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Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Future alternatives to finite elements for geotechnical modelling 1 Charles Augarde Mechanics Group School of Engineering Durham University

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Aim & Outline Aim: introduce mesh-free alternatives to finite elements I.Finite element methods – some drawbacks II.Alternatives through couplings III.Meshless methods a.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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Finite element methods 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Finite element methods 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 FE/BE Coupling – direct or by DD Elleithy et al. (2001) Structure=FE Foundation= BE Wang (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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 FE/BE Tunnel is FE, surrounding ground is BE Beer (2000), Swoboda et al. (1987) Alternatives through coupling

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 DE/BE modelling hydro-mechanical behaviour of jointed rock Wei & Hudson (1998) Alternatives through coupling

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Infinite and finite BE Beer et al. (2003) Alternatives through coupling

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Moving least squares Linear interpolation Quadratic interpolation Least squares (linear basis) Moving least squares (linear basis)

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Shape function for node i node i “Support” of node i MLS shape functions in 1D

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 In 2D these supports become circular MLS shape functions in 1D

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Meshless methods 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Meshless methods – some problems Complexity Even though we do not need a mesh we still need to know the influential neighbours of nodes

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Meshless methods – some problems 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Changing support Increasing size of nodal support r i = 3.0 r i = 1.125 r i = 2.0

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Cone penetrometer

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Cone penetrometer

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Cone penetrometer

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Cone penetrometer

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Layered soils Cone penetrometer Large strain plasticity

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 Meshfree method Scaled Boundary method +

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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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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∂ Workshop- Numerical modelling in geomechanics Newcastle - April 2009 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/charles.augarde charles.augarde@dur.ac.ukwww.dur.ac.uk/charles.augarde

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