Future alternatives to finite elements for geotechnical modelling

1. Future alternatives to finite elements for geotechnical modelling Charles Augarde Mechanics GroupSchool of Engineering Durham University

2. Aim & Outline • Aim: introduce mesh-free alternatives to finite elements • Finite element methods – some drawbacks • Alternatives through couplings • Meshless methods • Basics – how they work • Problems with meshless methods • Meshless methods for geomechanics to date • A new coupled meshless method for geomechanics • Conclusions

3. Finite element methods • 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) A number of commercial packages for geomechanics are now available: Plaxis, Oasys SAFE, Abaqus, LS-Dyna and others 2D robust & reliable 3D ???

4. 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.

5. 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).

6. 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?

7. 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

8. Alternatives through coupling 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)

9. Alternatives through coupling FE/BE Tunnel is FE, surrounding ground is BE Beer (2000), Swoboda et al. (1987)

10. Alternatives through coupling DE/BE modelling hydro-mechanical behaviour of jointed rock Wei & Hudson (1998)

11. Alternatives through coupling Infinite and finite BE Beer et al. (2003)

12. 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

13. Meshless methods Element-free Galerkin Meshless local Petrov-Galerkin Reproducing kernel particle Natural element hp-clouds • 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

14. Moving least squares Least squares (linear basis) Moving least squares (linear basis) Quadratic interpolation Linear interpolation

15. 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)

16. MLS shape functions in 1D node i Shape function for node i “Support” of node i

17. MLS shape functions in 1D In 2D these supports become circular

18. Meshless methods

19. 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

20. Meshless methods – some problems • Complexity • Even though we do not need a mesh we still need to know the influential neighbours of nodes

21. 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

22. Changing support ri = 3.0 ri = 1.125 ri = 2.0 Increasing size of nodal support

23. 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

24. Currently • 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 axisymmetricporoelasticity 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,

25. 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 ...

26. Cone penetrometer

27. Cone penetrometer

28. Cone penetrometer

29. Cone penetrometer

30. Cone penetrometer Large strain plasticity Layered soils

31. Meshfree method Scaled Boundary method +

32. Meshless method Scaled Boundary method + Permits non-linear material modelling Boundary difficulties Removes need for mesh (obviously) although some meshless methods require integration cells Does not permit non-linear material Models infinite boundaries Efficient

33. A new hybrid meshless method for geomechanics 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

34. 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

35. 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 listeningPapers available at www.dur.ac.uk/charles.augardecharles.augarde@dur.ac.uk