Advances in stabilized ﬁnite element and particle methods...

www.elsevier.com/locate/cma

Comput. Methods Appl. Mech. Engrg. 195 (2006) 6750–6777

Advances in stabilized finite element and particle methodsfor bulk forming processes

E. Onate *, J. Rojek, M. Chiumenti, S.R. Idelsohn, F. Del Pin, R. Aubry

International Center for Numerical Methods in Engineering, Universidad Politecnica de Cataluna, Gran Capitan s/n, 08034 Barcelona, Spain

Received 4 July 2004; received in revised form 28 October 2004; accepted 31 October 2004

Abstract

The paper describes some recent developments in finite element and particle methods for analysis of a wide range of bulk formingprocesses. The developments include new stabilized linear triangles and tetrahedra using finite calculus and a new procedure combiningparticle methods and finite element methods. Applications of the new numerical methods to casting, forging and other bulk metal form-ing problems and mixing processes are shown.� 2005 Elsevier B.V. All rights reserved.

Keywords: Bulk forming processes; Stabilized finite element method; Particle method; Particle finite element method; Mixing processes

1. Introduction

The development of efficient and robust numerical methods for analysis of bulk forming problems has been a subject ofintensive research in recent years [1–7]. Many of these problems require the solution of incompressible fluid flow situations(such as in mould filling problems) whereas in other cases (such as forging, rolling, extrusion, etc.) the numerical methodmust be able to account for the quasi/fully incompressible behaviour induced by the large plastic deformation. The solutionof these problems has motivated the development of the so called stabilized numerical methods overcoming the two mainsources of instability in the analysis of incompressible continua, namely those originated by the high values of the convec-tive terms in fluid flow situations and those induced by the difficulty in satisfying the incompressibility condition.

Different approaches to solve both type of problems in the context of the finite element method (FEM) have beenrecently developed [8]. Traditionally, the underdiffusive character of the Galerkin FEM for high convection flows has beencorrected by adding some kind of artificial viscosity terms to the standard Galerkin equations [8,9].

A popular way to overcome the problems with the incompressibility constraint in the FEM is by introducing a pseudo-compressibility in the continuum and using implicit and explicit algorithms ad hoc such as artificial compressibility schemes[10] and preconditioning techniques [11]. Other FEM schemes with good stabilization properties for the convective andincompressibility terms in fluid flows are based in Petrov–Galerkin (PG) techniques. The background of PG methodsare the non-centred (upwind) schemes for computing the first derivatives of the convective operator in FD and FV methods[8,9,12]. A general class of Galerkin FEM has been developed where the standard Galerkin variational form is extendedwith adequate residual-based terms in order to achieve a stabilized numerical scheme. Among the many FEM of this kindwe can name the Streamline Upwind Petrov Galerkin (SUPG) method [8,13–16], the Galerkin Least Square (GLS) method

0045-7825/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2004.10.018

* Corresponding author. Tel.: +34 9 3205 7016; fax: +34 9 3401 6517.E-mail address: [email protected] (E. Onate).URL: http://www.cimne.upc.es (E. Onate).

mailto:[email protected]

http://www.cimne.upc.es

E. Onate et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 6750–6777 6751

[8,17], the Taylor–Galerkin method [18], the Characteristic Galerkin method [19] and its variant the Characteristic BasedSplit (CBS) method [20,21], the pressure gradient operator method [22] and the Subgrid Scale (SS) method [23]. A goodreview of these methods can be found in [24]. Extensions of the CBS and SS methods to treat incompressible problemsin solid mechanics are reported in [25,26,58] and [27–29], respectively.

In this paper a different class of stabilized FEM for quasi and fully incompressible fluid and solid materials applicable toa wide range of bulk forming problems is presented. The starting point is the modified governing differential equations ofthe continuous problem formulated via a finite calculus (FIC) approach [30,31]. The FIC method is based in invoking theclassical balance (or equilibrium) laws in a domain of finite size. This introduces naturally additional terms in the differ-ential equations of infinitesimal continuum mechanics which are a function of the balance domain dimensions. The newterms in the modified governing equations provide the necessary stabilization to the discrete equations obtained via thestandard Galerkin FEM. One of the main advantages of the FIC formulation versus other alternative approaches (suchas mixed FEM, etc.) is that it allows to solve incompressible fluid problems using low order finite elements (such as lineartriangles and tetrahedra) with equal order approximations for the velocity and pressure variables [32–35]. The FIC formu-lation has been successfully used for analysis of fully or quasi incompressible solids [36,37].

The layout of the paper is the following. In the next section the basic FIC equations for incompressible flow problemsformulated in an Eulerian frame are presented. The finite element discretization is introduced and the resulting discretizedequations are detailed. A fractional step scheme for the transient solution is presented.

The stabilized Eulerian formulation is extended to account for thermal effects and the transport of the free surface whichare needed for mould filling processes.

The following sections outline the FIC formulation for analysis of quasi/fully incompressible solids using a Lagrangiandescription and linear triangles and tetrahedra. An explicit algorithm for integrating in time the equations of motion ofelasto-plastic solids in large-strain problems involving frictional contact is described. Examples of application to a castingproblem and some bulk forming processes are presented.

In the last part of the paper a Lagrangian formulation for fluid flow analysis is presented as a straightforward extensionof the formulation for solid mechanics. The procedure, called the particle finite element method (PFEM) [38–40], treats themesh nodes in the fluid and solid domains as dimensionless particles which can freely move, an even separate from the mainfluid domain, representing, for instance, the effect of liquid drops. A finite element mesh connects the nodes defining thediscretized domain where the governing equations are solved in the standard FEM fashion. The main advantage of theLagrangian flow formulation is that the convective terms do not enter in the fluid equations. The difficulty is howevertransferred to the problem of adequately (and efficiently) moving the mesh nodes. The final examples show that thePFEM is a promising method to solve mould filling and casting problems, material mixing processes and many otherbulk metal forming problems involving the interaction between solids and fluids which can be treated with the sameformulation.

2. FIC equations for viscous incompressible flows

The FIC governing equations for a viscous incompressible fluid can be written in an Eulerian frame of reference as[30–34]

Momentum

rmi �1

2hj

ormi

oxj¼ 0 in X. ð1Þ

Mass balance

rd �1

2hj

ord

oxj¼ 0 in X; ð2Þ

where

rmi ¼ qovi

otþ vj

ovi

oxj

� �þ op

oxi� osij

oxj� bi; ð3Þ

rd ¼ovi

oxii; j ¼ 1; nd . ð4Þ

Above X is the analysis domain which can evolve with time, nd is the number of space dimensions (nd = 3 for 3D prob-lems), vi is the velocity along the ith global axis, q is the (constant) density of the fluid, p is the absolute pressure (definedpositive in compression), bi are the body forces and sij are the deviatoric stresses related to the viscosity l by the standardexpression

6752 E. Onate et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 6750–6777

sij ¼ 2l _eij � dij1

3

ovk

oxk

� �; ð5Þ

where dij is the Kronecker delta and the strain rates _eij are

_eij ¼1

2

ovi

oxjþ ovj

oxi

� �. ð6Þ

The boundary conditions are written in the FIC approach as

njrij � tpi þ

1

2hjnjrmi ¼ 0 on Ct; ð7Þ

vj � vpj ¼ 0 on Cu ð8Þ

and the initial condition is vj ¼ v0j for t = t0.

Summation convention for repeated indexes in products and derivatives is used unless otherwise specified.In Eqs. (7) and (8) tp

i and vpj are surface tractions and prescribed velocities on the boundaries Ct and Cu, respectively, nj

are the components of the unit normal vector to the boundary and rij are the total stresses given by rij = sij � dijp.Eqs. (1) and (2) are obtained by invoking the classical balance equations in fluid mechanics in a domain of finite size and

retaining higher order terms [30]. The h0is in above equations are characteristic lengths of the domain where the balance ofmomentum and mass is enforced. In Eq. (7) these lengths define the domain where equilibrium of boundary tractions isestablished. In the discretized problem the hi coincide with a typical element dimension, as described in Section 5. Notethat by making hi = 0 in these equations the standard infinitesimal form of the fluid mechanics equations is recovered[8,9,24].

Eqs. (1)–(8) are the starting point for deriving stabilized FEM for solving the incompressible Navier–Stokes equationsusing equal order interpolation for the velocity and pressure variables [32–35]. Application of the FIC formulation tomeshless analysis of fluid flow problems using the finite point method can be found in [42].

Remark. In most metal forming processes the viscosity l is a non linear function of the strain rate and the yield stress ofthe material (which also depends on the temperature) [8,43,44]. This dependence adds another non linearity to the problem.

Remark. In Eqs. (1)–(7) X and Ct respectively denote the actual volume and the boundary of the domain where the gov-erning equations are solved at each instant of the forming processes. This domain is considered here to be fixed in space(Eulerian approach). Moving free surfaces, such as in the case of mold filling processes, can be modelled by using standardlevel set and volume of fluid (VOF) techniques [47,48]. An alternative is to use an arbitrary Lagrangian–Eulerian (ALE)description or even a fully Lagrangian description to follow the motion of the particles during the forming process. TheLagrangian description is typical in the case of solid mechanics problems (Section 6). A particular Lagrangian formulationfor fluid flow problems involving large motion of the free surface, named the particle finite element method, is presented inSection 8.

2.1. Stabilized integral forms

From the momentum equations it can be obtained [32–34]

ord

oxi’ hj

2ai

ormi

oxj; no sum in i; ð9Þ

where

ai ¼2l3þ qvihi

2; no sum in i. ð10Þ

Substituting Eq. (9) into Eq. (2) and retaining the terms involving the derivatives of rmi with respect to xi only, leads to thefollowing expression for the stabilized mass balance equation

rd �Xnd

i¼1

siormi

oxi¼ 0; ð11Þ

with

si ¼8l

3h2i

þ 2qvi

hi

!�1

. ð12Þ


The si’s in Eq. (12), when scaled by the density, are termed in the stabilization literature intrinsic time parameters.The weighted residual form of the momentum and mass balance equations (Eqs. (1) and (11)) is written asZ

Xdvi rmi �

hj

2

ormi

oxj

� �dXþ

ZCt

dvi rijnj � tpi þ

hj

2njrmi

� �dC ¼ 0; ð13ÞZ

Xq rd �

Xnd

i¼1

siormi

oxi

" #dX ¼ 0; ð14Þ

where dvi and q are arbitrary weighting functions representing virtual velocities and virtual pressure fields. Integrating byparts the rmi terms leads toZ

Xdvirmi dXþ

ZCt

dviðrijnj � tiÞdCþZ

X

hj

2

odvi

oxjrmi dX ¼ 0; ð15aÞZ

Xqrd dXþ

ZX

Xnd

i¼1

sioqoxi

rmi

" #dX�

ZC

Xnd

i¼1

qsinirmi

" #dC ¼ 0. ð15bÞ

We will neglect hereonwards the third integral in Eq. (15b) by assuming that rmi is negligible on the boundaries. Thedeviatoric stresses and the pressure terms in the first integral of Eq. (15a) are integrated by parts in the usual manner.The resulting momentum and mass balance equations areZ

Xdviq

ovi

otþ vj

ovi

oxj

� �þ odvi

oxjl

ovi

oxj� dijp

� �� dX�

ZX

dvibi dX�Z

Ct

dvitpi dCþ

ZX

hj

2

odvi

oxjrmi dX ¼ 0; ð16aÞZ

Xq

ovi

oxidXþ

ZX

Xnd

i¼1

sioqoxi

rmi

" #dX ¼ 0. ð16bÞ

In the derivation of the viscous term in Eq. (16a) we have used the following identity (prior to the integration by parts)

osij

oxj¼ 2l

oeij

oxj¼ l

o2vi

oxjoxj. ð17Þ

Eq. (17) is identically true for the exact incompressible limit ðovioxi¼ 0Þ.

2.2. Convective and pressure gradient projections

The computation of the residual terms can be simplified if we introduce now the convective and pressure gradient pro-jections ci and pi, respectively defined as

ci ¼ rmi � qvjovi

oxj;

pi ¼ rmi �opoxi

.

ð18Þ

We can express rmi in Eqs. (16a) and (16b) in terms of ci and pi, respectively which then become additional variables. Thesystem of integral equations is now augmented in the necessary number of equations by imposing that the residual rmi van-ishes (in average sense) for both forms given by Eq. (18). This gives the final system of governing equation asZ

Xdviq

ovi

otþ vj

ovi

oxj

� �þ odvi

oxjl

ovi

oxj� dijp

� �� dX�

ZX

dvibi dX�Z

Ct

dvitpi dC

þZ

X

hk

2

oðdviÞoxk

qvjovi

oxjþ ci

� �dX ¼ 0; ð19ÞZ

Xq

ovi

oxidXþ

ZX

Xnd

i¼1

sioqoxi

opoxiþ pi

� �dX ¼ 0; ð20ÞZ

Xdciq qvj

ovi

oxjþ ci

� �dX ¼ 0 no sum in i; ð21ÞZ

Xdpisi

opoxiþ pi

� �dX ¼ 0 no sum in i; ð22Þ


with i, j,k = 1,nd. In Eqs. (21) and (22) dci and dpi are appropriate weighting functions and the q and si weights are intro-duced for convenience.

We note that accounting for the convective and pressure gradient projections enforces the consistency of the formulationas it ensures that the stabilization terms in Eqs. (19)–(22) have a residual form which vanishes for the ‘‘exact’’ solution.Neglecting these terms lowers the accuracy of the numerical solution and it makes the formulation more sensitive tothe value of the stabilization parameters as shown in [34,36,37].

3. Finite element discretization

We choose C0 continuous linear interpolations of the velocities, the pressure, the convection projections ci and the pres-sure gradient projections pi over three node triangles (2D) and four node tetrahedra (3D) [8]. The linear interpolations arewritten as

vi ¼ N k�vki ; p ¼ Nk�pk;

ci ¼ N k�cki ; pi ¼ Nk�pk

i ;ð23Þ

where the sum goes over the number of nodes of each element n (n = 3/4 for triangles/tetrahedra), ð��Þk denotes nodal vari-ables and Nk are the linear shape functions [8].

Substituting the approximations (23) into Eqs. (19)–(22) and choosing the Galerkin form with dvi = q = dci = dpi = Ni

leads to following system of discretized equations

M _�vþH�v�G�pþ C�c ¼ f; ð24aÞGT�vþ L�pþQ�p ¼ 0; ð24bÞC�vþM�c ¼ 0; ð24cÞQT�pþ M�p ¼ 0. ð24dÞ

The form of the different matrices is given in the Appendix A.The solution in time of the system of Eqs. (24) can be written in general form as

M1

Dtð�vnþ1 � �vnÞ þHnþh�vnþh �G�pnþh þ Cnþh�cnþh ¼ fnþh; ð25aÞ

GT�vnþh þ Lnþh�pnþh þQ�pnþh ¼ 0; ð25bÞCnþh�vnþh þM�cnþh ¼ 0; ð25cÞGT�pnþh þ Mnþh�pnþh ¼ 0; ð25dÞ

where, for instance, Hnþh ¼ Hð�vnþhÞ, �vnþh are the velocities evaluated at time n + h and the parameter h 2 [0,1] (see Appen-dix A). The direct monolithic solution of Eqs. (25) is possible using an adequate iterative scheme. However, we have foundmore convenient to use a fractional step method as described in the next section.

3.1. Fractional step method

A fractional step scheme is derived by noting that the discretized momentum Eq. (25a) can be split into the two follow-ing equations

M1

Dtð~vnþ1 � �vnÞ þHnþh�vnþh � aG�pn þ Cnþh�cnþh ¼ fnþh; ð26Þ

M1

Dtð�vnþ1 � ~vnþ1Þ �Gð�pnþ1 � a�pnÞ ¼ 0. ð27Þ

In above equation ~vnþ1 is a predicted value of the velocity at time n + 1 and a is a variable whose values of interest arezero and one. For a = 0 (first order scheme) the splitting error is of order 0(Dt), whereas for a = 1 (second order scheme)the error is of order 0(Dt2) [45].

Eqs. (26) and (27) are completed with the following three equations emanating from Eqs. (25b)–(25d)

GT�vnþ1 þ Ln�pnþ1 þQ�pn ¼ 0; ð28aÞCnþ1�vnþ1 þM�cnþ1 ¼ 0; ð28bÞQT�pnþ1 þ Mnþ1�pnþ1 ¼ 0. ð28cÞ


The value of �vnþ1 obtained from Eq. (27) is substituted into Eq. (28a) to give

GT~vnþ1 þ DtGTM�1Gð�pnþ1 � a�pnÞ þ Ln�pnþ1 þQ�pn ¼ 0. ð29ÞThe product GTM�1G can be approximated by a Laplacian matrix, i.e.

GTM�1G ¼ 1

qL with Lab ¼

ZXe

$TN a$N b dX; ð30Þ

where Lab are the element contributions to L (see Appendix A).The steps of the fractional step scheme chosen here are:

Step 1. The fractional nodal velocities ~vnþ1 can be explicitly computed from Eq. (26) by

~vnþ1 ¼ �vn � DtM�1d ½Hn�vn � aG�pn þ Cn�cn � fn�; ð31Þ

where Md is the diagonal form of M obtaining by lumping the row terms into the corresponding diagonal terms.Step 2. Compute �pnþ1 from Eq. (29) as

�pnþ1 ¼ � Ln þ Dtq

L

� ��1

GT~vnþ1 � aDtq

L�pn þQ�pn

� �. ð32Þ

Step 3. Compute �vnþ1 explicitly from Eq. (27) as

�vnþ1 ¼ ~vnþ1 þ DtM�1d Gð�pnþ1 � a�pnÞ. ð33Þ

Step 4. Compute �cnþ1 explicitly from Eq. (28b) as

�cnþ1 ¼ �M�1d Cnþ1�vnþ1. ð34Þ

Step 5. Compute �pnþ1 explicitly from Eq. (28c) as

�pnþ1 ¼ �M�1d QT�pnþ1; ð35Þ

where Md is the lumped form of Mnþ1. A standard diagonal lumping procedure based in summing up the terms ofeach row has been used.

Note that all steps can be solved explicitly except for the computation of the pressure in Eq. (32) which requires to invertthe sum of two Laplacian matrices. This can be effectively performed using an iterative solution scheme such as the con-jugate gradient method.

Above algorithm has improved stabilization properties versus the standard segregation methods due to the introductionof the Laplacian matrix L in Eq. (32). This matrix emanates from the FIC formulation.

The boundary conditions are applied as follows. No condition is applied in the computation of the fractional velocities~vnþ1 in Eq. (31). The prescribed velocities at the boundary are applied when solving for �vnþ1 in step 3. The prescribed pres-sures at the boundary are imposed by making �pn equal to the prescribed pressure values when solving Eq. (32).

3.2. Stokes flow

Many metal forming processes can be simulated with the assumption that the convective terms are negligible (Stokesflow) [8,43,44]. The Stokes FIC formulation can be readily obtained simply by neglecting the convective terms in theNavier–Stokes formulation presented earlier. This also implies neglecting the convective stabilization terms in the momen-tum equations and, consequently, the convective projection variables are not larger necessary. Also the intrinsic timeparameters si take now the simpler form (see Eq. (12)):

si ¼3h2

i

8l. ð36Þ

We note again that in metal forming problems the viscosity l will typically be a function of the strain rate and the yieldstress of the material [8,43,44].

The resulting discretized system of equations can be written as (see Eqs. (24))

M _�vþ K�v�G�p ¼ f;

GT�vþ L�pþQ�p ¼ 0;

QT�pþ M�p ¼ 0.

ð37Þ


The algorithm of previous section can now be implemented.The steady-state form of Eqs. (37) can be expressed in matrix form as

K �G 0

�GT �L �Q

0 �QT �M

264375 �v

�p

�p

8><>:9>=>; ¼

f

0

0

8><>:9>=>;. ð38Þ

The system is symmetric and always positive definite and therefore leads to a non singular solution. This property holdsfor any interpolation function chosen for �v; �p and �p, therefore overcoming the Babuska–Brezzi (BB) restrictions [8].

A reduced velocity–pressure formulation can be obtained by eliminating the �p variables from the last row of Eq. (38)[37].

The FIC formulation for Stokes flows is applicable for analysis of quasi/fully incompressible solids. An analogous for-mulation based on solid mechanics concepts is derived in Section 6.

4. Thermal-coupled flows. Treatment of the free boundary

4.1. Thermal coupled flow

The effect of temperature can be easily introduced by solving the equation for heat transport coupled to the fluid flowequations.

The equation of balance of heat is written in the FIC formulation as [30]

r/ �hj

2

or/

oxj¼ 0; j ¼ 1; nd ; ð39aÞ

with

r/ :¼ �qco/otþ vi

o/oxi

� �þ o

oxkk

o/oxk

� �þ Q. ð39bÞ

In above / is the temperature, c and k are the specific heat and the thermal conductivity of the material, respectively, Q

is the heat source and hj are the characteristic length distances which are typical of the FIC formulation [30].Eq. (39a) is completed with the Dirichlet and Neumann boundary conditions for the heat problem. For details see

[30,46].The convective velocities vi in Eq. (39b) are provided by the solution of the fluid flow problem. As usual in metal forming

processes, the heat source Q is a function of the mechanical work generated in the flow of the material during the formingprocess. The temperature field affects in turn the flow viscosity via its dependence with the yield stress which is very sen-sitive to the temperature changes. The solution of the heat transfer equation is therefore fully coupled with that of the fluidflow problem [44].

4.2. Treatment of the free boundary motion

For the treatment of the free boundary we use a standard volume of fluid (VOF) technique, also known as pseudo-concentration method or level set technique [22,47,48]. In the VOF method the motion of the free boundary is followedby solving the following transport equation (written using the FIC formulation)

rb ��hj

2

orb

oxj¼ 0 in X; ð40aÞ

where

rb :¼ obotþ vk

oboxk

. ð40bÞ

In above b is an auxiliary variable which takes a value equal to one on the free surface and zero elsewhere. The solutionof Eq. (40a) in time in the discretized fluid-air domain allows to track the free surface which is characterized by the con-tours of b which have a unit value.

The underlined term in Eq. (40a) emerges from the FIC formulation [33]. Again, the �hj’s in Eq. (40a) are characteristiclength distances of the order of the element size. This term introduces the necessary stabilization in the transient solution ofEq. (40a).


4.3. Fully coupled algorithm for mould filling problems

The analysis of mould filling problems typically requires the solution of the coupled-thermal flow problem accountingfor the transport of the free surface.

The simplest explicit algorithm is as follows: For each time step:

• Compute the velocities and the pressure in the fluid ð�vnþ1i ; �pnþ1Þ using the fractional step scheme of Section 3.1.

• Compute the nodal temperatures (�/nþ1) solving Eq. (39a).• Compute the new position of the free surface (bn+1) solving Eq. (40a).

A number of implicit versions of the algorithm are possible and they all involve an iteration loop within each timestep until convergence for the flow variables, the temperature and the free surface position is found. For details see[47,48].

5. Computation of the characteristic lengths

The evaluation of the stabilization parameters is one of the crucial issues in stabilized methods. Excellent results havebeen obtained in all problems solved using linear tetrahedra with the characteristic length vector defined by

h ¼ hs

v

vþ hc

$vj$vj ; ð41aÞ

where v = jvj and hs and hc are the ‘‘streamline’’ and ‘‘cross wind’’ contributions given by

hs ¼ maxðlTj vÞ=v; ð41bÞ

hc ¼ maxðlTj $vÞ=j$vj; j ¼ 1; ns; ð41cÞ

where lj are the vectors defining the element sides (ns = 6 for tetrahedra).As for the free surface equation the following value of the characteristic length vector �h in Eq. (40a) has been taken

�h ¼ �hs

v

vþ �hc

$bj$bj . ð42aÞ

The streamline parameter �hs has been obtained by Eq. (41b) whereas the cross wind parameter �hc has been computed by

�hc ¼ max½lTj $b� 1

j$bj ; j ¼ 1; 2; 3. ð42bÞ

The cross-wind terms in Eqs. (41a) and (42a) account for the effect of the gradient of the solution in the stabilizationparameters. This is a standard assumption in most ‘‘shock-capturing’’ stabilization procedures [8,24].

6. FIC formulation for quasi/fully incompressible solids

6.1. Equilibrium equations

As usual the governing equations of solid mechanics are written in a Lagrangian reference frame. Following the argu-ments of Section 2 the equilibrium equations for a solid are written using the FIC technique as [37]

ri �hk

2

ori

oxk¼ 0 in X k ¼ 1; nd ; ð43Þ

where for the dynamic case

ri :¼ �qo2ui

ot2þ orij

oxjþ bi; j ¼ 1; nd . ð44Þ

In Eqs. (43) and (44) ui are the displacements, rij and bi are the stresses and the body forces, respectively and hk are char-acteristic length distances of an arbitrary prismatic domain where equilibrium of forces is considered.

Eqs. (43) and (44) are completed with the boundary conditions on the displacements ui

u � up ¼ 0 on C ð45Þ
i i u


and the equilibrium of surface tractions

rijnj � tpi �

1

2hknkri ¼ 0 on Ct. ð46Þ

In the above upi and ti are prescribed displacements and tractions over the boundaries Cu and Ct, respectively and ni are the

components of the unit normal vector.Note that for consistency with the fluid flow equations of previous section and differently from the tradition in solid

mechanics, the compression pressure has been taken as positive.

6.2. Stabilized pressure constitutive equations

For simplicity the treatment of the constitutive equations will be explained for the linear elastic model. The approachextends naturally to the non linear elasto-plastic/viscoplastic constitutive equations typical of metal forming problems.

As usual in quasi-incompressible problems the stresses are split into deviatoric and volumetric (pressure) parts

rij ¼ sij � pdij; ð47Þwhere dij is the Kronecker delta function. The linear elastic constitutive equations for the deviatoric stresses sij are written as

sij ¼ 2G eij �1

3evdij

� �; ð48Þ

where G is the shear modulus,

eij ¼1

2

oui

oxjþ ouj

oxi

� �and ev ¼ eii. ð49Þ

The constitutive equation for the pressure p can be written for an arbitrary domain of finite size of volume V as

1

Kpav ¼ �DV

V; ð50Þ

where K is the bulk modulus of the material and pav is the average value of the pressure over domain V.The value of pav can be approximated as [37]

pav :¼ 1

V

ZV

p dV ¼ p � hk

2

opoxkþOðhkÞ2; k ¼ 1; nd ; ð51Þ

where p is the pressure at an arbitrary point within the domain V and hk are characteristic lengths of such a domain.The ratio DV

V can be expressed as

DVV¼ ev �

hk

2

oev

oxkþOðhkÞ2; k ¼ 1; nd . ð52Þ

Substituting Eqs. (51) and (52) into Eq. (50) and neglecting second order terms in hk gives the FIC constitutive equation forthe pressure as

pKþ ev

� �� hk

2

o

oxk

pKþ ev

� �¼ 0; k ¼ 1; nd . ð53Þ

Note that for hk! 0 the standard relationship between the pressure and the volumetric strain of the infinitesimal theory(p = �Kev) is found.

For an incompressible material K!1 and Eq. (53) yields

ev �hk

2

oev

oxk¼ 0. ð54Þ

Eq. (54) expresses the limit incompressible behaviour of the solid. This equation is identical to Eq. (2) for incompressibleflow problems and there arises from the mass continuity condition [30,32].

By combining Eqs. (43), (44), (47), (48) and (53) a mixed displacement–pressure formulation can be written as

� qo2ui

ot2þ orij

oxiþ bi �

hk

2

ori

oxk¼ 0; ð55Þ

pKþ ev

� �� hk

2

o

oxk

pKþ ev

� �¼ 0. ð56Þ


From the observation of Eq. (56) we can obtain after some algebra [37]

o

oxk

pKþ ev

� �’ 3hj

8Gork

oxj. ð57Þ

Substituting Eq. (57) into (53) the mass balance equation can be written as

pKþ ev

� ��Xnd

i¼1

siori

oxi¼ 0 with si ¼

3h2i

8G. ð58Þ

In the derivation of Eq. (58) we have neglected the terms involving products hihj for hi 5 hj.The pressure gradient projections pi are introduced now as

pi � ri �opoxi

. ð59Þ

The final system of governing equations is

qo

2ui

ot2� orij

oxi� bi ¼ 0; ð60Þ

Dpkþ oðDuiÞ

oxiþXnd

i¼1

siopoxiþ pi

� �¼ 0; ð61Þ

ri �opoxiþ pi ¼ 0. ð62Þ

In Eqs. (60)–(62) we note the following:

(a) The stabilization terms have been neglected in Eq. (60). These terms were useful to derive the stabilized form of Eq.(61). However they are not longer necessary as the convective terms do not appear in the equilibrium equations.

(b) The constitutive equation for the pressure (Eq. (61)) has been written in an incremental form. This is more convenientfor non linear material behaviour typical of metal forming situations. Here, appropriate elasto-plastic or elasto-visco-plastic constitutive laws relating stresses and strains in incremental or rate forms must be used [1–5,43,44].

6.3. Finite element equations

The weighted residual form of the governing equations can be written as (after integration by parts of the relevant terms)ZX

duiqo2ui

ot2dXþ

ZX

deijrij dX�Z

Xduibi dX�

ZCt

duitpi dCt ¼ 0; ð63aÞZ

Xq

DpKþ oðDuiÞ

oxi

� �dXþ

ZX

Xnd

i¼1

oqoxi

siopoxiþ pi

� �" #dX ¼ 0; ð63bÞZ

Xdpisi

opoxiþ pi

� �dX ¼ 0 no sum in i; ð63cÞ

where again the si are introduced in Eq. (63c) for symmetry reasons.The finite element discretization of the displacements, the pressure and the pressure gradient projections is written by

expressions identical to Eqs. (23) with the nodal variables now being a function of the time t. Substituting the approxima-tions into Eqs. (63) and using the Galerkin form gives the following system of discretized equations

M€�uþ g� f ¼ 0; ð64aÞ

GTD�uþ CD�pþ bL�pþQ�p ¼ 0; ð64bÞ

QT�pþcM�p ¼ 0; ð64cÞ

where €�u is the nodal acceleration vector,

Cab ¼Z

Xe

1

kN aNb dX ð65Þ

is the pressure increment matrix,


g ¼Z

XBTrdX ð66Þ

is the internal nodal force vector and the rest of matrices and vectors are defined in the Appendix A. Note that the expres-sion of g of Eq. (66) is adequate for non linear analysis.

A four steps semi-explicit time integration algorithm can be derived as follows:

Step 1. Compute the nodal velocities _�unþ1=2 from Eq. (64a)

_�unþ1=2 ¼ _�un�1=2 þ DtM�1d ðf

n � gnÞ. ð67aÞStep 2. Compute the nodal displacements �unþ1

�unþ1 ¼ �un þ Dt _�unþ1=2. ð67bÞStep 3. Compute the nodal pressures �pnþ1 from Eq. (64b)

�pnþ1 ¼ �½Cþ bL��1½DtGT _�unþ1=2 � C�pn þQ�pn�: ð67cÞStep 4. Compute the nodal projected pressure gradients �pnþ1 from Eq. (64c)

�pnþ1 ¼ �cM�1d QT�pnþ1. ð67dÞ

In above, all matrices are evaluated at tn+1, (Æ)d denotes a lumped diagonal matrix and

gn ¼Z

Xe½BTr�n dX; ð68Þ

where the stresses rn are obtained by consistent integration of the adequate (non linear) constitutive law.Note that steps 1, 2 and 4 are fully explicit as a diagonal form of matrices M and cM has been chosen. The solution of

step 3 requires invariably the inverse of a Laplacian matrix. This can be an inexpensive process using an iterative equationsolution method (e.g. a preconditioned conjugate gradient method).

For the full incompressible case K =1 and C ¼ 0 in all above equations.The critical time step Dt is taken as that of the standard explicit dynamic scheme (see Section 6.5 and [9,37]).

6.4. Fully explicit algorithm

A fully explicit four steps algorithm can be obtained by computing �pnþ1 from step 3 in Eq. (67c) as follows:

�pnþ1 ¼ �C�1d ½DtGT _�unþ1=2 � ðCd � bLÞ�pn þQ�pn�. ð69Þ

Note that the explicit algorithm is not applicable in the full incompressible limit as the solution of Eq. (69) breaks downfor K =1 and C ¼ 0. The explicit form can however be used with success in problems where quasi-incompressible regionsexist adjacent to standard ‘‘compressible’’ zones. Examples of this kind are shown in the next section. In both cases thesemi-explicit and fully explicit schemes gave identical results with important savings in both computer time and memorystorage requirements obtained when using the explicit form.

6.5. Thermal coupled effects

The effect of temperature can be easily accounted for by solving the heat transfer equation formulated in a Lagrangianframe as

qco/ot� o

oxjk

o/oxj

� �� Q ¼ 0. ð70Þ

As usual, the source Q is dependent of the mechanical work generated during the forming process.Note that the convective terms do not enter into Eq. (70) This also eliminates the need to stabilize the numerical

solution.The coupled thermal-mechanical problem requires the computation of the temperature / at each time step using a tran-

sient solution scheme [47,48,50].


Remark. Above formulation is similar to that developed by Chiumenti et al. [27,29] and Cervera et al. [28] for analysis ofincompressible problems in solid mechanics using a sub-grid scale approach.

6.6. Computation of the intrinsic time parameter for quasi-fully incompressible solids

In solid mechanics applications it is usual to accept that all si are identical and constant within each element and givenby

sðeÞ ¼ 3ðhðeÞÞ2

8Gwith hðeÞ ¼ ½V ðeÞ�1=nd T ; ð71Þ

where V(e) is the element volume (or the element area for 2D problems). This expression for s(e) does not take into accountthe element distorsions along a particular direction during the deformation process.

The correct value of the shear modulus in the expression of s(e) is another sensitive issue as, obviously, for non linearproblems the value of G will differ from the elastic modulus. This fact has been identified by Cervera et al. [28] for nonlinear analysis of incompressible problems using linear triangles.

A useful alternative to compute s(e) for explicit non linear transient situations is to make use of the value of the speed ofsound in an elastic solid, defined by

c ¼ffiffiffiffiEq

s; ð72Þ

where E is the Young’s modulus. The stability condition for explicit dynamic computations is given by the Courant con-dition defined as [8]

DtðeÞ 6 DtðeÞc ¼hðeÞ

c; ð73Þ

where DtðeÞc is the critical time step for the element.Accepting that G ’ E

3for the incompressible case and using Eqs. (72) and (73) (assuming the identity in Eq. (73)) an

alternative expression for the element intrinsic time parameter in terms of the critical time step can be found as

sðeÞ ¼ ½DtðeÞc �2

q. ð74Þ

Eq. (74) shows clearly that the intrinsic time parameter varies across the mesh as a function of the critical time step foreach element.

7. Applications to casting and bulk metal forming processes

7.1. Aluminium casting simulation

A numerical simulation of an aluminium casting process is presented as a demonstration of the accuracy of the stabilizedformulation. The computations are performed with the finite element code VULCAN where the stabilized FEM presentedhas been implemented [56].

The analysis simulates the casting process of an aluminium (AlSi7Mg) specimen in a steel (X40CrMoV5) mould. Mate-rial behavior of aluminium casting has been modeled by a fully coupled thermo-viscoplastic model, while the steel mouldhas been modeled by a simpler thermo-elastic model [51]. Geometrical and material data were provided by the foundryRUFFINI. Fig. 1 shows the finite element mesh used for the part and the cooling system. The full mesh, including themould has 380.000 four node tetrahedra. The pouring temperature is 650 �C. Initial temperature for the mould is obtainedthrough a thermal die-cycling simulation. Fig. 2 shows the evolution of the mould temperature after 6 cycles. The coolingsystem has been kept at 20 �C. Filling evolution has been simulated as in a pressure die-casting process using the stabilizedVOF technique described in [47,48]. Fig. 3 shows different time steps of the simulation.

The final temperature field obtained after the filling simulation is taken as the initial condition for the solidification andcooling analysis. Temperature and liquid-fraction distributions during solidification are shown in Figs. 4 and 5, respec-tively. The heat transfer coefficient takes into account the air-gap resistance due to the casting shrinkage during thesolidification process. Fig. 6 shows volumetric and von Mises deviatoric stress distributions in a x–y section. The figuresalso show the air-gap between the part and the mould, responsible of a non-uniform heat flux at the contact interface.

Other examples of application of the stabilized formulation to casting problems can be found in [47–51].

Fig. 2. Temperature die-cycling.

Fig. 3. Filling evolution: pressure die-casting simulation.

Fig. 1. Finite element discretization of the aluminium casting.


Fig. 4. Temperature evolution during cooling phase.

Fig. 5. Liquid-fraction evolution during phase-change.

Fig. 6. Aluminium casting. Stress-trace and von Mises deviatoric stress indicator during phase-change (plane xy).



7.2. Side pressing of a cylinder

A cylinder 100 mm long with a radius of 100 mm is subjected to sidepressing between two plane dies. It is compressed to100 mm. The material properties are the following: E = 217 GPa, m = 0.3, q = 7830 kg/m3, r0 = 170 MPa, H = 30 MPa,friction coefficient = 0.2. The die velocity is assumed to be 2 m/s. Initial set-up is shown in Fig. 7. A quarter of a cylinderwas discretized with tri-linear hexahedra and linear tetrahedra meshes.

Fig. 8 shows the results obtained using the hexahedral mesh and a standard mixed formulation. The results show thedistribution of the effective plastic strain and pressure on the deformed shape. The sensitivity of the FIC results with

the expression of the intrinsic time parameter of Eq. (71) was studied by defining sðeÞ ¼ a ½hðeÞ�2G and solving the problem

for different values of a. The results obtained with the FIC method are shown in Figs. 9 and 10 for a = 0.1 and 0.03, respec-tively. The alternative expression for s(e) of Eq. (74) has also been studied. The results for this case are shown in Fig. 11.Quite a good agreement can be seen between the FIC solutions and the reference solution with the best results for the effec-tive plastic strain obtained for s(e) calculated according to Eq. (71) with a = 0.03. The results for the two alternative for-mulae for s(e) are similar, but those obtained using Eq. (71) seem to be slightly better. In any case, the results on thepressure distribution are quite insensitive to the value of s(e). This is also confirmed in Fig. 12b, which displays the distri-bution of the pressure along the line ABCDEA defined in Fig. 12a. A small perturbation can be seen at the sharp edges ofthe deformed body.

Fig. 7. Sidepressing of a cylinder: (a) initial tetrahedral mesh; (b) initial hexahedral mesh.

Fig. 8. Sidepressing of a cylinder, mixed formulation, hexahedral mesh: (a) effective plastic strain; (b) pressure distribution.

Fig. 9. Sidepressing of a cylinder, FIC algorithm (a = 0.1), tetrahedra, mesh of 22,186 elements: (a) effective plastic strain; (b) pressure distribution.

Fig. 10. Sidepressing of a cylinder, FIC algorithm (a = 0.03), tetrahedra, mesh of 22,186 elements: (a) effective plastic strain; (b) pressure distribution.

Fig. 12. (a) Definition of the line for comparison of pressure distribution; (b) pressure distribution along the line ABCDEA.

Fig. 11. Sidepressing of a cylinder, FIC algorithm (s(e) calculated according to Eq. (74)), tetrahedra, mesh of 22,186 elements: (a) effective plastic strain;(b) pressure distribution.



All the calculations here have been carried out using fully explicit version of the algorithm which is more efficient thanthe semi-explicit one with giving practically the same results.

7.3. Backward extrusion

Backward extrusion of a cylinder made of steel 16MNCr5 has been analysed using an axisymmetric formulation. This isa benchmark example of the finite element program for forming simulation MARC/Autoforge [54]. The tooling and billetgeometry are given in Fig. 13a. Initial material dimensions are the following: length 30 mm and diameter 30 mm. Thepunch of diameter 20 mm has a prescribed stroke of 28 mm. Material properties are as follows: Young’s modulusE = 3.24 · 105 MPa, Poisson’s coefficient m = 0.3, material density q = 8120 kg/m3, yield stress rY0 = 300 MPa and hard-ening modulus H = 50 MPa. Friction between the material and tools is defined by the Coulomb friction coefficient l = 0.1.

The simulation of the backward extrusion process was carried out with a particularization of the fully explicit FIC for-mulation of Section 6 for axisymmetric solids. Regeneration of the mesh was performed when element distorsion was exces-sive. Fig. 13b and c show the results in the form of the final deformed shape with the distribution of the effective plasticstrain obtained using quadrilaterals with a mixed formulation, and using triangles and the FIC algorithm, respectively. Theresults are in a good agreement with the solution given in [54]. This example demonstrates the efficiency of the FIC algo-rithm for simulation of bulk forming processes. Different stages of the forming process are shown in Fig. 14.

Fig. 13. Backward extrusion (a) geometry definition. Final deformed shape with effective plastic strain distribution; (b) solution with quadrilaterals andmixed formulation; (c) solution with triangles and the FIC algorithm.

Fig. 14. Backward extrusion—deformed shapes with effective plastic strain distribution at different stages of forming. Solution with triangles and the FICalgorithm.


7.4. Forming of a hose-clamp band

The stabilized formulation was applied to model the manufacturing of a hose-clamp band of steel AISI 409L (Fig. 15).The initial set-up of the tooling and band is shown in Fig. 16. A series of grooves are forged in the band by the roll passingover the band placed on the toothed punch. The band thickness is 0.7 mm and its width 8 mm. Plane strain conditions havebeen assumed. Material properties are as follows: Young’s modulus E = 2.1 · 105 MPa, Poisson’s coefficient m = 0.33,material density q = 7800 kg/m3, the true stress–true strain relationship is given by the power Ludwik–Nadai equationrY = 623(0.36822 · 10�2 + ep)0.1362 MPa, friction between the material and tools is defined by the Coulomb friction coef-ficient l = 0.1.

Fig. 15. A hose clamp.

Fig. 16. A hose clamp—initial set-up.

Fig. 17. A hose clamp—deformed shapes at different stages of forming with distribution of effective plastic strain.

Fig. 18. Detail of the deformed shape with finite element discretization and distribution of effective plastic strain.


Simulation was carried out using linear triangular elements and the fully explicit FIC formulation. As in the previousexample the meshes where regenerated when element distorsion was excessive. The purpose of the simulation was to checkif the expected groove depth and tooth height in the band were obtained. Fig. 17a and b show the results in the form of thedeformed shape with the distribution of the effective plastic strain at different stages of forming. The results are in goodagreement with the real process. Fig. 18 shows a detail of the deformed shape with finite element discretization and distri-bution of effective plastic strain. In this figure the obtained dimensions are compared with the required ones shown inbrackets. Effects of elastic springback are also clearly seen.

Remark. The examples presented show that the FIC/FEM formulation is an effective procedure for solving bulk metalforming problems involving full or quasi-incompressible situations. The key advantage of the FIC approach versus morestandard mixed FEM formulations is that it provides a natural theoretical framework for equal order finite elementinterpolations for the velocity and pressure variables, both in the context of implicit and explicit solution schemes. We notethe simplicity and effectiveness of the full explicit algorithm as demonstrated in the examples presented. The FICformulation reproduces also the best feature of the so called stabilized FEM method for incompressible problems, such asthe CBS scheme [20,21,25,26,58], the pressure gradient operator method [22] and the subgrid scale method [23] amongothers.

8. Lagrangian flows. The particle finite element method

8.1. The particle finite element method (PFEM)

The Lagrangian formulation is an excellent procedure for treating bulk forming processes involving the interaction offluids and solids using a unified formulation. An important advantage of the Lagrangian formulation is that both themotions of the solid and the fluid are defined in the same frame of reference and modelled with the some governingequations.

The Lagrangian fluid flow equations can be simply obtained by noting that the velocity of the mesh nodes and that ofthe fluid particles are the same. Hence the convective terms vanish in the momentum equations, while the rest of the fluidflow equations remain unchanged. The resulting governing equations have an identical form as those of the Stokes flowproblem, with the motion of the flow particles being referred now to a Lagrangian coordinate frame.

The FEM algorithms for solving the Lagrangian flow equations are very similar to those presented for incompressiblesolids in a previous section. Here we present a particular class of Lagrangian formulation called the particle finite elementmethod (PFEM, www.cimne.com/pfem) [38–41]. The PFEM treats the mesh nodes in the fluid and solid domains as dimen-sionless particles which can freely move an even separate from the main fluid domain representing, for instance, the effectof fluid drops. A finite element mesh connects the nodes defining the discretized domain where the governing equations aresolved in the standard FEM fashion.

The quality of the numerical solution depends on the discretization chosen as in the standard FEM. Adaptive meshrefinement techniques can be used to improve the solution in zones where large motions of the fluid or the structure occur.

http://www.cimne.com/pfem


A typical solution with the PFEM involves the following steps.

1. Identify the external boundaries for both the fluid and solid domains. This is an essential step as some boundaries (suchas the free surface in the fluid) may be severely distorted during the solution process including separation and re-enteringof nodes. The Alpha Shape method is used for the boundary definition (see Section 8.4).

2. Discretize the fluid and solid domains with a finite element mesh. For the mesh generation process we use and extendedDelaunay technique [52].

3. Solve iteratively the coupled Lagrangian equations of motion for the fluid and the solid domains. Compute the relevantstate variables in both domains at each time step: velocities, pressure and viscous stresses in the fluid and displacements,stresses and strains in the solid. The iterative solution scheme chosen is a particularization of the fractional step algo-rithm of Section 3.1 for a = 1. In summary the solution steps are the following.Step 3.1. Compute the predicted velocities (viz, Eq. (31))

~vnþ1;iþ1 ¼ �vn � DtM�1d ½K�vn �G�pn � fnþ1�. ð75Þ

Step 3.2. Compute �pnþ1;iþ1 from Eq. (32) for a = 1.Step 3.3. Compute explicitly �vnþ1;iþ1 from Eq. (33) with a = 1.Step 3.4. Compute �pnþ1;iþ1 explicitly from Eq. (35).Step 3.5. Solve for the motion of the solid. This can be performed by integrating the dynamic equations of motion in

the solid. Here the algorithm of Section 6 can be used, as it applies to both ‘‘compressible’’ and incompressiblematerials.

Step 3.6. Estimate a new position of the mesh nodes in terms of the time increment size as

xnþ1;iþ1j ¼ xn

j þ �unþ1;iþ1j Dt. ð76Þ

It is important to note that all matrices in steps 3.1–3.5 are evaluated at the last predicted configuration Xn+1,i.In steps 3.1–3.6 superindex i denotes the iteration within each time step.

Step 3.7. Check the convergence of the velocity and pressure fields in the fluid and the displacements, strains and stres-
ses in the solid. If convergence is achieved froze the final position of the mesh nodes and move to the next timeincrement (step 4), otherwise return to step 3.1 for the next iteration.
4. Go back to step 1 and repeat the solution process for the next time step.

Above algorithm can be found to be analogous to the standard updated lagrangian scheme typically used in non linearsolid mechanics problems [8,57].

Despite the motion of the nodes within the iterative process, in general there is no need to regenerate the mesh at eachiteration. In the examples presented in the paper the mesh in the fluid domain has been regenerated at each time step. Acheaper alternative is to generate a new mesh only after a prescribed number of converged time steps, or when the nodaldisplacements induce significant geometrical distorsions in some elements.

The boundary conditions are applied as described in Section 3.1.In the examples presented in the paper the time increment size has been chosen as

Dt ¼ minðDtiÞ with Dti ¼jvj

hmini

; ð77Þ

where hmini is the distance between node i and the closest node in the mesh.

Fig. 19. PFEM results for a large amplitude sloshing problem.


8.2. Treatment of contact between fluid and solid interfaces

The condition of prescribed velocities or pressures at the solid boundaries in the PFEM are applied in strong form to theboundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting

Fig. 20. (a) Filling of a 2D mould with a powder material using the PFEM. (b) Finite element mesh used for the computations at a certain instant. Thecircles indicate the nodes at the external and internal free surfaces.


solids. In some problems it is useful to define a layer of nodes adjacent to the external boundary in the fluid where thecondition of prescribed velocity is imposed. These nodes typically remain fixed during the solution process. Contactbetween liquid particles and the solid boundaries is accounted for by the incompressibility condition which naturally pre-

vents the liquid nodes to penetrate into the solid boundaries [38–41].

8.3. Generation of a new mesh

One of the key points for the success of the PFEM is the fast regeneration of a finite element mesh at every time step onthe basis of the position of the nodes in the space domain. In our work the mesh is typically generated using the so calledextended Delaunay tesselation (EDT) [38,52]. The EDT allows one to generate non standard meshes combining elements ofarbitrary polyhedrical shapes (triangles, quadrilaterals and other polygons in 2D and tetrahedra, hexahedra and arbitrarypolyhedra in 3D) in a computing time of order n, where n is the total number of nodes in the mesh. The C0 continuousshape functions of the elements are obtained using the so called meshless finite element interpolation (MFEM) [53].

8.4. Identification of boundary surfaces

The PFEM requires the correct definition of the boundary domain. Sometimes, boundary nodes are clearly distin-guished from internal nodes. In other cases, the total set of nodes is the only information available and the algorithm mustrecognize the boundary nodes.

Considering that the nodes follow a variable h(x) distribution, where h(x) is the minimum distance between two nodes,the following criterion has been used. For each two nodes (three nodes in 3D) a circle (a sphere in 3D) of radius equal to ah

containing the nodes is plotted. All nodes laying on a circle (sphere) with a radius greater than ah, not containing any othernode in the interior are considered as boundary nodes. In practice, a is a parameter close to, but greater than one. This cri-terion coincides with the Alpha Shape concept [39,55].

Fig. 21. Filling of a 3D mould with a casted metal using the PFEM. Colours indicate the temperature values. (For interpretation of color, the reader isreferred to the web version of this article).


The method also allows to identify isolated fluid particles (nodes) outside the main fluid domain. These particles aretreated as part of the external boundary where the pressure is fixed to the atmospheric value. We note that the ‘‘flying par-ticles’’ are in fact ‘‘points’’ with no mass which motion is followed by integrating the dynamic equations of continuummechanics for known values of the mass, the initial velocity and acceleration, gravity body forces and a prescribed zero(atmospheric) pressure.

8.5. Temperature coupled effects

Thermal–mechanical problems can be effectively treated with the PFEM. This requires solving for the temperature fieldat each time step, accounting for the couplings induced by the mechanical equations. The effect of temperature in themechanical problem is introduced via the constitutive equation in the usual manner.

Fig. 22. Simulation of a casting process using the PFEM. Cut through the mesh.


The form of the heat transfer equation is identical to that of Eq. (70). Recall that the Lagrangian formulation eliminatesthe convective term in the heat transfer equation and hence the FIC stabilization is here not needed [41].

Remark. The key advantage of the PFEM versus conventional particle methods is that it retains the best features of theFEM to solve problems in continuum mechanics using a variational approach. A standard finite element mesh is used todiscretize in space the problem variables and for solving the governing equations precisely as in the standard FEM. Thecombination of the lagrangian FEM with the Alpha-Shape approach for identification of the domain boundary at eachtime step and the frequent regeneration of the finite element mesh are the distinct features of the PFEM.

9. Applications of the PFEM to metal forming processes

9.1. Sloshing problem

The sloshing example shown in Fig. 19 illustrates the ability of the PFEM to simulate the flow of liquids within closedcavities with large motions of the free surface. This feature of the PFEM is essential to model the filling of moulds as shownin the next examples. More applications of the PFEM to sloshing problems are reported in [39].

9.2. Filling of moulds

Fig. 20 shows the results of the filling of a 2D mould cavity with a powder material using the PFEM. Note that theessential feature of the filling process are well reproduced. The finite element mesh used for the computation at a certain

Fig. 23. Analysis of the mixing of two fluids of different density with the PFEM.

Fig. 24. Mixing of particles in a fluid. Evolution of the particles during the mixing process.


instant is shown in Fig. 20b. This illustrates the fact that the PFEM is, in fact, a blending of particle and finite elementprocedures. Other applications of the PFEM to powder compaction problems are reported in [57].

Fig. 21 shows the different stages of the 3D simulation of the filling of a mould with a casted metal using the PFEM. Thepictures show the evolution of the casting during the filling process. The metal flows through the vertical mould first andthen fills the lower and upper lateral ducts, as expected. The gray colour intensities denote the different temperatures in themelt with a whiter colour indicating the cooler region. Note that the metal cools down as it progresses within the mould.

The example in Fig. 22 shows the simulation of the casting of a mechanical part using the PFEM. Note that the meshdiscretizing the casted region is progressively generated as the mould is filled. This is a distinct feature of the PFEM whichdistinguishes this method from more standard VOF techniques using Eulerian and ALE approaches.

9.3. Mixing problems

The PFEM is particularly suited for analyzing problems involving mixing of two fluids or that of a collection of particlesin a fluid. This is very useful to model a wide class of material forming processes [5,7].

A first example of this kind is shown in Fig. 23. The problem represents the mixing of two fluids of different density. ThePFEM allows to track the position of the fluid particles of lower density which mix within the second fluid.

Fig. 24 shows a last example of application of the PFEM to the mixing of a collection of particles within a containerfilled with a fluid of a higher density. Initially the particles are thrown into the container and mix within the fluid as shown.As time evolves the particles move up naturally towards the surface of the fluid due to their lower density.

The last two examples clearly show the possibilities of the PFEM for analysis of material mixing situations. Extensionsof the approach accounting for thermal-coupled effects and chemical reactive situations are straightforward [41].

10. Concluding remarks

The finite calculus form of the balance equations in mechanics is a good starting point for deriving stabilized finite ele-ment methods for solving a variety of bulk metal forming problems involving fluid flow and large deformation of solidsusing Euler and Lagrangian descriptions. Free surface effects and coupled thermal situations can be accounted for in astraightforward manner within the general algorithm. A fractional step scheme with intrinsic stabilization propertieshas been described. Applications of the Eulerian and Lagrangian methods presented range from mould filling, solidifica-tion and cooling situations in casting processes to forging and extrusion problems, among many others. The LagrangianPFEM formulation is very effective for the analysis of bulk metal forming problems and material design processes involv-ing filling of moulds, material mixing situations and complex fluid–structure interactions.


Acknowledgements

The authors thank the company QUANTECH ATZ, SA (www.quantech.es) for providing the code VULCAN for solu-tion of the casting problems presented. Thanks are also given to MIKALOR SA for providing data for the hose clampexample.

Appendix A

In Eqs. (24)

H ¼ Aþ Kþ K. ðA:1ÞIf we denote the node indexes with superscripts a, b, the space indexes with subscripts i, j, the element contributions to

the components of the arrays involved in Eqs. (24) and (A.1) are

Mabij ¼

ZXe

qNaN b dX

� �dij; Aab

ij ¼Z

XeqNaðuTrNbÞdX

� �dij;

Kabij ¼

ZXe

l$TN a$N b dX

� �dij; $ ¼ o

ox1

;o

ox2

;o

ox3

� �T

;

bK abij ¼

1

2

ZXeðhT$NaÞðquT$NbÞdX

� �dij; Gab

i ¼Z

Xe

oNa

oxiNb dX;

C ¼C1

C2

C3

264375; Cab

1ij¼ Cab

2ij¼ Cab

3ij¼ 1

2

ZXe½hT$N a�Nb dX

� �dij;

bLab ¼Z

Xeð$TN aÞ½s�$N b dX; ½s� ¼

s1 0 0

0 s2 0

0 0 s3

264375;

Q ¼ ½Q1;Q2;Q3�; Qabi ¼

ZXe

sioN a

oxiNb dX no sum in i;

CT ¼ ½C1; C2; C3�; bCab1 ¼ bCab

2 ¼ bCab3 ¼

ZXe

q2N aðuT$N bÞdX;

bM abij ¼

ZXe

siN aNb dX

� �dij; f a

i ¼Z

XeN afi dXþ

ZCe

N ati dC.

ðA:2Þ

It is understood that all the arrays are matrices (except f, which is a vector) whose components are obtained by groupingtogether the left indexes in the previous expressions (a and possibly i) and the right indexes (b and possibly j).

Note that the stabilization matrix K in Eq. (A.1) adds additional orthotropic diffusivity terms of value q hk ul2

.

References

[1] J.L. Chenot, E. Onate (Eds.), Modelling of Metal Forming Processes, Kluwer Academic, Dordrecht, 1988.[2] J.F.T. Pittman, O.C. Zienkiewicz, R.D. Wood, J.M. Alexander (Eds.), Numerical Analysis of Forming Processes, Wiley, Chichester, 1984.[3] J. Huetnik, F.P.T. Baaijens (Eds.), Simulation of Materials Processing: Theory, Methods and Applications, NUMIFORM 98, Enschede, Paıses

Bajos, 22–5 June, A.A. Balkema, Rotterdam, 1998.[4] M. Pietrzyk, J. Kusiak, J. Majta, P. Hartley, I. Pillinger (Eds.), Metal Forming 2000, A.A. Balkema, Rotterdam, 2000.[5] K. Mori, Simulation of Material Processing: Theory, Methods and Applications, A.A. Balkema, Lisse, 2001.[6] D.R.J. Owen, E. Onate, B. Suarez (Eds.), Computational Plasticity VII. Fundamentals and Applications, CIMNE, Barcelona, 2003.[7] S. Ghosh, J.C. Castro, J.K. Lee, in: Material Processing and Design: Simulation and Applications, NUMIFORM 2004, American Institute of

Physics, 2004.[8] O.C. Zienkiewicz, R.C. Taylor, A. Nithiarasu, The Finite Element Method, vol. 3, 6th ed., Elsevier, 2005.[9] C. Hirsch, Numerical Computation of Internal and External Flow, vol. 1, J. Wiley, London, 1988, vol. 2, 1990.

[10] J. Peraire, K. Morgan, J. Peiro, The simulation of 3D incompressible flows using unstructured grids, in: D.A. Caughey, M.M. Hafez (Eds.), J. Wiley,London, 1994 (Chapter 16).

[11] M. Storti, N. Nigro, S.R. Idelsohn, Steady state incompressible flows using explicit schemes with an optimal local preconditioning, Comput. MethodsAppl. Mech. Engrg. 124 (1995) 231–252.

[12] J.C. Heinrich, P.S. Hayakorn, O.C. Zienkiewicz, An upwind finite element scheme for two dimensional convective transport equations, Int. J. Numer.Methods Engrg. 11 (1977) 131–143.

http://www.quantech.es


[13] A. Brooks, T.J.R. Hughes, Streamline upwind/Petrov–Galerkin formulation for convection dominated flows with particular emphasis on theincompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259.

[14] T.J.R. Hughes, M. Mallet, A new finite element formulations for computational fluid dynamics: III. The generalized streamline operator formultidimensional advective–diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986) 305–328.

[15] M.A. Cruchaga, E. Onate, A finite element formulation for incompressible flow problems using a generalized streamline operator, Comput. MethodsAppl. Mech. Engrg. 143 (1997) 49–67.

[16] M.A. Cruchaga, E. Onate, A generalized streamline finite element approach for the analysis of incompressible flow problems including movingsurfaces, Comput. Methods Appl. Mech. Engrg. 173 (1999) 241–255.

[17] T.J.R. Hughes, L.P. Franca, G.M. Hulbert, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squaresmethod for advective–diffusive equations, Comput. Methods Appl. Mech. Engrg. 73 (1989) 173–189.

[18] J. Donea, A Taylor–Galerkin method for convective transport problems, Int. J. Numer. Methods Engrg. 20 (1984) 101–119.[19] R. Lohner, K. Morgan, O.C. Zienkiewicz, The solution of non-linear hyperbolic equation systems by the finite element method, Int. J. Numer.

Methods Fluids 4 (1984) 1043.[20] R. Codina, M. Vazquez, O.C. Zienkiewicz, A general algorithm for compressible and incompressible flow—Part III. The semi-implicit form, Int. J.

Numer. Methods Fluids 27 (1998) 13–32.[21] R. Codina, O.C. Zienkiewicz, CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as

stabilization parameter, Commun. Numer. Methods Engrg. 18 (2) (2002) 99–112.[22] R. Codina, O. Soto, A numerical model to track two-fluid interfaces based on a stabilized finite element method and the level set technique, Int. J.

Numer. Methods Fluids 4 (2002) 293–301.[23] R. Codina, Stabilized finite element approximation of transient incompressible flows using orthogonal subscales, Comput. Methods Appl. Mech.

Engrg. 191 (2002) 4295–4321.[24] J. Donea, A. Huerta, Finite Element Method for Flow Problems, J. Wiley, London, 2003.[25] O.C. Zienkiewicz, J. Rojek, R.L. Taylor, M. Pastor, Triangles and tetrahedra in explicit dynamic codes for solids, Int. J. Numer. Methods Engrg. 43

(1998) 565–583.[26] J. Rojek, O.C. Zienkiewicz, E. Onate, E. Postek, Advances in FE explicit formulations for simulation of metal forming processes, J. Mater. Process.

Technol. 119 (2001) 41–47.[27] M. Chiumenti, Q. Valverde, C. Agelet de Saracibar, M. Cervera, A stabilized formulation for incompressible elasticity using linear displacement and

pressure interpolations, Comput. Methods Appl. Mech. Engrg. 191 (46) (2002) 5253–5264.[28] M. Cervera, M. Chiumenti, Q. Valverde, C. Agelet de Saracibar, Mixed linear/linear simplicial elements for incompressible elasticity and plasticity,

Comput. Methods Appl. Mech. Engrg. 192 (49–50) (2003) 5249–5263.[29] M. Chiumenti, Q. Valverde, C. Agelet de Saracibar, M. Cervera, A stabilized formulation for incompressible plasticity using linear triangles and

tetrahedra, Int. J. Plastic. 20 (8–9) (2004) 1487–1504.[30] E. Onate, Derivation of stabilized equations for advective–diffusive transport and fluid flow problems, Comput. Methods Appl. Mech. Engrg. 151

(1998) 233–267.[31] E. Onate, Possibilities of finite calculus in computational mechanics, Int. J. Numer. Methods Engrg. 60 (1) (2004) 255–281.[32] E. Onate, A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation, Comput. Methods Appl.

Mech. Engrg. 182 (1–2) (2000) 355–370.[33] E. Onate, J. Garcıa, A finite element method for fluid–structure interaction with surface waves using a finite calculus formulation, Comput. Methods

Appl. Mech. Engrg. 191 (2001) 635–660.[34] E. Onate, J. Garcıa, S.R. Idelsohn, F. Del Pin, FIC formulations for finite element analysis of incompressible flows. Eulerian, ALE and Lagrangian

approaches, Comput. Methods Appl. Mech. Engrg., submitted for publication.[35] E. Onate, J. Arteaga, J. Garcıa, R. Flores, Error estimation and mesh adaptivity in incompressible viscous flows using a residual power approach,

Comput. Methods Appl. Mech. Engrg. 195 (2006) 339–362.[36] E. Onate, R.L. Taylor, O.C. Zienkiewicz, J. Rojek, A residual correction method based on finite calculus, Engrg. Comput. 20 (5/6) (2003) 629–658.[37] E. Onate, J. Rojek, R.L. Taylor, O.C. Zienkiewicz, Finite calculus formulation for incompressible solids using linear triangles and tetrahedra, Int. J.

Numer. Methods Engrg. 59 (2004) 1473–1500.[38] S.R. Idelsohn, E. Onate, F. Del Pin, A lagrangian meshless finite element method applied to fluid–structure interaction problems, Comput. Struct. 81

(2003) 655–671.[39] E. Onate, S.R. Idelsohn, F. Del Pin, R. Aubry, The particle finite element method. An overview, Int. J. Comput. Methods 1 (2) (2004) 267–307.[40] S.R. Idelsohn, E. Onate, F. Del Pin, The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking

waves, Int. J. Numer. Methods Engrg. 61 (2004) 964–989.[41] R. Aubry, S.R. Idelsohn, E. Onate, Particle finite element method in fluid mechanics including thermal convection–diffusion, Comput. Struct.

83 (17–18) (2005) 1459–1475.[42] E. Onate, C. Sacco, S. Idelsohn, A finite point method for incompressible flow problems, Comput. Visualizat. Sci. 2 (2000) 67–75.[43] O.C. Zienkiewicz, P.C. Jain, E. Onate, Flow of solids during forming and extrusion: some aspects of numerical solutions, Int. J. Solids Struct. 14

(1978) 15–38.[44] O.C. Zienkiewicz, E. Onate, J.C. Heinrich, A general formulation for coupled thermal flow of metals using finite elements, Int. J. Numer. Methods

Engrg. 17 (1981) 1497–1514.[45] R. Codina, Pressure stability in fractional step finite element method for incompressible flows, J. Comput. Phys. 170 (2001) 112–140.[46] E. Onate, M. Manzan, A general procedure for deriving stabilized space-time finite element methods for advective–diffusive problems, Int. J. Numer.

Methods Fluids 31 (1999) 203–221.[47] R. Codina, U. Schaefer, E. Onate, Mould filling simulation using finite elements, Int. J. Numer. Methods Engrg. 4 (1994) 291–310.[48] R. Codina, O. Soto, Finite element simulation of the filling of thin moulds, Int. J. Numer. Methods Engrg. 46 (1999) 1559–1573.[49] C. Agelet de Saracibar, M. Cervera, M. Chiumenti, On the Formulation of coupled thermoplastic problems with phase-change, Int. J. Plastic. 15

(1999) 1–34.[50] M. Cervera, C. AgeletdeSaracibar, M. Chiumenti, Thermo-mechanical analysis of industrial solidification processes, Int. J. Numer. Methods Engrg.

46 (1999) 1575–1591.


[51] C. Agelet de Saracibar, M. Cervera, M. Chiumenti, On the constitutive modeling of coupled thermo-mechanical phase-change problems, Int. J.Plastic. 17 (2001) 1565–1622.

[52] N. Calvo, S.R. Idelsohn, E. Onate, The extended Delaunay tesselation, Engrg. Comput. 20 (5/6) (2003) 583–600.[53] S.R. Idelsohn, E. Onate, N. Calvo, F. del Pin, The meshless finite element method, Int. J. Numer. Methods Engrg. 58 (2003) 893–912.[54] MARC/Autoforge User’s Guide 2.2, 1999.[55] H. Edelsbrunner, E.P. Mucke, Three dimensional alpha shapes, ACM Trans. Graphics 13 (1999) 43–72.[56] VULCAN, Finite element system for casting simulation, Quantech ATZ, Barcelona, Spain, 2003. Available from: <www.quantech.es>.[57] J.C. Cante, J. Oliver, C. Gonzalez, Powder forming simulation with the particle finite element method, in: Z.H. Yao, M.W. Yuan, W.X. Zhong (Eds.),

Computational Mechanics, CD-Rom Proceedings of the WCCM VI, Beijing, September 5–10, 2004.[58] J. Rojek, E. Onate, R.L. Taylor, CBS based stabilization in explicit solid mechanics, Int. J. Numer. Mech. Engrg. in press.

http://www.quantech.es

Advances in stabilized ﬁnite element and particle methods...

Documents

Transcript of Advances in stabilized ﬁnite element and particle methods...