A Finite Element Method for Animating Large Viscoplastic Flow

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A Finite Element Method for Animating Large
Viscoplastic Flow

• SIGGRAPH 2007

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Outline

• Abstract
• Finite Element Method
• Measuring Deformation
• Force Computation
• Plasticity Model
• Remesh
• Extracting the Mesh
• Results

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Abstract

• An extension to Lagrangian finite element methods
  for plastic deformation
• Explicitly update the linear basis function
  during simulation
• When basis function to become ill-conditioned,
  remesh the simulation domain
• Preserve original boundary

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Finite Element Method

• Measuring deformation
• Volumetric tetrahedral meshes
• Force computation
• First Piola-Kirchhoff stress
• It relates forces in the current configuration to
  areas in the reference configuration (material)
• Second Piola-Kirchhoff stress
• Both reference configuration (material)
• Plasticity model
• After plastic deformation, update basis function

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Measuring Deformation-1

• The stress at X in the material depends only on
  the deformation gradient
• Material coordinate X, world coordinate x
• F is a constant 3x3 matrix in tetrahedron
• FDsDm-1
• Ex In triangle
• dm1X1-X0, dm2X2-X0, ds1x1-x0, ds2x2-x0
• Dm with columns dm1 dm2, Ds likely

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Measuring Deformation-2

• Dm-1 is constant
• Tetrahedron is fixed in material coordinate
• F contains all information about deformation
• DsFDm
• Only need Force computation

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Force Computation

• First Piola-Kirchhoff stress P
• Force on node i gi-P(A1N1A2N2A3N3)/3
• AjNj are the area weighted normals (in material
  coordinates) of the faces of tetrahedron incident
  to node i
• Do not change during simulation

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Plasticity Model-1

• Plastic deformation
• eepee
• Plastic deformation strain ep elastic
  deformation strain ee
• Plastic strain reflects materials rest shape has
  been permanently distorted
• Strain deviation tensor
• e e-Tr(e)I
• FFe dot Fp
• Plastic update
• Update basis functions
• Recompute area-weighted face-normals

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Plasticity Model-2

• An elastic force (stress) attracts the total
  strain to the plastic strain
• Not the plastic strain to the total strain
• Plastic deformation depend on history of the
  total strains movement

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Examples

• Demo

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Remesh

• Consistent energy minimization
• Vertex positions and connectivity updates are
  performed to minimize the same quadratic energy
• Centroidal Voronoi Tessellations
• Optimal Delaunay Triangulations

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Centroidal Voronoi Tessellations

• Minimizing quadratic energy
• For a given set of vertices, compute their
  Voronoi diagram
• A mesh that minimizes energy has each vertex at
  the centroid of its own Voronoi cell

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Optimal Delaunay Triangulations

• The integral is taken over each 1-ring
• Underlaid v.s. overlaid

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Algorithm
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Setup Preprocessing

• Discrete skeleton
• Extract poles
• Local feature size
• Capturing both local thickness curvature of the
  shape
• Sampling

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Examples-1

• Pentium IV 3GHz
• 1000 vertices
• 50 iterations
• 16 second

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Examples-2

• Pentium IV 3GHz
• 36K vertices
• 2.1 second per iteration

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Extracting the Mesh

• Remove elements outside the surface
• Computing volume overlap
• Speed up hierarchical AABB
• Remove poorly shaped elements
• Add and remove elements to ensure that the
  surface is manifold

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Poorly Shaped Elements-1
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Poorly Shaped Elements-2
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Transferring Simulation Variables

• Stored per element
• Deformation gradient F
• Work hardening/softening variable a
• Finding all tetrahedron in old mesh it overlaps,
  new variable to be average weighted by the volume
  of overlap region
• Stored per node
• Velocities
• Barycentric interpolation
• Nearest neighbor

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Results-1
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Results-2

• Demo