Adaptive methods for multiscale fracture and surgical simulation 

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Adaptive methods for

multiscale fracture and surgical simulation

 

Stéphane Bordas & Pierre Kerfriden 

University of Luxembourg and Cardiff University

Imperial College London, London, UK

20140410

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2

•  A"small,"young,"dynamic"university"

•  3" languages" (English," German," French);" bilingual"and"trilingual"degrees" •  Strong"mathema>cs"and"Comp."Sc."" •  RUES:"3"professors"in"computa>onal" mechanics,"30"collaborators" •  Computa>onal"sciences"priority"1" •  Strong"local"industry" •  Strong"and"suppor>ve"na>onal"funding" •  7"EU"projects"in"engineering,"of"which"" RealTcut:"ERC"Star>ng"Grant"(Bordas)"

"

2"

•  A" large," established" university"

(1883)

"

•  95%"3"or"4*"at"RAE2008"in"Civil"

•  Over" 100" EU" projects" awarded" of" which"ITN:"INSIST"

•  M e c h a n i c s" R e s e a r c h :" 4 0" researchers,"14"faculty"members"

•  Advanced" manufacturing" and"

characterisa>on"

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_{Institute of Mechanics}

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& Advanced Materials

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1997-2003

MSc Geotechnical Engineering  

PhD. Damage Tolerance of Aerospace Structures (XFEM)

and Biofilm Growth

Chicago, Illinois

Advisor: Brian Moran Now vice-provost for faculty affairs at KAUST

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Post-doc 2003-2006 -

Meshless/XFEM Geomechanics

Lausanne

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Lecturer in

civil

engineering

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Prof. Stéphane Bordas, Director. Extended FEM/ Meshless Prof. Feodor Borodich Theoretical/Nano mechanics, contact, adhesion Prof. Pwt Evans Contact mechanics, tribology Dr. Paul Howson Transcendental eigenvalue problems Dr. Siva Kulasegaram Meshless methods Dr. Pierre Kerfriden Multiscale, model order reduction, fracture Prof. David Kennedy Eigenvalue problems, advanced numerical methods Prof. Bhushan Karihaloo Advanced materials, theoretical mechanics

Prof. Ray Snidle

Contact mechanics, tribology Dr. Hanxing Zhu Theoretical mechanics, cellular materials

Ins$tute of mechanics and advanced materials

Dr. Lars Beex

!

Multiscale methods 9

The ins$tute

•

6 professors, 6 lecturers/senior lecturers

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10 post-‐doc fellows

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17 PhD students

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~ £1.0M funding annually

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Theoretical & Computational

Tribology & Contact Mechanics

Experiments

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Mo$va$on: mulIscale fracture of engineering structures and materials

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Prac$cal early-‐stage design simula$ons (interac$ve)

[Allix, Kerfriden, Gosselet 2010]

Discretise

0.125 mm 50 mm

100 plies

courtesy: EADS

‣

Reduce the problem size while controlling the error (in QoI)

when solving very large (multiscale) mechanics problems

Discretise

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Mo$va$on: mulIscale fracture of engineering structures and materials

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!

Discre$za$on

!

➡

parIIon of unity enrichment

✓

(enriched) meshless methods

✓

level sets

!

➡

isogeometric analysis

➡

implicit boundaries

!

Model redu

_!

c$on

✓

mulI-‐scale

& homogen

isaIon

✓

algebraic m

odel reducI

on (using PO

D)

✓

Newton-‐Kry

lov, “local/g

lobal”, dom

ain

decomposiIon

12

!

Error control

!

✓

XFEM: goal-‐oriented error esImates

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used by CENAERO (Morfeo XFEM)

✓

meshless methods for fracture

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Institute of Mechanics

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Part 0. An adapIve method for fracture -‐

applicaIon to polycrystalline failure 

Ahmad Akbari, Pierre Kerfriden, Spaß 

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Adaptive method for fracture

➢

Introduction: Multiscale methods for Fracture

➢

Adaptive multiscale method

▪ Strategy

▪ Fine scale problem ▪ Coarse scale problem

• FE2 _method

• Adaptive mesh refinement

▪ Coupling fine and coarse discretisations ▪ Results

• L-shape problem

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erro

r

Coarse Element size

Di

scre

tiza

tio

n e

rro

r

Homogenisation error

Critical level of error

Error control in multiscale modelling

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Multiscale methods for Fracture

▪ Concurrent

▪ Non-concurrent

Damage zone is modelled by a macroscopic cohesive crack that homogenises the failure zone. V.P. Nguyen 2012

l

L

L/l >1

L

l

L/l >>1

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➢

Strategy:

• control the coarse scale

discretization error

Mesh refinement _{Hybrid method}

• control the

modelling error

FE

2

FE

2

FE

2

+ Domain

Decomposition

Method

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➢Microscale problem:

▪Orthotropic grains

!

▪Cohesive interface

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➢Macroscale problem:

▪ FE

2

Method

Based on averaging theorem

(computational homogenisation)

!

▪Adaptive mesh refinement

Error estimation by Zienkiewicz-Zhu-type recovery technique

Mesh refinement

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▪ The FE

2

Method

RVE

time step, i

Macroscale problem

Iteration, j

RVE problem

Iteration, k

❖ Shortcoming of the FE2_{Method :}

Lack of scale separation

RVE cannot be found in the

softening regime

!

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•

Error

estimation by Zienkiewicz-Zhu-type

recovery technique

➢

Coarse scale Adaptive mesh refinement

Element to refine Refined mesh

•Convergence criterion:

Coarse Scale: Adaptive mesh refinement

Error$due$to$the$ discre-sa-on$of$ neglected$$$

⌦

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⌦c

What is the solution for the FE

2

shortcoming:

“

Hybrid Multiscale Method”

•FE2 for non-critical region

(hierarchical multiscale)

!

•Domain decomposition for critical region (concurrent multiscale)

 

Fine-Coarse scales Coupling 

Critical region

FE

2

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Results: uni-axial tension

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Results: uni-axial tension

❖

100X (magnification of displacement)

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Results: uni-axial tension

❖

100X (magnification of displacement)

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Results: uni-axial tension

❖

100X (magnification of displacement)

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Results: uni-axial tension

❖

100X (magnification of displacement)

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Results: uni-axial tension

❖

100X (magnification of displacement)

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Verification

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Verification

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An adaptive multiscale method was developed for

discrete fracture in polycrystalline materials:

!

• _{An unstructured mesh is used for the coarse scale}

problem

!

• A local arc-length was used to control crack speed in the

fully resolved region.

!

• A recovery based error indicator was employed to limit

discretization error at each time step.

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Perspectives

• coarsening once the crack is open

• molecular dynamics at the fine scale

!

• real-life problems! :)

• coupling with algebraic model reduction

(POD)

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Part I. Streamlining the CAD-‐analysis transiIon 

Part II. Some advances in enriched FEM 

Part III. ApplicaIon to H cu\ng of Si wafers 

Part IV. InteracIve cu\ng sim.

 

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Part I. Streamlining the CAD-‐analysis transiIon 

Coupling, or decoupling?

 

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Mo$va$on: free boundary problems -‐ mesh burden

FEM

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3

CAD to Analysis

calculate

vM stress distribution

iterate

mesh 80%

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One would like to be able to use such a mesh

5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

(c)

Figure

5.27 – Approximation géométrique d’une microstructure contenant des inclusions

lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un

sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière E

B

.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

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Superimpose the geometry onto an arbitrary background mesh

5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

(c)

Figure

5.27 – Approximation géométrique d’une microstructure contenant des inclusions

lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un

sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière E

B

.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

95 5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

(c)

Figure

5.27 – Approximation géométrique d’une microstructure contenant des inclusions

lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un

sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière E

_B

.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

95 5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

(c)

Figure

5.27 – Approximation géométrique d’une microstructure contenant des inclusions

lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un

sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière E

_B

.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

95

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Compute interacIons between the geometry and the mesh

5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

(c)

Figure

5.27 – Approximation géométrique d’une microstructure contenant des inclusions

lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un

sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière E

B

.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

95 5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

(c)

Figure

5.27 – Approximation géométrique d’une microstructure contenant des inclusions

lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un

sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière E

_B

.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

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Perform the analysis

5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

(c)

Figure

5.27 – Approximation géométrique d’une microstructure contenant des inclusions

lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un

sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière E

B

.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

95 5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

(c)

Figure

5.27 – Approximation géométrique d’une microstructure contenant des inclusions

lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un

sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière E

_B

.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

95 5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

(c)

Figure

5.27 – Approximation géométrique d’une microstructure contenant des inclusions

lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un

sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière E

_B

.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

95 5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a)

(b)

Figure

5.28 – Champs de contraintes (a) et de déplacements (b).

Figure

5.29 – Approximation géométrique d’une microstructure contenant des inclusions

en forme de tore indépendamment de la taille du maillage ÉF.

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Implicit boundary method

• Immersed boundary method (Mi]al, et al. 2005)

• FicIIous domain (Glowinski, et al. 1994)

• Embedded boundary method (Johansen, et al. 1998)

• Virtual boundary method (Saiki, et al. 1996)

• Cartesian grid method (Ye, et al. 1999, Nadal, 2013)

Paradigm 1: Separate ﬁeld and boundary discreIsaIon

✓

Easy adapIve reﬁnement + error esImaIon (Nadal, 2013)

✓

Flexibility of choosing basis funcIons

• Accuracy for complicated geometries? BCs on implicit surfaces?

➡

An accurate and implicitly-‐deﬁned geometry from arbitrary

parametric surfaces including corners and sharp edges

(Moumnassi, et al. 2011)

5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a) (b)

Figure5.28 – Champs de contraintes (a) et de déplacements (b).

Figure5.29 – Approximation géométrique d’une microstructure contenant des inclusions en forme de tore indépendamment de la taille du maillage ÉF.

96

(a) (b)

Figure5.28 – Champs de contraintes (a) et de déplacements (b).

Figure_{5.29 – Approximation géométrique d’une microstructure contenant des inclusions} en forme de tore indépendamment de la taille du maillage ÉF.

96 5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a) (b)

(c)

Figure5.27 – Approximation géométrique d’une microstructure contenant des inclusions lenticulaires. (a) maillage grossier de l’approximation ÉF. (b) raﬃnement par un sous-maillage gradué (SMG) de niveau (n = 7) à l’intérieur de chaque élément de frontière EB.

(c) approximation de la géométrie indépendamment de la taille h du maillage.

95 5.2. Analyse de convergence en maillage non-conforme aux frontières courbes

(a) (b)

(c)

95

(a) (b)

(c)

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Ex: Moumnassi et al, CMAME DOI:10.1016/j.cma.2010.10.002

44 marchin_{g metho} d seed point(s) - requires one single global search

Level Set representation of a surface defined by a parametric function

•

Objectives

‣

insert surfaces in a structured mesh

-

without meshing the surfaces (boundary, cracks, holes,   inclusions, etc.)

-

directly from the underlying CAD model

-

model arbitrary solids, including sharp edges and vertices

‣

keep as much as possible of the mesh as the CAD model   evolves, i.e. reduce mesh dependence of the implicit  

boundary representation

‣

maintain the convergence rates and implementation simplicity of the FEM

• In order to reproduce the geometry accurately, significant mesh refinement is typi-cally needed;

• Because the whole boundary is defined using one single function, it is not straight-forward to locate and separate diﬀerent regions on ∂Ωhfor attribution of appropriate

boundary conditions;

• To eﬃciently approximate a curved domain, one generates a discrete approxima-tion of the scalar distance field φ by evaluating the funcapproxima-tion on a suﬃciently fine mesh, or by adaptive schemes like octree techniques to capture details of the domain boundary ∂Ωh. However, linear interpolation of the mesh values to approximate the

boundary is insuﬃcient for higher order analysis.

Figure 3: Approximation of an object with convex and concave boundaries with the same background mesh, resulting from Boolean combinations of half-spaces defined using analytically defined level set functions (8-planes and 3-cylinders). (a) The object is con-structed by a single level set resultant from Boolean operations (one scalar distance value is stored at each node). (b) shows the approximation by our new approach that preserves sharp features (eleven scalar distance values are stored at each node).

In the following section, we present a new approach to represent arbitrary regions using level set functions, which alleviates the pitfalls of the “single-level-set-description”.

11

Single Multiple level sets Advance by CRP Henri Tudor in 2011

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Institute of Mechanics and Advanced Materials

http://www.researcherid.com/rid/A-1858-2009

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mulIple level sets

!

•

single (len) versus mulIple (right)

!

45 Examples

• In order to reproduce the geometry accurately, significant mesh refinement is typi-cally needed;

• Because the whole boundary is defined using one single function, it is not straight-forward to locate and separate diﬀerent regions on ∂Ωhfor attribution of appropriate

boundary conditions;

• To eﬃciently approximate a curved domain, one generates a discrete approxima-tion of the scalar distance field φ by evaluating the funcapproxima-tion on a suﬃciently fine mesh, or by adaptive schemes like octree techniques to capture details of the domain boundary ∂Ωh. However, linear interpolation of the mesh values to approximate the

boundary is insuﬃcient for higher order analysis.

Figure 3: Approximation of an object with convex and concave boundaries with the same background mesh, resulting from Boolean combinations of half-spaces defined using analytically defined level set functions (8-planes and 3-cylinders). (a) The object is con-structed by a single level set resultant from Boolean operations (one scalar distance value is stored at each node). (b) shows the approximation by our new approach that preserves sharp features (eleven scalar distance values are stored at each node).

In the following section, we present a new approach to represent arbitrary regions using level set functions, which alleviates the pitfalls of the “single-level-set-description”.

11

(a) (b) (c)

Figure 12: (a) Conversion of four parametric functions into zero level sets. (b) Polygonal meshes extraction for the cutting method. (c) Approximated domain with sharp features.

it.

To obtain an accurate geometry description for domains with curved boundaries, we present in the following section two diﬀerent techniques: degenerated and graded sub-meshes which we shall name DSM and GSM, respectively.

5.4.1. Mesh refinement with degenerated sub-mesh (DSM)

We use the parametric information to generate the desired number of cut edges on the surface inside a boundary element EB which are tangent to this parametric surface (see

Figure 13). These cut edges are created by the corresponding zero level sets such that they are generated by a succession of analytically known level set planes p (x) = (x− x0)· n

that pass through the point x0 on the surface and defined by the normal n at this point.

Then we apply the cutting method to each boundary element EB by using these zero

level sets to create the sub-elements E∆. The next step is the classification of the

sub-elements into the interior boundary IB and exterior boundary OB to define the part of the approximate domain Ωh on the boundary B and the part of its boundary Γh (see

Figure 14).

5.4.2. Mesh refinement with graded sub-mesh (GSM)

The marching algorithm (cf. Section 4.3) benefit of a natural strategy to locate the narrow band from the all elements mesh, in which only the selected elements (i.e. ωi)

need to be used for refinement if desired. This is an attractive strategy to restrict local mesh refinement to boundary elements EB. This strategy will be used locally in EB and

27

Figure 17: A three-dimensional graded sub-mesh refinement of level (n = 6) inside a boundary element EB.

1. Subdividing EBbased on a linear (as in [23, 54]) or higher order (as in [40, 41])

description of the boundary.

2. Without subdividing EB as proposed in Ventura [55] using equivalent

poly-nomials. It is also possible to use the approach of Natarajan et al.[24, 25] based on the Schwarz Christoffel (SC) mapping of the interior/exterior polyg-onal areas to the unit disk. Another alternative is strain smoothing where domain integration is transformed into boundary integration as in [26]. The advantage of the latter is that it has the potential to be amenable to three dimensional cases, whereas the SC mapping technique remains restricted to two-dimensional problems. To use the SC mapping in 3D, the interior and outer parts of a boundary element could be integrated using strain smoothing and the SC mapping subsequently used to integrate along the boundary of the interior and exterior subregions. Since each of those boundaries is composed of the union of polygons, the SC mapping (or any other method to integrate numerically on polygons) can be used to compute the integral on each poly-gon. Note that strain smoothing modifies the variational principle so that the resulting stiffness matrix is usually not as stiff as that of the original finite

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Three-‐dimensional model problem

•

Laplace equaIon on a cube

•

convergence rates

➡

opImal

➡

requires proper Lagrange mulIplier space to eradicate spurious

oscillaIons

46

(a) (b)

Figure 29: Finite element solution of 3D Laplace model problem using implicit computa-tional domain. (a) implicit representation of the domain with sharp features, (b) illustrate the cut view of the solution uh_.

XFEM representation. These comparisons are shown in Figure 31. As can be seen, the analysis with conforming and non-conforming mesh yield nearly the same accuracy and convergence rates in the approximated energy and Lagrange multipliers. As to the enforcement of the Dirichlet boundary conditions, the accuracy and convergence rate are governed by the choice of the Lagrange multiplier space L∗

h. It is interesting to note that

all these numerical results for the case of non-conforming mesh are superior to the standard mixed method (naive approach), which yields oscillations of the Lagrange multipliers on the boundary.

8. Conclusions

We presented and validated a general method to carry out finite element analysis on arbitrary implicitly defined domains obtained from parametric surfaces. The input to the algorithm is the parametric description of the boundary of the object which is converted automatically and eﬃciently into implicit level set representations. The computational domain is then obtained by Boolean operations on those level set functions. A special adaptive numerical integration technique which uses the parametric description to increase

51 0.001 0.01 0.1 1 0.01 0.1 1 Relative Error h

Energy error FEM (R=0.75) LM error FEM (R=0.85) uh error FEM (R=1.83) (a) 0.001 0.01 0.1 1 0.01 0.1 1 Relative Error h

Energy error XFEM (R=0.75/1.15) LM error XFEM (R=0.92/1.44) uh error XFEM (R=1.42/2.09)

(b)

Figure 30: Convergence study results for the mixed formulation on unstructured tetra-hedral mesh: (a) analysis with a conforming mesh and FEM, (b) analysis with a non-conforming mesh and XFEM using the reduced Lagrange multiplier space.

0.001 0.01 0.1 1 0.01 0.1 1 Relative Error h

Energy error FEM (R=0.75) uh error FEM (R=1.83) Energy error XFEM (R=0.75/1.15) uh error XFEM (R=1.42/2.09) 0.001 0.01 0.1 1 0.01 0.1 1 Relative Error h LM error FEM (R=0.85) uh error FEM (R=1.83) LM error XFEM (R=0.92/1.44) uh error XFEM (R=1.42/2.09)

Figure 31: Comparison study between analysis with conforming mesh (Figure 30a) and non-conforming mesh (Figure 30b).

the geometrical faithfulness (thus decrease mesh dependence) was proposed. We showed that the resulting algorithm is adequate to describe objects with sharp features such as edges and corners.

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Institute of Mechanics and Advanced Materials http://www.researcherid.com/rid/A-1858-2009

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45 Geomety-  based   refinement

H-adaptive refinement based on error estimation

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45 Pixel/Voxel-‐based FEA on Cartesian grids (Valencia) 

Processing time

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!

Paradigm 2 : IGA

Couple Geometry and ApproximaIon

 

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3

Isogeometric analysis (with BEM)

Approximate the unknown ﬁelds with the same basis funcIons

( NURBS, T-‐splines … ) as that used to generate the CAD model

direct _calcula @on

meshing

calcula@on stress analysis

• Exact geometry.

• High order conInuity.

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3 KEY QUESTIONS FOR IGA

!

1. Generate a volume discretization using the surface geometry only?

!

2. Realistic solids can in general not be represented by only one volume (patch) and multiple patches must be glued together to avoid

“leaks” (Nitsche, T-splines, PHT-splines, RL/LR-splines)

!

3. Refinement must be done everywhere in the domain (T, PHT… splines)

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IGABEM

Domain

representaIon

Boundary

representaIon

Isogeometric Analysis with BEM

1. IGABEM with NURBS for 2D elasIc problems (Simpson, et al.

CMAME, 2011).

!

2. IGABEM with T-‐splines for 3D elasIc problems (Sco], et al.

CMAME, 2012).

!

3. IGABEM with T-‐splines for 3D acousIc problems (Simpson, et al.

2013 -‐ MAFELAP2013 TH1515).

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4

Knot vector

a non-‐decreasing set of coordinates in the parametric space.

B-‐spline basis func@on

!

NURBS basis func@on

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5

• ParIIon of Unity

!

• Non-‐negaIve

!

• p-‐1 conInuous derivaIves

• Tensor product property

!

No Kronecker delta property

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NURBS to T-‐splines

(NURBS geometry)

(T-‐splines geometry)

NURBS to T-‐splines

!

NURBS

• No waterIght geometry

• No local reﬁnement scheme

!

T-‐splines

• Local knot vector (as Point-‐

based splines)

• Global topology

www.tsplines.com

Y. Bazilevs, V.M. Calo, J.A. Co]rell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A. Sco], and T.W. Sederberg. Isogeometric analysis using T-‐splines. CMAME, 199(5-‐8):229–263, 2010.

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where and are ﬁeld point and source point respecIvely, and are displacement and tracIon around the boundary, and are fundamental soluIons.

IGABEM formulation

Regularised form of boundary integral equaIon for 2D linear elasIcity

DiscreIse the geometry and soluIon ﬁeld using NURBS

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Dam

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Propeller

Propeller: NURBS would require several patches -‐ single patch T-‐splines

Isogeometric boundary element analysis using unstructured T-‐splines

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!

Part II. Some recent advances in enriched FEM

 

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!

Handling discon@nui@es in isogeometric

formula@ons

!

with Nguyen Vinh Phu, Marie Curie Fellow

 

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PUM enriched methods

Discon@nui@es modeling

!

• IGA: link to CAD and

accurate stress ﬁelds

• XFEM: no remeshing

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Mesh conforming methods

!

• IGA: link to CAD and

accurate stress ﬁelds

(63)

PUM enriched methods (XIGA)

1. E. De Luycker, D. J. Benson, T. Belytschko, Y. Bazilevs, and M. C. Hsu. X-‐FEM in isogeometric analysis for linear fracture mechanics. IJNME, 87(6):541–565, 2011.

2. S. S. Ghorashi, N. Valizadeh, and S. Mohammadi. Extended isogeometric analysis for simulaIon of staIonary and propagaIng cracks. IJNME, 89(9): 1069–1101, 2012.

3. D. J. Benson, Y. Bazilevs, E. De Luycker, M.-‐C. Hsu, M. Sco], T. J. R. Hughes, and T. Belytschko. A generalized ﬁnite element formulaIon for arbitrary basis funcIons: From isogeometric analysis to XFEM. IJNME, 83(6):765–785, 2010. 4. A. Tambat and G. Subbarayan. Isogeometric enriched ﬁeld approximaIons.

CMAME, 245–246:1 – 21, 2012.

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Isogeometric cohesive elements

1. C. V. Verhoosel, M. A. Sco], R. de Borst, and T. J. R. Hughes. An

isogeometric approach to cohesive zone modeling. IJNME, 87(15):336–360, 2011.

2. V.P. Nguyen, P. Kerfriden, S. Bordas. Isogeometric cohesive elements for two and three dimensional composite delaminaIon analysis, 2013, Arxiv.

Knot inser@on

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Isogeometric cohesive elements: advantages

!

• Direct link to CAD

• Exact geometry

• Fast/straighvorward generaIon  

of interface elements

• Accurate stress ﬁeld

• ComputaIonally cheaper

!

• 2D Mixed mode bending test (MMB)

• 2 x 70 quarIc-‐linear B-‐spline elements

• Run Ime on a laptop 4GBi7: 6 s

• Energy arc-‐length control

V. P. Nguyen and H. Nguyen-‐Xuan. High-‐order B-‐splines based ﬁnite elements for

(67)

Isogeometric cohesive elements: 2D example

!

• Exact geometry by NURBS + direct link to CAD

• It is straighvorward to vary

(1) the number of plies and

(2) # of interface elements:

• Suitable for parameter studies/design

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Isogeometric cohesive elements: 2D example

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Isogeometric cohesive elements: 3D example with shells

!

• RotaIon free B-‐splines shell elements (Kiendl et al. CMAME)

• Two shells, one for each lamina

• Bivariate B-‐splines cohesive interface elements in between

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Isogeometric cohesive elements: 3D examples

!

•

cohesive elements for 3D meshes the same as 2D

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Isogeometric cohesive elements

!

• singly curved thick-‐wall laminates

• geometry/displacements: NURBS

• trivariate NURBS from NURBS surface(*)

• cohesive surface interface elements

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• Nitsche coupling -‐ NURBS-‐NURBS

Future work: model selecIon (conInuum, plate, beam, shell?)

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Model selec@on

• Model with shells

• IdenIfy “hot spots” -‐ dual

• Couple with conInuum

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!

Part III. ApplicaIon to mulI-‐crack propagaIon

with Danas Sutula, President Scholar

 

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Numerical Modeling of

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Physical process

Manufacturing process: SmartCut

TM

• H+_{ionizaIon
of
a
thin
surface
of
Si}

• Bonding to a handle-‐wafer (sIﬀener)

• High temperature thermal annealing

• NucleaIon and growth of caviIes ﬁlled with H₂

• Pressure driven micro crack growth

• Coalescence and post-‐split fracture roughness

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Objec@ves

Determine:

• micro crack nucleaIon points and direcIon

• mulIple crack paths unIl coalescence

• Ime to complete fracture

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Model

Modeling cavi@es by zero thickness surfaces

• disconInuiIes in the displacement ﬁeld

Linear elas@c fracture mechanics (LEFM)

• inﬁnite stress at crack Ip, i.e. singularity

staIsIcally distributed disconInuiIes

Cohesive interface with variaIon in surface energy

fracture criterion at the disconInuity Ip disconInuity subjected

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Discre@za@on: XFEM

Extended Finite Element Method (XFEM)

• Introduced by Ted Belytschko (1999) for elasIc problems

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Plate with 300 cracks -‐ ver@cal extension BCs

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Ver@cal extension of a plate with 300 cracks

Example #1

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Example #2

Mechanical spli\ng of a wafer sample

• Post-‐split roughness as a funcIon of micro crack distribuIon

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Example #2

Mechanical spli\ng of a wafer sample

• DiscreIsaIon (≈1mln. DOF, h

_e

= 150 nm)

Fracture control parameters

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Evaluation of stress intensity factors (SIF)

• The interaction integral (Yau 1980)

!

Crack growth criterion for mixed mode fracture

• Direction that maximises the energy release (Nuismer 1975)

!

Crack growth direction

• orthogonal to maximum hoop stress

!

(1) – from current solution (2) – known auxiliary solution

(88)

• Energy release rate w.r.t crack increment direction:

!

• The rates of the energy release rate are given by:

!

• where, in a discrete setting, the potential energy is:

!

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• The discrete potential energy:

!

• The discrete energy release rate:

!

• The rates of the energy release rate

!

, where

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• The discrete potential energy:

!

• The discrete energy release rate:

!

• The rates of the energy release rate

!

remote interaction

Crack growth: optimization of direction

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Updated directions:

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• Energy minimization w.r.t. to a finite crack propagation

• The growth direction is given by satisfying:

• Using the maximum hoop-stress criterion as initial guess

!

• Numerical examples:

!

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Mechanical splitting of a wafer

• Post-split roughness as a function of micro crack distribution • Consider a representative material sample

• BC: blade loading = fixed displacements (RHS) • 20 initial micro cracks within the damage zone

2 (mm) 1. 5 (mm) damaged zone (pre-existing flaws)

F

Physical experiment

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Mechanical splitting of a wafer

• Fracture path comparison: max-hoop crit. VS. energy min. • NOTE: non-uniform scaling of axis, y / x = 400

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Mechanical splitting of a wafer

• Comparison of post-split fracture surface roughness

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Application to Si-wafer splitting

Mechanical splitting of a wafer

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!

Part IV. ApplicaIon to surgical simulaIon

with InsItue of Advanced Studies (iCube, University of Strasbourg, France: Hadrien Courtecuisse), INRIA, SHACRA Team (Stéphane CoIn, ChrisIan Duriez); Karol Miller, UWA.

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[email protected], [email protected] RealTcut

iMAM

Surgical simulation (real time/interactivity)

Sim

Learning Planning Assistance

Precision

RealTcut

The ERC RealTcut project

‣

Reduce the problem size while controlling error in solving very large multiscale mechanics problems

complex microstructure

Courtecuisse et al. PBMB 2011

Discretise

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Approach

Concrete objec@ve: compute the response of organs during surgical

procedures (including cuts) in real Ime (50-‐500 soluIons per second)

99

Two schools of thought

‣

constant (me

➡

accuracy o.en controlled visually only

‣

model reduc(on or “learning”

➡

scarce development for biomedical problems

➡

no results available for cu9ng

Proposed approach: maximize accuracy 

for given computa(onal (me. Error control

A 4.30 A 10 7 2.6_⇥ U Q 0 0.02 0.04 0.06 0.08 0.1 0 20 40 60 80 100 120 140 U Q 15% 3.4 U Q xc Wc 1% 25 U

[Courtecuisse et al., MICCAI, 2013] Collabora(on INRIA

!

First implicit, interac@ve method   for cu\ng with contact

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Error control

!

• interac(vity

• space-‐(me discre(za(on?

• op(mize use of compute

resources

!

Complex geometries from

medical images

!

Topological changes & contact

!

Veriﬁca9on & Valida9on

Four main diﬃcul(es

RoI MRI RoI

segmenta9on

-RdI

Region of interest (RoI)

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6=

enriched zone

calculs oﬄine

calculs online: interac(vité

ac(on de l’instrument

généra9on solu(ons par(culières tri pré-‐opératoire

!

“mapping” spéciﬁque pa(ent ~10^3 snapshots POD O(10) fonc9ons espace réduit de pe(te dimension

!

approxima(on POD globale Local (FE) Local (FE) ~10^6 snapshots

!

enrichissement “pointe de coupe” r u

calcul champs asympto(ques

représenta(on locale

(102)

Results -‐ Dr Hadrien Courtecuisse, PhD INRIA

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OPEN SOURCE CODES

PERMIX: Mul(scale, XFEM, large deforma(on, coupled 2 LAMMPS, ABAQUS, OpenMP -‐

Fortran 2003, C++

MATLAB Codes: XFEM, 3D ISOGEOMETRIC XFEM, 2D ISOGEOMETRIC BEM, 2D MESHLESS DOWNLOAD @ h[p://cmechanicsos.users.sourceforge.net/

!

COMPUTATIONAL MECHANICS DISCUSSION GROUP

Request membership @

hep://groups.google.com/group/computa(onal_mechanics_discussion/about

104

TWO POST DOCS 

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the group… November 2012

thank you 

Application to Si-wafer splitting

Mechanical splitting of a wafer

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106 Ahmad Akbari Olivier Goury Haojie Liang _{Dr. Sundararajan} Natarajan Chang-Kye Lee Courtesy: PhD Comics Nguyen-Tanh Nhon

Dr. Robert Simpson Dr. Pierre Kerfriden

Chi Hoang

Xuan Peng

Daniel Paladim Danas

(107)

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Demos

• http://www.youtube.com/watch?

v=90NAq76mVmQ

• Solder joint durability

• http://www.youtube.com/watch?

v=Ri96Wv6zBNU

• http://www.youtube.com/watch?

v=1g3Pe_9XN9I

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Damage tolerance analysis directly from

CAD

• http://orbilu.uni.lu/handle/10993/11850

(113)

3

Isogeometric analysis

!

• P. Kagan, A. Fischer, and P. Z. Bar-‐Yoseph. New B-‐Spline Finite

Element approach for geometrical design and mechanical

analysis. IJNME, 41(3):435–458, 1998.

• F. Cirak, M. Or(z, and P. Schröder. Subdivision surfaces: a new

paradigm for thin-‐shell ﬁnite-‐element analysis. IJNME, 47(12):

2039–2072, 2000.

• Construc(ve solid analysis: a hierarchical, geometry-‐based

meshless analysis procedure for integrated design and analysis.

D. Natekar, S. Zhang,and G. Subbarayan. CAD, 36(5): 473-‐-‐486,

2004.

• T.J.R. Hughes, J.A. Coerell, and Y. Bazilevs. Isogeometric

analysis: CAD, ﬁnite elements, NURBS, exact geometry and mesh

reﬁnement. CMAME, 194(39-‐41):4135–4195, 2005.

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3

Isogeometric analysis

!

• P. Kagan, A. Fischer, and P. Z. Bar-‐Yoseph. New B-‐Spline Finite

Element approach for geometrical design and mechanical

analysis. IJNME, 41(3):435–458, 1998.

• F. Cirak, M. Or(z, and P. Schröder. Subdivision surfaces: a new

paradigm for thin-‐shell ﬁnite-‐element analysis. IJNME, 47(12):

2039–2072, 2000.

• Construc9ve solid analysis: a hierarchical, geometry-‐based

meshless analysis procedure for integrated design and analysis.

D. Natekar, S. Zhang,and G. Subbarayan. CAD, 36(5): 473-‐-‐486,

2004.

• T.J.R. Hughes, J.A. Co[rell, and Y. Bazilevs. Isogeometric

analysis: CAD, ﬁnite elements, NURBS, exact geometry and

mesh reﬁnement. CMAME, 194(39-‐41):4135–4195, 2005.

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1

!

R. Alwood, G. Cornes, “A polygonal finite element for plate bending

problems using the assumed stress approach”, International Journal for Numerical Methods in Engineering, 1(2): 135–149, 1969.

!

T. Belytschko, Y. Lu, L. Gu, “Element-free Galerkin methods”, International Journal for Numerical Methods in Engineering, 37: 229–256, 1994.

!

T. Belytschko, T. Black, “Elastic crack growth in finite elements with minimal remeshing”, International Journal for Numerical Methods in Engineering, 45: 601–620, 1999.

!

R. Mittal, G. Iaccarino, “Immersed boundary methods”, Annual Review of Fluid Mechanics, 37: 239–261, 2005.

!

G.R. Liu, K. Dai, T. Nguyen, “A smoothed finite element method for mechanics problems”, Computational Mechanics, 39: 859–877, 2007.

!

LB da Veiga, F Brezzi, LD Marini - Virtual Elements for linear elasticity

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1

F. Rizzo, “An integral equation approach to boundary value problems of classical elastostatics”, Quart. Appl. Math, 25(1): 83–95, 1967.

!

R. Glowinski, T. Pan, J. Periaux, “A fictitious domain method for Dirichlet problem and applications”, Computer Methods in Applied Mechanics and Engineering, 111(3-4): 283–303, 1994.

!

C. Song, J. Wolf, “The scaled boundary finite-element method–alias consistent infinitesimal finite-element cell method–for elastodynamics”, Computer

Methods in Applied Mechanics and Engineering, 147(3): 329–355, 1997.

!

R. Simpson, S. Bordas, J. Trevelyan, T. Rabczuk, “A two-dimensional

isogeometric boundary element method for elastostatic analysis”, Computer Methods in Applied Mechanics and Engineering, 209-212: 87–100, 2012.

!

Isogeometric boundary element analysis using unstructured T-splines

MA Scott, RN Simpson, JA Evans, S Lipton, SPA Bordas, TJR Hughes, TW Sederberg Computer Methods in Applied Mechanics and Engineering, 2013.

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1

!

E. Saiki, S. Biringen, “Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method”, Journal of Computational

Physics, 123(2): 450–465, 1996.

!

H. Johansen, P. Colella, “A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains”, Journal of Computational

Physics, 147(1): 60–85, 1998.

!

T. Ye, R. Mittal, H. Udaykumar, W. Shyy, “An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries”, Journal of Computational Physics, 156(2): 209–240, 1999.

!

M. Moumnassi, S. Belouettar, E. Bechet, S. Bordas, D. Quoirin, M. Potier Ferry, “Finite element analysis on implicitly defined domains: An accurate

representation based on arbitrary parametric surfaces”, Computer Methods in Applied Mechanics and Engineering, 200(5): 774–796, 2011.

Adaptive methods for multiscale fracture and surgical simulation