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

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 by a finite one

Motivation • Problem with the finite solution space: Scale u ( x ) X u ( x ) ≈ i u i ϕ i ( x ) δ ( x − x 0 )

Motivation • Problem with the finite solution space: Tessellation

Adaptive tessellation method • H-adaptive method • R-adaptive method • Dynamically adding or removing • Dynamically changing vertex vertices distribution and connectivity 1D 1D 2D 2D

Eulerian-on-Lagrangian method • Get applied for rods, cloth F Locking! Jumping! • Contact-oriented Lagrangian node EOL node

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 How to automatically evolve the dynamics? those material points?

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

System Lagrangian

Equation of motion ✓ ∂ L ◆ Euler-Lagrange equation ∂ L ∂ q − d = g dt ∂ ˙ q For positional variables r=[x, u]

Singularities in dynamic system x u F i = x i − x i +1 u i − u i +1 k F k � 1 ⌘ 0 x_i x_j u_i u_j

Constraint of material points • How to constrain material points? Fixed? region having a larger curvature should be more densely sampled . • Centroid Voronoi Tessellation

How to select the density function • Curvature aware density function

Constrained optimization • Formulation: r=[u, x] • Numerical solve • Simplify EoM by neglecting some terms • Turn EoM constraint into soft penalties • Solved by SQP

Is r-adaption enough? • Nyquist sampling theorem • Sampling frequency >= 2*signal frequency No DoF to fit the Works! contacting edge H-Adaption

H-adaption Node insertion Merging tiny segments for Efficiency • Node deletion Avoiding infinitely large stiffness •

Stiffness adjustment

Results

Excessive twisting

Rigid-body contacts

Knotting

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

Thanks! Q&A