold=∂znew
∂yold= 0. (3.22)
When solving for the inverse tranform using known values of xnew and ynew, the
values of xold and yold can be computed separately for ease of computational im-
plementation. First, xold should be computed using the Newton-Raphson method
since xnew is solely a function of xold. Then, ynew can be computed since ynew is a
50
function of xold and yold, where yold is the only unknown since xold was solved for
in the previous step. This approach only requires the derivatives ∂xnew
∂xoldand ∂ynew
∂yold
be used by the Newton-Raphson root-finding routine.
3.3.2 Finite-Volume Structural Mechanics Formulation
3.3.3 Finite Volume Linear Elastostatics
3.3.3.1 Discrete Governing Equations
The governing equation for conservation of momentum in linear elastostatic ma-
terials is provided in Section 2.2.4. The discrete form of this equations for an
arbitrary polyhedron in the computational domain is written as
−faces∑f
(µ∂ui∂xj
Sj
)= biV +
faces∑f
(µ∂uj∂xi
Sj + λ∂uk∂xk
Si
), (3.23)
where the terms on the left side of the equation are treated implicitly and terms
on the right side of the equation are treated explicitly. The difficulty of solving
this governing equation using fixed-point iterative schemes has been discussed [18]
where the following modification was proposed to improve solver issues
−faces∑f
[(2µ+ λ)
∂ui∂xj
Sj
]= biV +
faces∑f
[µ∂uj∂xi
Sj + λ∂uk∂xk
Si − (µ+ λ)∂ui∂xj
Sj
].
(3.24)
This modification serves to increase the diagonal dominance of the linear algebraic
system leading to the better solver performance.
Inertial contributions of linear elastic materials are accommodated by using the
rigid body 6DOF solver to compute the trajectory of the body geometry obtained
from the solution to the linear elastostatic problem.
51
3.3.3.2 Implementation of Boundary Conditions
Boundary conditions for the linear elastic problems of interest are prescribed trac-
tion and prescribed displacement.
In the case of prescribed displacement on a boundary face, the gradient terms
at that face are approximated by one-sided gradients using the prescribed face
displacement and the value at the polyhedron centroid.
In the case of prescribed traction on a boundary face, the following relation is
used:
Ti = µ∂ui∂xj
nj + µ∂uj∂xi
nj + λ∂uk∂xk
ni. (3.25)
Prescribed traction conditions can be enforced by substituting Equation 3.25 into
the discrete governing equations at the appropriate boundary faces.
In practice, for steady simulations, there should always exist at least one point
in the domain where displacement is prescribed to ensure the displacement solution
is not off by an arbitrary constant of integration. No such challenge arises for
unsteady simulations as the solid body will simply accelerate due to the applied
force imbalance.
3.3.4 Finite Volume Nonlinear Elastodynamics
3.3.4.1 Discrete Governing Equations
The governing equation for conservation of momentum in hyperelastic materials
using the Saint-Venant Kirchhoff constituent model is provided in Section 2.2.4.
The discrete form of these equations for an arbitrary polyhedron in the computa-
tional domain is written as
ρ∂2ui∂t∂t
V −faces∑f
(µ∂ui∂xj
Sj
)= biV +
faces∑f
(QSV Kij Sj
), (3.26)
where the terms on the left side of the equation are treated implicitly and terms
on the right side of the equation are treated explicitly.
52
Similar to the linear elastostatic equations, the discrete form of the governing
equations for SVK materials are modified for increased diagonal dominance as
ρ∂2ui∂t∂t
V −faces∑f
[(2µ+ λ)
∂ui∂xj
Sj
]= biV +
faces∑f
[QSV Kij Sj − (µ+ λ)
∂ui∂xj
Sj
]. (3.27)
Special attention must be given to the computation of the temporal displacement
gradient for the accommodation of the inertial contributions, as well as for the
transformation of the deformation gradient from the stress-reference configuration
to the meshed configuration at a given instance in time.
3.3.4.2 Accommodation of Inertial Contributions
Given uni = (xni − xn−1i ) in the solid domain, where xni is the i-th coordinate of the
spatial location at time n, the discrete form of the second derivative of uni can be
expressed as
∂2ui∂t∂t
≈np∑n=1
(fn2 xni )−
np∑n=1
(fn2 x
n−1i
). (3.28)
The following expression is then obtained through algebraic manipulation:
∂2ui∂t∂t
≈ fnp2 unpi + fnp2 xnp−1i +
np−1∑n=1
{(fn2 − fn+1
2 )xni}− f 1
2x0i . (3.29)
Equation 3.29 is useful in that it allows for an implicit treatment of inertial
contributions. The first term on the right side of the equations contains the un-
known state variable unpi and can be treated implicitly by adding the coefficient
fnp2 to the corresponding point coefficient. Every other term relies solely on the
history of mesh motion and can be added to the source term in the solid mechanics
solver.
53
3.3.4.3 Lagged Correction of Non-linear Terms
The definition of QSV Kij shows the SVK governing equations are non-linear in dis-
placement ui. Linearization is needed to numerically solve this system using any
linear solver.
While there are many ways to handle the non-linear terms in the model, this
work treats all non-linear terms explicitly. Non-linear terms are computed using
the displacement field provided by the most recent solution of the iterative non-
linear solver.
One interesting area of future work is an exploration of non-linear treatments
on numerical stability of the solution procedure. Implicit handling of some terms in
QSV Kij could provide increased stability and result in overall increases in solver per-
formance. The procedure described in Section 3.5 gives a framework for quantifying
the effects of linearization treatments on iterative solver stability and performance.
3.3.4.4 Quantification of Constitutive Model Selection
Continuity of traction dictates that (Sijnfluidj = T fluidi ) at any point on a fluid-solid
interface. This condition can be used not only to estimate the material properties
best suited for a particular constitutive model but also to quantify the error as-
sociated with selection of given constitutive models based on known observations.
Use of this condition requires knowledge of the solid’s stress-free geometry, the
loading applied to the solid surface, and the solid’s equilibrium geometry based on
the applied loading.
At each point on the surface of the material:
1. Compute deformation gradient between the reference and equilibrium con-
figurations Fij given
(Fij =
∂xequlibriumi
∂xstressFreej
).
2. Construct appropriate stress tensor Sij(Fij, φ1, φ2, ..., φk) as a function of Fij
and the k properties describing the material, (φ1, φ2, ..., φk).
54
3. Compute lp norm between vectors [S(F, φ1, φ2, ..., φk)nfluid] and T fluid.
Once the lp norm has been computed at each point, the following cost function
can be used in an optimization routine:
f cost(φ1, φ2, ..., φk) =∑
points on surface
{lp[T fluid, S(F, φ1, φ2, ..., φk)nfluid
]}. (3.30)
An optimization routine can be used to find values of (φ1, φ2, ..., φk) that min-
imize equation 3.30.
For leukocytes in this work, the cells are described using the Saint-Venant
Kirchhoff (SVK) constitutive model, assumed to be spheres of radius 4µm when
unstressed, and describe the equilibrium shape of a wall-adhered PMN using the
transformation given in Section 3.3.1.
The SVK stress tensor expressed in terms of F and material properties are
shown in Equation 2.51. In Equation 2.51, material properties µ and λ are func-
tions of the modulus of elasticity, E, and Poisson’s ratio, ν, as follows:
µ =E
2(1 + ν), (3.31)
λ =Eν
(1− 2ν)(1 + ν). (3.32)
The values of T fluid at each surface point is obtained by running a flow calcu-
lation of the deformed leukocyte body fixed at a specified location on the vascular
wall. The shear rate of the flow is chosen to match experimental data [53] and
the leukocyte is assumed to have no further deformation or velocity at the surface.
The obtained flow field is used to compute the traction at the surface of the cell.
Quantification of constitutive model selection error can be done by comparing
a given set of constitutive model and material properties with a reference cost
function value. The following reference cost function value is chosen due to its
indifference towards the constitutive model selection
55
f ref =∑
points on surface
[lp(T fluid, 0
)], (3.33)
where f ref > 0 if, and only if, there exists at least one point on the material surface
where T fluid 6= 0.
3.4 Proximity-based adaptive timestepping
A proximity-based adaptive timestepping approach was implemented to maintain
high temporal resolution throughout the simulations without sacrificing computa-
tional efficiency. This approach uses body trajectories to compute time-to-collision
for resolved bodies.
Consider two points, P a and P b, each of having an associated spatial location,
xi, and velocity, vi. The line-of-action, di, is then defined as
di = xbi − xai . (3.34)
Next, the frame of reference is shifted such that the velocity of P b, vbi , is identically
zero. The velocities in this frame are expressed as:
vbi = 0, (3.35)
va′
i = vai − vbi . (3.36)
The relative speed of the P a along the line of action is obtained as the magnitude
of the projection of va′i onto a unit vector along d. A characteristic time length
can be computed using the relative speed along d and the length of d:
vrel =1
|d|va′
i di, (3.37)
56
∆t∗ =|d|vrel
=|d|2
vai di. (3.38)
This relation can be used when vrel and |d| are both greater than zero. Non-
positive values of vrel may occur when the points are moving away from each other.
In such cases, ∆t∗ would not be considered while determining the new timestep
value. Values of |d| will only be zero in the event of collision, at which point the
finite-volume approximation used to solve the governing equations of the physical
systems may no longer be valid.
The characteristic timescale, ∆t∗, can be used to set a timestep that adequately
resolves the physics temporally. This work used a timestep of ∆t∗/N where N is an
integer. This value allows for N time steps before collision, assuming the velocities
remain unchanged. By analyzing all point-pairs in the domain, ∆t∗ is set to be
the minimum computed characteristic timescale. Bounds on the timescale will be
set, however, to ensure values of ∆t∗ are appropriate for the simulation.
3.5 Fourier Stability Analysis of Computational
Implementations
In the development of new computational methods, it is important to understand
the limits within which the method can be confidently relied upon. Moreover, it is
desirable to quickly estimate performance of a method without allocating the full
amount of computational resources required by the method.
Fourier stability analysis (FSA) of discrete computational operators provides
a framework to determine both stability and expected performance of a compu-
tational method. One drawback of the FSA approach, however, is the under-
lying assumption that all boundaries of the computational domain are periodic.
Nonetheless, FSA remains a powerful tool to explore method feasibility, despite its
inability to accommodate Dirichlet or Neumann boundary conditions.
Computational methods will rarely, if ever, outperform the estimates provided
57
by FSA in practice. This holds especially true when comparing performance esti-
mates to systems with Dirichlet or Neumann boundary conditions. As such, FSA
provides the best-case scenario stability estimate for a given method.
This section will demonstrate FSA using various governing equations represen-
tative of those often found in computational mechanics.
3.5.1 Analysis of 1D Laplace Equations
FSA will first be performed on the scalar Laplace’s equation in one dimension (1D)
expressed as
∂2φ
∂x2= 0. (3.39)
Laplace’s equation is the simplest example of an eliptic partial differential equation,
and provides great insight into many physical systems, including fluid dynamics
and heat conduction [106,107].
The finite difference approximation of second derivative centered in space using
uniform grid spacing gives
∂2φ
∂x2≈ φx+∆x − 2φx + φx−∆x
∆x2= 0. (3.40)
Solving for φx gives
φx =φx+∆x + φx−∆x
2. (3.41)
Four iterative computational solution strategies will be applied to the 1D scalar
Laplace’s equations and analyzed using FSA:
1. Jacobi method,
2. Relaxed Jacobi method,
3. Gauss-Seidel method,
4. Relaxed Gauss-Seidel method.
58
The Jacobi and Gauss-Seidel iterative methods are widely used in numerical
analysis. Details of these methods in general can be found in nearly all numerical
analysis textbooks.
3.5.1.1 Jacobi Method
Using the Jacobi method, the solution to any point is computed using lagged
values of the solution at every other point in the domain. Applying this approach
to solve for φx, the value of terms on the right side of the equation are set to the
solution from the previous iteration. The discrete iterative equation describing φx
is expressed as
φn+1x =
φnx+∆x + φnx−∆x
2, (3.42)
where n is the iteration number of the Jacobi procedure. Initial guess values are
used for φ0x to start the iterative process.
A solution roundoff error can be defined as
εnx = φnx − φnx, (3.43)
where φnx is the solution assuming finite mathematical precision and φnx is the
solution assuming infinite mathematical precision.
The roundoff error at each iteration can be expressed as
εn+1x =
εnx+∆x + εnx−∆x
2. (3.44)
Uniformly distributing N + 1 points in the domain of length L, such that
N∆x = L, the discrete Fourier series of εnx can be expressed as
εnx =N∑n=1
(εnke
iαx), (3.45)
α =πk
N, (3.46)
59
where α is the wavenumber per unit length.
Substituting Equation 3.45 into Equation 3.44 gives
N∑n=1
{eiαx
[εn+1k − εnk
2
(eiα∆x − e−iα∆x
)]}= 0. (3.47)
Since eiαx 6= 0, Equation 3.47 can only be satisfied, in general, if
εn+1k =
εnk2
(eiα∆x − e−iα∆x
), (3.48)
which can be further simplified as
εn+1k = εnk cos(α∆x). (3.49)
The amplification factor of solution error, Gk, of the iterative process for mode
k is defined as the ratio:
Gk :=εn+1k
εnk(3.50)
The spectral radius for each mode k, ρk, is defined as
ρk = |Gk| =∣∣∣∣ εn+1k
εnk
∣∣∣∣ . (3.51)
An iterative process will only converge if ρ < 1 for all k in the range (0 < α∆x ≤ π).
Substituting Equation 3.49 into Equation 3.51 shows ρk for the Jacobi method,
ρJk , to be
ρJk =
∣∣∣∣cos
(πk∆x
N
)∣∣∣∣ . (3.52)
Upon inspection, it is found that ρJk = 1 when k = 0 and k = N/∆x. Therefore,
Fourier stability analysis predicts the Jacobi iterative method will not converge
when attempting to solve the 1D Laplace’s equation due to the spectral radius
at k = N/∆x. The condition of k = 0 is never fully realized, in practice, as the
minumum value of k is determined by the discrete mesh spacing.
60
3.5.1.2 Relaxed Jacobi Method
Although the classic Jacobi method is found inadequate to solve the 1D Laplace’s
equation, the method benefits from relaxation strategies to help convergence to-
wards an error-free solution.
The relaxed Jacobi method is implemented by redefining φn+1x to be
φn+1x = ω
(φnx+∆x + φnx−∆x
2
)+ (1− ω)φnx, (3.53)
where the relaxation factor, ω, is a positive non-zero real-valued number that serves
to weight the updated value of φx. The system is considered underrelaxed when
ω < 1 and overrelaxed when ω > 1. It can be seen that choosing ω = 1 reduces
the iterative process to the classic Jacobi method.
Applying Fourier stability analysis to this system, the spectral radius of the
relaxed Jacobi method, ρRJk , is found to be
ρRJk =
∣∣∣∣εn+1k
εnk
∣∣∣∣ =
∣∣∣∣ωcos(πk∆x
N
)− ω + 1
∣∣∣∣ . (3.54)
The spectral radius, ρRJk , has two values of interest. The first is k = 0 where
ρ = |ω − ω + 1| = 1, (3.55)
which shows there is no damping of the zero wavenumber. The other value of
interest is k = N/∆x where
ρ = |−ω − ω + 1| = |1− 2ω| . (3.56)
Fourier stability analysis predicts the relaxed Jacobi iterative method, when
attempting to solve the 1D Laplace’s equation, will converge towards an error-free
solution when underrelaxed but diverge when overrelaxed.
61
3.5.1.3 Gauss-Seidel method
The Gauss-Seidel iteration procedure is similar to Jacobi, except that updated
values of φ are used as the solution progresses. This leads to φn+1x expressed as
φn+1x =
(φnx+∆x + φn+1
x−∆x
2
). (3.57)
Applying Fourier stability analysis to the Gauss-Seidel expression of φn+1x gives
εn+1k = εnk
(eiπk∆xN
2− e− iπk∆xN
), (3.58)
ρGSk =
∣∣∣∣εn+1k
εnk
∣∣∣∣ =
∣∣∣∣∣ eiiπk∆xN
2− e−i iπk∆xN
∣∣∣∣∣ =1√
5− 4 cos(iπk∆xN
) . (3.59)
which indicates the classic Gauss-Seidel method will converge when solving the 1D
Laplace’s equation since ρGSk < 1 for all values of necessary values of k.
3.5.1.4 Relaxed Gauss-Seidel Method
While the classic Gauss-Seidel method is shown to converge the 1D Laplace’s
equation, it is possible to increase the performance of the iterative process.
The spectral radius of an iterative procedure gives insight into not only solver
stability but also expected solver performance. Solver performance increases as
max(ρk) decreases for appropriate values of k. Section 3.6.1.2 explains the rela-
tionship between spectral radius and solver performance in greater detail.
As with the relaxed Jacobi method, φn+1x corresponding to the relaxed Gauss-
Seidel procedure is expressed as
φn+1x = ω
(φnx+∆x + φn+1
x−∆x
2
)+ (1− ω)φnx. (3.60)
Applying Fourier stability analysis to the relaxed Gauss-Seidel expression of φn+1x
gives
62
εn+1k = εnk
(ωe
iπk∆xN + 2(1− ω)
2− ωe− iπk∆xN
), (3.61)
ρRGSk =
∣∣∣∣εn+1k
εnk
∣∣∣∣ =
∣∣∣∣∣ωeiπk∆xN + 2(1− ω)
2− ωe− iπk∆xN
∣∣∣∣∣=
√ω4 [sin(2θ)− 2 sin(θ)]2 + [ω2(cos(2θ)− 2 cos(θ)) + 4(ω − 1)]2√
[−4ω cos(θ) + ω2 + 4]2,
(3.62)
θ = α∆x =πk∆x
N. (3.63)
Equation 3.62 can be used to sweep through values of ω and minimize max(ρk).
Figure 3.4 shows the distribution of ρk as a function of θ for various values of ω
while Figure 3.5 shows max(ρk) as a function of ω.
Figure 3.5 shows that the minimum value of max(ρk), corresponding with the
best estimated solver performance, for the relaxed Gauss-Seidel method is obtained
near ω = 2. In practice, the optimal value of ω also depends on the smallest
wavenumber present in the system. On an evenly spaced grid with N + 1 grid
points, the smallest realized wavenumber corresponds to θ = π/N .
63
Figure 3.4: Spectral radii of relaxed Gauss-Seidel applied to 1D Laplace’s equationas function of wavenumber
64
Figure 3.5: Maximum spectral radius of relaxed Gauss-Seidel applied to 1DLaplace’s equation as function of relaxation factor. Spectral radii are obtainedby sampling ρk at wavenumber π/64.
65
3.5.2 Analysis of 3D Steady Linear Elasticity Equations
The steady linear elastic equations in index notation are
∂σij∂xj
+ bi = 0, (3.64)
σij = µ
(∂ui∂xj
+∂uj∂xi
)+ λ
∂uk∂xk
δij. (3.65)
Assuming no body forces,
∂σij∂xj
= µ
(∂2ui∂xj∂xj
+∂2uj∂xi∂xj
)+ λ
∂2uk∂xi∂xk
= 0. (3.66)
Writing Equation 3.66 in x, y, and z dimensions, respectively, produces
µ
(∂2u
∂x2+∂2u
∂y2+∂2u
∂z2
)+ (µ+ λ)
(∂2u
∂x2+
∂2v
∂x∂y+
∂2w
∂x∂z
)= 0, (3.67)
µ
(∂2v
∂x2+∂2v
∂y2+∂2v
∂z2
)+ (µ+ λ)
(∂2u
∂x∂y+∂2v
∂y2+
∂2w
∂y∂z
)= 0, (3.68)
µ
(∂2w
∂x2+∂2w
∂y2+∂2w
∂z2
)+ (µ+ λ)
(∂2u
∂x∂z+
∂2v
∂y∂z+∂2w
∂z2
)= 0. (3.69)
Reorganizing the equations to combine all normal diffusion contributions shows
[µ
(2∂2u
∂x2+∂2u
∂y2+∂2u
∂z2
)+ λ
∂2u
∂x2
]+ (µ+ λ)
(∂2v
∂x∂y+
∂2w
∂x∂z
)= 0, (3.70)
[µ
(∂2v
∂x2+ 2
∂2v
∂y2+∂2v
∂z2
)+ λ
∂2v
∂y2
]+ (µ+ λ)
(∂2u
∂x∂y+
∂2w
∂y∂z
)= 0, (3.71)
66
[µ
(∂2w
∂x2+∂2w
∂y2+ 2
∂2w
∂z2
)+ λ
∂2w
∂z2
]+ (µ+ λ)
(∂2u
∂x∂z+
∂2v
∂y∂z
)= 0. (3.72)
These equations can now be discretized using the relaxed Jacobi iterative
method then transformed into Fourier space. Equation 3.70 will be used to show
this process.
A first-order and second-order central finite difference can each be written as
∂φ
∂x≈ φx+∆x,y,z − φx−∆x,y,z
2∆x, (3.73)
∂2φ
∂x2≈ φx+∆x,y,z − 2φx,y,z + φx−∆x,y,z
∆x2. (3.74)
In the n-th iteration of the point Jacobi iterative method, φ at the point of
interest will be that of step n while all other values of φ will be obtained from the
previous Jacobi step. As such,
∂φ
∂x≈φnx+∆x,y,z − φnx−∆x,y,z
2∆x, (3.75)
∂2φ
∂x2≈φnx+∆x,y,z − 2φn+1
x,y,z + φnx−∆x,y,z
∆x2. (3.76)
For the cross diffusion terms, finite difference approximations can be written
as
∂2φ
∂x∂y≈φnx+∆x,y+∆y,z − φnx+∆x,y−∆y,z − φnx−∆x,y+∆y,z + φnx−∆x,y−∆y,z
4∆x∆y. (3.77)
Substituting the finite differences into Equation 3.70 produces
67
[(2µ+ λ)
(unx+∆x,y,z − 2un+1
x,y,z + unx−∆x,y,z
∆x2
)+ µ
(unx,y+∆y,z − 2un+1
x,y,z + unx,y−∆y,z
∆y2
)+ µ
(unx,y,z+∆z − 2un+1
x,y,z + unx,y,z−∆z
∆z2
)]+ (µ+ λ)
(vnx+∆x,y+∆y,z − vnx+∆x,y−∆y,z − vnx−∆x,y+∆y,z + vnx−∆x,y−∆y,z
4∆x∆y
+wnx+∆x,y,z+∆z − wnx+∆x,y,z−∆z − wnx−∆x,y,z+∆z + wnx−∆x,y,z−∆z
4∆x∆z
)= 0.
(3.78)
Solving for un+1x,y,z and applying relaxation yields
un+1x,y,z = ω
{[(2µ+ λ)
(unx+∆x,y,z + unx−∆x,y,z
Ax∆x2
)+ µ
(unx,y+∆y,z + unx,y−∆y,z
Ax∆y2
)+ µ
(unx,y,z+∆z + unx,y,z−∆z
Ax∆z2
)]+ (µ+ λ)
(vnx+∆x,y+∆y,z − vnx+∆x,y−∆y,z − vnx−∆x,y+∆y,z + vnx−∆x,y−∆y,z
4Ax∆x∆y
+wnx+∆x,y,z+∆z − wnx+∆x,y,z−∆z − wnx−∆x,y,z+∆z + wnx−∆x,y,z−∆z
4Ax∆x∆z
)}+ (1− ω)unx,y,z,
(3.79)
Ax = 2
(2µ+ λ
∆x2+
µ
∆y2+
µ
∆z2
). (3.80)
The discrete Fourier transform for φ can be expressed as,
φnx,y,z → φnq,r,seiφqxeiφryeiφsz. (3.81)
For the sake of brevity while working with a multidimensional vector equation,
φnx,y,z will be denoted as the solution round-off error of φnx,y,z.
Applying the Fourier transform to Equation 3.78 and dividing through by
(eiφqxeiφryeiφsz) gives
68
un+1q,r,s = ω
{unq,r,s
[(2µ+ λ)
(eiφq∆x + e−iφq∆x
Ax∆x2
)+ µ
(eiφr∆y + e−iφr∆y
Ax∆y2
)+ µ
(eiφs∆z + e−iφs∆z
Ax∆z2
)]+ (µ+ λ)
(vnq,r,s
eiφq∆xeiφr∆y − eiφq∆xe−iφr∆y − e−iφq∆xeiφr∆y + e−iφq∆xe−iφr∆y
4Ax∆x∆y
+ wnq,r,seiφq∆xeiφs∆z − eiφq∆xe−iφs∆z − e−iφq∆xeiφs∆z + e−iφq∆xe−iφs∆z
4Ax∆x∆z
)}+ (1− ω)unq,r,s.
(3.82)
Which can be cleaned up as
un+1q,r,s = Bxu
nq,r,s + Cxv
nq,r,s +Dxw
nq,r,s, (3.83)
Ax = 2
(2µ+ λ
∆x2+
µ
∆y2+
µ
∆z2
), (3.84)
Bx =ω
Ax
[2µ+ λ
∆x2
(eiφq∆x + e−iφq∆x
)+
µ
∆y2
(eiφr∆y + e−iφr∆y
)+
µ
∆z2
(eiφs∆z + e−iφs∆z
)]+ (1− ω),
(3.85)
Cx =ω(µ+ λ)
4Ax∆x∆y
[ei(φq∆x+φr∆y) − ei(φq∆x−φr∆y)
− e−i(φq∆x−φr∆y) + e−i(φq∆x+φr∆y)
],
(3.86)
Dx =ω(µ+ λ)
4Ax∆x∆z
[ei(φq∆x+φs∆z) − ei(φq∆x−φs∆z)
− e−i(φq∆x−φs∆z) + e−i(φq∆x+φs∆z)].
(3.87)
69
Repeating this procedure for the y- and z-dimension produces
vn+1q,r,s = Byu
nq,r,s + Cyv
nq,r,s +Dyw
nq,r,s, (3.88)
wn+1q,r,s = Bzu
nq,r,s + Czv
nq,r,s +Dzw
nq,r,s, (3.89)
Ay = 2
(µ
∆x2+
2µ+ λ
∆y2+
µ
∆z2
), (3.90)
Az = 2
(µ
∆x2+
µ
∆y2+
2µ+ λ
∆z2
), (3.91)
By =AxCxAy
, (3.92)
Bz =AxDx
Az, (3.93)
Cy =ω
Ay
[µ
∆x2
(eiφq∆x + e−iφq∆x
)+
2µ+ λ
∆y2
(eiφr∆y + e−iφr∆y
)+
µ
∆z2
(eiφs∆z + e−iφs∆z
)]+ (1− ω),
(3.94)
Cz =AyDy
Az, (3.95)
Dy =ω(µ+ λ)
4Ay∆y∆z
[ei(φr∆y+φs∆z) − ei(φr∆y−φs∆z)
− e−i(φr∆y−φs∆z) + e−i(φr∆y+φs∆z)
],
(3.96)
70
Dz =ω
Az
[µ
∆x2
(eiφq∆x + e−iφq∆x
)+
µ
∆y2
(eiφr∆y + e−iφr∆y
)+
2µ+ λ
∆z2
(eiφs∆z + e−iφs∆z
)]+ (1− ω).
(3.97)
These three equations can be expressed simultaneously as
Bx Cx Dx
By Cy Dy
Bz Cz Dz
unq,r,s
vnq,r,s
wnq,r,s
=
un+1q,r,s
vn+1q,r,s
wn+1q,r,s
.The stability of this scheme is determined by the spectral radius of matrix G.
G(φq, φr, φs) =
Bx Cx Dx
By Cy Dy
Bz Cz Dz
. (3.98)
The spectral radius is defined as the maximum eigenvalue of the linear system.
Figures 3.6 and 3.7 show contour plots of the spectral radius as a function of φy
and φz when φx = 0 and φx = π, respectively. Figure 3.8 shows spectral radius
as a function of wavenumber when φx = φy = φz. The spectral radii profiles show
that for 3D linear elasticity the maximum spectral radius value may not occur at
the smallest wavenumber when using the relaxed Jacobi method. As such, care
must be taken when searching for the maximum value of spectral radius for a given
relaxation factor as the maximum may occur at any wavenumber present in the
discretized system.
71
Figure 3.6: Spectral Radius of relaxed Jacobi method applied to linear elasticityas function of φy and φz when φx ≈ 0.0
72
Figure 3.7: Spectral Radius of relaxed Jacobi method applied to linear elasticityas function of φy and φz when φx = π
73
Figure 3.8: Spectral radius of relaxed Jacobi method applied to linear elasticity asfunction of wavenumber when φx = φy = φz for various relaxation factors.
74
3.5.3 Fourier Stability of Analysis of Multi-Step Solution
Procedures
Fourier stability analysis typically assumes a single-step iterative method, meaning
a single, constant relaxation factor is used throughout the entire iterative proce-
dure. Work has been done showing the benefit of using multi-step solution pro-
cedures. This section briefly discusses the extension of a typical Fourier stability
analysis to such a procedure.
Assume an iterative procedure with stability matrix being a function of the
relaxation parameter, g(ω).
Given a two-step procedure, the analysis would provide
g(ω1)unq,r,s = u′
q,r,s, (3.99)
g(ω2)u′
q,r,s = un+1q,r,s, (3.100)
g(ω2)g(ω1)unq,r,s = un+1q,r,s. (3.101)
The effects of the stability matrices are multiplicative and the generalized form
for an m-step is [m−1∏i=0
g(ωm−i)
]unq,r,s = un+1
q,r,s. (3.102)
This relation allows the term∏m−1
i=0 g(ωm−i) to be used in analyzing multi-step
stability. Furthermore, the relaxation factors used in the multi-step procedure can
be optimized to improve code performance (as discussed in Section 3.6.1.2).
75
3.6 Miscellaneous High Performance Computing
Improvements
3.6.1 Linear Solver Performance Optimization
The possibility of multi-step Jacobi and Gauss-Seidel linear solvers opens up the
opportunity to explore the performance benefits associated with various sets of
relaxation factors for arbitrary numbers of steps. Given the nonlinear nature of
the physical system being solved, incremental improvements in linear solver per-
formance can have compounding effects on the overall performance of the compu-
tational tool developed through this work.
Two approaches to optimizing the linear solver performance are as follows:
1) an empirical-based approach that evaluates performance using the nonlinear
residual history and 2) a Fourier stability based approach that uses the spectral
radius of the numerical system to predict performance.
3.6.1.1 Empirically-based Performance Optimization
An initial attempt at optimizing linear solver performance is to find a combina-
tion of relaxation factors that minimize the slope of the solver error residual as a
function of iteration number. Figure 3.9 shows an example of such a residual plot.
As seen in the plot, the relationship becomes linear after many iterations. The
slope of this linear region can be computed and used as the cost function for the
optimization routine.
To help improve the performance of the optimization routine, radial basis func-
tions (RBFs) are used to develop a response surface of the function space. The
optimization routine is then used to find the global minimum of the response sur-
face. The optimal solution returned by the routine is evaluated using the linear
solver. The solution obtained from the linear solver is used to update the re-
sponse surface and re-run the optimization routine. This cycle is repeated until
the optimization routine converges on an optimal solution.
76
Figure 3.9: Sample residual profile showing solution root mean square (RMS) erroras a function of solver iteration number
RBF response surfaces provide an approximation to the unknown function gov-
erning system stability as a function of relaxation factors. The resulting function
approximation will pass exactly through the points used to compute the curve pa-
rameters since RBFs assume observe points to be exact [108]. While there many
forms of RBFs, this work uses the multiquadric radial basis function expressed as
φ(r) =√
1 + (εr)2, (3.103)
77
r = ||x− xi||, (3.104)
where ε is an adjustable constant set as the average norm between all observed
points, x is an arbitrary vector corresponding to a point in parameter space, xi is
the i-th observation point used to create the curve fit, and r is the norm computed
between x and xi.
Using the definition of φ, the curve-fit approximation is computed as
y(x)RBF, approx =N∑i=1
wiφi(||x− xi||) (3.105)
where N is the total number of observed points, φi is the RBF associated with the
i-th observation point, and wi is the weight associated with the i-th RBF.
The final curve-fit is a linear combination of the RBFs, φ, placed at each of the
observation points. This linearity allows the weights to be obtained using a linear
least-squares approach similar to that shown in Section B.1.
The linear least-squares approach allows the weights to be solved as
w = (XTX)−1XTyobserved, (3.106)
Xij =∂y(x)RBF, approxi
∂wj= φ(||xi − xj||). (3.107)
X will be a square matrix of rank m, where m is the number of observation points.
The performance benefit of using an RBF response surface as an approximation
of the response surface is in the ability to quickly evaluate function calls and the
possibility to analytically compute the gradient for use as input into the optimiza-
tion routine.
3.6.1.2 Fourier-based Performance Optimization
Let β0 be the maximum spectral radius of the a given solver configuration 0. It is
possible to compute the speedup, s, of any other solver configuration i as follows
78
s =ln(βi)
ln(β0). (3.108)
This equation is obtained from the relationship between spectral radius of sta-
bility matrix and solution error.
Beginning with the relationship
∣∣∣∣rn+1
rn
∣∣∣∣ = β, (3.109)
where rn is the error residual at n-th iteration of the linear solver. Given the
iterative nature of this equation, a multiplicative relationship can be shown to be
∣∣∣∣rn+α
rn
∣∣∣∣ = βα. (3.110)
Now assume that in reducing the residuals by a factor of 10 takes α/c and α
iterations for models 0 and a, respectively, the corresponding spectral radii are
related as
0.1 = βα/c0 = βαa . (3.111)
Taking the natural logarithm of each side produces
α
cln(β0) = αln(βa), (3.112)
which is simplified as
s =1
c=
ln(βa)
ln(β0), (3.113)
This analysis does assume, however, that each iteration requires the same amount
of computational work.
The ability to compute code speedup using spectral radii allows for the possibil-
ity of devising an optimization routine to maximize code performance, particularly
for multi-step solution procedures. It is possible to find a set of relaxation factors,
79
for a given multi-step procedure, that minimizes the spectral radius and, in turn,
maximizes code performance of the solver. However, developing such an optimiza-
tion scheme is no trivial task.
The first step in developing an optimization routine for multi-step solution pro-
cedures is obtaining the spectral radius for an arbitrary set of relaxation factors.
There has been some work done in finding spectral radii and optimal relaxation
factors for linear solvers of elliptic PDE systems [109, 110]. While these studies
provided analytic approaches to the system optimization, the analysis was per-
formed for scalar PDEs in 1D and 2D. The work being presented deals with 3D
vector PDEs of elliptic and parabolic nature.
Section 3.5.2 shows the Fourier stability analysis of the static 3D linear elasticity
equations and presents an expression for the stability matrix G as a function of the
wavenumber in each spatial dimension. Any arbitrary combination of wavenumbers
will produce three eigenvalues for the 3x3 matrix G. The spectral radius of the
system is the maximum eigenvalue of all possible values for matrix G with φ in
the range [φmin, π].
One approach to obtaining the spectral radius of G is to sweep through possible
values of φ to find the maximum system eigenvalue; this approach, however, is time
consuming and computationally inefficient.
A second approach is to analytically derive the maximum eigenvalues by finding
values of φ that satisfy the condition
∂G
∂ei= 0, (3.114)
where ei is the i-th system eigenvalue.
A third approach is to use a numerical optimization routine to find the maxi-
mum system eigenvalue. Using a metaheuristic optimization approach (e.g., tabu
search, simulated annealing, particle swarm) it may be possible to quickly locate
the global minimum. This third approach will be used in this work due to its ease
of implementation.
The second, and final, step in developing an optimization routine for multi-
80
step solution procedures is finding the set of relaxation factors that minimize the
spectral radius of matrix G, thus increasing speedup. This step of the routine
will use the subroutine developed in step one to compute the spectral radii, then
compute speedup by using Equation 3.108 and the spectral radius of a reference
solver.
For illustrative purposes, this optimization routine was performed on a two-step
solver using the model system described in Section 3.5.2.
Figure 3.10 shows a plot of the computed performance increase for a given pair
of relaxation factors; any combination of relaxation factors predicted to diverge
is prescribed a speedup value of zero. All remaining speedup values are made
negative so that global minimum of the system corresponds to maximum predicted
speedup. The speedup values in this plot are obtained using a simulated annealing
optimization routine.
81
Figure 3.10: Estimated performance increase of two-step relaxed Jacobi methodapplied to 3D linear elasticity equations. Two-step methods show the potential of78% increase in performance over the single step relaxed Jacobi method.
Chapter 4Verification, Validation, and Results
4.1 Fixed Shape Leukocyte Rolling
The PMN rolling algorithm presented in Section 3.3.1 allows the model PMN to
move along the near-wall region in a manner that captures complex flow-structure-
biochemistry interactions.
Figures 4.1, 4.2, and 4.3 show the trajectory of a point on the PMN surface. The
PMN translational and rotational velocities are set to 120 µm/µs and 120 rad/µs,
respectively. Assuming a perfect sphere in pure rolling leads to both velocities
having the same magnitude equal to that of the shear rate. The trajectory plots
do, however, show some slipping at wall-adjacent points. The slippage is caused
by the analytic transform.
The oscillation of the surface point is indicative of the PMN surface rotating
about the body centroid.
For this case, the centroid height on the undeformed sphere is set to zero and
the PMN centroid is moving in the positive-x direction. Figure 4.1 shows the path
of the point throughout the entire simulation. The point path is smooth and much
longer on the top of the PMN than the bottom. Figure 4.2 shows the point does
spend more time in forward motion than backwards motion. Figure 4.3 shows
the point does spends equal time in upward motion and downward motion. The
smooth trajectory and sustained time along the top of the PMN allows for stable
83
Figure 4.1: Trajectory of point on the surface of rolling PMN.
localized modeling of biochemistry.
Applying the rolling algorithm to every point belonging to the PMN compu-
tational mesh enables mesh motion while retaining the mesh topology needed to
accommodate localized modeling of biochemical bonds.
84
Figure 4.2: X coordinate of surface point vs time during PMN rolling.
85
Figure 4.3: Y coordinate of surface point vs time during PMN rolling.
86
4.2 Finite-Volume Structural Mechanics
4.2.1 3D Linear Elastostatics
Linear elastostatics, while ill-suited for cellular FSBI problems in the microvascula-
ture, provides great insight into the ability to solve structural mechanics problems
using a baseline solver designed for fluid mechanics. Testing of the linear elastic-
ity implementation is done using classical beam problems for which linear elastic
analytic solutions exist.
4.2.2 Stretching of Prismatic Beam under Self-Weight
Consider a 3D rectangular beam as seen in Figure 4.4. With gravity acting in the
positive-z direction, the beam will experience the body force b = [0, 0, ρg]; where
ρ is the material mass density and g is the acceleration due to gravity.
For the prismatic beam test cases, notional properties of steel are used in the
simulations; modulus of Elasticity, E, set to 200 GPa and Poisson’s ratio, ν, set to
0.3. The beam dimensions are set to 0.3 meters, 0.3 meters, and 2.4 meters in the
x, y, and z directions, respectively. Grid spacing is chosen to be uniform in every
direction.
In this configuration, the boundary conditions are
σz(x, y, L) = 0,
σx = σy = 0,
τxy = τyz = τzx = 0.
(4.1)
Using the direct integration approach, the stress field in the rectangular beam
is found to be
87
Figure 4.4: Pictorial representation of prismatic beam stretching under self-weight.
σz(x, y, z) = ρg(L− z),
σx = σy = 0,
τxy = τyz = τzx = 0.
(4.2)
Assuming symmetry about the surface at (z = 0) and no rigid-body motion (i.e.,
center of beam does not move and axes does not rotate) at point C (x = y = z = 0)
gives the solution
u(x, y, z) =νρgx
E(z − L),
v(x, y, z) =νρgy
E(z − L),
w(x, y, z) = − ρg2E
[z2 − 2Lz + ν(x2 + y2)
].
(4.3)
The results of this test case are shown in Figure 4.5. Necking of the beam does
88
occur at z = 0 which is consistent with the symmetry about the z = 0 surface.
Grid convergence results using the l2-norm of σzz for this case are shown in
Figure 4.6. The lp-norm on an arbitrary function a is described in Equation 4.4 for
an arbitrary value of p. The order of convergence for the linear elasticity solver was
found to be 1.45. While the discrete operators described in Section 3.1 are second-
order accurate on a block mesh with uniform grid spacing, the boundary condition
implementation is less than second-order. Proper implementation of second-order
accurate boundary conditions will then return a second-order accuracy for the
entire solver.
lp =(∑
(|aobserved − aexact|))1/p
(4.4)
Figure 4.5: Deformation of bar due to self weight.
89
Figure 4.6: Grid convergence plot for stretching of beam due to self weight.
4.2.3 Flexure of Prismatic Beam Due to End Loading
Given the same geometry and material properties as the self-weight case, consider
a 3D rectangular beam subject to end loading T as shown in Figure 4.7. T = 90N
is applied at the shear center of the (z = L) cross-section to ensure the body is
not subject to any torsion.
In this configuration, the boundary conditions are
90
Figure 4.7: Pictorial representation of prismatic beam experiencing flexure due toend loading T .
u(x, y, 0) = v(x, y, 0) = w(x, y, 0) = 0,
by(0, 0, L) =T
dV=
T
∆x∆y∆z,
σz(x, y, L) = 0,
σx(x, y, z) = σy(x, y, z) = 0,
τxy(x, y, z) = 0.
(4.5)
Assuming a harmonic stress function, as presented by Sadd [111], the stress
field in the rectangular beam is found to be
91
τxz(x, y, z) =νA2T
2(1 + ν)π2Ix
∞∑n=1
(−1)n
n2
sin2nπxA
sinh2nπyA
coshnπBA
,
τyz(x, y, z) =T
2Ix
(B2
4− y2
)+
νT
6(1 + ν)Ix
[3x2 − A2
4− 12A2
4π2
∞∑n=1
(−1)n
n2
cos2nπxA
cosh2nπyA
coshnπBA
],
σz(x, y, z) = − TIxy(L− z),
Ix =AB3
12.
(4.6)
where Ix is the area moment of inertia of the cross-sections normal to the z-direction
and T is the magnitude of the end loading. More detailed description of the analytic
solution technique can be found in Sadd [111].
The results of this test case are shown in Figure 4.8. The prismatic beam flexes
as expected. Grid convergence results using the l2-norm of σzz for this case are
shown in Figure 4.9 and reproduce the 1.45 order of convergence shown by the
self-weight case.
92
Figure 4.8: Deformation of bar due to flexure traction loading at bar end.
93
Figure 4.9: Grid convergence plot for flexure of beam due to end loading.
94
4.2.4 Saint-Venant Kirchhoff Hyperelasticity
The implementation of hyperelastic structural mechanics is built upon the linear
elastic functionality. Addition of the explicitly-treated nonlinear terms arising from
the SVK constitutive model can be tested using steady rigid body rotation cases.
4.2.4.1 Hyperelastic Sphere undergoing Rigid Body Rotation
Consider a 3D sphere centered at c=[cx, cy, cz] as shown in Figure 4.10.
Figure 4.10: Surface mesh of sphere.
A rotation θ is applied at the sphere center about an axis pointing the positive
z-direction passing through the sphere center. The displacement of any point on
the body due to the rotation θ is
95
u
v
w
=
(cos θ − 1) − sin θ 0
sin θ (cos θ − 1) 0
0 0 0
x− cx
y − cy
z − cz
. (4.7)
Equation 4.7 can be used to show E′′ij to be
E′′
ij = 0.5(F′′
kiF′′
kj − δij) = 0. (4.8)
It can also be shown that any hyperelastic strain measure will be identically
zero for any steady rigid body rotation. As such, this case is used to ensure the
correct implementation of hyperelastic constitutive models.
The displacements computed using Equation 4.7 are set as the boundary con-
ditions at the surface of the sphere. The l2-norm of the stress field computed in
the body was zero for arbitrary values of θ.
4.3 Single Body Simulations
A number of single body cases are presented using scales relevant to the microvas-
cular environment.
4.3.1 Impulsively Started Rigid TC in Uniform Flow
An important case to test the semi-implicit rigid body 6DOF motion solver is an
impulsively started rigid body in uniform low Reynolds number flow, as shown in
Figure 4.11.
The flow-induced drag on a rigid TC in low Reynolds number uniform flow is
computed using Stokes’s law
F (t)stokes = 6πµr[uflow − u(t)TC ], (4.9)
where µ is the dynamic viscosity of the fluid, r is the sphere radius, and uflow is
96
Figure 4.11: Pictorial representation of computational domain and boundary con-ditions for rigid TC in uniform flow.
the velocity of the uniform flow field. Solving Newton’s law of motion gives the
expected velocity profile for the rigid sphere
du(t)TC
dt=
6πµr
ρV[uflow − u(t)TC ], (4.10)
∴ u(t)TC = uflow(
1− e−6πµrtρV
). (4.11)
where ρ and V are the density and volume of the TC, respectively.
The time to steady-state can be found using Equation 4.11 solving for t0.99
where [u(t0.99)TC/uflow] = 0.99
t0.99 =−ρV6πµr
ln(0.01), (4.12)
Setting µ = 1 pgµm·µs , ρ = 1 pg
µm3 , and r = 8µm, time to steady-state is found
to be t0.99 ≈ 65.5µs. The value of t0.99 is less than the stability limit of the
97
semi-implicit 6DOF solver allowing timestep to be chosen based solely on desired
temporal resolution.
This case was run using the presented FSBI solver with UInf = 1.4× 10−4 µmµs
.
Figure 4.12 shows the flow domain at t = 10µs. Figure 4.13 shows the 6DOF
solver results alongside the profile given by Equation 4.11. The two trajectories
are seen to be in agreement.
Figure 4.12: Angled view of TC and slice of flow field along the centerline of thecomputational domain at t = 10µs. Slice is colored by velocity magnitude andflow is in the +x direction. Gradients in the velocity field indicate the TC has notyet reached Uinf
98
Figure 4.13: Analytic and computed velocity profiles of rigid TC in uniform flowas function of time.
4.3.2 Rigid TC in Linear Shear Flow
Another case to explore the ability of the semi-implicit 6DOF solver is a rigid body
in linear shear flow, as shown in Figure 4.14.
The linear shear flow will induce translation and rotation on the TC. In fact,
the TC will generate lift in the direction of increasing velocity due to the shear
[112,113].
Setting µ = 1 pgµm·µs , ρ = 1 pg
µm3 , r = 8µm, and γ = 50s−1, this case was simulated
using the presented FSBI solver. The flow field computed by the simulation is
99
Figure 4.14: Pictorial representation of computational domain and boundary con-ditions for rigid TC in linear shear flow.
shown in Figure 4.15. The results of this case are shown in Figures 4.16 and
4.17. This case makes use of the body cyclic conditions proposed in [29]. The
vertical dashed lines in Figure 4.16 denote the end of the discretized computational
domain. The cyclic conditions allow for an effective flow length of 480µm despite
using a computational domain of length 120µm. The case ended when the rigid
TC collided with the top wall of the computational domain. The effects of the top
wall can be seen in Figure 4.17 by the increase in signal noise towards the end of
the simulation.
100
Figure 4.15: Angled view of TC and slice of flow field along the centerline of thecomputational domain at t = 1ms. Slice is colored by velocity magnitude and flowis in the +x direction. Gradients in the velocity field are due to the prescribedflow shear rate.
101
Figure 4.16: Trajectory of TC centroid in linear shear flow. Dashed vertical linesrepresents the end of the modeled computational domain. Cyclic conditions areused to increase the effective flow length.
102
Figure 4.17: Velocity profile of TC centroid in linear shear flow. Simulation endsas TC collides into the top wall of the computational domain. Wall effects appearin the profile as signal noise towards the end of the simulation.
103
4.3.3 Rigid Helical Microswimmer
One exciting application of this work is the simulation of rigid helical microswim-
mers. Much experimental work has been done on the design, fabrication, and
actuation of helical microswimmers [88,90] which can be enhanced by appropriate
modeling techniques.
The presented FSBI solver is well-suited to simulate helical microswimmers
and demonstrate this ability. Two simulation are performed for a rigid helical
microswimmer was placed in a quiescent fluid. In the first simulation, a constant
torque due to the rotation of a magnetic field aligned with the helical axis is applied
to the microswimmer. In the second simulation, the microswimmer is subject to a
constant angular velocity. Flow is induced by the microswimmer due to the no-slip
condition at the fluid-solid interface. The parameters used are listed in Table 4.1
and Figure 4.19 shows the resulting microswimmer. Effects of frequency step-out,
loss of propulsion due to the microswimmers inability to remain in sync with the
input signal, are ignored in these simulations.
104
Figure 4.18: Schematic of helical microswimmer with labeled geometric entities.Helical geometries are defined by helix angle (θ), helix radius (R), filament radius(r), helix pitch (λ), and number of turns (n)
105
Figure 4.19: Mesh of the helical microswimmer used in the simulations performedin this work. Geometric properties of this helix are found in Table 4.1
106
Figure 4.20: Mesh of the helical microswimmer and slice of the meshed fluid domainused in the simulations performed in this work. Points on the interface of the fluidand helix domains match exactly as conformal meshing is used at the interface.
107
4.3.3.1 Constant Torque
In this simulation, a constant torque per unit volume of 1.95× 10−2pg/µm/µs2 is
applied to a rigid helical microswimmer. The model presented by Abbott [90] pro-
vides an expected relationship between angular velocity and translational velocity,
which is expressed in Equation 4.13. This expected relationship is obtained using
Purcell’s helical motion propulsion matrix [114] and Lighthill’s approximation of
drag on thin rigid helical bodies [115],
u =−baω, (4.13)
a = 2πnR
(γp cos2(θ) + γn sin2(θ)
sin(θ)
), (4.14)
b = 2πnR2(γp − γn) cos(θ), (4.15)
γp =2πµ
ln(
0.36πRr sin θ
) , (4.16)
γn =4πµ
ln(
0.36πRr sin θ
)+ 0.5
. (4.17)
With constant torque acting on the body, the expected angular and transla-
tional velocity profiles are
dω(t)
dt=TaxialIaxial
(4.18)
Table 4.1: Properties of Helical Microswimmer
Pitch, λ [µm] 2Filament Radius, r [µm] 0.1Helix Angle, θ [rad] tan−1(π)Helix Radius, R [µm] 1Number of Turns, n [-] 5
108
Figure 4.21: Helix in flow and slice of flow field along the centerline of the compu-tational domain at t = 97µs. Slice is colored by the z-components of velocity. Redindicates flow coming out of the page and blue indicated flow going into the page.
ω(t) =TaxialIaxial
t (4.19)
Setting µfluid = 1 pgµm·µs , ρfluid = 1 pg
µm3 , the computed trajectory of the mi-
croswimmer centroid is shown in Figure 4.22 and the flow field computed by the
simulation is shown in Figure 4.21. The velocity profile computed using the pre-
sented FSBI solver gives a linear profile which favorably agrees with the Abbott
model expectations. The profile slopes describe the acceleration of the body. The
ratio of the computed acceleration and the Abbott model expectation is approxi-
mately 1.45. This deviation can likely be attributed to the Lighthill drag approx-
imation.
109
Lighthill approximated the viscous drag on a helix assuming stokeslets along
the helical path. This approach may not accurately capture the effect a helical
segment may have on the effective drag of a near-by segment. More work can be
done to explore potential relationships between the magnitude of deviation from
theory and helical geometric parameters.
Figure 4.22: Comparison of computed axial velocity profile and predictions basedon the Abbott model for a helical microswimmer undergoing constant torque.These models are in agreement with a slight difference in the profile slope. Thisdifference is likely caused by the idealized viscous drag approximation in the Abbottmodel.
110
4.3.3.2 Constant Angular Velocity
In this simulation, a rigid helical microswimmer is subjected to a constant angular
velocity of 10 revolutions per second about the microswimmer’s helical axis. Using
the same fluid and geometric properties as the constant torque simulation, the
computed normalized velocity profile of the microswimmer is shown in Figure
4.23. The velocity profile shown is normalized by the steady-state axial velocity
predicted by the Abbott model. The FSBI solver found the normalized steady-
state velocity to be approximately 0.86. As with the constate torque case, the
velocity profile’s deviation from the value predicted by the Abbott model is likely
attributed to the drag approximation used by Abbott.
111
Figure 4.23: Comparison of computed normalized axial velocity profile for a helicalmicroswimmer undergoing constant angular velocity. Velocity is normalized by theAbbott model prediction. The steady-state velocity asymptotes to approximately0.86.
4.3.4 Magnetically Actuated Hyperelastic Microbead in Cou-
ette Flow
Another area of interest is the design and modeling of magnetically actuated de-
formable DDMRs.
For this case, an elastic body was placed near a wall with the nominal PMN
shape presented in Section 3.3.1, as show in Figure 4.24.
112
Figure 4.24: Pictorial representation of computational domain for wall-adjacenthyperelastic microbead in Couette flow.
The FSI traction interface condition was enforced on the entire body surface.
The body was tethered in place using a volumetric body force representative of
magnetic forces. The tethering body force per unit volume is described in Equation
4.20 and applied to each polyhedron inside the deformable body.
bmagi =1
V body,tot
∫S
(µ∂ui∂xj− pfaceδij
)dSj, (4.20)
where the integral is computed at the body surface, ui is the i-th component of
the flow velocity, pface is flow pressure at the surface, µ is the fluid viscosity, and
V body,tot is the total body volume.
This case was run with a timestep of ∆t = 10µs, modulus of Elasticity of
E = 0.504Pa, Poisson’s ratio of ν = 0.133, flow shear rate of 50s−1, the un-
deformed microbead is sphere of radius 4µm, the microbead shape is initialized
using the PMN transform described in Section 3.3.1, and the centroid of the trans-
formed microbead is set to a height of 2.25µm. Figure 4.25 shows the equilibrium
113
microbead shape and a slice of the flow field over the body along the center of the
computational domain. Figure 4.26 shows a comparison of the body shapes and
location at various points in time during the simulation. The final shape has a flat
bottom surface and has evolved to an equilibrium shape best suited for this flow
region.
In addition to obtaining the equilibrium shape, this case provides a glimpse
into the development of active controllers for microbead design. The body force
obtained from Equation 4.20 can be used with Equations 2.73 and 2.74 to esti-
mate the required magnetic field needed to hold the bead in place. Furthermore,
Equations 4.20, 2.70, 2.71, 2.73, and 2.74 can be used together to determine the
field needed to move the bead along a prescribed trajectory.
Figure 4.25: Side view of the hyperelastic microbead and slice of flow field along thecenterline of the computational domain. The microbead has reached its equilibriumshape. Flow is in the +x direction
114
Figure 4.26: Comparison of initial and final shape of wall-adjacent magnetic mi-crobead in linear shear flow. The final shape has a flat bottom surface and hasevolved to an equilibrium shape best suited for this flow region. Flow is from leftto right.
4.4 Multiple Body Simulations
4.4.1 Free-Flowing Rigid TC and Wall-Adhered Hypere-
lastic PMN in Couette Flow
The FSBI solver presented has been developed to handle an arbitrary number of
bodies with differing structural mechanics treatments. This capability is demon-
strated by using the unified solver to simultaneously model a rigid TC and hyper-
elastic PMN in Couette flow in the same computational domain.
115
A wall-adhered PMN is placed in the near-wall region and modeled using the
SVK hyperelastic constitutive model. A rigid TC is placed upstream of the PMN.
Figure 4.27 shows a pictorial representation of the computational domain.
Figure 4.27: Pictorial representation of computational domain for free-flowing rigidTC and wall-adhered hyperelastic PMN in Couette flow.
For simplicity, zero-displacement Dirichlet conditions are enforced on PMN
faces near the wall and the FSI traction matching interface condition is enforced
on all other PMN faces. The TC trajectory is solved using the 6DOF solver.
The zero-displacement condition on the near-wall PMN faces, while non-physical,
allows for an exploration of the hyperelasticity solver’s ability without requiring
the fine mesh resolution necessary to resolve the flow between the wall and fixed
body.
Figure 4.28 shows the cell shapes and positions at various timepoints through-
out the simulation. The rigid TC translates and rotates as expected in shear flow.
The PMN deforms in a shearing motion and begins to exhibit signs of rolling along
the endothelial surface. Figure 4.29 gives a side view of the computational domain
116
at t = 9, 320µs to better show the computed shapes.
Figure 4.28: Rigid TC and wall-adhered hyperelastic PMN at various times duringthe simulation. The bodies are colored by magnitude of displacement.
More work is needed to fully capture the PMN rolling motion using this FSBI
formulation. One approach is to use the localized biochemistry model presented
in Chapter 2 to solver for all interactions between wall-adjacent PMN and the
117
Figure 4.29: Rigid TC, wall-adhered hyperelastic PMN, and slice of flow fieldalong centerline of computational domain at t = 9, 320µs. The slice is colored bymagnitude of flow velocity.
endothelial wall. This approach, however, may significantly increase the compu-
tational cost of these simulations as the mesh on the surface of the endothelial
wall would need to be refined for adequate spatial resolution of the biochemical
interactions. Another approach is to develop a different biochemistry model that
is not dependent on the resolution of the computational mesh at the wall (e.g., a
biochemistry model based primarily on distance from the wall). Either approach
may allow wall-adjacent PMN to roll along the endothelial wall. Nonetheless, this
dissertation provides much of the framework needed to model that phenomenon.
118
4.4.2 Rigid Two-cell Aggregate Formation Simulations
After demonstrating all of the new functionality implemented into the CellCFD-
PSU framework, the rigid two-cell ADH1 case from [29] was re-run to ensure the
new FSBI solver could reproduce the results.
In this case, a free-flowing rigid TC interacts with a wall-adhered rigid PMN.
An initial collision occurs between the bodies, followed by a period of transient
biochemical interactions, and finally aggregate formation. An aggregate is declared
to be formed once the TC has no motion relative to the fixed PMN.
The parameters used to initialize the system are found in Table 4.2. The PMN
rolling model was used with translational and rotational velocities set to 0 µm/µs
and 0 rad/µs, respectively, to fix the PMN at the wall. A simple moving average
using 75 sample points was used to smooth the velocity signal in post-processing
to better show when aggregate formation occurs.
Table 4.2: ADH1 Case Parameters
Parameter Value
Equilibrium Dissociation Rate, k0off [s−1] 0.3
Equilibrium Association Rate, k0on [s−1] 3× 10−3
Equilibrium Spring Length, λ [µm] 0.05
Flow Shear Rate, γ [s−1] 150
Spring Constant, s [N/m]: 2× 10−3
Transition Spring Constant, sts [N/m]: 1× 10−3
TC Molecular Surface Density (ICAM-1), nICAML [N/m2]: 13× 10−12
PMN Molecular Surface Density (LFA-1), nLFAL [N/m2]: 13× 10−12
Domain Size [µm]:X 60Y 32Z 42
Tumor Cell Initial Centroid [µm]:X 18Y 10Z 21
PMN Initial Centroid [µm]:X 30Y 2.5Z 21
119
Figure 4.30 shows the results of the simulation. The results first show a sharp
decrease in TC velocity indicative of a collision event. During this collision, there
is some biochemical interacting shown by small amounts of bond formation and
breakage. TC velocity begins to increase as the cell rolls over the PMN surface
due to fluid forcing.
Figure 4.30: TC velocity and number of bonds as a function of time. Initial dropin velocity indicates collision between TC and PMN. TC-PMN aggregate formswhen TC velocity reaches zero.
As the TC rolls over the PMN, significant biochemical interaction takes place
and formation of long lasting bonds occur. An interaction threshold is eventually
surpassed and the TC velocity once again decreases until the TC has no motion
120
relative to the PMN. The TC-PMN aggregate is formed once the TC velocity
reaches zero. The TC velocity does slightly oscillate about zero once the aggregate
has been formed. This velocity oscillation can be attributed to the continued
formation and breakage of individual bonds and fluid forcing acting on the TC.
Chapter 5Conclusions and Future Work
5.1 Conclusion
A computational tool has been developed to model flowing cellular systems and
has been applied to direct numerical simulation of microvascular flows with a vision
towards personalized medicine. This tool couples CFD, computational structural
mechanics, 6DOF motion, and surface biochemistry, in the context of interface-
resolved cell geometry, to provide a detailed model of the heterogeneous blood
flow microenvironment. This tool can be used to study drug-mediated cellular
interactions in the vasculature and to design magnetically actuated DDMRs with
targeting capabilities.
Structural mechanics models were introduced, along with accompanying FSBI
coupling conditions for each of the models. The semi-implicit 6DOF motion solver
uses the iterative nature of many flow solvers to increase stability of the motion
solver. The iterative implicit procedure developed allowed for orders-of-magnitude
speedup over explicit 6DOF motion solvers. The PMN rolling model provides
a computationally straightforward approach to capture complex flow-structure-
biochemistry interactions of wall-adhered PMNs. The rolling model can be tuned
to match experimentally observed phenomena. The finite-volume discretization of
the hyperelastic Saint-Venant Kirchhoff constitutive model allows for large defor-
mation modeling of arbitrary bodies in microvascular flow. This hyperelasticity
122
implementation shows it is possible to model systems including both large strains
and large rigid body motions. Both structural mechanics models were implemented
into an existing flow-solver showing the ability to backfit other existing codes with
structural modeling capability.
DDMR designs were simulated using this tool, showing great promise for the
future of DDMR computational analysis. A rigid helical microswimmer was placed
in a quiescent fluid and actuated with both a prescribed torque and a prescribed
angular velocity, reproducing the effects of a rotating magnetic field. The results
of the helical microswimmer simulation were in agreement with theoretical predic-
tions. A hyperelastic microbead was placed in the near-wall region of a Couette
flow and held in place with a volumetric body force reproducing the effects of an
applied magnetic field. The microbead evolved to an equilibrium shape due to the
applied fluid forcing and volumetric body force.
Multiple linear solver analysis methods were presented. Fourier stability analy-
sis (FSA) allows for performance evaluations of computational systems with apriori
knowledge of the underlying governing equations and discretization strategy. An
empirically-based performance optimization routine was also presented requiring
no knowledge of the underlying governing equations or discretization strategy.
5.2 Future Work
Various aspects of this research can benefit from further development.
One area of potential benefit is exploring ways to fully capture the PMN rolling
motion using the hyperelasticity structural modeling implementation. Figure 4.28
shows the onset of PMN rolling, however, more work is needed to better model
the biochemical interactions between the PMN and the endothelium wall. Use of
the localized biochemistry model would require greatly increased spatial resolution
along the endothelial wall and smaller timesteps to ensure the bond physics are
adequately resolved. Such an increase would significantly impact runtime of the
computational tool and likely render runtimes intractable.
123
A second area for further exploration is the implementation of hyperelastic
models that do not depend on the slow-process assumption. More general hypere-
lastic models would have the potential advantage of better stability of the physical
system and may allow for arbitrarily large timesteps. However, rearranging the
general hyperelastic governing equations into the unified equation form presented
in Section 2.2.4 is not immediately obvious.
A third area of exploration is using this tool to better understand swarm con-
trol of DDMRs in microvascular environments. It is possible to determine the
magnetic actuation force and torque needed to move a DDMR along a prescribed
trajectory using Equations 4.20, 2.70, and 2.71. Equations 2.73 and 2.74 can
then be used to solve the inverse problem of determining the necessary magnetic
field. Furthermore, solving the inverse magnetic field problem provides a path to-
wards the development of active controllers capable of providing localized DDMR
instructions using a single global signal.
Appendix A
A.1 SI Scaling of Microvascular System
Mass: picogram [pg]
Length: micrometer [µm]
Time: microsecond [µs]
1 picogram = 10−12 grams = 10−15 kilograms
1 micrometer = 10−6 meters
1 microsecond = 10−6 seconds
Velocity:
1[ms
]=
(1m · 106µm
1m
)(1
1s· 1s
106µs
)= 1
[µm
µs
](A.1)
Density:
1
[kg
m3
]=
(1kg · 103g
1kg· 1012pg
1g
)(1
1m· 1m
106µm
)3
=1015
1018
[pg
µm3
]= 10−3
[pg
µm3
] (A.2)
125
Viscosity:
1
[kg
m · s
]=
(1kg · 103g
1kg· 1012pg
1g
)(1
1m· 1m
106µm
)(1
1s· 1s
106µs
)=
1015
1012
[pg
µm · µs
]= 103
[pg
µm · µs
] (A.3)
Force (N):
1
[kg ·ms2
]=
(1kg · 103g
1kg· 1012pg
1g
)(1m · 106µm
1m
)(1
1s· 1s
106µs
)2
=1021
1012
[pg · µmµs2
]= 109
[pg · µmµs2
] (A.4)
Pressure (Pa):
1
[kg
m · s2
]=
(1kg · 103g
1kg· 1012pg
1g
)(1
1m· 1m
106µm
)(1
1s· 1s
106µs
)2
=1015
1018
[pg
µm · µs2
]= 10−3
[pg
µm · µs2
] (A.5)
Surface Density:
1
[molecule
m2
]= (1 molecule)
(1
1m· 1m
106µm
)2
= 10−12
[molecule
µm2
](A.6)
Association Constant:
1
[1
M · s
]= 1
[liter
mole · s
]= 10−3
[m3
mole · s
]= 10−3
(1m · 106µm
1m
)3(1
1s· 1s
106µs
)(1
1 mole
)=
1015
106
[µm3
mole · µs
]= 109
[µm3
mole · µs
] (A.7)
Dissociation Constant:
1
[1
s
]=
(1
1s· 1s
106µs
)= 10−6
[1
µs
](A.8)
126
Spring Constant (N/m):
1
[kg
s2
]=
(1kg · 103g
1kg· 1012pg
1g
)(1
1s· 1s
106µs
)2
=1015
1012
[pg
µs2
]= 103
[pg
µs2
] (A.9)
Base SI units of Boltzmann Constant (J/K):
1
[m2 · kgK · s2
]=
(1kg · 103g
1kg· 1012pg
1g
)(1m · 106µm
1m
)2(1
1s· 1s
106µs
)2(1
K
)=
1027
1012
[µm2 · pgK · µs2
]= 1015
[µm2 · pgK · µs2
](A.10)
Therefore, Boltzmann Constant:
1.38064852× 10−23
[m2 · kgK · s2
]= 1.38064852× 10−8
[µm2 · pgK · µs2
](A.11)
Appendix B
B.1 Gradient Reconstruction using Least Square
Optimization
B.1.1 Derivation
In finite-volume formulations, it is often necessary to reconstruct the gradient
field of a scalar. One approach to doing this is using least-squared optimization.
Referring to the computational stencil shown in Figure B.1, a Taylor expansion of
variable Φ at point N taken about point P yields:
ΦN = ΦP + δxNPδΦ
δx
∣∣∣∣P
+ δyNPδΦ
δy
∣∣∣∣P
+ δzNPδΦ
δz
∣∣∣∣P
+H.O.T, (B.1)
where δγNP = γN − γP and higher-order terms (H.O.T) are neglected. As with
every Taylor series expansion, the approximation error is the sum of the higher
order terms neglected higher-order terms. Therefore, the error eN can be computed
as
eN = ΦP − ΦN + δxNPδΦ
δx
∣∣∣∣P
+ δyNPδΦ
δy
∣∣∣∣P
+ δzNPδΦ
δz
∣∣∣∣P
. (B.2)
For numerical stability, it is common practice to use a weighted error by multi-
plying ek with some weighting function wk (where k is the index of the neighboring
cell). For this work, an inverse distance weighting function will be used. This
weighting function is written as
128
Figure B.1: Uniform rectilinear computational compact stencil.
wk =1√
(δxkP )2 + (δykP )2 + (δzkP )2. (B.3)
Next, the square of the weighted error between the cell-center P and each of
it neighbors are summed and this value is used as the objective function to be
minimized through the optimization process.
Fobj =k∑
(wkek)2, (B.4)
where k represents the index of the neighboring cells. In the 2D stencil shown in
B.1, k = N,S,W,E.
The goal is to compute the gradient values at point P that minimize Fobj.
Assuming Fobj is convex, we can find the global minimum with respect to the
gradient where
δ∑
(wkek)2
δ δΦδx
∣∣P
=δ∑
(wkek)2
δ δΦδy
∣∣P
=δ∑
(wkek)2
δ δΦδz
∣∣P
= 0. (B.5)
129
Beginning with the derivative with respect to δΦδx
∣∣P
gives
δ∑
(wkek)2
δ δΦδx
∣∣P
=∑(
w2k
δe2k
δ δΦδx
∣∣P
)= 0, (B.6)
where the derivative of e2k can be easily computed as
δe2k
δ δΦδx
∣∣P
= 2ekδxkP . (B.7)
Substituting this derivative back into the minimized objective function gives
∑(w2kekδxkP
)=∑[
w2k
(ΦP − Φk + δxkP
δΦ
δx
∣∣∣∣P
+ δykPδΦ
δy
∣∣∣∣P
+ δzkPδΦ
δz
∣∣∣∣P
)δxkP
]= 0.
(B.8)
Some algebra allows this summation to be expressed as
δΦ
δx
∣∣∣∣P
∑(w2kδx
2kP
)+δΦ
δy
∣∣∣∣P
∑(w2kδxkP δykP
)+δΦ
δz
∣∣∣∣P
∑(w2kδxkP δzkP
)=∑[
w2kδxkP (Φk − ΦP )
].
(B.9)
This new form can be rewritten as
axδΦ
δx
∣∣∣∣P
+ bxδΦ
δy
∣∣∣∣P
+ cxδΦ
δz
∣∣∣∣P
= fx, (B.10a)
ax =∑(
w2kδx
2kP
), (B.10b)
bx =∑(
w2kδxkP δykP
), (B.10c)
cx =∑(
w2kδxkP δzkP
), (B.10d)
fx =∑[
w2kδxkP (Φk − ΦP )
]. (B.10e)
Repeating this technique for derivatives of Fobj with respect to δΦδy
∣∣P
and δΦδz
∣∣P
130
yields the following equations, respectively:
ayδΦ
δx
∣∣∣∣P
+ byδΦ
δy
∣∣∣∣P
+ cyδΦ
δz
∣∣∣∣P
= fy, (B.11a)
ay =∑(
w2kδxkP δykP
), (B.11b)
by =∑(
w2kδy
2kP
), (B.11c)
cy =∑(
w2kδykP δzkP
), (B.11d)
fy =∑[
w2kδykP (Φk − ΦP )
], (B.11e)
azδΦ
δx
∣∣∣∣P
+ bzδΦ
δy
∣∣∣∣P
+ czδΦ
δz
∣∣∣∣P
= fz, (B.12a)
az =∑(
w2kδxkP δzkP
), (B.12b)
bz =∑(
w2kδykP δzkP
), (B.12c)
cz =∑(
w2kδz
2kP
), (B.12d)
fz =∑[
w2kδzkP (Φk − ΦP )
]. (B.12e)
This system of equations can be expressed in matrix form as:
ax bx cx
ay by cy
az bz cz
δΦδx
δΦδy
δΦδz
P
=
fx
fy
fz
. (B.13)
Now, inverting the 3x3 matrix on the LHS gives the three components of the
∇Φ at cell-center P.
131
B.1.2 Boundary Conditions
B.1.2.1 Dirichlet BC
Implementation of the Dirichlet BC is straight forward. The distances used are
computed from the cell-center P to the boundary face and ΦBC is used on the
RHS.
B.1.2.2 Extrapolation BC
If Φ is not specified on the boundary face, the value of Φ may be extrapolated to
the face using a Taylor series expansion. Therefore, we can write Φ on boundary
face b as
Φb = ΦP + δxbPδΦ
δx
∣∣∣∣P
+ δybPδΦ
δy
∣∣∣∣P
+ δzbPδΦ
δz
∣∣∣∣P
. (B.14)
Then we can compute (Φb − ΦP ) as
Φb − ΦP = δxbPδΦ
δx
∣∣∣∣P
+ δybPδΦ
δy
∣∣∣∣P
+ δzbPδΦ
δz
∣∣∣∣P
. (B.15)
In this method, the RHS will be treated explicitly. For the extrapolation BC,
the values of ∇Φ used to compute (Φb − ΦP ) should be the gradient values from
the previous iteration.
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Vita
Byron J. Gaskin
Byron Gaskin grew up in Miami, Florida where he graduated from MonsignorEdward Pace High School in 2007. It was during high school that Byron was firstintroduced to computer programming. Upon completion of high school, Byronattended Florida International University in Miami where he obtained a Bache-lor’s Degree in Mechanical Engineering. During his undergraduate career, Byronconducted undergraduate research in computational fluid dynamics and worked asan intern at the Penn State Applied Research Laboratory.
Byron began his Master’s degree at Penn State in Mechanical Engineering inAugust 2012 and completed the degree in August 2014. He completed a Mas-ter’s thesis titled “Coupled Flow-Biochemistry Simulations of Dynamic Systems ofBlood Cells”.
After completion of his Master’s degree, Byron started work on his doctor-ate in Bioengineering in Fall 2014. Byron was, in Spring 2018, selected for theEmerging Leaders in Data Science Fellowship at the National Institute of Allergiesand Infectious Diseases (NIAID) at the National Institutes of Health (NIH). Uponcompletion of the doctorate, Byron will begin his 12-month data science fellowshipat NIAID in Rockville, MD.
Byron is a member of Sigma Phi Epsilon Fraternity and a founder of the OmegaGamma chapter of Theta Tau Fraternity.
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