Cosserat Rod with rh- Adaptive Discretization Jiahao Wen 1 , Jiong - - PowerPoint PPT Presentation

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Cosserat Rod with rh- Adaptive Discretization Jiahao Wen 1 , Jiong - - PowerPoint PPT Presentation

Mar 05, 2023 371 likes •643 views

Cosserat Rod with rh- Adaptive Discretization Jiahao Wen 1 , Jiong Chen 1 , Nobuyuki Umetani 2 , Hujun Bao 1 , Jin Huang 1 1 Zhejiang University 2 The University of Tokyo Background Discretization Approximate infinite dimensional solution space

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Cosserat Rod with rh- Adaptive Discretization

Jiahao Wen1, Jiong Chen1, Nobuyuki Umetani2, Hujun Bao1, Jin Huang1

1Zhejiang University 2The University of Tokyo

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Background

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Discretization

Approximate infinite dimensional solution space by a finite one

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Motivation

  • Problem with the finite solution space: Scale

δ(x − x0) u(x) u(x) ≈ X

i uiϕi(x)

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Motivation

  • Problem with the finite solution space: Tessellation
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Adaptive tessellation method

  • H-adaptive method
  • Dynamically adding or removing

vertices

  • R-adaptive method
  • Dynamically changing vertex

distribution and connectivity

1D 2D 1D 2D

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Eulerian-on-Lagrangian method

  • Get applied for rods, cloth
  • Contact-oriented

F

Locking! Jumping!

EOL node Lagrangian node

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Eulerian-on-Lagrangian method

  • Get applied for rods, cloth
  • Contact-oriented
  • Only EOL node can move
  • Limited in adaptivity
  • More to consider: intricate geometry
  • Our extension
  • Unify both EOL and Lagrangian node
  • Drive the motion of material points in more flexible ways

How to correctly formulate the dynamics? How to automatically evolve those material points?

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Moving mesh discretization

  • Focus on Cosserat rod
  • Easy to bend and twist
  • Accuracy is largely limited by its initial discretization
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System Lagrangian

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Equation of motion

For positional variables r=[x, u] Euler-Lagrange equation ∂L

∂q − d dt ✓∂L ∂ ˙ q ◆ = g

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Singularities in dynamic system

x u x_i u_i x_j u_j

Fi = xi − xi+1 ui − ui+1

kFk 1 ⌘ 0

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Constraint of material points

  • How to constrain material points? Fixed?
  • Centroid Voronoi Tessellation

region having a larger curvature should be more densely sampled.

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How to select the density function

  • Curvature aware density function
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Constrained optimization

  • Numerical solve
  • Simplify EoM by neglecting some

terms

  • Turn EoM constraint into soft

penalties

  • Solved by SQP
  • Formulation: r=[u, x]
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Is r-adaption enough?

  • Nyquist sampling theorem
  • Sampling frequency >= 2*signal frequency

H-Adaption

No DoF to fit the contacting edge Works!

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H-adaption

Node insertion Node deletion Merging tiny segments for

  • Efficiency
  • Avoiding infinitely large stiffness
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Stiffness adjustment

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Results

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Excessive twisting

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Rigid-body contacts

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Knotting

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Future works

  • Better prior or posterior knowledge to drive dynamics of material

points

  • Accurate handling of equation of motion
  • R-adaption in a higher dimensional space
  • Energy smoothness
  • Customized numerical solver
  • Statics-dynamics separation
  • H-adaption guided by multiscale theory
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Thanks!

Q&A