Direct image-analysis methods for surgical simulation and mixed - - PowerPoint PPT Presentation

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Direct image-analysis methods for surgical simulation and mixed - - PowerPoint PPT Presentation

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Direct image-analysis methods for surgical simulation and mixed meshfree methods Jack S. Hale Marie Curie Postdoctoral Fellow Research Unit in Engineering Science Universit du Luxembourg Collaborators: Stephane P. A. Bordas, Universite du

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Direct image-analysis methods for surgical simulation and mixed meshfree methods

Jack S. Hale Marie Curie Postdoctoral Fellow Research Unit in Engineering Science Université du Luxembourg

Collaborators: Stephane P. A. Bordas, Universite du Luxembourg/Cardiff University Pierre Kerfriden, Cardiff University Juan J. Rodenas Garcia, Universidad Politecnica de Valencia

  • Pedro M. Baiz, Imperial College London

Alejandro Ortiz-Bernardin, Universidad de Chile Christian J. Cyron, Yale University

Université Pierre-et-Marie-Curie Institut Jean le Rond d’Alembert 26th May 1400

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An Overview

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The problems and my research

Mesh distortion in hyperelastic problems

Numerical locking in incompressible elasticity, plates and shells, Cosserat elasticity

Enriched implicit boundary methods Meshfree methods Mixed variational forms Patch projection techniques Direct image- analysis transition for surgical simulators.

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Project 1: Direct image to mesh analysis for surgical simulation

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Surgical Simulation

Model-order reduction Enriched and implicit boundary methods High Performance Computing Real-time simulation

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RealTCut and SHACRA Inria

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COLONIX, OSIRIX

How can we move from an image…

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…or perhaps a series of images…

Source: COLONIX, OSIRIX

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Pipelines to analysis

Acquire images Segment images Mesh Surfaces Mesh Volume Perform analysis

Traditional

Acquire images Segment images Perform analysis

Implicit Boundary

NURBS

Implicit Explicit

}

}

{

Implicit

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SLIDE 11

Nested Octree

Og Od M Discretisation Geometry

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SLIDE 12

How to transfer geometric information back to the discretisation?

V pd

hd (Od)

V pg

hg (Og)

pd > pg hd > hg pg = 1 Enrichment

V pd

hd (Od)

  • E[V pg

hg (Og)]

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SLIDE 13

Octree Level 5/Level 3

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Surface

6a2e86c

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Features

  • 2D and 3D problems on the same code-path.
  • parallel hybrid MPI/TBB assembly and solution.
  • fast and robust computational geometry using CGAL.
  • automatic Delaunay tessellation of integration subdomains.
  • completely separate representation of discretisation and geometry via

nested octree data structures.

  • constructs to represent soft and hard segmentations of image data.
  • implicit representation via level-sets, inside-outside functions.
  • independent hpe-type adaptivity on discretisation and geometry.
  • fast refinement and coarsening operations.
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Outlook

  • We are developing a cartesian grid implicit

boundary/enriched finite element method toolkit within deal.ii.

  • By uncoupling discretisation and geometry we

hope to produce a method particularly suited to image-based analysis.

  • Many challenges ahead, particularly with

imposition of Dirichlet boundary conditions on hyperelastic soft-tissue model.

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Project 2: Mesh-based and Meshfree Partitions

  • f Unity
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Partition of Unity

uh(x) =

N

  • i=1

φiui

N

  • i=1

φi = 1

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Finite elements

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 φ

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Mesh-based Partition of Unity

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Finite Element Method

  • Reference

Mesh

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Meshfree-based Partition of Unity

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Meshfree partition of unity is constructed in global space

Good:

  • improved approximation

properties.

  • high-order continuity.
  • less dofs for given accuracy.
  • less sensitive to poor quality

node distributions.

  • eases mesh generation.
  • Not so good:
  • expensive.
  • extra flexibility also means extra

complexity:

  • integration.
  • adaptivity/error measures.
  • mathematical proof.
  • stability.
  • boundary conditions.
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Maximum-entropy basis functions

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 φ

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0.0 0.2 0.4 0.6 0.8 1.0 x −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 u

uh u ui

0.0 0.2 0.4 0.6 0.8 1.0 x −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 u

uh u ui

uh(x) =

N

  • i=1

φiui

RBF Maximum-Entropy

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Maximum-entropy basis functions

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Maximum-entropy basis functions

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 0.30 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.06 0.12 0.18 0.24 0.30 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

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Maximum-entropy basis functions

uh ∈ H1

0(Ω)

All basis functions associated with interior nodes vanish on the convex hull. Vital for easy imposition of Dirichlet boundary conditions.

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The problem.

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The problem: numerical locking

Problems with small parameters crop up nearly everywhere! Hyperelasticity Incompressible fluid flow Plates and Shells Cosserat elasticity and probably many more…

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Nearly-incompressible elasticity

λ = νE (1 + ν)(1 − 2ν) Find uh ∈ Uh such that

  • Ω

C(uh) : (v) dΩ =

  • Γ

f · v dΩ ∀v ∈ H1

0(Ω)

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Locking

102 103 dofs 10−1 100 eL2

t = 0.1 t = 0.01 t = 0.001

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Locking

Inability of the basis functions to satisfy the constraint imposed whilst still having adequate approximation properties

||u − uh|| ≤ C(λ)hp||u||

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p-refinement

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p-refinement

102 103 104 dofs 10−6 10−5 10−4 10−3 10−2 10−1 100 eH1(θ)

p = 1 p = 2 p = 3 p = 4 p = 5

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Mixed variational formulation

Find uh ∈ Ph and ph ∈ Ph such that: Looking better

2µ

  • Ω

(uh) · (v) dΩ +

  • Ω

ph · v dΩ =

  • Ω

f · v dΩ v Uh

  • Ω

· uh q dΩ 1

  • Ω

phq dΩ = 0 q Ph

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Great, so the problem is solved right?

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LBB Stability

inf

q∈Ph sup v∈Uh

  • Ω q∇ · vdΩ

||v||1||q||0/R ≥ βh > 0

A question of balance

Ph Uh

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Stability

(a) Bad approximation (b) Good approximation

Image: Marie E. Rognes

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MINI element

Arnold, Brezzi, Fortin 1984 Uh Ph Linear + Bubble Linear

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The Volume Averaged Nodal Pressure Method (VANP)

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Questions I will answer…

  • 1. Can we produce a stable pair of spaces for the mixed

formulation using meshfree approximation schemes?

  • 2. Can we eliminate the pressure space to produce a generalised

displacement method?

  • 3. Can we produce a general scheme, which works for arbitrary

spaces of meshfree basis functions and even finite element basis functions?

  • 4. How does enrichment, in the manner of the MINI bubble, affect

the convergence and stability?

  • 5. Does it work in 3D?
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Questions I can answer….

  • Exactly how to construct meshfree basis functions,

including high-order reproducing radial basis functions and maximum-entropy basis functions.

  • How to accurately integrate the weak form.
  • How to implement the method in algorithmic form.
  • And of course any other questions you may have…
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  • 1. Can we produce a stable pair of spaces for the

mixed formulation using meshfree approximation schemes?

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C T (Ω) N s N b C T (Ω) N s N b∗

  • Fig. 1: Schematic representation of a two-dimensional simplicial tessellation for the VANP method.

(a) Enhanced node set N +, and (b) enhanced node set N ∗.

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Spaces

Uh := [ME(Ω; Nh, ρ)]2 Ph := CG1(Ω; Th)

For simplicity of the exposition, not required

uh(x) =

N

  • i=1

φiui ph =

M

  • i=1

Nipi

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Saddle-point problem

  • Ω

BT DB dΩ u +

  • Ω

BT mN p dΩ p =

  • Ω

ΦT

uf dΩ

  • Ω

N T

p mT B dΩ u − 1

λe

  • Ω

N T

p N p dΩ p = 0

B =   

∂φux ∂x1 ∂φuy ∂x2 ∂φux ∂x2 ∂φuy ∂x1

   m = 1 1 0T D =   2µ 2µ µ  

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  • 2. Can we eliminate the pressure space to produce a

generalised displacement method?

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Saddle-point problem

A B BT −C u p

  • =

f

  • How can we get rid of p?
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Projection

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Many methods include a projection or ‘softening’

Enhanced Assumed Strains (EAS) Reduced Integration Mixed Interpolation of Tensorial Components Smoothed Finite Element Method (SFEM)

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N

  • b=1
  • Ω

NpamT Bb dΩ ub − 1 λ

M

  • a=1
  • Ω

Npa dΩ pa = 0

N

  • b=1
  • Ωa

NpamT Bb dΩ ub − 1 λ

M

  • a=1
  • Ωa

Npa dΩ pa = 0 For every pressure node: Restrict integration domain to local domain: Re-arrange to get equation for every pressure dof: pa = −λ

N

  • b=1
  • Ωa NpamT Bb dΩ
  • Ωa Npa dΩ
  • ub
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Results

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  • 1. Can we produce a stable pair of spaces for the

mixed formulation using meshfree approximation schemes?

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Timoshenko Beam

y x P

100 mm 200 mm

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Timoshenko Beam

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −7 −6 −5 −4 −3 −2 −1

ln(1/h) ln(energy norm of the error)

MINI VANP−T1+/T1−m VANP−T1*/T1−m 1 1

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Leaky-lid cavity flow

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0.0 0.2 0.4 0.6 0.8 1.0 y −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ux

LPP MaxEnt 11 × 11 MINI 10 × 10 MINI 30 × 30

Velocity across P − P

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0.0 0.2 0.4 0.6 0.8 1.0 x −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 uy

LPP MaxEnt 11 × 11 MINI 10 × 10 MINI 30 × 30

Velocity across Q − Q

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  • 3. Can we produce a general scheme, which works

for arbitrary spaces of meshfree basis functions?

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Timoshenko Beam

0.2 0.6 1 1.4 1.8 −7 −6 −5 −4 −3 −2 −1

ln(1/h) ln(energy norm of the error)

VANP−T2+/T1−m VANP−T2*/T1−m VANP−T2/T1−m VANP−T2+/T1−r VANP−T2*/T1−r VANP−T2/T1−r 1 2

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  • 4. How does enrichment, in the manner of the MINI

bubble, affect the convergence and stability?

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MINI* element

Kim and Lee, 2000, 10.1023/A:1018973303935 Uh Ph

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C T (Ω) N s N b C T (Ω) N s N b∗

  • Fig. 1: Schematic representation of a two-dimensional simplicial tessellation for the VANP method.

(a) Enhanced node set N +, and (b) enhanced node set N ∗.

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Constrained Block

MINI First-order MaxEnt Full Bubbles

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Constrained Block

MINI First-order MaxEnt Half Bubbles

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Constrained Block

MINI No bubbles

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  • 5. Does it work in 3D?
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(a) (b) (c) (d) (e) (f) (g) (h) (i)

  • Fig. 17: Compression of a constrained block: Nodal pressure variable for (a) MINI element, (b)

VANP-T +

1 /T1-m, (c) VANP-T ∗ 1 /T1-m, (d) VANP-T + 2 /T1-m, (e) VANP-T ∗ 2 /T1-m, (f) VANP-T2/T1-

m, (g) VANP-T +

2 /T1-r, (h) VANP-T ∗ 2 /T1-r, and (i) VANP-T2/T1-r.

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Summary

  • Stable meshfree methods are developed by

mimicking existing inf-sup stable finite element methods.

  • The auxiliary pressure variable is eliminated using

a volume-averaged nodal pressure technique.

  • Currently extending to hyperelasticity, where the

robust nature of the meshfree shape functions will be a great advantage.

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Acknowledgements

  • Jack S. Hale would like to thank:
  • The support of the FNR through the AFR Marie

Curie COFUND scheme.

  • The partial support of the Université du

Luxembourg.

  • FONDECYT Chile Foreign Scholar Grant.
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Questions?