Issues with the Material Point Method for the modelling of geotechnicalprocesses, and how to solve them

C.E. Augarde, Y. Bing, T.J. Charlton, W.M. Coombs, M. CortisDepartment of EngineeringDurham University, UK

M.J.Z. Brown, A. Brennan, S. Robinson.Civil EngineeringUniversity of Dundee, UK.

ABSTRACT: The Material Point Method (MPM) for solid mechanics was first proposed by Sulsky and co-workers in the 1990s. Since then it has been developing a growing band of followers not least because of itsability to handle large deformation problems with ease. This feature has more recently come to the notice ofgeotechnical researchers who have plenty of problems to solve involving large deformations. It is clear fromrecent publications, however, that many geotechnical researchers have found difficulties with the use of theMPM in a number of areas. In this paper we visit three of these problem areas and highlight solutions we havedeveloped. It is to be hoped that this can remove some of the roadblocks to the use and further development ofthe MPM for geotechnical problems in future.

1 INTRODUCTION

The problems that have to be solved by geotechni-cal engineers are wide-ranging, covering tunnelling,foundations, slopes and many other areas, howeverthere are common features of these problems that pro-vide challenges to numerical modellers in particu-lar. Principally these are material and geometric non-linearity. The former has been recognised as crucialfor accurate modelling from before the developmentof computational geotechnics, in the recognition thatplasticity is a vital part of any constitutive model forsoil. Indeed material non-linearity is likely to featurein the majority of the papers at this conference. Ge-ometric non-linearity has received less attention todate largely because material non-linearity is crucialto a wider range of geotechnical problems than geo-metric non-linearity. When using this term, we meanthe ability to model large deformations and the useof strain definitions which are no longer linear withdisplacement, as opposed to infinitesimal strain mea-sures for standard analyses.

The finite element method (FEM) remains themethod of choice for most geotechnical numericalmodelling, and with good reason. However, there areissues with its use for the class of problems mentionedabove, i.e. in the modelling of large deformation prob-lems. If large deformations are to be modelled then

any mesh-based method (the FEM included) will re-quire an update of the mesh during a stepped non-linear solve to avoid the inaccuracies associated withdistorted elements. Any change of mesh will then re-quire a mapping of state variables from the old to thenew mesh. While both of these actions bring poten-tial errors the FEM has been successfully adapted forlarge deformation problems via modifications suchas the Arbitrary Lagrangian Eulerian (ALE) methodand there are other examples which do much thesame thing, such as the Coupled Eulerian-Lagrangian(CEL) method; indeed some of these techniques areavailable in commercial software such as ABAQUS,and have been used for geotechnical problems (e.g.Kim et al. (2015)). Having said this, there is a schoolof thought that says adherence to mesh-based meth-ods places a restriction of the development of numeri-cal modelling, and there are many examples of mesh-free methods being developed, although most incur agreater computational cost to the FEM for standardproblems at present (Heaney et al. 2010).

In this paper we are concerned with a relative new-comer to computational geotechnics, called the Ma-terial Point Method (MPM) which seems to offer ad-vantages over the standard FEM for large deforma-tion problems, without some of the complexities ofrival approaches mentioned above. The MPM is ofkey and current interest in geotechnics having fea-

tured in Alonso’s 2017 Rankine Lecture, and its usein geotechnics having been the feature of a major con-ference in the same year (Anura3D MPM ResearchCommunity 2017). In this paper we highlight a num-ber of issues that geotechnical modellers might facewhen using the MPM and present recent solutions de-veloped by the authors.

2 THE MATERIAL POINT METHOD

The MPM for solid mechanics was developed from anearlier method for fluids (the FLIP method) by Sulskyand co-workers (Sulsky et al. 1994). It is usually de-scribed in an explicit form where it is used for time-stepping analyses of problems with inertia and accel-eration. It is however equally possible to formulatethe MPM in implicit form (Guilkey and Weiss 2003)for quasi-static problems (usually of most interest ingeotechnics) where to deal with non-linearity a totalapplied action is split into a number of substeps, inwhich each requires the solution of a linear systeminvolving a stiffness matrix and an unknown vector ofdisplacements.

The method is often referred to as an Eulerian-Lagrangian method (Muller & Vargas 2014) but thisis not really correct; there are aspects of the methodthat make it look that way. In fact the calculations arejust Lagrangian. In the MPM, a problem domain isdefined with a set of material points (MPs, sometimesalso referred to as particles). These MPs carry allthe information relating to that location in the prob-lem domain throughout a calculation, i.e. total dis-placement, strain, stress and if necessary other statevariables required by a constitutive model. All cal-culations are however carried out on a finite elementmesh (often referred to as a background grid) to whichdata are mapped back and forth from the materialpoints. The feature in the MPM that is most attrac-tive to those wishing to model large deformation isthat the deformation of the problem domain is repre-sented at the material points only and the backgroundgrid can be discarded after each time step. This meansnever having to calculate on a distorted grid, and alsomeans that a regular structured grid can be reusedeach time, avoiding the overhead linked to unstruc-tured mesh generation. A secondary but important as-pect of the MPM, from the implementation point ofview, is that much finite element technology can beseamlessly transferred to the MPM, especially itemssuch as constitutive models and basis functions, re-ducing the overhead of code development and, per-haps, providing some confidence in the use of themethod.

An important problem dealt with in the literaturewith the standard MPM is grid-crossing instability,which is a numerical artefact linked to a deformationpattern in which a material point leaves one grid ele-ment and enters another. In the standard MPM, wheredomain volume (or mass) is concentrated at a mate-

rial point, this leads to a sudden change in stiffnessof a grid element which causes non-physical numer-ical effects. This problem has been tackled by de-veloping variants of the standard MPM where eachmaterial point carries a spatially-defined and poten-tially deforming volume (or mass) with it. Methodsinclude the Generalised Interpolation MPM (Barden-hagen and Kober 2004, Charlton et al. 2017) andCPDI methods (Sadeghirad et al. 2011, Nguyen et al.2017). While these variants effectively reduce grid-crossing instabilities they lead to additional complexi-ties in calculations (especially the CPDI methods) andall suffer from the same issues which are covered inthis paper.

3 THE MPM FOR GEOTECHNICS

It is clear there is keen and current interest in the useof the MPM in geotechnics as evidenced by an in-creasing number of papers describing its use for vari-ous geotechnical problems, for instance soil-structureinteraction (Ma et al. 2014) and slope stability (Zabalaand Alonso 2011), and a recent conference was de-voted to MPM and geotechnics (Anura3D MPM Re-search Community 2017). In addition there are a num-ber of survey papers, e.g. Sołowski and Sloan (2015).The MPM has also been developed to model coupledproblems important in geotechnics, for example Jas-sim et al. (2013). The following sections cover threeissues affecting use of the MPM or its variants forgeotechnical analysis. The discussion relates primar-ily to our experiences with implicit MPM codes forelastic and elasto-plastic statics problems rather thanexplicit MPM with dynamics.

3.1 Essential Boundary Conditions

Essential (or Dirichlet) boundary conditions (BCs)are necessary to fully define a boundary value prob-lem. They do not (usually) enter the weak form de-scription of the problem but are subsidiary conditionsthat have to be incorporated. In the standard FEM, es-sential BCs are usually imposed directly, i.e. those de-grees of freedom with essential BCs are treated dif-ferently, or the stiffness matrix is amended, as de-scribed in standard texts, e.g. Potts and Zdravkovic(1997). This is possible due to the Kronecker deltaproperty of standard FE shape functions which de-livers an interpolation of nodal unknowns. In con-trast most weak-form based meshless methods, suchas the element-free Galerkin method (Belytschkoet al. 1994) have basis functions, often derived frommoving least squares, that do not possess the Kro-necker delta property and hence lead to approxima-tions rather than interpolations. Consequently essen-tial BCs have to be imposed indirectly (Fernandez-Mendez and Huerta 2004).

The MPM (in all of its different flavours as men-tioned above) has a problem with essential BCs of a

(a) (b)

(c)

footing

(d)

Figure 1: (a) slope stability problem geometry; (b) MPM solution requiring collinear grid and domain bound-aries; (c) ideal MPM model with no requirement of collinear gird and domain geometry; (d) deformed domain.Red dots are the material points in each case.

slightly different nature. Consider Fig. 1 which showsa typical 2D plane strain discretisation of a slope sta-bility problem, in which the vertical boundaries aresubject to an essential boundary condition of zerodisplacement normal to the boundary while the bot-tom horizontal boundary is subject to an essentialBC of full fixity or zero normal displacement. Pro-viding the essential boundaries align exactly with thebackground grid (Fig. 1(b)), these conditions can beapplied in the MPM directly just as in the standardFEM. All calculations are carried out on the back-ground grid and therefore as the domain boundarycoincides with edges of elements in the backgroundgrid for these three boundaries, there is exact co-incidence with imposing essential BCs on the grid.However, if the problem domain boundary does notcoincide with edges of elements in the backgroundgrid, an essential boundary condition cannot be ap-plied this way. The example so far has concentratedon essential BCs which act as fixities in a boundaryvalue problem, however geotechnical problems oftenrequire non-homogeneous essential BCs such as forthe modelling of the footing shown at the top of theslope in Fig. 1. In this case, while it might be possibleto arrange for the initial grid to align with the startingposition of the footing, subsequent loading steps (inthis case increments of applied displacement to thefooting) would require grids to follow the predictedmovements exactly, thus losing a key advantage of theMPM in that a regular grid, not connected to the do-

main geometry, can be used. While we now present asolution to this problem, in fact, Fig. 1 will be used toillustrate the other problems with the MPM on whichthis paper focusses in later sections. The issue out-lined above is linked to a large body of emergingliterature in computational solid mechanics on non-matching mesh methods which, as the name suggests,are numerical methods where solutions are calculatedwithout matching the discretisation mesh to the prob-lem domain, examples include immersed FE meth-ods and the Finite Cell method (Ramos et al. 2015,Schillinger et al. 2012). In the case of the MPM, wehave developed a new method for the imposition ofessential BCs which removes any need for alignmentof mesh edges and problem domain boundaries. It isbased on Kumar and co-workers’ implicit boundaryapproach (Kumar et al. 2008) and can be viewed asa form of penalty method of applying the essentialBCs. In the method, essential boundaries are definedby signed distance functions φj where φj < 0 indi-cates the exterior and φj > 0 the interior of the prob-lem domain for the jth boundary. Essential boundary(or Dirichlet) functions dj(φj) are then defined foreach boundary, which are equal to zero on the bound-ary and rise to unity a small distance δ on the domainside of the boundary. dj are often simple discontinu-ous quadratic functions in φj . An example is shown

in Fig. 2 where the Dirichlet function is given by

d =

0, φ < 0

1−(1− φ

δ

)20 ≤ φ ≤ δ

1 φ > δ

. (1)

At points where more than one essential boundary isactive, i.e. at a domain corner, then the product of thed(φj) forms the essential boundary function and ingeneral we write the net essential boundary at a pointas

Dk =∏j

d(φj), (2)

where k refers to the component of displacement de-fined at that boundary. Dk are then the componentsof the diagonal matrix [D] used to redefine the trialfunctions for the grid elements as

u′ = [D]u+ ua (3)

where [D] = diag(D1, . . . ,Dnd) and nd is the dimen-

sionality. In Eqn 3, u is the standard approximationfor displacement in a finite element while ua is theessential boundary condition (i.e. zero for a fixed de-gree of freedom or non-zero for a prescribed displace-ment). Within the narrow band adjacent to the implicitboundary, the first term in Eqn 3 will be suppressedvia the matrix [D], enforcing the essential boundarycondition in the second term. Substituting these trialfunctions and some suitable test functions into thestandard weak form for equilibrium leads to expres-sions for the stiffness matrices of elements contain-ing essential boundary conditions. Fig. 3 shows a gridelement (cell) cut by an essential boundary. For thiselement the net stiffness matrix would be

[kE] = [K1] + ([K2] + [K2]T ) + [K3], (4)

where [K1] is the standard finite element stiffness ma-trix for the part of the element occupied by the prob-lem domain and is obtained through the summationof the MP contributions, while [K2] and [K3] containDirichlet functions and their derivatives. The addi-tional stiffness matrices [K2] and [K3] are effectivelypenalty terms imposing the essential BC crossing theelement and their components are calculated by nu-merical integration in the bandwidth δ as shown inFig. 3. It is straightforward to implement essentialBCs which cut an element at an angle with respect tothe element global coordinates; a transformation ma-trix is applied to components of the matrices forming[K2] and [K3]. An example of the use of the approachis given in Fig. 4 where a rigid footing penetrates aunit distance into an elastic square domain of side 3units. The roller sides and base are imposed as im-plicit homogeneous essential BCs while the footingis an implicit non-homogeneous essential BC. Fulldetails of this approach for implementation of essen-tial boundaries in the MPM are given in Cortis et al.(2017).

Figure 2: A Dirichlet function in 1D.

Figure 3: A single grid element (cell) crossed by anessential boundary condition

3.2 Tracking domain boundaries

Fig. 1(d) serves to illustrate another problem withthe MPM for geotechnical problems, that of track-ing evolving boundaries to a problem domain and ap-plication of traction (Neumann) boundary conditions.The former is particularly troublesome given thatgeotechnical engineers would like to use the MPM forproblems in which this can be a crucial output, e.g. akey prediction for long runout landslides is the finalsurface profile of the disturbed material (e.g. Wanget al. (2016)). To date, researchers wishing to trackfree surfaces in an MPM analysis have been limited todetermining the location of the edges of material pointindividual volumes which exist in GIMP and CPDImethods but not in the standard MPM. The best thatthese can deliver are piecewise linear edges (CPDI2)or stepped approximations (GIMP and CPDI1). As re-gards the application of traction boundary conditionsthis has been attempted in the past by placing loadsat material points, which is clearly an inaccurate rep-resentation, particularly when wishing to exploit thecapabilities of the MPM for large deformation prob-lems.

In recent research undertaken at Durham, Bing(2017) describes a new method which can deliverboth accurate tracking of domain boundaries and ap-plication of surface tractions in any variant of theMPM for arbitrary large deformations. A local cu-bic B-spline interpolation is used to represent theboundary based on a set of defined boundary mate-rial points. These can be additional material points(with near-zero volumes) placed on the boundaries oran outer layer of standard material points in the phys-

uy=0.05

3

3

x

y

(a) (b)

Figure 4: Rigid footing penetrating into an elastic domain: (a) problem definition, (b) deformation predictionusing the MPM. (From Cortis et al. (2017))

ical domain. Spline segments are fitted between adja-cent sampling points to calculate the B-spline controlpoints and a suitable knot vector determined complet-ing the description of the B-spline curve. Althoughfitting a curve globally to the boundary (Piegl & Tiller1997) would result in higher continuity, it would notbe capable of reproducing sharp corners which areimportant features in many geotechnical problem do-mains. Local fitting, as used here, constructs curvesin a piecewise fashion so that only local data are usedat each step, so a fluctuation in data only affects thecurve locally. As regards spline order, a local cubicinterpolation has a simpler formulation than the localquadratic interpolation (Piegl & Tiller 1997) and nospecial cases or angle calculations are needed.

Once an accurate boundary representation is inplace we can consider how to apply boundary con-ditions. Essential boundary conditions on B-splineboundaries can be imposed with the implicit bound-ary method described above, with a few minor alter-ations. The application of tractions to a B-spline de-fined boundary is also straightforward with the onlycomplexity arising in the integration required, as willbe outlined now. In the standard FEM, nodal forcesf t consistent with a surface traction t on a sur-face dΩ are obtained as

f t =

∫∂Ω

[M ]TtdΩ, (5)

where [M ] contains the standard finite element shapefunctions. So for the case of the B-spline boundarieswe have to consider how to carry out this integration.A pth-degree B-spline curve can be integrated nu-merically by using (p− 1)th order Gauss quadrature.However, the local coordinate of 1D Gauss quadra-ture has a range of [−1,1], whereas, the local coor-dinate of a B-spline curve has positive values only,therefore mapping between these two systems is re-quired to complete the integration (in addition to the

Figure 5: Physical and parametric spaces for numeri-cal integration of boundary conditions.

standard map between the local and global coordi-nates, or physical space for isoparametric FEs). To al-low for this, an additional space, called the parent do-main is introduced over which quadrature takes place(Hughes, Cottrell, & Bazilevs 2005). Fig. 5 shows anillustration of the three spaces: the physical space,x, the parametric space, ξ, and the parent domain,ξ. In the physical space, the boundary geometry is de-fined in global coordinates. The parametric space con-tains the knots (local coordinates) which run along thecurve and the parent domain is simply a local sys-tem where ξ ∈ [−1,1] on which numerical integra-tion is performed. To carry out the integration, the B-spline curve segment is pulled back from the physicalspace to the parametric space, i.e. the local coordi-nates (ξj and ξj+1) of the start and the end point of thesegment are identified by using their global coordi-nates. A linear transformation between the parent do-main, ξ ∈ [−1,1], and the parametric space, [ξj, ξj+1]maps the locations of Gauss points between these twospaces. In order that the integration can be completed,a Jacobian mapping is required between the parentand physical spaces between the Gauss quadraturelengths, which are defined over the local coordinatesξ ∈ [−1,1], and the physical space. Due to the use ofa two local coordinate systems, the Jacobian contains

Figure 6: Cantilever beam geometry.

(a) Final deformation of the ma-terial points.

(b) Boundary visualisation aftereach load step.

Figure 7: Cantilever beam deformation.

two components

JB =

dC

dξ

=

dC

dξ

dξ

dξ(6)

where C is the B-spline curve defining a segment inthe physical boundary. Applying Gauss quadrature to(5), we obtain

f t ∼=ngp∑i=1

[Mi]Tti ||JBi||wi, (7)

where ngp is the number of Gauss points used to in-tegrate over the segment within the background gridelement, wi is the weight associated with Gauss pointi, ||(·)|| denotes the L2 norm of (·) and in this casethe L2 norm of the boundary Jacobian, JB, whichmaps the length of the boundary between the parentand physical spaces.

To illustrate the use of this method Fig. 6 showsan elastic cantilever beam of length 10 m and depthof 2 m is subjected to a constant pressure of 1500Pa applied along the top boundary; a traction whichremains perpendicular to the top surface of the can-tilever throughout the analysis. A background gridwith 1.5 m by 1.5 m elements is used, and the problemdomain is discretised using 896 uniformly distributedstandard material points. The outer layer of the mate-rial points are identified as the problem boundarieswhich are approximated using B-splines. The ini-tial discretisation and the final deformed cantileverbeam are shown in Fig. 7a. The advantage of this ap-proach is that the boundaries can be tracked after eachload step without plotting out all the material points(see Fig. 7b) and the deformed shape appears to besuccessfully captured by the B-spline approximation.Further examples and full details of this approach to

accurately track domain boundaries and apply trac-tion boundary conditions in the MPM are given inBing et al. (2018) and Bing (2017).

3.3 Volumetric Locking

In the MPM the material points are integration pointsfor the calculation of grid element stiffness, and sincethey are allowed to convect through the grid, they can-not provide the accuracy of an equivalent number ofproperly placed Gauss points. Consequently it is nor-mal to use many more material points per grid ele-ment than the number required for accurate numericalintegration. Combining this with the fact that the gridis usually comprised of low order elements, e.g. bilin-ear quadrilaterals in 2D, means that the method is sus-ceptible to volumetric locking (resulting in over-stiffbehaviour) when modelling near-incompressible ma-terials. This volumetric locking is caused by excessiveconstraints placed on an element’s deformation. Thatis, the constitutive model will require near-isochoricbehaviour at the integration (or material) point’s lo-cation within the element and each of these pointsplaces a constraint on the deformation of the element.A common technique to avoid volumetric locking infinite element methods is to use higher order elementswith reduced Gaussian integration. However, this isnot viable in MPMs since it is not known how manymaterial points will be in any given element at a givenload step. In the context of finite deformation solidmechanics, a number of formulations have been pro-posed to overcome volumetric locking in finite ele-ments. (A review can be found in de Souza Neto et al.(2008)). The issue of volumetric locking in MPMs hasreceived little attention to date with the notable excep-tion of Mast et al. (2012) who investigated the issue ofkinematic (volumetric and shear) locking in the stan-dard MPM and developed a complex multi-field vari-ational principle based approach which introduces in-dependent approximations for the volumetric and thedeviatoric components of the strain and stress fields.

For the MPM we have instead adopted the Fapproach hitherto applied to the standard FEM byde Souza Neto et al. (1996) for the following rea-sons: (i) unlike mixed approaches it does not intro-duce any additional unknowns into the linear system,(ii) it is simple to implement within existing finite el-ement codes (and therefore also the MPMs), (iii) theapproach can be used with any constitutive model and(iv) it does not introduce any additional tuning param-eters into the code. In the F approach applied to stan-dard FEM, the volumetric and deviatoric componentsof the deformation gradient are sampled at differentlocations. The deformation gradient becomes

Fij =

(det(F 0

ij)

det(Fij)

)1/nD

Fij, (8)

where nD is the number of physical dimensions and

F 0ij is the deformation gradient obtained from the de-

formation field at the centre of the element. Thereforethe volumetric component of the deformation gradi-ent for all of the Gauss points within an element isobtained from a single point, thus relaxing the vol-umetric constraint on the element when the materialbehaviour is near incompressible.

For the MPM, where we have large deformations,we adopt the incremental equivalent of (8), giving theF deformation gradient increment as

∆Fij =

(det(∆F 0

ij)

det(∆Fij)

)1/nD

∆Fij, (9)

where ∆F 0ij is the volumetric component of the de-

formation gradient increment. It is straightforward tomodify the standard material point method by replac-ing ∆Fij with ∆Fij in the finite deformation formula-tion. This is because the shape functions are directlyadopted from the finite element basis. However, it ismore appropriate to use the geometric centre of thematerial points located within a given finite elementrather than the centre of the element. This is due totwo key reasons:

1. when a single material point is used to integratethe background grid cell the F deformation gra-dient, (9), equals the standard deformation gra-dient; and

2. when a background grid cell is only partiallyfilled with material points the volumetric be-haviour is centred on the physical region.

To demonstrate the performance of the MPM withthe F approach results are presented for the analy-sis of a smooth square rigid footing bearing onto a3D weightless elasto-plastic domain. Due to symme-try only a quarter of the physical problem is mod-elled and the footing has a half width of 0.5 m andthe simulated domain is 5 m in length in each di-rection. The same material properties were adoptedas (de Souza Neto et al. 2008) for their plane strainanalysis of a rigid footing. The smooth footing wasdisplaced vertically (z-direction) by 0.002m over 200loadsteps and roller boundary conditions were im-posed on the sides and the base of the domain. All ofthe boundary conditions were imposed using the im-plicit boundary method discussed above. A relativelycoarse regular background grid of tri-linear hexahe-dral elements with h = 0.2 m was used to analysethe problem and the physical domain was discretisedusing 8 standard material points per background gridcell (125,000 material points in total). The force ver-sus displacement response for the standard and F ma-terial point methods are shown in Fig. 8. The standardformulation locks and predicts an over-stiff responsewhereas the F formulation reaches a limit load, asexpected for this type of analysis. Due to the smallimposed displacement, material points do not cross

Figure 8: 3D footing: force displacement response forstandard and F MPMs. (From Coombs et al. (2018)).

.

Figure 9: 3D footing: minor principal stress for (i)standard and (ii) F MPMs. (From Coombs et al.(2018)).

between background grid cells and both formulationsgive a smooth response. The minor principal (mostcompressive) stress distribution at the end of the anal-ysis for the two formulations are shown in Fig. 9. Thestandard material point formulation contains spuriousstress oscillations caused by volumetric locking. Inparticular, the column of material points underneaththe footing oscillate between tensile and compressivestress states. The F formulation stress distributionshown in Fig. 9 (ii) demonstrates the correct com-pressive region underneath the footing, as shown bythe blue-shaded particles. Full details of this approachto deal with locking in the MPM and other variantssuch as the GIMP method are given in Coombs et al.(2018).

4 CONCLUSIONS

In this paper we have attempted to summarise a num-ber of issues with the use of the MPM that may af-flict geotechnical analyses more than other applica-tions for which it may be used. The literature to dateshows that while these challenges have often been ev-ident, many researchers have avoided tackling themhead-on, especially in the area of imposition of es-sential boundary conditions. We have presented four

issues but have also presented solutions developed inour research group at Durham University. It is to behoped that these solutions will provide useful to oth-ers attempting geotechnical modelling with the MPMin future.

ACKNOWLEDGEMENTS

Some of the work described in this paper has been un-dertaken for the UK EPSRC funded project Seabedploughing: modelling for infrastructure installation(EP/M000362/1 & EP/M000397/1). The thirds authorwas supported by an EPSRC DTA Studentship duringPhD study (grant ref EP/K502832/1) which has con-tributed to parts of this research.

REFERENCES

Anura3D MPM Research Community (2017). First Interna-tional Conference on the Material Point Method for Mod-elling Large Deformation and SoilWaterStructure Interac-tion. http://mpm2017.eu/home.

Bardenhagen, S. & E. Kober (2004). The generalized interpola-tion material point method. CMES- Computer Modeling inEngineering and Sciences 5(6), 477–495.

Belytschko, T., Y. Y. Lu, & L. Gu (1994). Element-free Galerkinmethods. International Journal for Numerical Methods inEngineering 37, 229–256.

Bing, Y. (2017). B-spline based boundary method for the mate-rial point method. MScR thesis, Durham University, UK.

Bing, Y., M. Cortis, T. Charlton, W. Coombs, & C. Augarde(2018). B-spline based boundary conditions in the materialpoint method. Computer Methods in Applied Mechanics andEngineering. Under review.

Charlton, T. J., W. M. Coombs, & C. E. Augarde (2017). iGIMP:An implicit generalised interpolation material point methodfor large deformations. Computers & Structures 190, 108–125.

Coombs, W., T. Charlton, M. Cortis, & C. Augarde (2018). Over-coming volumetric locking in material point methods. Com-puter Methods in Applied Mechanics and Engineering. Ac-cepted for publication.

Cortis, M., W. M. Coombs, C. E. Augarde, M. J. Brown,A. Brennan, & S. Robinson (2017). Imposition of essentialboundary conditions in the material point method. Interna-tional Journal for Numerical Methods in Engineering.

de Souza Neto, E., D. Peric, M. Dutko, & D. R. J. Owen (1996).Design of simple low order finite elements for large strainanalysis of nearly incompressible solids. International Jour-nal of Soilids and Structures 33, 3277–3296.

de Souza Neto, E., D. Peric, & D. Owen (2008). Computationalmethods for plasticity: Theory and applications. John Wiley& Sons Ltd.

Fernandez-Mendez, S. & A. Huerta (2004). Imposing essentialboundary conditions in mesh-free methods. Computer Meth-ods in Applied Mechanics and Engineering 193, 1257–1275.

Guilkey, J. & J. Weiss (2003). Implicit time integration for thematerial point method: Quantitative and algorithmic compar-isons with the finite element method. International Journalfor Numerical Methods in Engineering 57, 1323–1338.

Heaney, C., C. Augarde, A. Deeks, W. Coombs, & R. Crouch(2010). Advances in meshless methods with application togeotechnics. In Proc. NUMGE Trondheim, pp. 239–244.

Hughes, T., J. Cottrell, & Y. Bazilevs (2005). Isogeometric anal-ysis: Cad, finite elements, nurbs, exact geometry and meshrefinement. Computer Methods in Applied Mechanics andEngineering 194(39), 4135 – 4195.

Jassim, I., D. Stolle, & P. Vermeer (2013). Two-phase dynamicanalysis by material point method. International Journal forNumerical and Analytical Methods in Geomechanics 37(15),2502–2522.

Kim, Y., M. Hossain, D. Wang, & M. Randolph (2015). Numer-ical investigation of dynamic installation of torpedo anchorsin clay. Ocean Engineering 108(Supplement C), 820 – 832.

Kumar, A. V., S. Padmanabhan, & R. Burla (2008). Implicitboundary method for finite element analysis using non-conforming mesh or grid. International Journal for Numeri-cal Methods in Engineering 74(9), 1421–1447.

Ma, J., D. Wang, & M. Randolph (2014). A new contact algo-rithm in the material point method for geotechnical simu-lations. International Journal for Numerical and AnalyticalMethods in Geomechanics 38(11), 1197–1210.

Mast, C. M., P. Mackenzie-Helnwein, P. Arduino, G. R.Miller, & W. Shin (2012). Mitigating kinematic lockingin the material point method. Journal of ComputationalPhysics 231(16), 5351–5373.

Muller, A. L. & E. A. Vargas (2014). The material point methodfor analysis of closure mechanisms in openings and im-pact in saturated porous media. In 48th U.S. Rock Mechan-ics/Geomechanics Symposium, 1-4 June, Minneapolis, Min-nesota. American Rock Mechanics Association.

Nguyen, V. P., C. T. Nguyen, T. Rabczuk, & S. Natarajan (2017).On a family of convected particle domain interpolations inthe material point method. Finite Elements in Analysis andDesign 126(Supplement C), 50 – 64.

Piegl, L. & W. Tiller (1997). The NURBS Book (2Nd Ed.). NewYork, NY, USA: Springer-Verlag New York, Inc.

Potts, D. & L. Zdravkovic (1997). Finite element analysis ingeotechnical engineering: Theory. Thomas Telford.

Ramos, A., A. Aragn, S. Soghrati, P. Geubelle, & J.-F. Molinari(2015). A new formulation for imposing Dirichlet boundaryconditions on non-matching meshes. International Journalfor Numerical Methods in Engineering 103(6), 430–444.

Sadeghirad, A., R. M. Brannon, & J. Burghardt (2011). A con-vected particle domain interpolation technique to extend ap-plicability of the material point method for problems involv-ing massive deformations. International Journal for Numer-ical Methods in Engineering 86(12), 1435–1456.

Schillinger, D., M. Ruess, N. Zander, Y. Bazilevs, A. Duster, &E. Rank (2012). Small and large deformation analysis withthe p- and b-spline versions of the finite cell method. Com-putational Mechanics 50(4), 445–478.

Sołowski, W. T. & S. W. Sloan (2015). Evaluation of mate-rial point method for use in geotechnics. International Jour-nal for Numerical and Analytical Methods in Geomechan-ics 39(7), 685–701.

Sulsky, D., Z. Chen, & H. Schreyer (1994). A particle method forhistory-dependent materials. Computer Methods in AppliedMechanics and Engineering 118, 179–196.

Wang, B., M. A. Hicks, & P. J. Vardon (2016). Slope failure anal-ysis using the random material point method. GeotechniqueLetters 6(2), 113–118.

Zabala, F. & E. Alonso (2011). Progressive failure of aznalcllardam using the material point method. Geotechnique 61(9),795–808.