An arbitrary Lagrangian-Eulerian finite element ... - Semantic Scholar

Computer methods in applied mechanics and engineering ELSEVIER

Comput.

Methods Appl. Mech. Engrg. 162 ( 1998) 79-106

An arbitrary Lagrangian-Eulerian three-dimensional

finite element method for solving free surface flows

Azzeddine Soulaimani”‘*, Yousef Saadb “Ecole de technologie supkieure, Dkpartement de @tie mkanique, 1100 Notre-Dame Ouest, Montreal, PQ, H3C IK3, Canada hUniversity of Minnesota, Computer Science Department, 4-192 EEfCSci Building, 200 Union Street S.E., Minneapolis, MN 55455, USA Received

15 May 1995

Abstract This paper discusses numerical solution of unsteady three-dimensional free surface flows. The governing equilibrium equations are written in the framework of the Arbitrary Lagrangian-Eulerian kinematic description. The corresponding variational formulation is established afterwards. Since the variational problems are nonlinear with respect to the moving coordinates, a second-order approximate variational problem is derived after a consistent linearization of the referential motion. Stability of the discrete formulations is ensured with the help of a new stabilization method. A robust preconditioned GMRES algorithm is then used to solve the resulting set of nonlinear equations. Finally, the computational algorithms are assessed through numerical studies of various problems: a large sloshing flow in a threedimensional reservoir, a discharge flow from a reservoir, simulation of a liquid vortex produced inside a cylindrical container with a disk 0 1998 Elsevier Science S.A. All rights reserved. rotating at the bottom and a three-dimensional practical hydraulic problem.

1. Introduction

Free boundary problems are very common in real life and in industrial processes. Cruyer [l] reported over 3300 references covering 1200 topics from almost all branches of continuum mechanics. The difficulties inherent to these problems are very challenging for mathematical and computational analyses. From the computational point of view, which is of interest to us in this paper, one has to identify the unknown part of the boundary and its kinematics and to solve the strong coupling between the boundary motion and the dynamics of the continuum. Numerical methods of solving free boundary problems may be classified into three categories: Lagrangian, Eulerian and Arbitrary Lagrangian-Eulerian methods. In the first group of methods, any point of the free boundary moves with the fluid velocity. The so-called Line Segments Method [2-41 and the Marker Particles method [5] lie in this category. These methods are very accurate but frequent remeshings are required. In the Eulerian kinematic description based methods, the actual position of the points of the free boundary are localized in a fixed mesh. The equations of the fluid are solved in a larger domain than that really occupied by the fluid, the interface between the dry and wet regions has to be distinguished by using, for instance, the notion of the characteristic function (e.g. Volume of Fluid method of Hirt and Nichols [6], Soulaimani [7]). In principle, these methods can be employed for general free and moving boundary problems, however, the identification of the interface needs a refined mesh in order to obtain a sufficient accuracy. Arbitrary Lagrangian-Eulerian methods (ALE) are intended to combine the respective advantages of the previous methods. The computational domain is completely occupied by the fluid but it can be animated with its proper motion. At the free or moving

* Corresponding

author.

0045.7825/98/$19.00 0 1998 Elsevier Science S.A. All rights reserved. PII: SOO45-7825(97)00330-7

80

A. Soulaimani, Y. Saad I Comput. Methods Appl. Mech. Engrg. 162 (1998) 79-106

boundary, the mesh points velocities are related to the fluid velocities in order to satisfy an additional kinematic boundary condition. However, inside the domain one can in principle define an arbitrary motion in the sense that it is independent of the fluid one. The advantage from this procedure is to move continuously the mesh of the domain as the free boundary moves, thus avoiding any mesh collapsing or frequent remeshings [8-151. It is worth noting, however, that in most of the works which employed the ALE formulation, the question of accurately solving the geometrical nonlinearities due to the presence of a free surface has not been clearly addressed. Therefore, most of the numerical methods which have been used are in fact restricted to a first-order accuracy. Soulaimani [ 131 and Soulaimani et al. [ 141 developed a Newton type algorithm to get the second-order accuracy. The ultimate objective of this work is the development of a three-dimensional code capable of solving various free surface problems encountered in hydraulic structure design (e.g. penstocks, spillways, etc.); in the near future it will be extended to simulate some MHD processes. For this purpose, we choose to use the ALE concept for the kinematic description, the finite element method for space discretization, an implicit scheme for time discretization and finally all the equations are solved in a coupled fashion; the primary unknowns being the pressure, the fluid velocity and the mesh velocity. Our contribution in the present paper concerns the following important issues: a stabilized Galerkin formulation, the use of a robust iterative procedure for solving the resulting nonlinear system of equations and establishment of benchmark tests to easily validate the code. It is, indeed, well known that when discretizing the Galerkin formulation of the Navier-Stokes equations care should be exercised to avoid undesirable oscillations, for pressure in the low Reynolds regime and for the whole system in the advection dominated regime. The first kind of instabilities can be avoided by choosing compatible velocity and pressure finite element approximations, i.e. to satisfy the inf-sup condition [ 16,171. Whereas in the later case, instabilities can be avoided by adding some diffusive mechanism such as the effect induced in the streamline Petrov Galerkin method and Galerkin Least Squares method [ 181. Using the Mini element of Arnold et al. [ 191, Soulaimani et al. [20], were the first to prove, by using the static condensation of the internal degrees of freedom, that the effect of local bubble functions is in fact a regularization of the original system which consists of replacing the null block-pressure of the discrete Navier-Stokes equations by a smoothing operator. Later, bubble functions were proven to induce a consistent streamline artificial diffusions effect as in SUPG [21,22]. On the other hand, our experience with the Mini-element showed that the static condensation requires the storage of the internal degrees of freedom, along with the local element matrices associated to the bubbles [20], in order to get the consistent element residual vectors and matrices. Furthermore, it was also experienced that the treatment of bubbles could be quite time consuming especially if the residuals have to be computed very frequently as it is the case when iterative solvers, such as the GMRES algorithm [23], are used. In the same spirit as in [20,21], we develop a variant of the bubble-stabilized Galerkin method for general unsteady advective-diffusive systems which avoid the above drawbacks of the Mini-element. Besides its stable behavior, it will be shown that this formulation can be indeed well married with iterative solvers such as the preconditioned GMRES algorithm. An outline of the paper is as follows. In Section 2 the ALE kinematic description is reviewed. The Navier-Stokes equations are written for the referential representation using, respectively, a Cartesian and a covariant frame. In Section 3 we discuss a possible choice of the referential motion. In Section 4, variational methods using the Galerkin method are presented. In Section 5 an original approach for stabilizing the Galerkin formulation is proposed. In Section 6 we describe a general procedure for linearizing the variational formulations with respect to the moving coordinates. In Section 7 we discuss the use of the GMRES solution algorithm along with the ILUT preconditioning. In Section 8 some numerical results are presented in order to assess the proposed methods and algorithms.

2. The arbitrary

Lagrangian-Eulerian

kinematic

description

Arbitrary Lagrangian-Eulerian kinematic description (or Referential) has proved to be very suitable for simulating some free boundary problems encountered in fluid and solid mechanics (free surface flows, large deformation problems, . . .). Unlike the Lagrangian and the Eulerian description, the computational domain may be animated with its proper motion. In this situation it is possible to track the free boundary motion while maintaining a fairly regular mesh.

81

A. Soulaimani, Y. Saad I Comput. Methods Appl. Mech. Engrg. 162 (1998) 79-106

Let $8 denote the material body which occupies an open region of Rnd, where nd is the space dimension (Fig. 1). A fixed Cartesian reference system (el, e2, . . . , end) is chosen such that the Cartesian coordinates of points of .B noted by X=(X,,X, ,..., X,,,). The notation 0 stands for another open and bounded part of Rnd with smooth boundary r’. A motion of .%I is noted by 4 and the spatial velocity and acceleration vectors are, respectively, defined by

(2.1) and &,

du f) = 5 (x7 t)

(2.2)

with x = 4(X, t) being the position of point X at time t. Any point f of 0 will occupy a position x = A(_?,t) at time t following the referential motion A, with velocity: w(x, t) =

z

(a, t) .

We will use the following convention in our notation: every function defined over 0 will be denoted by a superposed hat; for instance, w@, t) = w(A(i, t), t) = ti(Xn, t). An updated referential configuration A,(R) is then completely determined by the velocity field I$(.?, t) or by the gradient deformation tensor F(f, t), whose components are given by

Any point x of A,(O) n +#I)

is a new position of two points X and P being related by the following

a=A-‘(&X,t),t)=P(X,t). The mapping follows:

9(X, t) represents

(2.3) in fact the relative

motion

of 93 over R Eq. (2.3) may also be written

x = A(P(X, t), t) a straightforward

differentiation

relations:

as

(2.4) of (2.4) with respect to t leads to

(1(x, t) = w(x, t) + P(i, t) *Z(f, t)

Fig. 1. Sketch of the material

(2.5)

and referential

configurations.

A. Soulaimani, Y. Saab I Comput. Methods Appl. Mech. Engrg. 162 (1998) 79-106

82

where $3, t) represents the relative velocity; the components of which are expressed system, which is defined as (kl, i,, . . . , knd) with ii = (kj/~$)ej, by

in the moving

reference

alkf Ci(& t) = dt (X, t) . We also need the expression follows:

of the vector acceleration

of the point X at time t as a function

off.

It is derived as

e

a(x, t) =

tqa,t) = g

(i, t)

and using the chain rule differentiation. ,. a^($ t) =

$

we get

(2, t) + (C$, t) * $)ti(i, t) .

Using (2.5), Eq. (2.6) is then rewritten

(2.6)

as

n

B(~,t)=~(iQ)+,(V

. (Ii - l+))

where 6 stands for the gradient 2.1. Conservation

equations

We now consider material equilibrium p(x, 04x,

operator

.6]zq.f,t)

(2.7)

with respect to the referential

in the Cartesian

i coordinates.

inertial frame

the formulation of conservation principles using the referential is expressed in the spatial description by the classical relation,

j_?($,t)&(i, div;

The momentum

0 = P(X, t)f(x, 0 + div, a(~, r)

(2.8)

where p is the density, f is the body force and u is the Cauchy stress tensor. Since the actual configuration of the material body at time t may be unknown, rewrite Eq. (2.8) in a well-defined domain such as the referential one. Making above equation can be expressed in terms of P by

where

concept.

t) = fi_f, t)&i,

denotes

t) + div, &,

the divergence

operator

it is then more convenient to a change of coordinates, the

t) with respect

(2.9) to f, j

is the determinant

of F, j = det F and

p = ja(p -I)’ is the Piola-Kirchhoff stress tensor of the first kind. This, however, does not have the same meaning as in the material description. In the referential kinematic description the momentum conservation equations become, by substituting (2.7) into (2.9), as follows: (2.10) It is also possible to derive description, which reads

the expression

of the mass conservation

equation

in a referential

kinematic

(2.11) 2.2. Conservation

equations

in the noninertial frame

Instead of using the fixed Cartesian reference system, it may be thought useful to express the component vectors in the covariant reference system. For the sake of completeness only, we present in this section the conservation equations in the covariant noninertial frame. We do not actually implement this formulation, since we believe that the Cartesian representation is easier and more practical for the problems we have solved so far. Again, this presentation is intended for the researchers who may find an interest in using it for some particular problems (e.g. turmachinery problems involving different domains moving at different speeds). For instance,

A. Soulaimani,Y. Saad I Compur.Methods Appl. Mech. Engrg.

162 (1998) 79-106

83

Ogawa and Ishiguro [15], used the covariant representation of the Navier-Stokes equations for solving flows around moving boundary bodies. Therefore, the acceleration can be decomposed respectively into the relative, the Coriolis and the frame accelerations: a@, t) = P(& t) * (ii’@, t) + 29&f,

t) +aQ,

t))

(2.12)

where

iqf,t)=V*ti(i,t) a^$, t)

=

$

(f, t)

-

ii’@, t)

=

$ (2, t) + 6&

t)

1

a”(_?, t) is the acceleration as seen by an observer attached to the noninertial frame. The notation eij the covariant derivative of j in the Z direction: &;-y); = ((i * fwi

stands for

+ y;&&

where

yg.2,t)

=

a'x, ai, a.fi a;,ax,

are the Christofell symbols. Using Eq. (2.12), the momentum and mass conservation equations are rewritten as (2.13) and (2.14) wheref=p-’

.E

2.3. Constitutive equations It still remains to define the constitutive laws of the continuum. We restrict ourselves to incompressible (p = constant) and Newtonian fluids for which the Cauchy stress tensor is linear with respect to the velocity gradient tensor: V(X, t) = -pz + /.&(Vu+ (Vu)‘)

(2.15)

where p represents pressure, /.L is the viscosity and Z is the identity tensor. Making use of the change of coordinates, Eq. (2.15) is rewritten as 42, t) = -@Z + /L(% . P + (VU * P)‘) REMARKS

(2.16)

2.1.

-

in the case of Cartesian inertial frame, the independent variables are U = (u, p, w) whereas in the noninertial frame the primary unknown variables would be U = (C, p, W); - for a frame with a rigid rotational motion, Eq. (2.12) is reduced to the classical expression: a(x, t) = I’@, t) . (?,, + (K:jFVi),jl* %!%U))

[-A;W

dfi =o

(5.9)

Let us now draw the following important comments: (i) The variational problem (5.9) may be rewritten as [W*(A,U

+A,U,,

- 9)) + M’, *K,U,,l da - lrW.~~dy-~1,1~~*(W).~(W)(U))dn=O

e

(5.10) where the operator Z* is nothing but the adjoint operator associated to 2, i.e. z*(W)

=AbW-

(AfW),, - (K:jW,i),j .

This formulation has a strong similarity with the SUPG integral form is added a pertubation-like term which enrichment of the finite element approximations. (ii) It is worth noting that the solution U b depends support functions. Let U b be the set of nld = neq X nb

and GLS formulations. Indeed, to the standard Gale&in in the present development emanates from the local only on the projection of the residual 92(U) on local local degrees of freedom associated to Ub and [apI the

A. Soulaimani,

Y. Saad I Comput. Methods Appl. Mech. Engrg.

162 (1998) 79-106

87

basis of polynomial bubbles, this has nld X nZd dimension, that is the finite element representation of Ub is given by Ub = [@l(U)” Hence, the algebraic system of Eqs. (5.7) gives

KJ)”=

-[Joe

[@l’c4,[@1

-I

+A;[@l,i - (K,[@l ,j),i) d’

1

.

[I

RP

[@]‘%!(U) dR

1

Therefore, the matrix D=

I Re

(5.11)

[@I’@,[@1 +A,[@l,i - (K,[@l .j),i>dfl

has to be nonsingular to find a unique solution U b, however a slight modification of the definition of D as shown later (see Eq. (5.17) can ensure existence of its inverse. This matrix is nothing but the discrete representation of the original partial differential operator 2 on the subspace of local support functions. Let us define .&i(U)=

[@]‘%?(U) da

I f2e

Thus, Ub = -[@ID-‘&(U).

We can rewrite the variational problem (5.10) as: Find U such that for all weighting function W, IR

[We (A,U + A;U,, - 9) + kVi . (KijUJ d0 - I,-W*~Jdy-~~o/!Z*(W),@]Di&?(U)dO=O.

r

(5.12) (iii) It is important to note that the variational problems (5.2) and (5.9) are perfectly equivalent, therefore the existence and the stability of their solutions are subjected to the same conditions. Furthermore, the introduction of internal degrees of freedom in the finite element approximations leads to the local conservation result given by Eq. (5.6). For the incompressible Navier-Stokes equations and for moderate Reynolds numbers, the above stability conditions are resumed to be the coercitivity and the inf-sup (Brezzi-Babuska) conditions. Thus, the trial and the weighting functions have to belong to appropriate discrete finite element spaces to ensure stability and convergence with mesh refinements. On the other hand, for three-dimensional problems, first-order finite element approximations are more suitable than higher-order ones mainly for implementation and cost-memory efficiency reasons. In the case of the Navier-Stokes equations, the simplest finite element continuous approximations which satisfy inf-sup condition are those of the Mini-element [19]. These approximations are, respectively, linear for the pressure and linear plus a bubble function for the velocity. Thus, this kind of approximations falls into the same framework as (5.2). It was shown previously how the bubble functions can be handled at the element level using a static condensation procedure, given formally by Eq. (5.8). Our numerical experience with the Mini-element has shown that not only the spurious pressure instabilities are avoided but the element is also quite stable for relatively high Reynolds numbers. However, the use of the Mini-element can be cumbersome in terms of memory in the case of large scale problems. Also, the static condensation procedure can be time consuming especially when iterative solvers are used. We want, in the sequel, to modify the variational problem (5.9) in order to avoid the above drawbacks associated with the bubble functions. We propose the following formulation:

I

R

[We (AJ

+AiU,; - S) + IV, *(KijlJj)l dLI - I, W. Ss dy - c

e

i,. Y*(W)* T~?~(U>d.0 = 0 (5.13)

where

A. Soulaimani, Y. Saad ! Comput. Methods Appl. Mech. Engrg. 162 (1998) 79-106

88

7’ = measure(0’)[ @]D - ’ [@I’ .

(5.14)

The rb matrix is then defined from the choice of the bubble functions and is dependent upon the original operator Y and also on the element geometry. The dimension of rb can be verified to be nld X nld. To get more insight into the proposed formulation (5.13), let us briefly recall the original version of the Galerkin-Least Square (GLS) formulation associated to (5.1) in the case where A, = 0, this reads [21] W- (A,U,, -

(K,jU,j),i - S) da - jrW.~~dy+CI,,~(A:W,-(K;W,),j).7W(~)dn=0. e

(5.15)

For the incompressible Navier-Stokes equations and up to the correct definition of the stabilization matrix r, GLS (or generalized SUPG) formulation can be proved stable for relatively high Reynolds numbers. There are indeed some empirical guidelines for the choice of the matrix r but a sound theoretical basis is still lacking. Douglas and Wang [24] and later Franca and Frey [25] modified slightly the GLS formulation by changing the sign of K:, in the last perturbation integral form,

I, W- [Aiyi - (K,U,,),; - .%Ida

-

Ir

W. 9s dy + c I,.. [AfFV, + (K;,IV,),,] .6%(U) da = 0. e

(5.16)

The only justification in choosing between formulations (5.15) and (5.16) respectively, is the stability behavior which has been proven, for the incompressible Navier-Stokes equations, to be superior for (5.16). On the other hand, and in the light of our previous development, we see that the consistent sign of Kij tends to be a plus sign rather than a minus sign as in the GLS formulation. Furthermore, the proposed formulation (5.13) provides a methodology for designing the stabilization matrix 7’. Indeed, for every set of bubble or local-support functions corresponds a matrix rb; the best one would improve the stability [26]. For instance, for the incompressible Navier-Stokes equations with fixed coordinates, only the components of velocity can be enriched as in the Mini-element by setting nb = 1 and [@I = diag{Nb, Nb, Nb, 0) with the bubble function Nb = &t[( 1 - 5 - 77- 5) and 5, q, IJ are the local element coordinates. Since the matrices Ai are symmetric and (almost) divergence free in the case of the incompressible Navier-Stokes equations, i.e. Ai.i = 0, then the matrix D is reduced to D=

I ne

([@]‘A,[@] + [@l~iKij[@l,j)d’

here D is in fact a singular diagonal matrix D = diag(d, d, d, 0) we define D -’ as D -’ = diag( 1 /d, 1/d, 1 /d, 0). Note also in this case that D-l is independent on the advection matrices Ai’s. Thus, for the convection dominant regime the stability may need to be reinforced for the local problem, unless A,, is a dominant positive-definite matrix. Another definition of D could be given by D=

J ne

([@l’(A,[@l +Ai[@l,i - (Kij[@l,j>,i>+A:[@l,i . pAj[@l ,j)dfi

(5.17)

with p = h/a, h is a typical mesh size and a is a norm of a typical reference velocity. On the other hand, the stabilization matrix D given by (5.17) is symmetric positive definite for symmetric advective-diffusive systems (i.e. for symmetric and divergence free Ai’s and symmetric positive Kij’s). 5.2. Unsteady case Consider now the advective-diffusive A,,UT+L%‘(U*)-9=0

with 9(u*)

=A&

- (K,U;,),,

multidimensional system of neq equations having the form (5.18)

89

A. Soulaimmi, Y. Saad I Comput. Methods Appl. Mech. Engrg. 162 (1998) 79-106

In order to build a variational formulation we proceed as before by using choosing weighting and trial functions which are defined respectively by

the Gale&in

method

w*=w+wb

and by

(5.19)

U*=U+Ub. Thus, the variational

problem

reads: find U and Ub such that for all weighting

[W.(A,U;+A,UT-9)+W’;(K,U:)]dLh

IR

II

functions

W and Wb,

W.TYddy=O

(5.20)

and

i ne

[Wb *(A,Uf: +A,UP, - (K,U;,),,)

where C%?(U)is the local residual,

da = -

Wb4?.(U)dfI i RF

i.e.

9?(U) =AOU,, +A;U.i - (K,U,,),,

- 9.

(5.21)

For time discretization we use either the implicit first order Euler scheme or the implicit scheme, i.e. the time derivative f; of any function f is approximated, in general, by f;, = CXJ” - 2

second-order

Gear

a;f”_’

wheref” is the value of the function at the nth time step (i.e. the current time is t = n At) and C+, i = 0, 1, . . . , are time step dependent coefficients. Thus, using time discretization, the solution Ub,” at the current time step n is given as a function of the solutions at the previous time steps by Wb. (q,A,,Ubsn +A;lJ>” i RC

- (KijUf;“),,) da = -

I ne

Wb * %@I”) da +

IP

Wb .A (y,Ub,n-i do 0 I (5.22)

Therefore,

iR

the Galerkin

[WqAoU;t

formulation

+A,UP;”

+A,U;

defined by Eq. (5.20) is exactly equivalent

- 9) + N’, *(K;jU;j)] d.0 - I, We Ps dy - 2 i,< X*(W). e

1 0

Ub,” dR (5.23)

where Ub,” IS computed from Eq. (5.22). Thus, one This may be very cumbersome in terms of memory, order to overcome this shortcoming of the enriched modify it as we did in the steady case. We start approximate U b,n by

has to store all the Ubxnmi solutions in order to obtain U”. particularly for large scale three-dimensional problems. In Galerkin formulation and also to enhance its stability, we by dropping the terms U :” along with ajUb~n-i and we

U b.n = &qU”) with 7’ = measure(On’)[@]D-‘[@]’ and D=

to

I oe

Therefore,

[@l’(@o[@l +Ai[@l,; - (Kij[@],i>.;)df2.

the proposed

formulation

reads

A. Soulaimani, Y. Saad I Compuc. Methods Appl. Mech. Engrg. 162 (1998) 79-106

90

[W~(A,U~+AiV~-9)+W,~(KjjU~j)]dL!_ REMARK

I

Ir

W.SSddy

3*(W) 3+%?(ZP) do = 0. 5.2. A fully space-time

variational

(5.24) formulation

[W.(A,U,, +AiU,; - 9) + Wi *(K,u,j)I do”-

4i

generalizing

(5.13) reads

Ir ~‘9~dy dt- 2 i,. c%'~(W)*~~L%%.J)d~dt =0 ,I e " (5.25)

with T?(W)

= (AiW),, - (AfW),, - (K:jWl),j

4 = n X It,, I,+ 1[, 4 = r X It,, t,+ 1[ and W(. , t,) = W). , t,+ , ) = 0. Many tions can then be derived from (5.25).

6. Consistent

special

semi-discrete

formula-

linearization

The motion of referential domain and/or the presence of a free boundary lead to geometrical nonlinearities which are expressed into the conservation equations and into the variational problem through the gradient deformation tensor. These set of highly nonlinear equations need to be solved by an iterative procedure. For the Newton iterative procedure of this kind, one needs to perform the consistent linearization in order to obtain the consistent Jacobian matrix which guarantees an asymptotically quadratic convergence rate. This may be properly performed in the variational formulation by using a first-order approximation of the referential motion. For the sake of simplicity, we shall show how to obtain the approximate variational problem associated to the Gale&in formulation (4.6). Indeed, all the integral forms involved in Eq. (4.6) depend on the gradient deformation tensor F, which is determined after integrating the differential equation: dx dt

=

IiqP, t)

(6.1)

with appropriate initial conditions. Let us write a simple relation between

the actual coordinates

and the referential

ones:

x=f+&X which ti represents the coordinates displacement. In the iterative procedure this will represent a small perturbation displacement if the referential domain R is chosen as the most updated configuration of the actual domain (Fig. 2). In other words, fl will be in fact constituted by the updated mesh at the current time step. The perturbation vectors 6x represent the distance separating the two domains. Hence, a first-order approximation of the gradient deformation tensor in the neighborhood of identity tensor (which is of dimension nd X nd) Z is simply:

F=Z+Qisx. The first-order

approximations

(6.2) of its determinant

and its inverse are respectively

J=l+div&=l+6J

expressed

by (6.3)

and by

F-‘=I-v&=I+SF-‘. Furthermore,

an approximation

(6.4) of the variation

of w is related to 6x by

91

A. Soulaimani, Y. Saad I Comput. Methods Appl. Mech. Engrg. 162 (1998) 79-106

4 (Q) ;

with t -0

Fig. 2. The referential domain and its configuration after a small motion.

Sw=a&x

(6.5)

where LY= s At with At the time step and Euler implicit scheme or s = 3/Z for the Also, to the fluid velocity and pressure Sp. The approximate variational problem

[W .(A,64, +.iiMJi) =-

(I n

+ W.i*itij tXIj] dR

+

[W*(&l,U~,+ 6AiU,,)+ W,;*6Kijl.Ij] do

[w.(A,U,, +@I; - 9+ W',*iiijU,jldfl -

where &,

the coefficient s takes, for instance, the value s = 1 for the first-order second-order Gear scheme. are associated first order variations denoted, respectively, by 6cl and reads: find 6U = (sll, Zip, 6~) such that

=A,(Z,U);

Ai =Ai(Z,U)

;

(6.6)

Eij= K,(Z, U) ;

and

aA0

6Ao=-.tiU; au

aKij

aA;

iui=-jpJ;

SKij=-&XJ.

The matrices &IO, &Ii and SK, can be indeed expressed analytically using in particular the approximations (6.2)-(6.5). This will help to get a more accurate Jacobian matrix or any approximation of it that can be used as a preconditioner.

7. Solution algorithms 7.1. Introduction Using time and space discretizations, algebraic equations of the form:

the variational

Y({u}“~‘-’ ,{su}“9i = -{ .B}>