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Organization of Talk

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. Organization of Talk. Motivation & Background Efficient Solution Scheme

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Organization of Talk

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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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