A stabilized Runge–Kutta, Taylor smoothed particle hydrodynamics algorithm for large deformation problems in dynamics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2012)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4324

A stabilized Runge–Kutta, Taylor smoothed particlehydrodynamics algorithm for large deformation problems

in dynamics

Thomas Blanc*,† and Manuel Pastor

ETS de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Spain

SUMMARY

Numerical simulation of large deformation and failure problems present a series of difficulties when solvedusing mesh based methods. Meshless methods present an interesting alternative that has been explored inthe past years by researchers. Here we propose a Runge–Kutta Taylor SPH model based on formulating thedynamic problem as a set of first-order PDEs. Two sets of nodes are used for time steps n and n C 1=2,resulting on avoiding the classical tensile instability of some other SPH formulations. To improve the accu-racy and stability of the algorithm, the Taylor expansion in time of the advective terms is combined witha Runge–Kutta integration of the sources. Finally, as boundaries change during the process, a free surfacedetection algorithm is introduced. Copyright © 2012 John Wiley & Sons, Ltd.

Received 28 April 2011; Revised 13 November 2011; Accepted 13 February 2012

KEY WORDS: dynamics; localization; Runge–Kutta Taylor SPH; tensile instability; viscoplasticity; largedeformation problems

1. INTRODUCTION

Engineers have to predict the behavior of structures under service loading and the conditions underwhich failure will take place, the latter being important because, once known, failure mechanismsprovide information on where to reinforce the structure. In other cases, such as failure of largeslopes, post failure analysis allows to estimate the run out of the sliding mass, its depth, and veloc-ity. Frequently, failure is a dynamic process, and even in cases where triggering occurs under quasistatic loads, the analysis has to take into account accelerations. In addition to the nonlinear materialbehavior, structures present geometric nonlinearities, especially in cases where fracture occurs.

Large deformation problems can be analyzed using either Lagrangian or Eulerian frameworks. Inthe former case, nodes are attached to the moving material while in the latter the material nodes flowin a fixed grid. Concerning discretization techniques, there exist mesh-based methods, such as finiteelements, and meshless methods, such as SPH and material point methods, just to mention a few.

Lagrangian finite elements have been extensively used in the past to analyze failure and largedeformation problems. In most occasions, the models have been cast in terms of displacements(or velocities). Such displacement based formulations are known to present problems related to(i) overestimating failure loads or locking, (ii) providing solutions depending on mesh structureand alignment, and (iii) not being able to provide accurate results when shocks are propagatingbecause of numerical diffusion and dispersion problems. It is important to note that the effect ofmesh alignment prevents obtaining the correct failure mechanism.

*Correspondence to: Thomas Blanc, Department of Applied Mathematics and Informatics, ETS de Ingenieros deCaminos, Universidad Politécnica de Madrid, Spain.

†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

T. BLANC AND M. PASTOR

Mixed finite elements formulated using displacements (or velocities) and pressures as primalvariables improve the locking behavior but still are not totally satisfactory concerning the last twoproblems [1]. Enhanced strain methods provide an excellent solution for both locking and meshalignment problems [2]. However, if they are used for coupled problems such as occurs in satu-rated geomechanics, special stabilization techniques have to be used when using the same order ofinterpolation for displacement and pore pressures [3, 4].

As an alternative, mixed stress displacement (or velocities) methods present a very good accu-racy concerning the determination of limit loads and failure mechanisms. We can mention herethe pioneering work of Cantin et al. [5], and Zienkiewicz and Boroomand [6]. Again, mixed stressdisplacement formulations have to fulfill Babuska–Brezzi conditions, otherwise they will not be sta-ble. The interested reader can find in Codina’s work [7] a detailed description, together with somestabilization techniques.

It is worth mentioning the stabilization technique proposed by Cervera et al. [8, 9], which allowsthe use of stress and displacements as primal variables.

One such technique is the Taylor–Galerkin method in combination with a formulation of thedynamic problem as a system of first-order equations, which has been shown to be stable byCodina [7]. The Taylor–Galerkin method was introduced independently by Donea [10] andLohner et al. [11], and applied to fluid dynamics problems by Peraire et al. [12], Donea et al. [13],and Quecedo and Pastor [14]. The interested reader can find a detailed description in the text byZienkiewicz and Taylor [15]. The Taylor–Galerkin method, which had been introduced to solvecomputational fluid dynamics problems, has been extended to other application fields, such aselectromagnetic wave problems [16] and solid dynamics [17–20].

Taylor–Galerkin algorithms formulated in terms of velocities and stresses provide good propaga-tion properties and mitigates locking and mesh alignment problems, and therefore can be appliedto localized failure computations without the limitations exhibited by the classical displacementformulations [19].

On the other hand, meshless Lagrangian methods such as the SPH can deal in a natural mannerwith large deformation and failure problems, without the difficulties encountered in mesh-basedmethods where mesh refinement has to be performed.

The SPH is the first meshless method that has been proposed. It was applied to model astro-physical problems [21, 22]. From there, it was extended to classical hydrodynamics problems [23].Today SPH is used in many areas, among which it is worth mentioning magnetohydrodynamics[24], multiphase flows [25], viscous flows [26], quasi-incompressible flows [27, 28], flows throughporous media [29], metal-forming [30], impact problems [31], elastic dynamics problems [32], fastlandslide propagation [33, 34], and fluid structure interactions [35]. Recently, and for the first time,SPH has been applied to soil problems involving soil–water interaction [36] and failure [37].

Concerning the disadvantages and difficulties presented by the SPH method, we can mention (i)the boundary deficiency problems that can be solved by applying a normalization to the smoothedhydrodynamics method [38], and (ii) the tensile instability which appear in dynamics problems withmaterial strength [39–41].

The purpose of this paper is to explore how an extension of the Taylor–Galerkin model to SPHcan provide accurate results in failure and large deformation problems under dynamic conditions.We have considered some important aspects, such as:

(i) Describing localized failure under dynamic conditions.(ii) Showing how the proposed method avoids the tensile instability exhibited by some SPH

formulations.

The use of Taylor SPH methods was introduced first by Blanc [42], and Herreros and Mabssout [43],and Blanc and Pastor [44]. In our case, being conscious of the importance of viscoplastic strain onthe stability of the Taylor SPH model, we decided to extend the algorithm using a Runge–Kuttascheme for the viscoplastic sources. This was first introduced by Quecedo [45] in the framework ofshallow water equations.

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2012)DOI: 10.1002/nme

A RK-TAYLOR SPH ALGORITHM FOR LARGE DEFORMATION PROBLEMS IN DYNAMICS

The paper is structured as follows. First, we will recall succinctly the proposed balance of momen-tum and constitutive equations that will be used. The second part of the paper presents the two-stepRunge–Kutta Taylor SPH model. Finally, we have selected some representative cases where theperformance of the algorithm can be assessed.

2. MATHEMATICAL MODEL

The mathematical model used in this work consists of (i) the balance of momentum equation and(ii) a suitable constitutive model describing soil behavior. It has been used by Mabssout et al. inthe past [17–20] to solve finite elements soil and solid dynamics problems within the framework ofsmall deformation theory.

We will recall, for the sake of completeness, the main equations and assumptions of the model,which will be formulated using an updated Lagrangian framework. The basic idea is that at eachtime station we use the actual configuration as a reference of a Lagrangian analysis. After all rele-vant increments of this time step are obtained, the coordinates of the nodes are updated, providinga new configuration that will be used as initial in the following time step. This updated Lagrangianapproach has been used in the past by other authors (see for instance [46]). The equations refer to thepresent configuration, which is assumed to be constant during the global time step. This, of course,is an approximation, and all the differential operators found in the text correspond to time tn. Asin other texts, we have decided to avoid complexity in notation, dropping the super index n fromall operators.

Concerning objective rates of the involved variables, we will use the Jaumann–Zaremba rate ofthe Cauchy stress tensor � , and the rate of deformation tensor d, given by

rσD

Dσ

DtCσ.ω�ω.σ (1)

And

dD gradsymv (2)

Or

dij D1

2

�@vi

@xjC@vj

@xi

�, (3)

where we have introduced the antisymmetric tensor ω, which is usually referred to as the spintensor, with components given by

!ij D1

2

�@vi

@xj�@vj

@xi

�. (4)

(i) The model is cast in terms of velocities and stresses, using the balance of momentum equation,which is written as

divσC b D �dv

dt, (5)

where σ is the Cauchy stress tensor, b the body forces, � the density and v the velocity vector.(ii) The material behavior can be described in several alternative ways depending both on the

type of the material and on the velocity at which load is applied. The behavior of soils, especiallyunder dynamic conditions, depends on strain rate. Viscoplastic constitutive laws provide a suitableframework within which most features of behavior can be described. Moreover, models based onviscoplasticity do not present the ill-posedness nature exhibited by classical plasticity based models,where wave propagation velocities become imaginary in the softening regime [47].

Among all alternative formulations, we have chosen that of Perzyna, where the relation betweenthe stress tensor and the strain tensor is given by

r� D De W .d� dvp/ (6)



Above, De is the elastic constitutive tensor, d the rate of deformation tensor, and dvp its viscoplas-tic component. We have assumed an additive decomposition of the rate of deformation tensor intoelastic and viscoplastic components. The viscoplastic component of the rate of deformation tensoris given by [48]

dvp D �m h� .F /i . (7)

In Equation (7)

- the symbol h: : :i represents the Macaulay brackets:

²h�i D � if � > 0D 0 otherwise

- � is the fluidity parameter- m is a unit norm tensor characterizing the direction of the plastic flow- �.F / is an arbitrary function

We will choose for �.F /as

�.F /D

�F �F0

F0

�N, (8)

whereN is a model parameter and F a function describing a convex surface in the stress space. Thevalue F0 characterizes the stress below which no viscoplastic flow occurs.

To complete the Perzyna model, the function F has to be defined. In this paper, we will use thesurface determined by the von Mises yield criterion.

Von Mises yield criterion depends only on the second invariant of the deviatoric stress tensor, andis written as

f D q � Y D 0, (9)

where q is related to J2 as q Dp3J2 and Y is a measure of the material cohesion.

We will choose F as

F D q (10)

and the initial size of the yield surface will be given by

F0 D Y0. (11)

The size of the yield surface will vary according to a suitable hardening/softening law. Here wewill assume that the size of the yield surface will be proportional to the increase of the equivalentdeviatoric plastic strain, N"vp

dY0

dN"vpDH . (12)

We will also assume an associated flow rule, which means that the plastic potential surface g.σ/D 0coincides with the yield surface F .σ, internal variables/D 0 and in consequence

g D q and mD@g

@σ. (13)

Using Jauman stress rate the constitutive relation can be written as

dσ

dtCσ

dω

dt�

dω

dtσD De W .d� dvp/ . (14)

Then Equation (3) is inserted into Equation (14) to relate the stress tensor to the velocity vector.In the case of two-dimensional plane strain problems, bringing together Equations (3), (5), and

(14), we obtain a system of equations that can be divided into two parts as:



(i) Constitutive relations, which is the first system of equations

@

@t

0B@�11�22�12�33

1CA� @

@x1

0B@D11v1D12v1D33v2D41v1

1CA� @

@x2

0B@D12v2D22v2D33v1D42v2

1CAD�

0BB@D11 P"

vp11CD12 P"

vp22C P!1k�1k C P!1k�k1

D12 P"vp11CD22 P"

vp22C P!2k�2k C P!2k�k2

D33 P"vp12C P!2k�1k C P!1k�k2

D41 P"vp11C D42 P"

vp22CD44 P"

vp33

1CCA,

(15)where Dij are the components of the plane strain elastic matrix

De DE

.1C �/ .1� 2�/

0B@1� � � 0 �

� 1� � 0 �

0 0 .1� 2�/ = 2 0

� � 0 1� �

1CA

(ii) Balance of linear momentum, which is the second system of equations

@

@t

�v1v2

��

@

@x1

��11 = �

�12 = �

��

@

@x2

��12 = �

�22 = �

�C

�b1b2

�D

�0

0

�. (16)

Systems (15) and (16) can be cast as a system of first-order hyperbolic PDEs where the source termsare originated by the viscoplastic strains.

It is important to remark that both Equations (15) and (16) include divergences of fluxes, whichare first-order space derivatives of the unknowns. They have the same nature of the convective termsfound in classical convective problems, and we will refer to them as ‘convective’ terms. In the samemanner, we can see in both systems source terms.

3. NUMERICAL MODEL: A TWO-STEP RUNGE–KUTTA TAYLOR SPH ALGORITHM: I.DISCRETIZATION IN THE TIME DOMAIN

We will describe here the proposed numerical model that is based on a time discretization using aTaylor series expansion of the convective terms and a fourth-order Runge–Kutta scheme to integratethe source terms. It is important to note that this part of the model is the same as was proposedby the authors in the past to study fast landslide propagation problems where strong source termsappeared [45].

Once we have discretized the equations in time, we will perform the space discretization, asdone in the Taylor–Galerkin method where finite elements are used. Here, we will do the spacediscretization using the SPH technique.

This section will be devoted to present the discretization in time, where we have introduced asmall variation in the original scheme, which allows to circumvent the generation of spurious zeroenergy modes of deformation.

Instead of considering both systems (15) and (16) together, we will deal with them separately.Concerning the former, Equation (15) can be written in a conservative form [49] as

@�

@tC divfD Ns, (17)

where

- � is the vector of unknowns- f is the advective flux tensor- Ns is the source vector

Equation (17) will be discretized in time using the two-step Runge–Kutta Taylor–Galerkinscheme. It consists of discretizing the source terms Ns using a Runge–Kutta fourth-order schemeand the convective terms divf using the two-step Taylor–Galerkin introduced by Peraire et al.[12] and Peraire [50] within the framework of finite elements to solve computational dynamicsproblems. There, the time discretization process is based on a forward-time Taylor series expan-sion of time derivative of the vector of unknowns followed by a space discretization using the



Boubnov–Galerkin method. It is called ‘two-step’ because an intermediate step is added to solvethe system of equations at the time nC .1=2/ and to decrease the computational time [12].

First, the vector of unknowns � in Equation (17) is expanded in time using a Taylor series up tosecond order

�nC1 D �nC�t@�

@t

ˇ̌̌ˇn

C1

2�t2

@2�

@t2

ˇ̌̌ˇn

, (18)

where

- �n is the vector of unknowns at time tn

- �t is the time step

The first-order derivative of the unknowns with respect to time can be calculated usingEquation (17) as

@�

@t

ˇ̌̌ˇn

D .Nsn � divnf/ . (19)

To obtain the second-order time derivative of Equation (18), the two-step Taylor–Galerkin procedureconsiders an intermediate step between tn and tnC1. The aim of the first time step is to calculate thesolution at time tnC1=2. This step is followed by a second one that brings the solution to tnC1.

The first step results in

�nC1=2 D �nC�t

2.Nsn � divnf/ , (20)

which allows calculating the advective flux tensor fnC1=2 and the vector of sources NsnC1=2 at timetnC1=2. In fact the values of fnC1=2 and NsnC1=2 are calculated using �n+1=2 and the value of thefluxes and the sources found in Equation (15).

Considering a Taylor series expansion of first order of the flux and source vectors, we have

fnC1=2 D fnC�t

2

�@f@t

�n

NsnC1=2 D NsnC�t

2

�@Ns@t

�n. (21)

Differentiating with respect to time Equation (19), we have

@2�

@t2

ˇ̌̌ˇn

D@

@t.Nsn � divnf/. (22)

From Equation (21), the fluxes and sources time derivative are obtained as�@f@t

�nD

2

�t

�fnC1=2 � fn

��@Ns@t

�nD

2

�t

�NsnC1=2 � Nsn

�. (23)

Incorporating (23) into (22), the second-order time derivative of the vector of unknownsresults in

@2�

@t2

ˇ̌̌ˇn

D2

�t

�NsnC1=2 � Nsn � divnC1=2fnC1=2 � divnfn

�. (24)

Substituting now the expressions obtained for the first-order (19) and the second-order (24) timederivatives in the Taylor series expansion (18) results in

�nC1 D �nC�t�NsnC1=2 � divnC1=2fnC1=2

�. (25)



Equations (20) and (25) correspond, respectively, to the first and second steps of the two-step Taylor–Galerkin method. Both equations need to be discretized in space, which in theTaylor–Galerkin method is accomplished using finite elements.

The Taylor–Galerkin method has been shown to work well, providing a very good compromisebetween computational cost and accuracy except in cases where the source terms are important.Quecedo et al. [45] pointed out the instabilities in a two-step Taylor–Galerkin algorithm caused bythe existence of strong sources, for which they proposed to use a splitting scheme [45, 51, 52]. Herewe have used an adaptation of the splitting scheme proposed by Quecedo et al. [45]. This schemehas been used by Mabssout et al. in their Runge–Kutta Taylor–Galerkin scheme [20].

The method consists of decomposing the problem on the following steps:

- The ordinary differential equation

@�@tD Ns

�.x, tn/D �n

³) �Source1,n (26)

- The homogeneous, pure advection problem

@�@tC divfD 0

�.x, tn/D �Source1,n

³) �adv,nC1 (27)

- And, finally, on the ordinary differential equation

@�@tD Ns

�.x, tn/D �adv,nC1

³) �Source2,nC1 D �nC1. (28)

The splitting scheme is of second order and can be written in a compact form as

�nC1 D L

��t

2

�ıAdv .�t/ ıL

��t

2

��n, (29)

where

- Adv.�t/ is the advective operator based on the two-step Taylor–Galerkin scheme

Adv W �adv,nC1 D �Source1,n ��t � divnC1=2fnC1=2. (30)

In Equation (30), the divergence of the advective tensor, divfnC1=2 is calculated from �nC1=2,which is provided by Equation (20) of the conventional two-step Taylor–Galerkin method, inwhich the source term has been eliminated

�adv,nC1=2 D �Source1,n ��t

2.divnf/ (31)

- L.�t/ is the source operator.We will use a variant of the Runge–Kutta scheme proposed by Jameson et al. [53] to solve

the ordinary Equations (26) and (28), which is detailed below

L.�t/ W �Source D �initialC�t Nsn, (32)

where

Nsn D1

6

hs.1/C 2s.2/C 2s.3/C s.4/

iNsn D

1

6

hs��.1/

�C 2s.�.2//C 2s

��.3/

�C s.�.4//

i. (33)



With

- �initial D �n for Equation (26) and �initial D �adv,nC1 for Equation (28)- �.1/ D �initial

- �.2/ D �initialC �t2s.1/

- �.3/ D �initialC �t2s.2/

- �.4/ D �initialC �t2s.3/.

Mabssout et al. [20] have applied to the one-dimensional problem a first-order Runge–Kutta split-ting scheme and observed that the splitting scheme converges to the exact solution for the velocityand with an error less than 0.4% for the stress.

Concerning the second part of the first-order PDEs system of Equation (16), we will separatefrom the classical Runge–Kutta Taylor–Galerkin scheme described above. The new variation pro-posed avoids the hourglass mechanism of deformation observed when using the original algorithm,as will be illustrated in the section devoted to examples and applications.

Equation (16) will be discretized as follows:

- First the Runge–Kutta splitting scheme described in the last section is used to separatelyintegrate both source and advective terms.

- Then the velocity is obtained at an intermediate step, t D n C 1 = 2, using Equation (31) :�adv,nC1=2 D �Source1,n � �t

2.divnf/.

- Finally, the velocity is calculated at time t D nC 1 using the following scheme:

�nC1 D 2�nC1=2 � �n. (34)

4. DISCRETIZATION IN THE SPACE DOMAIN: THE SPH METHOD

4.1. Introduction: basic concepts and notation

Once the problem has been discretized in time, we will discretize it in space using the SPH method.We will first recall here some basic concepts, together with the notation used for the sake of com-pleteness. SPH is based on the integral approximation of a given function �.x/ and its derivatives.The starting point is the operational definition of the Dirac delta singular distribution ı.x � x0/ as

�.x/D

Z�

��x0�ı.x � x0/dx0. (35)

Dirac delta singular distribution can be approximated as the limit of a sequence of regulardistributions depending on a characteristic length h and a kernel W fulfilling the unity property as

�.x/� < � .x/ > D

Z�

�.x0/W�x � x0, h

�dx0. (36)

Equation (36) is the starting point to construct SPH approximations, where regular distributions areused to approximate the value of a function. Above, we have used < �.x/ > to denote the integralapproximation of �.x/. The accuracy of SPH approximations depends on the properties of the ker-nel W.x � x0, h/. Indeed, the sequence of regular distributions depending on h should tend in thelimit h ! 0 to the Dirac delta singular distribution. The functions used as kernels are required tofulfill a series of conditions that ensure obtaining in the limit the Dirac delta (normalization, com-pact support, positiveness, being monotonically decreasing, and symmetry). Here we have used theGaussian kernel introduced by Gingold and Monaghan [54]. Concerning the integral representationof the spatial derivatives of a given function �.x/ they are given by

< grad�.x/ >DZ�

�grad�.x0/

W.x � x0, h/dx0. (37)



Applying Gauss theorem and taking into account that kernels have a compact support, we obtain

< grad�.x/ >��Z�

��x0�� gradW.x � x0, h/dx0. (38)

Equation (38) shows how the spatial gradient of a function can be determined from the values ofthe function and the derivative of the kernel function W , rather than from the derivatives of thefunction itself.

From the integral representation of functions and derivatives, SPH approximations are built intro-ducing the concept of ‘particles’ or ‘nodes’, to which information concerning field variables andtheir derivatives is linked. But indeed they are nodes, much in the same way that those found in thefinite elements or the finite differences, and the quality of the approximation depends on the spacingbetween such nodes. It is well known that the SPH approximation of a function �.x/ at a point I isgiven by the expression

�I D

NXJD1

mJ

�J� .xJ /W.xI � xJ , h/, (39)

where �I is the value of the function approximated at point I ,mJ D�VJ�J is the mass associatedto node J , �VJ being the associated volume and �J the density at it. The kernel W .xI � xJ , h/depends on the distance between points I and J together with the parameter h. Taking into accountthat the kernel has compact support, the summation results on

�I D

NPXJD1

mJ

�J�.xJ /WIJ , (40)

where we have introduced the notation WIJ denoting

WIJ DW.xI � xJ , h/. (41)

Using a similar reasoning, the particle approximation for the spatial derivative of a function can beobtained as:

grad�I DNPXJD1

mJ

�J�.xJ /gradWIJ , (42)

where we introduced

xI � xJ D xIJ

rIJ D jxI � xJ j

� DrIJ

h

gradWIJ DxIJ

h � rIJ

@WIJ

@�

and

grad�I D< grad'.xI / >P

. (43)

It is most important to remark that when the support domain of a particle extends out of the problemdomain, a problem referred to as the boundary deficiency problem appears. It arises because thesmoothing kernel function W is truncated by the boundary. Because particles do not exist beyondthe boundary, there is an insufficiency of particles in the summation process. Among the alternativemethods to solve this problem, it is worth mentioning the Corrective Smoothing Particle Method,which has been introduced to overcome this problem [38]. This method is based on the Taylor



series expansion and gives the normalization of the kernel and of the particle approximation nor-mally used in the conventional SPH method. Using the corrective smoothing particle method, theparticle approximation of a function is obtained in one-dimensional problems as

�I D

NPPJD1

mJ�J'.xJ /WIJ

NPPJD1

mJ�JWIJ

. (44)

In a multidimensional space, the method is more complex, and requires solving a system

A�D r , (45)

which in two dimensions particularizes to

AD

�A11,I A12,I

A21,I A22,I

�I �D

��1,I

�2,I

�I r D

�r1,I

r2,I

�(46)

with

- �˛,I the particle approximation of the first derivative of � with respect to dimension ˛

- Aˇ˛,I DNPPJD1

mJ�J.x˛,J � x˛,I/WIJ ,ˇ

- rˇ ,I DNPPJD1

mJ�JŒ�.xJ /� '.xI/WIJ ,ˇ

and where

WIJ ,ˇ [email protected] � xJ , h/

@xˇ. (47)

To obtain the particle approximation of the first derivative of �, the inverse matrix, A�1, of Aneedsto be calculated and then

�D A�1r (48)

from where

@�

@x1

ˇ̌̌ˇI

D A�111,I �

NPXJD1

mJ

�JŒ�.xJ /� �.xI/

@WIJ

@x1CA�112,I �

NPXJD1

mJ

�JŒ�.xJ /� � .xI/

@WIJ

@x2

@�

@x2

ˇ̌̌ˇI

D A�121,I �

NPXJD1

mJ

�JŒ�.xJ /� �.xI/

@WIJ

@x1CA�122,I �

NPXJD1

mJ

�JŒ�.xJ /� ' .xI/

@WIJ

@x2

.

(49)In the SPH method, the smoothing length h defines the influence area of the kernel function at agiven particle. To keep a sufficient number of particle interactions, the smoothing length of eachparticle must be updated, otherwise the number of particles in the influence domain of a given par-ticle could decrease below a critical number or increase too much, resulting in both cases in a lossof accuracy [55]. Several methods exist to evolve the smoothing length. For dynamic problems, theevolution of the smoothing length must take into account both the space and the time. We will usehere the method proposed by Benz et al. [56], which consists of updating the smoothing length as

dh

dtD�

1

d

h

�

d�

dt, (50)

where d is the number of dimensions.



In Equation (50), the time derivative of the density, �, is approximated with the continuity densityapproach in which the time derivative of the density is closely related to the relative velocity betweena given particle and its neighboring particles

D�I

DtD

NXJD1

mJ v˛,IJ@WIJ

@x˛,I, (51)

wherev˛,IJ is the ˛ component of the relative velocity of particle Iwith J : v˛,IJ D v˛,I � v˛,J

4.2. A note on smoothed particle hydrodynamics tensile instability

Numerical methods present in some cases instabilities, such as those found in mixed finite elementsfor incompressible problems when using the same order of interpolation for displacements and pres-sures, or the tensile instability of SPH, analyzed by Swegle et al. [57]. In the case of SPH, Dykaet al. [58, 59] provided further insight.

According to Belytchsko et al. [60], there are two main types of instabilities, related to consti-tutive behavior in the case of rate independent materials, and the tensile instability. In the casesconsidered in this paper, the material is of viscoplastic type, and thus, only the latter type can occur.It has been shown by Belytschko et al. [60] that tensile instability occurs in Eulerian kernels, butnot in Lagrangian kernels.

It is also worth mentioning the work on stability made by Rabczuk et al. [46] and Samaniegoet al. [61], which continue the analysis of Belytschko et al. [60]. The former contribution considersLagrangian kernels, and presents an updated Lagrangian formulation based on frame invariant ratesof the Cauchy stress.

In addition to the analyses conducted by Dyka et al. [58, 59] and Randles and Libersky [62], itis worth mentioning that of Belytschko et al. [60] for stress integration points using both Eulerianand Lagrangian kernels. They concluded that for the former, and using quartic splines with smallsupport, there was no tensile instability, which appeared only when increasing the support size. Inthe case of Lagrangian kernels with stress points, no tensile instability appears.

We have used an Eulerian kernel approximation, which has the disadvantage of presenting thetensile instability, and the advantage of avoiding the lack of versatility described by Belytscko et al.[60] of Lagrangian kernels because of the invariance of neighbouring particles and the distortion ofthe domain of influence. In consequence, the use of a classical SPH scheme will result on tensileinstability unless special techniques are used such as the introduction of stress points as proposedby Dyka et al. [58, 59] and Randles and Libersky [62, 63]. The purpose of this paper is to presentan alternative to the use of stress points for numerical integration, with two sets of master and slavenodes being used. Primal variables here are velocities and stresses, that is, we are using a mixedmethod in which the framework of finite element computations [18, 19] has been shown to providebetter accuracy both for localization and specially for shock propagation.

4.3. Material and auxiliary sets of smoothed particle hydrodynamics nodes

The model requires an SPH grid arrangement similar to that proposed by Randles and Libersky [63].We will use an initial staggered node arrangement, which consist of a double set of SPH nodes.We call the nodes of the first set the material SPH nodes, and the nodes of the second set theauxiliary SPH nodes. It is interesting to note that auxiliary nodes play the same role than Gausspoints in the finite element Taylor–Galerkin scheme. Thus, each variable of the model is defined atboth material and auxiliary SPH nodes. However, the material is only represented by the SPH nodes.In consequence, the total mass of the material, Mtotal, is

Mtotal D

NXID1

mI , (52)

where mI is the mass of each SPH material node and N the number of SPH material nodes.



To approximate functions on the SPH nodes, only information coming from the SPH auxiliarynodes will be used and, vice versa, to approximate functions on the SPH auxiliary nodes, only infor-mation coming from the SPH nodes will be used. Our initial SPH grid can be schematized as in theFigure 1.

The governing equations of our model are those described in the mathematical model (Equations(15), (16), and (17)). These equations have been discretized in time using the Runge–KuttaTaylor–Galerkin method described in the preceding Sections to get Equations (30), (31), (32), and(33). Then, functions and derivatives in Equations (30) and (31) will be approximated using thecorrective SPH method. Thus, in the SPH grid, the vector of unknowns � is approximated at aparticle I as

�I D

NPPJD1

mJ�J�JWIJ

NPPJD1

mJ�JWIJ

, (53)

where

- �J are the value of the unknowns at particle J- J D ¹1, 2, 3, � � � ,NPº is the set of particles in the support domain of particle I

The divergence of the flux tensor will be approximated at particle I of a two-dimensionalspace as

divfI D@fx1,I

@x1C@fx2,I

@x2(54)

from where we obtain

@fx1,I

@x1D A�111,I �

NPXJD1

mJ

�JŒfx1,J � fx1,I

@WIJ

@x1CA�112,I �

NPXJD1

mJ


@WIJ

@x2

@fx2,I

@x2D A�121,I �

NPXJD1

mJ


@WIJ

@x1CA�122,I �

NPXJD1

mJ


@WIJ

@x2

. (55)

In Equations (54) and (55) we have used the notation

- fx1,I the value of the first component of the advective flux tensor at particle I- fx2,I the value of the second component of the advective flux tensor at particle I- A�1

ˇ˛,I are the component matrix of the inverse matrix of A for particle I

Figure 1. SPH grid for the proposed model.



with

Aˇ˛,I D

NPXJD1

mJ

�J.xˇ ,J � x˛,I /WIJ ,ˇ (56)

and

- x˛,I is the ˛ coordinate of the position vector of particle I- x˛,J is the ˛ coordinate of the position vector of particle J- WIJ ,ˇ D

@W.xI�xJ ,h/@xˇ

4.4. Updating of the particle position

The spatial configuration of the SPH nodes is updated by calculating the new position of the nodesat time t D nC 1:

xnC1SPH Node D xnSPH NodeC�t � �

nC1=2SPH Node. (57)

The position of the auxiliary SPH nodes is updated by averaging the new position of the SPH nodesxnC1SPH Node for always having an auxiliary SPH node in the middle of four SPH nodes (Figure 2)

xnC1SPH aux D1

4

4XjD1

xnC1SPH Node j . (58)

It is important to note that other alternatives, such as obtaining the auxiliary nodes as the centroidsof the Delaunay triangulation of the SPH material nodes can be easily implemented.

4.5. Details of the algorithm

The proposed algorithm allows passing from �nnode to �nC1node using an intermediate calculation of�nC1=2aux where �nnode is the vector of unknowns in the SPH nodes at time t D n and �nC1=2aux is thevector of unknowns at the SPH auxiliary nodes at time t D nC 1 = 2. The algorithm can be dividedin the following steps:

5. BOUNDARY CONDITIONS

The problem being solved is a set of hyperbolic first-order PDEs formulated in terms of stressesand velocities. The number of boundary conditions to apply is related to the characteristics enteringthe domain. In general, we will have boundaries where (i) tractions are known, (ii) velocities are

Figure 2. Update of the spatial configuration of the SPH particles.



known, (iii) the incident wave is known, and (iv) the waves will leave the domain without spuriousreflections. Here, we will consider only the first two types. Of course, conditions can be combinedas in the case where we prescribe the normal velocity and the tangential component of thesurface traction.

One special case of interest is that of free surfaces where the applied tractions are zero. In classi-cal finite elements it is immediate to apply such conditions. The model we present here deals withcases where the boundaries may change, and therefore, both the nodes belonging to them and thenormal vectors will change accordingly.

In conclusion, the boundary conditions at fixed horizontal or vertical boundary walls can beapplied directly to boundary nodes, because the nodes lie always on these surfaces. On the contrary,a preliminary step is needed in the case of free surfaces. The procedure involves three steps:

- The detection of SPH nodes located on the free-surface- The calculation of the normal to the free-surface at the boundary SPH nodes- Applying the traction-free boundary conditions

In the literature, there exist a number of algorithms of free-surface detection for SPH applications.It is worth mentioning the work of Dilts on free-surface detection for two-dimensional SPH applica-tions [64], the extension of this method to three dimensions introduced by Haque and Dilts [65], andthe algorithm proposed by Marrone et al. [66] who implemented a free-surface detection algorithmfor complex two-dimensional and three-dimensional flow simulation.

Here we propose a modification of the Marrone et al. method, which is based on the steps listedabove. The description of the implemented algorithm is given in the Appendix.

To show the performance of our boundary detection algorithm, we will present a simple example,including both wall and free boundary nodes, which is sketched in Figure 3(a) below. The nodes weare searching are those not belonging to either left or bottom walls, where we will assume velocitieshave been prescribed.

Using the algorithm described above, we obtain the free boundary from the set of nodes hav-ing zero interactions within their respective scanning regions and not belonging to the walls(Figure 3(b)).

Next, the normal to the detected free-surface is calculated using our method proposed above(Figure 4(a)). To compare with the normal to the free-surface given by the method of Marrone et al.[66], we also approximate the normal using their method (Figure 4(b)).

It can be seen how the proposed correction improves the quality of the normals, which is a keyfactor to apply boundary conditions on a deforming domain.

6. EXAMPLES AND APPLICATIONS

6.1. Introduction

The proposed algorithm presents a series of advantages, which are the following:

(i) It eliminates the SPH tensile instability in large deformation problems, and avoids spuriouszero energy modes of deformation.

(ii) It presents good wave propagation properties in large deformation problems.(iii) It is useful to model localized failure problems in dynamics where sharp shear bands are

incepted.

To illustrate those aspects, we have selected the following examples:

(i) A hanging bar where a zero energy mode is avoided.(ii) Propagation of a wave on an elastic one-dimensional bar.

(iii) Failure of an extremely low cohesive vertical slope under gravity..(iv) Failure of a vertical slope under constant loading.(v) Failure of a shallow stratum under a wide strip footing: bearing capacity test.



Figure 3. (a) Geometry of the validation test for free-surface detection. (b) Classification of all the SPHnodes into two groups: Subsets 1 and 2.

Figure 4. Normal to the free-surface approximated by (a) our method and (b) the method of Marrone et al.

6.2. A hanging bar: avoiding zero energy modes of deformation

The proposed model prevents their inception because in our algorithm:

- The stress on the SPH nodes at time nC1 is calculated using the Runge–Kutta Taylor–Galerkinscheme described in Step 10a of Table I.

- The velocity is calculated using Equation (36).

To assess how the proposed scheme eliminates the spurious zero-energy mode, we will consider thecase of a hanging beam subjected to gravity (Figure 5).

The calculations have been made with two different algorithms:

(i) With a complete the Runge–Kutta Taylor SPH: both systems of equations (Equation (15) forthe stress variable and (16) for the velocity variable) have been solved with the Runge–KuttaTaylor–Galerkin scheme

(ii) With our algorithm described in detail in Table I.

The results are presented in Figure 6.In Figure 6 the vertical velocity field and the deformed mesh obtained with both algorithms

after 0.2 s is presented. The comparison of the results shows clearly that a spurious zero-energymode occurs with the complete Runge–Kutta Taylor SPH algorithm and that the problem disap-pears with the algorithm described in Sections 3 and 4. Our proposed algorithm gives a constantvertical velocity field whereas the other algorithm provides an oscillating vertical velocity field.



Table I. Steps of the algorithm.

Step 1 Calculation of Nsnnode

Nsn D 16

hs��nnode

.1/�C 2s

��nnode

.2/�

C2s��nnode

.3/�C s

��nnode

.4/�i

Step 2 Calculation of �Source1,nnode �

Source1,nnode D �nnodeC

�t2 Ns

n

Step 3 Approximation of �Source1,naux �

Source1,naux I D

NPPJD1

mJ�J�

Source1,nnodeJ WIJ

NPPJD1

mJ�J

WIJ

Step 4 Calculation of fnnode and fnaux Equation (15)

Step 5 Calculation of divfnaux

@faux,x1 ,I

@x1D A�1

11,I �NPPJD1

mJ�JŒfx1,node J � fx1,aux I

@WIJ@x1

CA�112,I �

NPPJD1


@WIJ@x2

@faux,x2 ,I

@x2D A�1

21,I �NPPJD1


@WIJ@x1

CA�122,I �

NPPJD1


@WIJ@x2

Step 6 Calculation of �adv,nC1= 2aux �

adv,nC1= 2aux D �

Source1,naux � �t

2 .divnfaux/

Step 7 Approximation of �adv,nC1= 2node �

adv,nC1= 2node D

NPPJD1

mJ�J�adv,nC1= 2

aux WIJ

NPPJD1

mJ�J

WIJ

Step 8 Calculation of fnC1= 2node and fnC1= 2aux Equation (15)

Step 9 Calculation of divfnC1= 2node

@fnode,x1 ,I

@x1D A�1

11,I �NPPJD1

mJ�JŒfx1,aux J � fx1,node I

@WIJ@x1

CA�112,I �

NPPJD1


@WIJ@x2

@fnode,x2 ,I

@x2D A�1

21,I �NPPJD1


@WIJ@x1

CA�122,I �

NPPJD1


@WIJ@x2

Step 10a Calculation of σadv,nC1node σadv,nC1

node D σSource1,nnode ��t � divFnC1= 2node

Step 10b Calculation of �adv, nC1node �

adv, nC1node D 2�

adv,nC1=2node � �nnode

Step 11 Calculation of Nsnnode

Nsn D 16

hs��

adv,nC1.1/node

�C 2s

��

adv,nC1.2/node

�C2s

��

adv,nC1.3/node

�C s

��

adv,nC1.4/node

�iStep 12 Calculation of �nC1node �nC1node D �

adv,nC1node C �t

2 Nsn

Step 13 Application of the boundary conditions

Step 14 Updating the particle position xnC1SPH Node D xnSPH NodeC�t � �

nC1= 2SPH Node and

xnC1SPH aux D14

4PjD1

xnC1SPH Node j



Figure 5. Sketch of the hanged vertical beam and boundary conditions.

Figure 6. Vertical velocity field and deformed mesh according to the velocity variable (The factor ofdeformation of the mesh is 200).

6.3. One-dimensional elastic bar

This case study consists of a shockwave that propagates on an elastic one-dimensional bar. The baris 1-m long, has a unit section and is fixed at its left end (Figure 7).

The velocity and stress are initially equal to zero. On the right extremity of the bar the imposedvelocity is Vt (Figure 7).

The elastic bar has the following material properties: the density is �D 2000 kg/m3 and the elas-tic modulus is E D 8 � 107 Pa. The bar is discretized by 100 SPH elements and 101 SPH nodes. Thedistance between two consecutive SPH nodes is �x D 0.01m. The time step is �t D 0.00005 s.



Figure 7. Sketch of the elastic one-dimensional bar and velocity shockwave.

To assess the stability of the algorithm the velocity history at SPH nodes 11 (x D 0.1 m/ can beobserved (Figure 8):

The velocity shockwave propagates well along the bar and no instability appears. Indeed thevelocity wave is reflected when it reaches the left end of the bar and becomes negative. The ampli-tude of the wave is constant over time. No numerical dispersion nor numerical diffusion appear.Observing the horizontal stress at SPH node 11, the same conclusion can be reached (Figure 9).

In Figures 8 and 9 we can observe that small oscillations appear at the extremity of waves in thevelocity history and in the stress history. These oscillations are present because the distance betweentwo consecutive SPH nodes is not constant over time because of the update of the SPH nodes posi-tion. Because the distance, �x, varies, the number of Courant is not always equal to 1 but it isclose to 1.

In Figure 10 the displacement history at SPH node 11 is represented.The displacement of SPH node 11 oscillates between 0 and 0.001 m. This is because the velocity

alternates between 1 m/s and �1 m/s as a result of the reflection of the velocity shockwave. Theoscillation of the displacement of the SPH nodes is better illustrated when plotting the deformed bar(Figure 11)

The size of the bar increases until t D 0.005 s because of the imposed velocity (Figure 7). Thenthe velocity wave is reflected on the extremities of the bar because both extremities are fixed. The

Figure 8. History of the velocity at SPH node 11.

Figure 9. Horizontal stress history at SPH node 11.



Figure 10. Horizontal displacement history at SPH node 11.

Figure 11. Deformed bar (Factor �20).

SPH nodes plotted in blue are located initially at the center of the elastic bar. Figure 11 shows clearlythe oscillation of the SPH nodes. The blue shadowed area delimits the oscillating displacement ofthe blue SPH nodes.

This case study has been used to test the stability of the Runge–Kutta Taylor SPH algorithm. Themodel can be used for one-dimensional problems in elastic regime. The updating of the SPH nodesposition introduces a slight oscillation of the stress and velocity but these oscillations are not signif-icant and do not disturb the algorithm stability. The displacements of the SPH nodes are calculatedand represent well the reality. Finally, the algorithm presents good wave propagation properties.

6.4. Failure of an extremely low cohesive vertical slope under gravity

In this example we will study the failure of an extremely low cohesive vertical slope because ofgravity. The vertical slope is modeled using a bidimensional approximation under plane strain con-ditions. The vertical slope is 2-m high and 4 m long. The soil is only subjected to its own weightbecause of gravity forces. The vertical slope is discretized by 861 SPH nodes and 800 SPH elements.The applied boundary conditions are (Figure 12):

- On the left-hand side, 1, the nodes can only move along the vertical axis:vx D 0 I �xy D 0- On the bottom, 2, the nodes can only move along the horizontal axis: vy D 0 I �xy D 0- The right-hand side, 3, and the top, 4 are the free-surface, thus �n D 0 I � D 0



Figure 12. Sketch of the vertical slope and the applied boundary conditions.

The boundary conditions have been chosen such that the vertical slope could be subjected to largehorizontal deformations.

The material parameters are the following: elastic modulus: E D 8 � 107 Pa, Poisson’s ratio� D 0.3, and density �D 2000 kg/m3.

The material is viscoplastic. The parameters of the Perzyna’s model and of the von Mises yieldcriterion are: fluidity parameter � D 1 s�1; exponent N D 1; initial size of the yield surfaceY0 D 2 � 10

3 Pa; and the softening modulus equal to 0: H D 0 Pa.To eliminate possible oscillations because of the application of gravity and to reach a stationary

state, an artificial viscosity characterized by the factor is employed. In the work of the author in[42], it has been shown that the value of 50 is the most appropriate to reach as soon as possible thestationary state The gravity forces become²

b1 D�50 � vxb2 D�˛t � kgk � 50 � �y

, (59)

where ˛t is a time coefficient to apply the gravity forces following a ramp curve.The material has an extremely low cohesion (Y0 D 2 � 103 Pa/ and in consequence it deforms like

a viscous fluid. The part of the vertical slope where most deformations occur is the right, which isclose to the free-surface 3 (Figure 13). It is important to note that the grid of quadrilaterals depictedin the figure is just a device to improve the understanding of the displacement field, providing abetter idea of distortion.

At the end of the calculation the free-surfaces 3 and 4are almost a smoothed continuous linebecause the right-bottom corner of the vertical slope is the area where the displacements are bigger.At time t D 35 s, the right-bottom corner has moved 1.5 m from its original position (Figure 14).It is worth mentioning that the factor of mesh deformation in Figure 14 is equal to 1. Looking atthe evolution of the deformation mesh (Figure 14), we can point out the rotation of some part of thevertical slope. For instance on the right-part of the slope at around 1 m high, the squares formed byfour SPH nodes have rotated in the clockwise direction.

From the results obtained in this example we can conclude that the Runge–Kutta Taylor SPH isable to threat large deformation problems. The vertical slope is subject to deformations of the orderof 150%.

Figure 13. Deviatoric viscoplastic strain at the end of the calculation.



Figure 14. Displacement of the vertical slope plotted on the deformed mesh (Factor of deformation �1).

6.5. Failure of a vertical slope under constant loading

In this section the failure of a vertical cut under constant loading is studied. An analytical solutionexists for the failure load of vertical slope and therefore it will give us an idea of the accuracy theRunge–Kutta Taylor SPH algorithm. The vertical cut is modeled using a bidimensional approxima-tion under plane strain conditions. The vertical cut is 10 m high and 10 m long. The footing is 5 mwide. Unlike the finite elements it is not required to model the footing (Figure 15). The boundaryconditions in the area of the footing will represent the displacements because of the loading onthe footing.

The boundary conditions are (Figure 16):

- On the left-hand side 1 the nodes can only move along the vertical axe, thus vx D 0 and�xy D 0

- On the bottom 2 the nodes are fixed, vx D 0 and vy D 0- The right-hand side, 3, is the free-surface, thus �n D 0 I � D 0



Figure 15. Sketch of the vertical cut and its discretization.

Figure 16. Boundary conditions of the vertical cut.

- On the right part of the top, 4, the vertical velocity is imposed to represent the displacementof the footing: vy D Vt and �xy D 0

- On the left part of the top, 5: vy D 0 and �xy D 0

The modulus of the velocity Vt increases constantly over time (Figure 16).The material parameters are the elastic modulus:E D 1�105 Pa, the Poisson’s coefficient � D 0.35

, and the density �D 2000 kg/m3.The material is viscoplastic. The parameters of the Perzyna’s model and of the von Mises yield

criterion are the fluidity parameter, � D 2 s�1; the model parameter, N D 1; and the initial sizeof the yield surface Y0 D 200 Pa. In the first step the softening modulus is equal to 0: H D 0 Pato compare the failure load calculated by the Runge–Kutta Taylor SPH algorithm to the analyticalfailure load. In the second step the softening modulus isH D�1 � 103 Pa.

The analytical failure load,Panalytical, for a von Mises material is given by

Panalytical D2bY0p3

. (60)

In the Runge–Kutta Taylor SPH algorithm, the failure load is calculated as

Pcalculated D1

NP

NP�1XID1

1

2.�22,I C �22,IC1/ kxI ,IC1k , (61)



Figure 17. Load–displacement curves: analytical and computational solutions.

Figure 18. Evolution of the viscoplastic deformation in the vertical cut: (a) perfect viscoplastic case and (b)case with softening.

where

- NP are the 11 SPH nodes located under the footing- SPH nodes I and I C 1 are consecutive SPH nodes of the boundary surface- �22,I is the vertical tension on SPH node I- xI ,IC1 is the vector formed by SPH nodes I and I C 1



To compare the results obtained with the new Runge–Kutta Taylor SPH algorithm to the analyticalsolution, the failure load is plotted as a function of the displacement at the base of the rigid footing(Figure 17).

In the perfect plastic case where no softening occurs, the load–displacement relationship of theSPH gives a response in accordance to the analytical solution of the problem. The curve rises lin-early until 1000N and then shows nonlinear behavior, which corresponds to plastic loading, until itreaches the collapse load. The analytical failure load is 1154 Nand the estimated failure load givenby our model is between 1171 N and 1267 N. In the case with softening, the curve observed inthe load–displacement relationship graph corresponds to the response imposed by the constitutivemodel. Indeed once irreversible viscoplastic deformations occur, the material starts to soften andthen the failure load decreases.

The failure of the material is shown by the evolution of the deviatoric viscoplastic strains in thevertical cut (Figure 18). The viscoplastic deformations are accumulated in a sharp shear band clearlydefined. The shear band has an orientation of 45°, which corresponds to the solution of the problem.In the first part of the calculation, there is no difference between the case of perfect viscoplasticity

Figure 19. Displacement contours on the deformed mesh (factor of deformation �2): (a) perfect viscoplasticcase and (b) case with softening.



and the case with softening. However, at the end of the calculation the shear band is better definedin the case with softening.

The failure of the vertical cut is well illustrated when plotting the displacement contours on thedeformed mesh (Figure 19). It two discontinuities appear where failure occurs: at the middle ofthe top border and at the middle of the right border. The right-upper triangle of the vertical cut istranslated along the failure line represented by the shear band. The failure is sharper in the case withsoftening than in the perfect viscoplastic case.

This case study shows how accurate the Runge–Kutta Taylor SPH algorithm is. The model isable to represent the sharp failure of a vertical cut under constant loading of a rigid footing. Theviscoplastic deformations reach 73% in the case of perfect viscoplasticity and 89% in the case withsoftening. Thus, the algorithm is useful for doing a geotechnical analysis of a vertical cut in largedeformation theory.

6.6. Failure of a shallow stratum under a wide strip footing: bearing capacity test

In this section the Runge–Kutta Taylor SPH algorithm is used to study a shallow stratum under awide strip footing. The case study is known as the bearing capacity test and an analytical solutionexists for the failure load. Thus, this example is useful to examine the accuracy of our algorithm.The shallow stratum is modeled using a bidimensional approximation under plane strain conditions.The stratum is 3 m depth and 14 m wide. The footing is 2 m wide. As in the last section it is notnecessary to model the footing but only the shallow stratum. Because there is a vertical symmetricaxis located at the middle of the stratum, only the right part of the stratum is taken into account(Figure 20). The boundary conditions in the area of the footing will represent the displacementsbecause of the loading on the footing.

The boundary conditions are (Figure 20):

- On the left-hand side 1 and on the right-hand side 3 the nodes can only move along thevertical axe, thus vx D 0 and �xy D 0

- On the bottom 2 the nodes are fixed, vx D 0 and vy D 0- The right part of the top, 4, is the free-surface, thus �n D 0 I � D 0- The left part of the top 5 is the part of the stratum below the footing, thus vy D Vt and�xy D 0.

The modulus of velocity Vt increases constantly over time (Figure 21):

Figure 20. Sketch of the shallow stratum.

Figure 21. Imposed velocity on the area situated under the footing.



The material parameters are the elastic modulusE D 2.5�107 Pa, the Poisson’s coefficient � D 0.4and the density �D 2000 kg/m3.

The material is viscoplastic. The parameters of the Perzyna’s model and of the von Mises yieldcriterion are the fluidity parameter � D 2 s�1, the model parameter N D 1, and the initial size ofthe yield surface Y0 D 15, 000 Pa. In the first step the softening modulus is equal to 0: H D 0 Pato compare the failure load calculated by the Runge–Kutta Taylor SPH algorithm to the analyticalfailure load. In a second step the softening modulus is H D�1 � 106 Pa.

Figure 22. Load–displacement curve: analytical and computational solutions.

Figure 23. Evolution of the viscoplastic deformation in the shallow stratum.



The analytical failure load,Panalytical, for a von Mises material is given by

Panalitycal D.2C �/p3� b � Y0. (62)

In the Runge–Kutta Taylor SPH algorithm, the failure load is calculated as in the example of thevertical cut. To examine the accuracy of our model, the load-displacement relationship is plotted in(Figure 22)

The load–displacement relationship curve gives a response that converges to the analytical solu-tion of the problem. The curve rises linearly until 50, 000 N and then shows nonlinear behavior,which correspond to plastic loading, until it reaches the collapse load. The analytical failure load is89, 027Nand the estimated failure load given by our model is 90, 650N.

The failure of the shallow stratum is shown by the evolution of the deviatoric viscoplastic strains(Figure 23). The viscoplastic deformations start to accumulate along a shear band (time t D 0.4 s/.Then the shear band is reflected on the left side of the shallow stratum. The reflected shear bandfollows a circular line (timet D 1 s and t D 1.53 s/. The second shear band develops until reachingthe free-surface (time t D 2.18 s and t D 3.2 s/. The mechanism of failure of this example is

Figure 24. Displacement contours on the deformed mesh (Factor of deformation �2).



complicated because of the reflection of the straight shear band in a new circular shear band. TheRunge–Kutta Taylor SPH is able to reproduce such mechanism.

The different steps of the mechanism of failure can be observed in Figure 24. First the left-uppertriangle failed. The boundary condition on the left border imposed this block to move down and thusit presses the shallow stratum. The pressure causes the failure of the second bloc along a circularline. The second block is pushed up.

This example shows that the Runge–Kutta Taylor SPH algorithm is able to reproduce complicatedmechanisms of failure. Viscoplastic deformations are intercepted in two sharp shear bands, whichhave different orientations and shapes. The deformations reach 6%. Although the deformations aresmaller than in other examples, large deformations are calculated in this example.

7. CONCLUSIONS

Failure of geomaterials under dynamic condition presents several important limitations, such asnumerical diffusion and dispersion, volumetric locking, dependence on mesh alignment, and useof elements with the same order of interpolation for the displacements and the pressures, amongothers. Here, we proposed a novel SPH algorithm that is based on a first-order system of PDEsinvolving velocities and stresses. The scheme is similar to the Taylor–Galerkin algorithm, whichwas introduced by the authors for soil dynamics problems, but in this case, the special discretizationis performed via the SPH method.

The main advantage of the proposed method is the better accuracy because of using stresses anddisplacements as nodal variables, which avoid the tensile instability. The proposed algorithm hasbeen assessed against a set of cases ranging from elastic propagation of waves in a bar to failureof a vertical slope in a softening cohesive soil. From the examples we have presented, it can beconcluded that:

(i) The proposed model has good propagation properties in the dynamics range.(ii) It avoids the tensile instability in a natural manner.

(iii) It provides good results for geotechnical problems.(iv) It provides a good definition of shear bands on softening materials.(v) It is useful to analyze large deformations problems.

The authors have shown in the paper a series of examples where localization appears. What isimportant to notice is that, by using a viscoplastic model, the dependence on the discretizationlength (element size in finite elements or distance between nodes here) is avoided. This spuriousdependence on node spacing appears when using classical elastoplastic models, and it is avoidedby using improved descriptions of the continuum, such as Cosserat or nonlocal models. In all thecases considered in this paper, Perzyna’s viscoplasticity has been used. It is well known (and theauthors have illustrated it in [18,19]) that the shear bandwidth does not tend to zero as the mesh sizedecreases, keeping a constant value, which can be considered as an internal length depending onlyon constitutive parameters of the model.

Finally, an improved algorithm for obtaining boundary nodes and normal vectors has beenproposed.

APPENDIX

7.1. Detection of boundary nodes

The boundary detection algorithm used here comes from the work of Marrone et al. [66], and con-sists of using the renormalization matrix proposed by Randles and Libersky [62] and then used byChen et al. in the Corrective Smoothed Particle Hydrodynamics method [38], to approximate thenormal to the surface. The renormalization matrix is

A.xI /D

"XJ

mJ

�JrWJ .xI/˝ .xJ � xI/

#�1. (63)



And the approximation of the normal for a SPH node I is given by

n.xI /D�.xI /

j� .xI/jwith �.xI/D�A .xI/

XJ

mJ

�JrWJ .xI/. (64)

From the approximated normal, a scan region is defined for each particle (Figure 25). Then thealgorithm checks if particles are located inside the scan region of particle I [66].

To define the scan region, the algorithm calculates:

- The coordinates of point T , which is located at a distance h of the particle I in the normaldirection (Figure 25).

- The unit vector nT perpendicular to n (Figure 25).

The conditions to assess if particle I belongs or not to the free-surface are8̂̂<ˆ̂:8J 2N

hjxIJ j>

p2h, jxJT j< h

i) I … F

8J 2NhjxIJ j<

p2h, jn � xJT j C jnT � xJT j< h

i) I … F

otherwise ) I 2 F

. (65)

If the first condition is true, the particle J is situated in the dark grey part of the scan region (S1).If the second condition is fulfilled, the particle J is in the light grey. If any particle J is inside thescan region of the particleI , then the particle I does not belong to the free-surface, F and particleI is placed in subset 2. If no particle is within the scan region, the concerned particle I is assignedto subset 1. If an SPH node belongs to the subset 1 but velocity boundary conditions are applied onit, this node passes from subset 1 to subset 2 because it is not situated on the free-surface but onfixed or imposed velocity boundary. At the end of the first step of the algorithm the SPH nodes areseparated into two groups:

- Particles of subset 1 form the free-surface F- Particles of subset 2 are located inside the problem domain or situated on a boundary, which

does not have the properties of the free-surface.

7.2. Computing unit normal vectors to free surface

The second step of the algorithm consists of calculating the unit normal vector to the free-surfacedetected in the first step. Marrone et al. used the approximation of the normal to each SPH nodegiven by the Equation (6). However, in our case, this approximation of the normal is not accurateenough. The reasons for the inaccuracy that we have obtained with Equation (6) could be the choiceof the smoothing function and the small smoothing length employed in our implementation. Toovercome this problem, we implemented the following algorithm to determine the normal to thefree surface.

Figure 25. Sketch of the scan region.



Figure 26. Method to calculate the normal to the free-surface.

The unit normal vector to the free-surface at particle I ,nI , is calculated obtaining the mean aver-age of the vectors, fIJ , perpendicular to the vector, pIJ formed by particle I and particle J . Theparticle J , which steps in the calculation of the average, must belong to the free-surface and to thesupport domain of particle I (Figure 26)

nI D1

NP

NPXJD1

fIJ , (66)

where NPis the set of SPH nodes neighbors of particle I and belonging to the free-surface.Vectors fIJ are obtained from pIJ

pIJ D xJ � xI (67)

ensuring that they are pointing outwards.

ACKNOWLEDGEMENTS

The authors would like to gratefully acknowledge the support of the Spanish Ministry of Science and Tech-nology (Project GEODYN BIA2009-14225-C02-01), the Spanish ministry of Education for the FPU grantawarded to the first author, and to the EC (Project SAFELAND FP7-ENV-2008-1 226479).

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