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Efficient Multilevel Subdivision-Based Thin Shell Elements for Blood Cell Modeling

This talk explores the development of multilevel shell elements using subdivision-based techniques for accurate and efficient modeling of thin structures like red blood cells. The presentation covers the motivation, solution schemes, geometric representation, advantages, and applications of these elements for large-scale simulations. It discusses the challenges in modeling thin shells accurately and showcases the benefits of using hierarchical simulation techniques and the MGPCG solver for robust solutions. The talk also delves into boundary conditions, validation, and the modeling of blood flow, emphasizing the importance of employing innovative computational approaches for realistic simulations.

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Efficient Multilevel Subdivision-Based Thin Shell Elements for Blood Cell Modeling

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

  2. Organization of Talk • Motivation & Background • Efficient Solution Scheme • Multilevel Shell Element • Constraints & Convergence • Applications to Blood Cell Modeling

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

  4. Example: Flow of Blood Cells 8mm

  5. Shell: Modes of Deformation Initial Membrane (A) Bending ( )

  6. Geometric Representation:Subdivision Surfaces • Smooth geometry is associated with a coarse control mesh • Control mesh geometry defined by nodal positions

  7. Subdivision Advantages • Compact representation • Guaranteed continuity

  8. 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

  9. Subdivision Thin Shell Element • Displacement is a function of adjacent nodes • Basis defined by local topology • Semi-local basis functions • Rotation-Free • Triangular Element:[Cirak et. al. 01]

  10. 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)

  11. Hierarchical Simulation • Considering a hierarchy of discretizations simultaneously can increase solution efficiency

  12. MGPCG Solver(Multigrid-Preconditioned Conjugate Gradient) • Combines efficiency of Multigrid with robustness of Conjugate Gradient algorithm

  13. Interlevel Transfer (I) • Need to transfer solution refinements between levels Displacements Forces

  14. 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

  15. Result: Plate with Inhomogeneous Material Properties • Near O(N) performance

  16. Ordinary Extraordinary Super- Extraordinary Element Characterization

  17. Element Distribution • Super-extraordinary (red) • Extraordinary (green) • Ordinary (blue) Increasing subdivision 

  18. Super Extraordinary Element Parameterization • Linear transformation to 4 child sub-regions

  19. Multilevel Data Structure Parent Edge Primary Child

  20. Boundary Representation • Non-closed geometry has geometric boundaries • Subdivision requires a neighborhood of “ghost faces”

  21. 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]

  22. 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!

  23. Explicit Boundary Modelingand Discretization • Ghost faces incorporated into model design • Improved design flexibility • Significantly improvedconvergence • Allows for extraordinarypoints along boundary

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

  25. bn-1 bn b1 a b2 b4 b3 Evaluation at Control Mesh Vertex Location • Limit positions and tangents of control mesh vertices are expressed as sums of neighboring vertex displacements

  26. Common Boundary Conditions • Directional constraint • Rotation constraint • Simple support: combination of 3 independent direction constraints • Clamped support: combination of simple support and rotation constraint

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

  28. 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

  29. 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 DV=0

  30. Motivation:Blood 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

  31. 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]

  32. Cell Membrane Geometry • Sample data sets [Fung 72] • Reconstruction Technique [Hoppe et. Al. 94]

  33. Graded Cell Mesh Family

  34. 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

  35. Simulation Approach • STM tip modeled as point application of force with no slip • Table modeled as frictionless plane • “Spin” restricted

  36. Point Load Animations Volume Conservation w/o Volume Conserv.

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

  38. 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

  39. Simulation Results

  40. Attack of the Killer Blood Cell

  41. 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

  42. 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.

  43. 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

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

  45. Questions ?

  46. Internal Constraints

  47. Law of Virtual Work • Leads to P.D.E. + Boundary Conditions

  48. Kirchhoff Assumption • Lines initially normal to the middle surface before deformation remain straight and normal to middle surface after deformation

  49. Graph Paper Example

  50. Example: Deformable Mirror Nanolaminate mirror image courtesy NASA/JPL

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