laxwendroff_updatedlagrangian

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Computational Mechanics manuscript No.(will be inserted by the editor)

Eulerian SPH formulation for large distortion problems

Yolanda Vidal1, Javier Bonet2, Antonio Huerta1

1 Laboratori de Calcul Numeric (www-lacan.upc.es), Departament de Matematica Aplicada III, Universitat Politecnica deCatalunya, Jordi Girona 1, Barcelona, Spain.

2 Civil and Computational Engineering Center (C2EC), University of Wales Swansea, Singleton Park, Swansea, U.K.

Received: date / Revised version: date

Abstract The objective of this paper is to presentan Eulerian formulation which is capable to solve prob-lems with extremely large distortions. Smooth particlehydrodynamics with a total Lagrangian formulation arewell-suited for solving large distortion problems in solids.Nevertheless, in problems with severe distortions a La-grangian formulation will require updates of the refer-ence configuration. However a standard updated Lagrang-ian formulation suffers the presence of zero-energy modesthat are more likely to be activated. The classical SPHformulation defining a fixed support in the laboratoryfor each particle, and thus recomputing neighbors at

each time-step (i.e. updated neighbor search) that willbe called here Eulerian formulation, presents tension in-stability, see for instance [12] and [3]. In this paper anew Eulerian formulation based in a conservative form ofthe continuum equations is proposed. This new approachprecludes the numerical instabilities existent in standardupdated Lagrangian and Eulerian formulations. Somenumerical examples demonstrate the capability of theproposed formulation to solve fast-transient dynamicsproblems with extremely large distortions.

1 Introduction

The impact of a fluid-filled container to a structure at ve-locities of the order of several hundred meters per secondmay result in numerous complex phenomena, such as thetearing of the container followed by fluid leaking throughthe openings. The simulation of such phenomena usingtraditional tools such as the finite element method is ex-tremely difficult: large deformations in the fluid, sloshingeffects in the container, the multiplication of impacts and

the cracking of solids are complex and costly to model Partially supported by MCYT, Spain. Grant Contract:

DPI2007-62395

with a method based on a fixed mesh, particularly be-cause of remeshing problems. The acknowledgement ofthis problem triggered the development of what is calledmeshless methods.

Among these methods, the smoothed particle hydro-dynamics (SPH) method was one of the first proposed. Ithas a very wide range of applications. For instance, theLagrangian SPH code has been applied successfully tohigh strain problems [11], the SPH algorithm has beenincorporated into a standard Lagrangian code such asEPIC [9], and it has also been used for impact prob-lems [14,10]. The method, however, has the following

two major drawbacks: it can lend itself to numerical in-stabilities, and its consistency is guaranteed only if thediscretization is formulated very carefully. These ques-tions are widely discussed in the literature. For example,consistency problems were gradually resolved thanks tonormalizing methods [8,10] and, more recently, the Cor-rected Smooth Particle Hydrodynamics method allows toobtain linear consistency in the interpolation of the func-tion and in the interpolation of the gradient (see [2]).Numerous solutions were also proposed to eliminate sta-bility problems. See [7,15,1] for a detailed study of sta-bility problems.

The total Lagrangian SPH is much more stable thanthe usual updated Lagrangian equation and is also moreefficient. The shape functions are determined on the ini-tial configuration and calculated once and for all, andthe determination of neighbors, which is a very time-consuming task, needs to be carried out only during thefirst time step. This approach is clearly unsuitable forthe modeling of continua undergoing very large strainsbecause the environment of a particle can change dras-tically during the simulation, but it works well for mod-erate strains. Vidal et al. [16] proposed an intermedi-ate model that consists in updating nodal positions onlyfrom time to time rather than at every time step. Thislast method enables the simulation of large strains whileavoiding tensile instability. However, a stable Eulerianformulation capable to model extremely large distortion


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2 Yolanda Vidal et al.

problems is still missing. This is the purpose of this pa-per.

The remainder of the paper is structured as follows.Firstly, the total Lagrangian corrected SPH formulation

is revised for large strains dynamic problems. In Sect. 3the problem is reformulated in conservative form. Thiswill be the key idea of the paper. Sect. 4 studies theSPH formulation of the Lax-Wendroff two-step methodwhich is a second-order accurate, explicit, time-steppingalgorithm. Sect. 5 describes how to introduce a locallycontrolled dissipation in the Lax-Wendroff method. Theproposed modification combines the good resolution of-fered by a second-order time scheme and the capabil-ity to damp out the non-physical oscillations. Finally inSect. 6, some numerical tests are shown demonstratingthe performance of the proposed methodology.

2 Total lagrangian corrected SPH (CSPH)

This section will not be devoted to develop or discussLagrangian corrected SPH formulation in detail. For anexcellent reference see [3]. Here some basic notions willbe recalled in order to introduce the notation and theapproach employed in following sections.

We consider the motion of a continuum defined bya mapping between a reference volume V and a currentvolume v(t) of the form x = (X, t). Here, X is thematerial coordinate in the reference configuration and xdenotes the position of particle X at time t.

The Jacobian of the transformation is the deforma-tion gradient tensor, F, and is given by

F =x

X, or Fij =

xiXj

.

The determinant ofF is denoted by J and relates thevolume elements in the reference and current configura-tions. That is, ifdV = dX1dX2dX3 and dv = dx1dx2dx3,then dv = JdV. Since mass is conserved, the density inthe current configuration , needs to adjust itself to ac-

count for volumetric changes. Therefore, dm =dv = 0dV, where 0 is the density in the reference configura-

tion, and consequently, the conservation of mass reducesto J = 0.

The material velocity, v(X, t), and the linear mo-mentum per unit of reference volume, p(X, t), are givenas

v(X, t) =x

t, p(X, t) = 0v.

For reversible problems in elastodynamics, it is sufficientto consider p(X, t) and F(X, t) as problem variables forwhich standard conservation laws can be derived as it is

shown in sections 3.1 and 3.2.Let us consider a discretized body using SPH parti-

cles. The mapping between initial (or reference) and

current positions can be approximated using SPH ap-proximation as,

xb = (Xb, t) = c Vc CXb Xc

xc.The deformation gradient can be evaluated now in a cer-tain particle b in terms of the current positions just tak-ing derivatives as

Fb = 0 =c

xc GT

c(Xb), (1)

where 0 indicates the gradient respect to the initialconfiguration, xc is the current position of particle c andwhere the functions G contain the corrected kernel gra-dients at the initial configuration, that is,

Gc(Xb) = Vc 0Xb Xc

,

where 0 is a corrected gradient to ensure linear com-pleteness as shown in [2].

In order to find general equations for the internalforces using a Lagrangian corrected SPH formulation,consider the equation of the internal virtual work inthe reference configuration in terms of the first Piola-Kirchhoff tensor, P,

wint = V0 P : FdV0 b V0

b Pb :

Fb. (2)

The variation of the virtual deformation gradient emergesfrom equation (1) as

Fb =c

vc GT

c(Xb),

where after substituting into (2) leads to the expressionof the internal virtual work

wint b

V0b Pb :

c

vc GT

c(Xb)

=c

vc

b

V0b Pb Gc(Xb).

This expression allows the vector of internal forces cor-responding to a certain particle a to be identified as:

Ta =b

V0b Pb Ga(Xb). (3)

It is important to observe that in (3) the kernel deriva-tives, Ga(Xb), are fixed in the reference configurationand therefore they do not depend on the current posi-tions of the particles. This implies that corrections areonly calculated at the beginning reducing the computa-tional cost.


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Eulerian SPH formulation for large distortion problems 3

3 Conservation equations

3.1 Convservation of linear momentum

The conservation of linear momentum for an arbitraryreference volume V is expressed in integral form as (nobody forces):

d

dt

V

p dV =

V

t dA,

where t denotes the traction vector in the current con-figuration associated to an element of area dA in the ref-erence configuration. The stress tensor that relates thistraction vector to the unit normal N in the reference

configuration is the first Piola-Kirchhoff stress tensor P,

t = PN.

Substituting this expression into the momentum conser-vation law and making use of Gauss theorem leads tolocal momentum conservation law or equilibrium equa-tion as,

p

t P = 0 . (4)

Here, denotes the gradient operator in the reference

space, i.e. X . Note the partial time derivative ofp(X, t) in the equation above is taken at constant Xand hence it denotes a material time derivative.

3.2 Conservation of the deformation gradient

A conservation equation for the deformation gradient, F,can be easily derived by noting that the time derivativeofF is related to the linear momentum vector, p = 0v,by:

F = (p/

0)X

. (5)

We note that the corresponding integral form, obtainedvia the application of the Gauss theorem on the righthand side term, gives the average rate of change of thedeformation gradient in a reference volume in terms ofthe velocity vectors on the surface surrounding this vol-ume as:

d

dt

V

FdV =

V

1

0pNdA.

This can be considered as a generalization of the volu-metric conservation law, or continuity equation, used forfluids.

3.3 Conservation-Law formulation

The conservation laws for linear momentum (4) and de-formation gradient (5) can be combined into a single

system of first order equations as,

U

t+

FiXi

= 0 ,

where

U =

p1p2p3F11F12F13

F21F22F23F31F32F33

,F

i =

P1i(F)P2i(F)P3i(F)i1p1/

0

i2p1/

0

i3p1/

0

i1p2/

0

i2p2/

0

i3p2/

0

i1p3/

0

i2p3/

0

i3p3/

0

, for i = 1, 2, 3 .

In particular, in a total Lagrangian setting (no bodyforces) the conservation-law system reads,

0

v

tX P = 0

FtXv = 0.

In an updated Lagrangian setting where the referenceconfiguration is the position at step k the conservation-law system then reads,

(J1c 0v)

tk (J

1c PF

kT) = 0 (6a)

G

tkv = 0 ; F = GF

k, (6b)

where k and k are the gradient and divergence oper-

ators at reference k, i.e.

xk . It is noteworthy that fork = 0 the algorithm is total Lagrangian and for k = n itbecomes Eulerian which is the most interesting case.

4 SPH formulation of the Lax-Wendrofftwo-step method

4.1 Scalar conservation equation

For sake of simplicity let us present the SPH formulationof the Lax-Wendroff two-step method by considering the

1D conservative equation

U

t+

F

x= 0.


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In a finite differences approach the Lax-Wendroff two-step method reads

Un+1/2a =1

2

(Una+1 + Una1)

t

2

(Fna+1 Fna1)

2x

(7a)

Un+1a = Una t

(Fn+1/2a+1 F

n+1/2a1 )

2x(7b)

where a represents a node in the finite difference grid. Forfinite differences this algorithm is second order accurateand possesses the so-called unit CFL property (it is con-ditionally stable under the condition CFL 1). Recallthat CFL stands for the initials of Courant, Friedrichsand Lewy authors of the paper [5]. The Lax-Wendrofftwo-step method has been also used in finite elements.In [6] an analysis of the classical time-stepping schemesis presented. It is investigated the stability and accuracyproperties of the Lax-Wendroff two-step method com-bined with the Galerkin finite element formulation. Itcomes out that when using a diagonal mass matrix thediscrete equations obtained in 1D on a uniform meshof linear elements are identical to those obtained withsecond-order central differences. The scheme is secondorder accurate and possesses the so-called unit CFL prop-erty. It is noteworthy that the Lax-Wendroff two-stepmethod is equivalent to the Lax-Wendroff one-step meth-od for the linear convection equation. However, for non-linear PDEs, systems of PDEs, and two and three di-mensional physical spaces, the Lax-Wendroff two-step

method is much easier to apply than the one-step ver-sion. The explicit Lax-Wendroff two-step method canbe used in a straightforward manner to solve nonlin-ear PDEs, systems of PDEs, and multidimensional prob-lems.

From expressions (7a) and (7b) emerges the Lax-Wendroff method for the CSPH method as

Un+1/2a = Sa(Un)

t

2

F

x

na

(8a)

Un+1a = Una t

F

x

n+1/2

a(8b)

where a now represents a particle, the derivatives areapproximated using CSPH approximation and the func-tion Sa is a smoothing function such that Sa() = when is lineal. A detailed description on how to choosethis smoothing function is given in the next section. TheLax-Wendroff CSPH method using linear reproducibil-ity and a dilation parameter /h 1 resembles the Lax-Wendroff method for linear finite elements with a diag-onal mass matrix. So, in this case, the method is sec-ond order accurate and possesses the so-called unit CFLproperty.

The accuracy of the numerical schemes can be as-sessed by comparing numerical and exact damping andphase values. The relative phase errors are depicted inFigure 1.

Fig. 1 Relative phase error in pure convection of the LaxCSPH method (left) and the Lax-Wendroff CSPH method(right).

STABILITY ANALYSIS (Finite Differences equivala SPH agafant suport de la window function tal quenoms tres partcules dintre (la prpia i dues venes)?????

4.2 Smoothing function

The smoothing function Sa is defined such that Sa() =, when is a lineal function. Lets start with

Sa() = (1 )a + b

Vref

b bWb(xref

a ),

and lets impose that

Sa()

a= 0.

That is,

(1 ) + Vrefa Wa(xrefa ) = 0

which leads to,

= 11 Vrefa Wa(x

refa )

, (1) = Vrefa Wa(xrefa )

1 Vrefa Wa(xrefa )

,

so the smoothing function must be defined as

Sa() =

b V

refb bWb(x

refa ) V

refa aWa(x

refa )

1 Vrefa Wa(xrefa )

.

4.3 Algorithms

The total Lagrangian Lax-Wendroff algorithm emergesfollowing equations (8a) and (8b) for the velocity anddeformation gradient,

Fn+1

2a = Sa(F

n) +t

2

Xb

V0b vnb 0Wb(x

0a)

vn+1/2a = Sa(v

n) t

2ma

Xb

V0a V0b P(F

nb )0Wa(x

0b)

vn+1a = v

na

t

ma

Xb

V0a V0b P(F

n+1/2b )0Wa(x

0b)

Fn+1a = F

na + t

Xb

V0b vn+1/2b 0Wb(x

0a)

x

n+1

a = x

n

a + tv

n+1/2

a .

Observe that the position of the particles is updated usingthe leapfrog method. From the previous equations and from


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Eulerian SPH formulation for large distortion problems 5

equations (6a) and (6b) with k = n, it readily comes availablethe Eulerian Lax-Wendroff algorithm,

Gn+ 1

2a = I+

t

2 Xb Vnb v

nb nWb(x

na)

Fn+ 1

2a = G

n+ 12

a Sa(Fn)

vn+ 1

2a = Sa(v

n)t

2ma

Xb

V0a Vnb J

1

n,bP(Fnb )F

nbTnWa(x

nb )

vn+1a = v

na

t

ma

Xb

V0a Vnb J

1

n,bP(Fn+ 1

2

b )FnbTnWa(x

nb )

Gn+1a = I+ t

Xb

Vnb vn+ 1

2

b nWb(xna)

Fn+1a = G

n+1a F

na

xn+1a = x

na + tv

n+12

a .

5 Locally controlled dissipation

The method to locally introduce a controlled dissipation usesthe concepts of flux-corrected transport, see references [4]and [13]. The proposed method combines the good resolutionoffered by a high-order scheme together with the capability todamp out the non-physical oscillations of a low-order scheme.For sake of simplicity the 1D case is considered as in section4.1.

Recall from (8b) that the second step of the CSPH Lax-Wendroff method reads at a given particle a,

Un+1a = Una t

F

x

n+1/2a

,

which can be rewritten in a general form as,

Un+1 = Un + U, (9)

where U is the increment of the unknowns obtained for thegiven scheme at time t = tn. Our aim is to obtain a U ofas high an order as possible without introducing overshoots.To this end, we rewrite equation (9) as

Un+1 = Un + Ul + (Uh Ul), (10a)

Un+1

= Ul

+ (Uh

Ul

). (10b)

Here Uh and Ul denote the increments obtained bysome high- and low-order shemes, respectively, whereas Ul

is the ripple-free solution at time t = tn+1 of the low-orderscheme. The idea behind FCT is to limit the second term onthe right-hand side of equation (10b). That is, for a givenparticle a,

Un+1a = daUla + (1 da)(U

ha U

la),

where da is chosen in each particle in such a way that thehigh-order scheme is reduced to low-order near oscillations.The second order Lax-Wendroff scheme is used as high-order

scheme. The low-order method is defined as

Ula = Una +

Xb

MCabUnb M

LaaU

na

!ma

MLaa MCaa,

where the mass matrices in the previous equation are definedas

MCab =

XcmcNa(xc)Nb(xc) M

Laa =

XcmcNa(xc).

When da = 0 the second order Lax-Wendroff scheme is used.When da = 1 a low order scheme is used. The value of da ischosen as

da = min (1,Xmax2v

x2

b

)

where X is an adjustable parameter. In this way, an effectivesensor is constructed by considering the second derivative ofthe velocity.

When dealing with multidimensional problems the localmodulation coefficient is computed and applied in each di-rection.

6 Numerical examples

6.1 Example 1: Propagation of a cosine pro?le

A simple 1D problem is proposed to illustrate and comparethe performance of the proposed schemes. The pure convec-tion equation

ut + aux = 0 (11)

is solved over the spatial interval ]0, 1[ considering the follow-ing initial:

u(x, 0) =

1

2(1 + cos ((x x0)/)) if |x x0| ,

0 otherwise

and boundary condition: u(0, t) = 0 for t 0, where x0 = 0.2and = 0.12.

The exact solution of equation 11 with a=1 correspondsto the translation to the right of the initial profile at unitspeed. Figure 2 compares the numerical solutions obtainedat time t=0.6 using a particle distribution of 50 uniformlydistributed particles and different values of the CFL num-ber. The problem is solved using the Lax CSPH method andthe Lax-Wendroff CSPH method. Figure 2 shows that Lax-

Fig. 2 Propagation of a cosine profile: comparison betweenthe exact solution (dotted line) and the Lax and Lax-

Wendroff CSPH solutions.

Wendroff scheme exhibits a better phase accuracy than theLax scheme. Note that the Lax scheme cannot be operatedwith CFL> 1 and that the Lax-Wendroff scheme cannot beoperated with CFL> 2. Moreover, the Lax-Wendroff schemesshows a phase lead at CFL= 2. These findings are fully con-sistent with the phase error diagrams in Figure 1.

6.2 Example 2: 1D bar test

The one dimensional bar test proposed in [7] is reproduced.A simple elastic bar (Youngs modulus, E=200e9 Pascals) isfixed at the left end, A, and the right quarter of the bar isgiven an initial velocity of v0 = 5m/s thus putting the bar


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Fig. 3 1D bar problem statement.

in tension initially. The density of the bar is =7833 Kg/m3.Standard SPH methods cannot solve this problem due totension instability that immediately develops.

The problem is solved using a uniform distribution ofparticles. As shown in Figure 3 the corrected SPH particledistribution is very coarse with only 40 uniform particles.A comparison between the solutions obtained with the totalLagrangian and Eulerian approaches can be seen in figure 4.The displacement and velocity history of the end particle, B,is plotted. Total time computed is 5e 4s.

0 1 2 3 4

x 104

5

4

3

2

1

0

1

2

3

4

5

x 105

time

0 1 2 3 4

x 104

5

4

3

2

1

0

1

2

3

4

5x 10

5

time

Fig. 4 Total Lagrangian (left) and updated Lagrangian(right). Results using 40 particles and an FCT parameterof 5.5e07 (updated method). The black line represents dis-placement history of the right end (point B) and the blueline represents its velocity history scaled by a factor of 1e5in order to compare it with the displacement history in thesame graph.

6.3 Example 3: punch test

7 Concluding remarks

This contribution proposes a new Eulerian formulation forthe corrected SPH method. The main idea is based in aconservative form of the continuum equations. First, a SPHformulation of the Lax-Wendroff two-step scheme is devel-oped. Afterwards, a method which introduces a locally con-trolled dissipation in the Lax-Wendroff method is described.The proposed modification combines the good resolution of-fered by a second-order time scheme and the capability todamp out the non-physical oscillations. This new approach

precludes the numerical instabilities existent in standard up-dated Lagrangian and Eulerian formulations. The results inthe analyzed examples, both academical and practical, demon-strate the performance of the proposed approach.

References

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