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Hierarchical Multi-Resolution Finite Element Model for Soft Body Simulation
Matthieu Nesme, François Faure, Yohan Payan
2nd Workshop on Computer Assisted Diagnosis and SurgeryMarch 2006, Santiago de Chile
TIMC & GRAVIR laboratories
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Hierarchical Multi-Resolution Finite Element Model for Soft Body Simulation
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Motivation
To model soft bodies for surgical training or per-operative simulators
(EPIDAURE-INRIA)
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Goal
Deformable model allowing:– Interactivity
speed robustness
– Accuracy accurate deformations and forces Finite Element dynamics = fast deformation propagation
– Patient specific mesh
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Hierarchical Multi-Resolution Finite Element Model for Soft Body Simulation
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Goal
Deformable model allowing:– Interactivity
speed robustness
– Accuracy accurate deformations and forces Finite Element dynamics = fast deformation propagation
– Patient specific mesh
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Outline
Previous work– Interactive deformable models– Patient specific models
Contribution– Deformable Octree Mesh– Hierarchical Finite Element
Results
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Previous Work - Interactive deformable models
Speed / Fast computation– Multiresolution [Debunne01,Wu01,Dequidt05]
– FE Linear computation of large displacements [Etzmuß03,Müller04,Nesme05]
[Debunne01]
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Background
Dynamical Propagation / Static convergence
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Hierarchical Multi-Resolution Finite Element Model for Soft Body Simulation
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Background
How many time steps or iterations are necessaryto propagate deformation to all nodes ?
?
Dynamical Propagation / Static convergence
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Hierarchical Multi-Resolution Finite Element Model for Soft Body Simulation
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Background
How many time steps or iterations are necessaryto propagate deformation to all nodes ?
?
Explicit solvers #steps=O(#nodes)
Iterative implicit solvers#iterations=O(#nodes)
Iterative static solvers
Dynamical Propagation / Static convergence
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Previous Work - Interactive deformable models
Fast dynamical propagation / static convergence
– Multigrid solver [Terzopolous88,Wu04]
– Hierarchical FE [Grinspun02,Capell02]
Better compromise speed / propagation
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Previous Work - Patient specific models
1. Scan data voxels
2. Segmentation tagged voxels
3. Surface meshing triangles
4. Volume meshing tetrahedra, hexahedra
5. FEM modeling simulation
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Previous Work - Patient specific models
1. Scan data voxels
2. Segmentation tagged voxels
3. Surface meshing triangles
4. Volume meshing tetrahedra, hexahedra
5. FEM modeling simulation
4 generic volume mesh + matching
[Couteau00,Clatz05]
– Slow, specific cases, degenerate mesh
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Previous Work - Patient specific models
1. Scan data voxels
2. Segmentation tagged voxels
3. Surface meshing triangles
4. Volume meshing tetrahedra, hexahedra
5. FEM modeling simulation
4 & 5 Meshless method [Desbrun95,Müller04]
– Not interactive
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Previous Work - Patient specific models
1. Scan data voxels
2. Segmentation tagged voxels
3. Surface meshing triangles
4. Volume meshing tetrahedra, hexahedra
5. FEM modeling simulation
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Contribution
Deformable Octree Mesh 3 & 4 & 5
– Does not need for volume or surface mesh
directly segmented scan– Multiresolution
Hierarchical FE– Fast propagation
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Octree Mesh
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Octree Mesh
Decompose the bounding box of the object into an octree
FEM modeling on cells
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Octree Mesh
Decompose the bounding box of the object into an octree
FEM modeling on cells Interpolate the displacement within cells
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Octree Mesh
Decompose the bounding box of the object into an octree
FEM modeling on cells Interpolate the displacement within cells Add degrees of freedom where needed
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Octree Mesh
Precompute a maximal resolution
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Octree Mesh
Precompute a maximal resolution Multiresolution with virtual tree structures
– Mechanics– Rendering– (collision mangement, …)
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Octree Mesh
Compute cell properties using filling ratio and material properties
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Octree Mesh
Compute cell properties using filling ratio and material properties
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Octree Mesh
Compute cell properties using filling ratio and material properties
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Mechanics
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Mechanics
Classical Finite Element formulation
Elements Interpolations functions
NODAL
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Hierarchical FE Bases
Define functions using a hierarchical basis Goal: to give a better propagation
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Hierarchical FE Bases
Define functions using a hierarchical basis Goal: to give a better propagation
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Hierarchical FE Bases
Define functions using a hierarchical basis Goal: to give a better propagation
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Hierarchical FE Bases
Hierarchical positions
hierarchical nodal
P0 P1
P2P3
ΔP4
ΔP5
ΔP6
ΔP7 ΔP8
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Hierarchical FE Bases
Hierarchical positions
hierarchicalnodal
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Hierarchical FE Bases
Functions influencing an element
hierarchicalnodal
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Hierarchical FE Bases
Propagation in nodal
Explicit
Implicit / static
step 0
iteration 0
initial deformation
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Hierarchical FE Bases
Propagation in nodal
step 0 step 1
iteration 0 iteration 1
Explicit
Implicit / static
initial deformation propagated deformation
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Hierarchical FE Bases
Propagation in nodal
step 0 step 1 step 2
iteration 0 iteration 1 iteration 2
Explicit
Implicit / static
initial deformation propagated deformation
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Hierarchical FE Bases
Propagation in nodal
step 0 step 1 step 2 step 3
iteration 0 iteration 1 iteration 3iteration 2
Explicit
Implicit / static
initial deformation propagated deformation
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Hierarchical FE Bases
Propagation with hierarchical functions
Explicit
Implicit / static
step 0
iteration 0
initial deformation
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Hierarchical FE Bases
Propagation with hierarchical functions
step 0 step 1
iteration 0 iteration 1
direct interaction
Explicit
Implicit / static
initial deformation propagated deformation
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Hierarchical FE Bases
Propagation with hierarchical functions
step 0 step 1
iteration 0 iteration 1
direct interaction interpolation+
Explicit
Implicit / static
initial deformation propagated deformation interpolated deformation
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An implementation
Linear strain and stress tensors Linear interpolation inside cubic elements Static solver using Conjugate Gradient
in hierarchical coordinates
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Results - Propagation
Evolution of the solution during a static solving
– Pull blue node– Measure displacement of green node
force
Displacement
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Results - Propagation
Evolution of the solution during a static solving
– Pull blue node– Measure displacement of green node
force
Displacement
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Results - Propagation
Evolution of the solution during a static solving
– Pull blue node– Measure displacement of green node
force
Displacement
real-time
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Results - Propagation
Static solver with changing boundary conditions 10 CG iterations
nodal / hierarchical
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Results – Global convergence
Number of CG iterations for global convergence
Example 1 Example 2
force
gravity
Number of elements composing the beam
1 8 64 512
Example 1 nodal 1 13 55 146
hierarchical 1 11 27 47
Example 2 nodal 8 50 87 198
hierarchical 6 24 37 52
(in number of CG iterations)
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Conclusion & Perspective
Deformable octree mesh– does not need for surfacic or volumic mesh
Hierarchical FE– fast propagation
Hierarchical corotational for large displacements Automatic adaptative criteria Hierarchical boundary conditions
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Hierarchical Multi-Resolution Finite Element Model for Soft Body Simulation
Thank you
Matthieu Nesme, François Faure, Yohan Payan
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An implementation
Difference with classical nodal FE:– Computations of K take into account all ancestors cells
Functions at the same level of the considered element
Functions at an upper level from ancestors
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An implementation
Integration
Range definition of the functions
ancestor
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Previous Work - Interactive deformable models
Force computation (deformation, stress)
Dynamics Time integration
Multiresolution models
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Previous Work - Dynamics
Explicit methods– Fast computation of one step– Slow propagation
Implicit methods [Baraff98]
– Stable– Solve a system– Longer step– Propagation / computational time
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Previous Work - Force computation
Precomputation [Cotin96]– Precomputed inverse matrix– Quasi-static solver– Small displacements
Large displacements– Non-linear strain tensor [Debunne01,Picinbonno03]
– Corotational [Etzmuß03,Müller04]
Material laws [Terzopoulos88,Hauth03]
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Previous Work - Adaptativity
Multiresolution– Allocate ressources where they are most profitable
Several meshes at several levels of detail [Debunne01]– Difficult meshes building– No mathematical proofs
Precomputed progressive meshes [Wu01]– Difficult to adapt meshes during the animation
Octree Mesh [Dequidt05]– Problems at join between several levels of detail
Hierarchical [Grinspun02,Wu04] Better compromise speed / propagation
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Hierarchical FE Bases
Hierarchical positions
P0
P1
P2
P3
ΔP4
ΔP5
ΔP6
ΔP7
ΔP8
P0
P1
P2
P3
(P0+P1)/2+ΔP4
(P1+P2)/2+ΔP5
(P2+P3)/2+ΔP6
(P3+P0)/2+ΔP7
(P0+P1+P2+P3)/4+ΔP8hierarchical nodal
P0 P1
P2P3
ΔP4
ΔP5
ΔP6
ΔP7 ΔP8
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Hierarchical FE Bases
Define functions using a hierarchical basis Goal: to give a better propagation
Hierarchical functionsNodal functions
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Octree Mesh
Compute cell properties using filling ratio and material properties
Image par image, virer surfacique, commencer par finnest->coarsest + multires et fleche
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Results - Propagation
Evolution of the solution during a static solving
– Pull blue node– Measure displacement of green node
force
Displacement
real-time
Relative error