Multilevel, Subdivision-Based, Thin Shell Finite Elements: Development and Application to Red Blood Cell Modeling Seth Green University of Washington Department of Mechanical Engineering December 12, 2003

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Multilevel, Subdivision-Based, Thin Shell Finite
Elements Development and Application to Red
Blood Cell ModelingSeth GreenUniversity of
WashingtonDepartment of Mechanical
EngineeringDecember 12, 2003
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Organization of Talk

• Motivation Background
• Efficient Solution Scheme
• Multilevel Shell Element
• Constraints Convergence
• Applications to Blood Cell Modeling

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Motivation

• Accurate and efficient numerical simulation of
  thin structures
• Large deformation effects
• Interaction with other simulations(solids and
  fluids)

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Example Flow of Blood Cells
8mm
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Shell Modes of Deformation
Initial Membrane (?A) Bending (?? )
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Geometric RepresentationSubdivision Surfaces

• Smooth geometry is associated with a coarse
  control mesh
• Control mesh geometry defined by nodal positions

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Subdivision Advantages

• Compact representation
• Guaranteed continuity

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Shell Finite Elements

• Thick Shells
• Simple to implement
• Inaccurate for thin bodies
• Prone to shear locking
• Thin Shells
• Accurate for thin bodies
• Mathematically involved
• Strict requirements on representation used for
  FEA C1 H2

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Subdivision Thin Shell Element

• Displacement is a function of adjacent nodes
• Basis defined by local topology
• Semi-local basis functions
• Rotation-Free
• Triangular ElementCirak et. al. 01

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Large Scale Simulation

• Adding degrees of freedom (DOF) increases
  solution accuracy
• Direct solution schemes are not practical for
  large problems(10K DOF)
• Iterative schemes efficiency is related to the
  condition number of the system
• Condition number grows as O(N2)

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Hierarchical Simulation

• Considering a hierarchy of discretizations
  simultaneously can increase solution efficiency

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MGPCG Solver(Multigrid-Preconditioned Conjugate
Gradient)

• Combines efficiency of Multigrid with robustness
  of Conjugate Gradient algorithm

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Interlevel Transfer (I)

• Need to transfer solution refinements between
  levels

Displacements
Forces
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Interlevel Transfer (II)

• We chose the subdivision matrix S for
  interpolation retains displacement field
• By work equivalency, restriction operator is ST
  (simple to compute)
• Subdivision now used for
• Geometric Representation
• Deformation
• Numerical Preconditioning

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Result Plate with Inhomogeneous Material
Properties

• Near O(N) performance

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Element Characterization
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Element Distribution

• Super-extraordinary (red)
• Extraordinary (green)
• Ordinary (blue)

Increasing subdivision ?
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Super Extraordinary Element Parameterization

• Linear transformation to 4 child sub-regions

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Multilevel Data Structure
Parent Edge
Primary Child
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Boundary Representation

• Non-closed geometry has geometric boundaries
• Subdivision requires a neighborhood of ghost
  faces

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Prior Work Approach toBoundary Conditions

• Ghost faces are created automatically to create
  splines along edges Schweitzer Duchamp 96
• This criteria is required to hold during
  deformation
• Cirak 01

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Prior Work Inconsistency

• Prior constraints are inconsistent with clamped
  and simply supported b.c.s
• Do not allow finite curvatures at boundary
• Couple membrane and bending deformation
• Sufficient, but not necessary!

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Explicit Boundary Modelingand Discretization

• Ghost faces incorporated into model design
• Improved design flexibility
• Significantly improvedconvergence
• Allows for extraordinarypoints along boundary

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Constraint Approach

• Boundary conditions are expressed as constraints
• Constraints are discretized and applied pointwise
  to geometric boundaries, or interior of the body

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Evaluation at Control Mesh Vertex Location

• Limit positions and tangents of control mesh
  vertices are expressed as sums of neighboring
  vertex displacements

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Common Boundary Conditions

• Directional constraint
• Rotation constraint
• Simple support combination of 3 independent
  direction constraints
• Clamped support combination of simple support
  and rotation constraint

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Validation

• Belytschko et. al. obstacle course
• Result O(N2) Convergence in displacement error!

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Convergence of Uniformly Loaded Flat Plat with
Clamped Boundaries
Prior Approach Our Approach
Err 1e-6 1e-3 1
Elem 1 10 100 1000 10e3
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Volume Conservation Constraint

• Divergence theorem is used to compute volume as a
  surface integral using the smooth limit surface
• Generalized displacements are constrained such
  that DV0

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MotivationBlood Flow Simulation

• Treatments of blood as a homogeneous fluid are
  not accurate
• A micro-structural model is required
• Simulations involving many blood cells
  interacting with their surroundings demands the
  most efficient computational techniques

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Red Blood Cell Model

• Flexible thin membrane bounding an incompressible
  fluid center
• Large resistance to changes in area
• Small resistance to bending deformations
• Change in shape with constant volume induces
  changes in area
• Bending stiffness must be included Eggelton
  and Popel 98

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Cell Membrane Geometry

• Sample data sets Fung 72
• Reconstruction Technique Hoppe et. Al. 94

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Graded Cell Mesh Family
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Point Load Deformation Experiment

• Red blood cell lays on flat workspace
• Tip of a scanning tunneling microscope (STM) used
  to deform cell membrane
• Reaction forcerecorded

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Simulation Approach

• STM tip modeled as point application of force
  with no slip
• Table modeled as frictionless plane
• Spin restricted

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Point Load Animations
Volume Conservation
w/o Volume Conserv.
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Micropipette Aspiration

• Early experiment to determine red blood cell
  properties
• Pressure drop causes cell to deform inside of
  pipette

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Simulation Approach

• Pipette modeled as thin rigid cylinder
• Constant vertical load inside of pipette volume
• Pipette mouth discretized into several points
• Two step quasi-equilibrium solution procedure
• Tangential sliding prediction
• Positional correction

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Simulation Results
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Attack of the Killer Blood Cell
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Contributions

• Unified framework for multilevel quadrilateral
  and triangular elements
• 2nd order accurate boundary conditions
• Efficient multilevel solution algorithm
• Application to bio-sciences cell membrane
  simulation

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Conclusion

• Multilevel, subdivision-based thin shell finite
  elements allow engineers to simulate large
  deformations of thin bodies efficiently and
  accurately in a variety of physical contexts.

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Future Work

• Biologically inspired material models
• Formal proof of convergence of boundary
  conditions
• Blood flow simulation involving many
  simultaneously deforming cells
• Application to problems in which bending
  deformations have been ignored

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Acknowlegements

• Committee
• Audience
• NSF Information Technology Research Program
• The Boeing Company
• Ford Motor Company

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Questions
?
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Internal Constraints
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Law of Virtual Work

• Leads to P.D.E. Boundary Conditions

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Kirchhoff Assumption

• Lines initially normal to the middle surface
  before deformation remain straight and normal to
  middle surface after deformation

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Graph Paper Example
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Example Deformable Mirror
Nanolaminate mirror image courtesy NASA/JPL
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Example Buckling
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MechanicsApproach
Law of virtual displacements leads to a P.D.E.
Boundary Conditions
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Finite Element Method

• Model is discretized into elements
• Deformations restricted to the form u(x,y)S
  Ni(x,y) vi
• Nodal displacements have a geometric
  interpretation

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Thin Shells

• Thin shells are bodies well described by a
  middle surface

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Multigrid

• Leverages smoothing principle
• Combines fast solutions on coarse mesh with
  accurate solutions on fine mesh to increase
  solution efficiency
• Cons
• Difficult to tune properly

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Compact Representation
Retains smoothness under large deformations
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Cell Modeling Discussion

• Multilevel subdivision elements are an efficient
  and geometrically robust approach to modeling
  cell membrane deformations with emergent physical
  behavior
• Lack of an implementation of a robust material
  model prevented further validation during this
  study

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• Model is defined as the limit of a sequence of
  smooth refinements of a coarse initial mesh

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Subdivision Surfaces

• Generalize splines to arbitrary topology meshes
  with guaranteed smoothness under deformation
• Popular schemes Loop (triangular) and
  Catmull-Clark (quadrilateral)

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Geometric RepresentationSubdivision Refinement
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DiscussionConstrained Systems
K Cu fC 0 lg

• May be made positive definite via novel inner
  product of Bramble and Pasciak 88
• Multilevel representation of K
• Perturbation of K
• Estimated scaling of l vs. u

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DiscussionConstrained Systems

• May be made positive definite via novel inner
  product of Bramble and Pasciak 88
• Multilevel representation of K
• Perturbation of K
• Estimated scaling of l vs. u

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RotationConstraint

• r n ? e
• ConstraintNr 0
• Use basis function derivatives inside elements
• Use tangent masks at vertices
• Nn ?

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Clamped Constraint

• Clamped boundary No Displacement No Rotation
• Displacement x,y,z ? 3 constraints
• Rotation ? 1 constraint
• Clamped ? 4 constraints

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Red Blood Cell Modeling

• Geometry Fung et. al. 72
• Material Evans Skalak 80
• Bending stiffness must be included Eggelton
  and Popel 98

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Red Blood Cell Modeling Paradigms

• Human red blood cells consist of a bilayer of
  lipids and spectrin polymer surrounding a plasma
  (fluid) center
• Continuum vs. Microscale modeling