Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Future alternatives to finite elements for geotechnical modelling 1 Charles Augarde

Workshop- Numerical modelling in geomechanics Newcastle - April 2009

Future alternatives to finite elements for geotechnical

modelling

1

Charles AugardeMechanics Group

School of Engineering Durham University

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Aim & Outline

Aim: introduce mesh-free alternatives to finite elements

I. Finite element methods – some drawbacks

II. Alternatives through couplings

III. Meshless methodsa. Basics – how they work

b. Problems with meshless methods

c. Meshless methods for geomechanics to date

d. A new coupled meshless method for geomechanics

IV. Conclusions

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Widely and routinely used in geomechanics. Plenty of evidence for that in today’s presentations

Some history

•Simpson. "Finite elements applied to problems of plane strain deformation in soils" PhD thesis, University of Cambridge 1973

•Zienkiewicz,Pande & Naylor at Swansea

•Smith at Manchester (...gap...)

•Potts and Zdravković book on geotechnical FE (1999)

Finite element methods

A number of commercial packages for geomechanics are now available:

Plaxis, Oasys SAFE, Abaqus, LS-Dyna and others

2D robust & reliable

3D ???

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The basics you know

You need a mesh – but structured or unstructured?

Choose elements – but some are better than others

Materials: deals well with nonlinear material models, can deal with mixed problems, e.g. consolidation. Many other variants, e.g. thermo-hydro models, unsaturated soils but restricted to academia and in-house codes, I think.

Soil-structure interaction: fine, can include tunnel linings, footings, nails, reinforcement etc.

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The basics you may have forgotten: the maths needed to understand what comes later

Approximation of displacements is based on the use of interpolation functions (shape functions)

Nodal displacements are the unknowns we seek. Shape functions allow us to write down how things vary throughout elements

The stiffness matrix is found by expressions integrated over each element (that’s why we need a continuous expression which the shape functions give us).

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Some drawbacks of finite elements

The need to generate a mesh

In 2D no problem, Delaunay triangulation, advancing front

In 3D potential future problems

– Multi-stage analyses necessary for non-linear materials

– Adaptive analyses – where the mesh is changed to reduce error or to take account of changing geometry (e.g. large deformations/strains)

– Ambition for 3D ever increasing, billions of nodes in a model?

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If we don’t use FEs what do we use?

What other robust options are there?:

• Finite difference (FD), e.g. FLAC

• Discrete element (DE) modelling, e.g. Itasca PFC

• Boundary elements (BE)

• Meshless methods

What is needed for your problem? Leads us to coupled methods

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FE/BE

Coupling – direct or by DDElleithy et al. (2001)

Structure=FEFoundation= BEWang (1992)

Vibration from trains in tunnels (Shell FE for the tunnel, BE for the surrounding soil)Andersen & Jones (2006)

Alternatives through coupling

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FE/BE

Tunnel is FE, surrounding ground is BE

Beer (2000), Swoboda et al. (1987)


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DE/BE

modelling hydro-mechanical behaviour of jointed rockWei & Hudson (1998)


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Infinite and finite BE

Beer et al. (2003)


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Meshless methods

Appear to be the most solid choice for a future competitor to finite elements

Able to do everything finite elements can do. Not limited to certain problems/materials

Why bother? No mesh is needed - only a distribution of nodes

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Meshless methods

There are many proposed meshless methods for solid mechanics (and hence geomechanics) out there

Here I concentrate on those which use a certain approach for their shape functions namely …

… moving least squares (MLS): the most popular methods used in solid mechanics take this approach

The key is approximation rather than interpolation

We will see this causes problems later on

Element-free Galerkin

Meshless local Petrov-Galerkin

Reproducing kernel particle

Natural element

hp-clouds

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Moving least squares

Linear interpolation

Quadratic interpolation

Least squares (linear basis)

Moving least squares (linear basis)

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Moving least squares shape functions

But, shape functions do not possess the (FE) property of equalling one at the node with which they are associated (the “delta” property)

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x

Shape function for node i

node i

“Support” of node i

MLS shape functions in 1D

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x

In 2D these supports become circular

MLS shape functions in 1D

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Meshless methods

domain boundary

Zones around each node where weighted shape functions are non-zero

Test radius shown shaded around one node, contributions to integration at node 1 occur where test radius overlaps with non-zero shape functions from nodes

1

2

3

4

5

2

3

4

5

1

Once we have shape functions things proceed much as finite elements

Except that we have no elements over which to carry out integrations to form the terms in the stiffness matrix

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Meshless methods – some problems

(This is the, “however”, slide)

Essential boundary conditions (e.g. points fixed to supports) cannot be imposed directly as with FE

xi x

iuiu

xuh

• At a node and fixities must be imposed on

• So we cannot simply set values of as we would do in the FEM

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Complexity

Even though we do not need a mesh we still need to know the influential neighbours of nodes

domain boundary

1

2

3

4

5

Point under consideration

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Too many possible tweaks for the user

Choice of size of shape function support? How far away from the node does it have influence

Weight function. How rapidly does the influence of a node diminish as you move away from it?

What distribution of nodes? Uniform nodal arrangements sometimes hide problems

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Changing support

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Increasing size of nodal support

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

ri = 3.0ri = 1.125

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

ri = 2.0

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Meshless methods

How soon might we sort these problems out? Active research at the moment but mainly generic solid mechanics

Will meshless methods ever challenge FE methods in geomechanics? Possibly: because of the particular problems we wish to model: 3D, non-linear materials, large deformations and large strains

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Meshless methods are starting to make an appearance in geomechanics research

Praveen Kumar et al. (2008) use the EFG method to model unsaturated flow through a rigid porous medium with applications in contaminant transport modelling.

Ferronato et al. (2007) presents a model of axisymmetric poroelasticity for prediction of subsidence over compacting reservoirs using the MLPG method

Kim & Inoue (2007) present modelling of 2D seepage flow through porous media using the basic EFG method

Vermeer et al. (2008) provide a range of convincing examples of the use of a Material Point Method (MPM )for geotechnics,

Currently

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A new hybrid meshless method for geomechanics

Research project underway at Durham to develop a coupled meshless method for geomechanics including a new large strain anisotropic plasticity model

Motivation – unbounded domains in geomechanics

What sorts of problems? Large deformation, large strain, materially nonlinear.

Applications? CPT, piles, NATM ...

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Cone penetrometer

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Cone penetrometer

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Cone penetrometer

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Cone penetrometer

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Layered soils

Cone penetrometer

Large strain plasticity

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Meshfree method

Scaled Boundary method

+

Ground surface Tunnel

Meshfree zone

Infinite scaled boundary zone

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Permits non-linear material modelling

Boundary difficulties

Removes need for mesh (obviously) although some meshless methods require integration cells

Meshless method

Scaled Boundary method

+

Does not permit non-linear material

Models infinite boundaries

Efficient

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Currently ...

Elasto-plasticity implemented in the meshless region

How to allow meshless region to “evolve” during an analysis.

Mapping SB region values to revised meshless zone

Coupling a large strain meshless region to a small strain scaled boundary region

A new hybrid meshless method for geomechanics

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Conclusions

Finite elements are now dominant but will this remain the case?

Who will make the move to meshless? Plaxis are working on a moving point method (MPM) to commercialise

Clear role for researchers to sort out current problems and present a robust formulation

Role for developers: to become acquainted with meshless methods and how they differ from FE methods

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Acknowledgements

Dr Claire Heaney, Research Associate

Xiaoying Zhuang, PhD student

Will Coombs, PhD student

Prof. Roger Crouch, Professor of Civil Engineering

and other members of the mechanics group at Durham

Thank you for listening

Papers available at www.dur.ac.uk/[email protected]

Documents

Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Future alternatives to finite elements for geotechnical modelling 1 Charles Augarde