Beyond Bouncing Boxes

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Beyond Bouncing Boxes

• Fast, yet realistic, deformation and fracture
• James F. OBrien
• University of California, Berkeley

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Before We Start...

• Complex MATH Complex CODE
• Complex MATH Slow CODE

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Problem Setup

• Lagrangian Formulation
• Where is space did this material move to?
• Eulerian Formulation
• What material is at this location in space?

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Problem Setup

• Lagrangian Formulation
• Where is space did this material move to?

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Lagrangian Formulation

• Deformation described by mapping from local to
  world coordinates

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Strain

• Strain measures deformation
• Only geometric
• Example simple strain in a bar

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Strain

• Greens strain tensor
• Vanishes when not deformed
• Only measures deformation
• Does not depend on the coordinate system

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Strain

• Greens strain tensor

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Strain Rate

• Time derivative of Greens strain tensor
• Measures rate of deformation
• Used for internal damping

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Stress

• Stress determines internal forces

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Stress due to Strain
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Stress due to Strain Rate
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Energy Potentials
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Discretization

• Transition from continuous model to something we
  can actually use...

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Finite Element Method

• Disjoint elements tile material domain
• Derivatives from shape functions
• Nodes shared by adjacent
  elements

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FEM Discretization

• Solid volumes
• Tetrahedral elements
• Linear shape functions

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FEM Discretization

• Each element defined by four nodes
• m - location in material/local coordinates
• p - position in world coordinates
• v - velocity in world coordinates

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Element Shape Functions

• Barycentric coordinates
• Invert to obtain basis matrix

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Material Derivatives

• World pos. as function of material coords.
• Derivative w.r.t material coordinates

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Recall
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Recall
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Node Forces

• Combine derivative formula w/ equation for
  elastic energy
• Integrate over volume of element
• Take derivative w.r.t. node positons

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Modal Analysis
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Deformation
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Sound
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Modal Decomposition (multiple steps)

• Linearize non-linear system

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Modal Decomposition (multiple steps)

• Consequences of linearization
• No local rotations

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Modal Decomposition (multiple steps)
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Modal Decomposition (multiple steps)
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Analytical Solutions
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Fast Computations

• Only a pair of complex multiplies per times step
• No stability limit on step size
• Jump to arbitrary point in time
• Only keep useful modes

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• Further information
• www . cs . berkeley . edu / b-cam