Hierarchical Multi-Resolution Finite Element Model for Soft Body Simulation Matthieu Nesme, François Faure, Yohan Payan 2 nd Workshop on Computer Assisted.

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


2

Motivation

To model soft bodies for surgical training or per-operative simulators

(EPIDAURE-INRIA)


3

Goal

Deformable model allowing:– Interactivity

speed robustness

– Accuracy accurate deformations and forces Finite Element dynamics = fast deformation propagation

– Patient specific mesh


4

Goal

Deformable model allowing:– Interactivity

speed robustness

– Accuracy accurate deformations and forces Finite Element dynamics = fast deformation propagation

– Patient specific mesh


5

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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Background

How many time steps or iterations are necessaryto propagate deformation to all nodes ?

?



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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



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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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1. Scan data voxels





4 generic volume mesh + matching

[Couteau00,Clatz05]

– Slow, specific cases, degenerate mesh


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1. Scan data voxels





4 & 5 Meshless method [Desbrun95,Müller04]

– Not interactive


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1. Scan data voxels






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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


FEM modeling on cells Interpolate the displacement within cells


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Octree Mesh


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



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Octree Mesh



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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 positions

hierarchical nodal

P0 P1

P2P3

ΔP4

ΔP5

ΔP6

ΔP7 ΔP8


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hierarchicalnodal


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Functions influencing an element

hierarchicalnodal


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Propagation in nodal

Explicit

Implicit / static

step 0

iteration 0

initial deformation


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step 0 step 1

iteration 0 iteration 1

Explicit

Implicit / static

initial deformation propagated deformation


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step 0 step 1 step 2

iteration 0 iteration 1 iteration 2

Explicit

Implicit / static



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step 0 step 1 step 2 step 3

iteration 0 iteration 1 iteration 3iteration 2

Explicit

Implicit / static



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Propagation with hierarchical functions

Explicit

Implicit / static

step 0

iteration 0

initial deformation


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step 0 step 1


direct interaction

Explicit

Implicit / static



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step 0 step 1


direct interaction interpolation+

Explicit

Implicit / static

initial deformation propagated deformation interpolated deformation


40

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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force

Displacement


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force

Displacement

real-time


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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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Thank you

Matthieu Nesme, François Faure, Yohan Payan

[email protected]


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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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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


52

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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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 functionsNodal functions


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Octree Mesh


Image par image, virer surfacique, commencer par finnest->coarsest + multires et fleche


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force

Displacement

real-time

Relative error

Hierarchical Multi-Resolution Finite Element Model for Soft Body Simulation Matthieu Nesme, François Faure, Yohan Payan 2 nd Workshop on Computer Assisted.

Documents

forces finite element

goal deformable model

fem modeling simulation

soft bodies

degenerate mesh slide

nesme05 debunne01 slide

specific cases

volume meshing tetrahedra