Acta Mechanica 139, 129 - 142 (2000) ACTA MECHANICA �9 Springer-Verlag 2000

Bifurcation indicators

M. Jamal, H. Elasmar, B. Braikat, Casablanca, Maroc and E. Boutyour, Metz, France and B. Cochelin, Marseille, France and N. Damil, Casablanca, Maroc and M. Potier-Ferry, Metz, France

(Received July 22, 1998)

Summary. In this paper we propose bifurcation indicators for linear or nonlinear eigenvalue problems. These indicators are the determinants of a reduced stiffness matrix. They measure the intensity of the response of the system to perturbation forces. The numerical computation of the indicators is done by a direct method and by an Asymptotic Numerical Method.

1 Introduction

The detection of bifurcation is one of the most interesting problems in structural mechanics. A large number of techniques, using iterative incremental methods, have been proposed [1] to [9]. The most popular ones consist in inspecting the determinant or the smallest pivot of the tangent matrix.

In this paper, we propose new indicators to detect bifurcation. These indicators are scalar functions, defined along the fundamental branch, the roots of which correspond to bifurca- tion points. These functions measure the intensity of the response to one or several perturba- tion forces. More exactly, the indicator is a determinant of a matrix of order k, k being the number of perturbation forces introduced in the system. The coefficients of this matrix are solutions of linear equations.

In the case of one perturbation force, this indicator has been used to detect stationary bifurcation for elastic problems [10], [11] and in fluid mechanics [12]. A similar idea has also been used to detect Hopf bifurcation in solids and in fluid mechanics [13], [14].

The computation of these indicators will be based on an Asymptotic Numerical Method [15]. The solution of the obtained problems is sought in the form of power series. The coeffi- cients of these series are numerically computed by a classical finite element method.

To illustrate this method, we consider the elastic and plastic buckling beams problem with variable cross-section that leads to linear and nonlinear eigenvalue problems, respectively.

2 Definition and computation of bifurcation indicators

In a number of bifurcation problems, such as buckling in solid mechanics [16], the equations that determine bifurcation points on a fundamental branch Uf(A) can be obtained by lineari- zation of the governing equations around a point (A0, Uf(A0)). The resulting eigenvalue problem is then governed by a linear operator La that we suppose, for the sake of conve-

130 M. Jaraal et al.

nience, self-adjoint and that can vary nonlinearly with respect to the bifurcation parameter ), (knowing that the case of a non-selfadjoint operator is possible, see [12], [14], [17]). We sup-

pose that, for ~ = ~0 the operator is regular (i.e., L0 = La0 is invertible). A bifurcation occurs when the operator La is singular and the resulting eigenvalue problem can be written as:

= 0 , (1)

where U is the eigenvector and X is the corresponding eigenvalue.

In the following we propose bifurcation indicators to characterize the bifurcation points. Namely, we introduce, in a fictitious way, k forces of perturbat ion fJ of intensity A#3 in the right-hand side of Eq. (1). Let AUo "~ be the response to the force f*~ for ~ = X0:

Lo Uo : (2)

The response U = uf(x) + AU to these perturbations satisfies

k

Lx AU : 2 a#Jf j , (3) j = l

z~U- ~ AmAUo m, ~U0 'z = 0 , for n = l t o k , (4) m = l

where A "~, in the supplementary orthogonali ty conditions Eq. (4), are arbitrary scalars. The symbol < . , . > is a given scalar product. The definition equations (3) and (4) mean that the perturbation lies in the subspace generated by the forces fJ, but we do not prescribe the num- bers A#J that exactly determine the perturbat ion force in that subspace. By contrast, we have to prescribe k conditions on AU, as in Eq. (4). The bifurcation points then correspond to A#~(,~) = 0 for all m.

We can solve the Eqs. (3) and (4) with respect to AU and A# "~, by assuming that the k numbers A "~ are given. Since the system is linear and well-posed, the solution linearly depends on the A ' ' . Thus we can write A/2 ~ in the form

{Ap} : [C] {A},

t { A p } = { A # ' A#2 ,Apk}, t { A } = { A 1,A 2, ,A~} - (5)

The coefficients GiJ(i,j = 1 to k) of the square matrix G, depend on the parameter IX. The matrix G has an obvious interpretation in elasticity. Indeed, the numbers A m are measures of the displacement field, while the numbers A#J represent the intensity of the perturbat ion forces. Thus, the matrix G is a reduced stiffness matrix. The bifurcation values are such that A#'~(,~) = 0, for m = 1 to k. So, according to Eq. (5), this is possible if and only if the deter- minant of the system equation (5) vanishes:

det (G(X)) = 0. (6)

In this paper we consider the determinant Eq. (6) as the bifurcation indicator. In solid mechanics the criterion Eq. (6) is quite natural and means that the reduced stiffness matrix G is singular.

In order to define the matrix elements G~J(i, j = i to/~), we inject the expression of A# ~ in terms of G as it is in Eq. (5) into Eq. (3) and then we search the solution of Eq. (3) in the form:

k

= E (7) 77~1

Bifurcation indicators 131

The vectors A U "~ and the coefficients GiJ(i, j = 1 to k), are then solutions of the following equations for m = 1 to k:

k L a A U "~ = ~ G ~ f ~ , (8 I)

i-1

(AU "~-AU0 "~ ,AUo ~ } = 0 , for n = l t o k . (9)

Equations (9) are obtained by substitution of Eq. (4). Hence, the bifurcation indicator is given by Eq. (6), where the coefficients G ij are solutions of the linear equations (8) (9).

In the case of only one perturbation A p l f 1 of intensity Z~# 1 and of direction fa, we obtain the following bifurcation indicator under the form that had been introduced in [10], [12]:

det (G(A)) = G 11 = (z~U01' z~U~ (10'1 {Lx-lf 1, AU01} �9

The zeros of this function G n give the eigenvalues of the initial problem Eq. (1). In what follows, the numerical computation of the indicator will be done in two different

ways. First, we can directly apply formula Eq. (10) for various values of the parameter ~. Nevertheless, this computation requires much computer time because the operator LA has to be inverted for many values of ~. In this paper this computation will be used as a reference.. This procedure will be referred to as the direct method. The second way of computation relies; on the Asymptotic Numerical Method (ANM) that allows to get the solutions of Eqs. (8) and (9) in terms of power series with respect to the parameter A. So, by using this perturbation. method, we shall be able to get the required solutions, for a range of values of ~, by only inverting one matrix L0.

Indeed, the ANM is a class of algorithms which solve nonlinear problems, associating: asymptotic expansions with numerical discretization methods. In the present paper it allows us to seek solution branches V(A) = (AU "~, U ira) (A) of the problem Eqs. (8) and (9) in the shape of truncated power series expansions with respect to the parameter A:

v( ;9 = v0 ~ ;~v~ + ~2�89 + . . . + ~PG.

In this way we transform the problem Eqs. (8) and (9) into a set of recurrent linear problems which admit the same tangent operator L0. Hence this method allows us to systematically compute a large number of terms of the series which provides a quantitative description of the solution path in a large neighborhood of the starting point. In many cases [18], [15], [19], it has been established that the computational cost of such a series is equivalent to that of a modified Newton-Raphson step, As the series generally have a finite radius of convergence, the whole solution path Aum(~) and Gi'~(A) of Eqs. (8) and (9) has to be computed step by step. To do that, a continuation ANM technique has been proposed [15]: the new starting point V0 is the end point of the previous step. Cochelin [20] has proposed a simple criterion to define the end of the step. It is based on the difference between two successive orders which must be smaller than a given accuracy parameter e, which leads to an estimate of the step length

t I IV~l l ) "

This formula defines the maximal value X,, of the path parameter, the new starting point is then given by V0 = V(A,~). Comparisons between the ANM and more classical algorithms can be found for instance in [15] for nonlinear geometrical elasticity, in [21] for plasticity, in

132 M. Jamal et al.

[22] for visco-plasticity and in [17] for Navier-Stokes equations. This method, in case of/~ = 1, has been applied to seek bifurcation points along nonlinear equilibrium paths and within elastic shell stability [11] or fluid mechanics [12], [14]. It has been coupled with an algorithm to compute the bifurcating branches [11]. In the beam buckling problems that are considered in this paper only one step is needed.

3 Beam buckling problems in elasticity and plasticity

We shall apply this idea of bifurcation indicator to elastic and plastic beam buckling prob- lems. In this section we seek the most simple variational formulation of these problems in such a way, that the operator Lx depends on A in the most simple possible fashion.

3.1 Elastic beam buckling problem

The classical equations governing the buckling of an elastic beam of length 1 subjected to a compressive load F can be written as

( d V(x) - o, (11) dx 2 E I ~U-zz j + F dec2

where E and I are Young's modulus and quadratic moment, respectively, and V(x) is the lat- eral displacement. We assume that E and I are constant and that the beam is simply sup- ported. Equation (11) is a linear eigenvalue problem, V(x) is the eigenmode and F the corre- sponding eigenvalue. This eigenvalue problem can be written in the variational formulation

L;,U = LU - AL~U = O, (12)

where F = ),, U = V, and L, Lo are linear operators defined by:

1 1

\ dx 2 d x 2 / d x , <LoV, 6V> = ~ 6 dx dx. (13) 0 0

Let us note that many eigenvalue problems in elastic stability can be written in the framework Eq. (12), where L and L~ represent the elastic and the geometrical stiffness, respectively.

3.2 Plastic beam buckling problem

We consider an elastoplastic beam of length l, with a variable section S(x) and a quadratic moment I(x), loaded by a longitudinal compressive force F (Fig. 1). The stress ~(x), assumed to be uniaxial, is related to the force F by

- F = �9

F i ..................................................................... g, ..................................................................... ~ - F

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... ~" Fig. 1. Beam with a variable X cross-section

Bifurcation indicators 133

Moreover, we suppose that the stress-strain relationship is given by a power law (Ramberg Osgood type law) in the form

where E is the Young's modulus, crv is the elastic limit and n is a hardening coefficient supposed to be an odd integer. The tangent modulus is then given by

Et &7 E \ (7 v /

In the framework of these hypotheses one can readily establish that the critical load F and the buckling mode V are solutions of the eigenvalue problem

d' ( d'v(x) dx' E~(x,F) r(x) dx~ ) + F d~ - 0

This equation is nonlinear with respect to the compressive force F. These equilibrium equa- tions can be derived from a Lagrangian functional L given by

L(V, M) = k [M dx2x2 2EtI(x) 2 \~fxJ J dx, ~5(L(V, M)) = 0 VSV, 5M, (14) 0

M being the bending moment. For the sake of simplicity we introduce the following non- dimensional parameters

(So) a - ~ S o ' g ( x ) = \~)/ , ( 15 )

where So is a special cross-section (minimal or maximal). Hence with the help of Eq. (15) the Lagrangian Eq. (14) becomes

f r a y v' s l L(V, M, A) o L ~zz 2EI(x) {1 + A '~ lg(x)} - ~ \dx/] ~ 0j dx,

and the nonlinear eigenvalue problem writes in the framework

L~U = LU - A n - I L p U - AL~,U = O, (16)

where the unknown U is a mixed variable, of components (V, M), and L, Lv and L~ are linear operators defined by:

1

ILu. : f / d ' v 0

) EI(x) 5M dx,

1 1

0 0

In this formulation, the operator L represents the elastic stiffness expressed in a mixed flame.- work, the geometric stiffness L~ is almost the same as in elasticity and the operator L v accounts for the loss of stiffness due to the plastification of the structure.

134 M. Jamal et al.

4 Algorithm to compute the indicators in elasticity

In order to calculate numerically the bifurcation indicators, we use the Asymptotic Numerical Method (ANM) to solve the problems Eqs. (8) and (9). We have recalled that within a linear buckling analysis, the operator La linearly depends on the load in the form (see Eqs. (12), (13))

La = L ALo. (17)

With an operator like Eq. (17), Eqs. (8) and (9) are written in the form

k L A U "~ - AL~AU m = ~ G{'~f ~ , (18.1)

i-1

( a u m - z~u0 "~, LzSUo~> : 0 , for ~ = 1 t o k . (18.2)

Note that, for the sake of simplicity, the scalar product (u, v} is substituted by (u, Lv) with respect to Eqs. (4) and (9). The solutions (AU "~, Gir~ 0 of Eq. (18) are sought in the form of integro-power series around (AUo ~, Go ~'~ = 5,i,~) with respect to A [15]

A U m = AUo m § AAU1 m q- A2AU2 m + . . . , (19.1)

G im = Go im + AG1 i m + A2G2 i m + . . . . (19.2)

Injecting these expansions Eq. (19) into Eq. (18), we get a set of linear problems satisfied by the vectors AUp "~ and the scalars Gp i'~,

k LAUp m = S~z2Up-1 + E Gpi~f i , (20.1)

i-1

(LAUp "~, AU0 ~} = 0, for n = i to k. (20.2)

By combining Eq. (20.1) and Eq. (20.2), we can eliminate A U S ~ from the Eq. (20.2). So we get a system of k equations (n = 1 to k) and of k u n k n o w n s Gpim(i =- 1 to ~:) that has the f o r m

k

Gp {'~ {f{, AU0 ~} : -(L~AU~Til, AUo~). (21) i :1

The right-hand sides of Eq. (21) depend only on AU~_ 1 that has been computed at the pre- vious order. When the Gp i~ has been obtained, one can deduce the value of the vectors AUp ~

solving Eq. (20.1). This is achieved by a classical finite element method. The discretization of the problem Eq. (20.1) then leads to

[<] {~} = {G}, (22)

where {Avp} is the nodal unknown, [K~] is the stiffness elastic matrix (which is the same for all these linear problems) and {Fp} is a column vector depending on the already computed terms. Further information can be found in the Appendix. This method can be easily imple- mented in an existing finite element code. The only difficulty is to define the column vectors {Fp} that is the right-hand side of Eq. (20.1). We refer to [15] for details of implementation in a similar framework. The computation time to get these {Fp} up to 15 th o r 20 th order is about 50% of the computation time to compute and decompose the stiffness matrix, see [15]. The computation cost to solve Eq. (22) is due, first to assembly and to the decomposition of the

Bifurcation indicators 135

Table 1. From [17], the computing time of ANM. Comparison between linear analysis, linear buckling analysis and one step ANM

Linear ANM ANM ANM Linear buckling elasticity order 5 order 10 order 15 analysis

Ratio 1 1.18 1.34 1.53 7.57 time/elastic time

matrix, second to the computation of the right-hand sides {Fp}. This computational cost has been evaluated in many papers [18], [19], [22], [17] and such an evaluation will not be repeated here. For instance in [18], the computation time to get 5, 10, 15 terms of the series has been evaluated and compared to a linear elastic analysis and to a linear buckling analysis in the case of plate buckling problem discretized by 4 812 degrees of freedom (see Table 1). In the

case of two perturbation forces this computing time should be greater; for instance at order 5, it should be 1 + 2 x 0.18 = 1.36. So, it appears that the present method is interesting from that point of view, as compared with classical eigenvalue computation method. Let us note that we only invert one matrix if the critical load lies in the radius of convergence of the series Eq. (19.2). Applications of this method will be given in Sect. 6.

5 Algorithm to compute the indicators in plasticity

Let us suppose that the eigenvalue problem Eq. (1) can be written in the form

LaU = L U - a lL , U - a2LoU = 0, (231 )

where the parameters al and au are related to the bifurcation parameter A. In the plastic beam buckling problem, we have al = A ~-1 and a2 = A. The linear operators L, Lp and Lo represent the elastic part, the plastic part and the geometrical part, respectively. Using the framework Eq. (23), Eqs. (8) and (9) are written in the form

k L A U m _ a lLpAU "~ - a2L~,AU m = ~ G'i'~ f i , (24.111

i--1

(AUm - AU0 m, LAUo ~) = 0, for n = 1to k. (24.2;,

As in Eq. (18), we have modified the scalar product in Eq. (9). In this case the solutions;

(AU "~, G i'~) of Eq. (24) are sought in the form of double integro-power series around (AUo "~,

Go i'~ = 5i,~) with respect to al and a2:

A U m = AUo "~ + alAU~o + a2AU~l + ala2AU~l + . . . (25) a ,,~im im G im= Go ~ + a~G[;~ + 2"~o,1 + ala2Gl,1 + . . .

Injecting the above expansions Eq. (25) in Eq. (24), we get a set of linear equations which are, satisfied by the vectors AU~,~ and the scalars G ~ ,

k

LAU~,~n : ZpAUp-l,n + L~AO~,n-1 + E Opm, n f i , (26.1) i=1

<LAU~,~, AUo~) = 0, for n = 1 to k. (26.2)

136 M. Jamal et al.

As in Eq. (21) we can drop the vectors AU~,~ from Eq. (26.2) and get a system of k equations

and of k unknowns @,~ (i = 1 to/~):

k E ira i Gp,~( f , AUo ~) = -{LvzgU,~_I,,~ , AUo ~} - (L~AU~,~z_ 1 A U o ~ } . i=1

(27)

Thus, the resolution o f the problem Eq. (24) is reduced to a succession of linear problems,

so that each problem may be solved knowing the solutions at previous orders. The solution of

these linear equations is computed by a classical finite element method, in the same way as in

the previous part.

6 Application: Buckling of beams

6.1 Elas t ic buck l ing

In this section we consider the buckling of a simply supported elastic beam of circular cross-

section, with a radius of R = 1 ram, a length of 1 = 100 mm and a Young 's modulus of

E = 2.155 Mpa. The interval [0, l] has been devided into 100 beam elements. Let us recall that

the fundamental branch is U/(A) = 0 and that Euler 's formula gives the analytical expression

of the buckling loads:

EIm2~r 2 [/'m - - - 12

where m is the number of eigenvalues.

In Table 2 these formulas have been compared to the direct calculus of bifurcat ion indica-

tor Eq. (10) and to the A N M computa t ion of Eqs. (8) and (9) with respectively one per turba-

t ion and two perturbat ions. In this case ),0 = 0. Let us remark that the bifurcat ion indicator

Eq. (10) gives almost the same buckling loads as the analytical formula, the slight difference is

due to spatial discretization.

The numerical results are presented in Table 2 and in Fig. 2. In some cases, the first bifur-

cation load is obta ined with a very high accuracy, even with one per turbat ion force and with

a small order (order 6). The computa t ion with two forces is always better than the one with

Table 2. Comparison between analytical formulae, direct computation of Eq. (10) and ANM computa- tion of Eqs. (8), (9) with one force at orders 6, 10 and ANM computation of Eqs. (8), (9) with two forces at orders 9, 15. Of course, the indicator with one perturbation force is not the same as the one with two forces, but they have the same roots

Bifur- Euler Bifurcation Bifurcation indicator (6) Bifurcation indicator (6) cation formula indicator (10) with one perturbation with two perturbations loads direct method ANM computation ANM computation

of (8, 9) of (8, 9)

first 155.031 383 4015 155.031 383 938 79

second 620.125 533 6060 620.125 546 921 49

third 1395.282450613 I395.2826034030 fourth 2480.502134424 2480.5029928657

155.036 367 075 66 order 6 155.031 383 938 79 order 9 155.031 410 722 11 order 10 155.031 383 938 79 order 15

620.420 006 437 30 order 9 620.134 171 929 36 order 15

Bifurcation indicators 13 7

det

-2

-4

a 0 500

ANM order 5 /

/

25"" 1000

Direct computation

i t f f

1500 2000 2500 X

3000

det

i ~ANM orde [ lO

Direct computation

l ~ r

r r x b 0 500 1000 1500 2000 2500 3000

Fig. 2.a Comparison between direct computation of Eq. (10) and ANM computation of Eqs. (8), (9) with one force at order 5; b Comparison between direct computation of Eq. (10) and ANM computation of Eqs. (8), (9) with two forces at order 10

one force. In Fig. 2 (a) we can see that the function A -+ det (G(A)) has some poles. Wi th one

per turbat ion, one pole between two consecutive roots can be found. So, in the case of one per-

turbat ion, the range of validity of the A N M approximat ion cannot include more than one

eigenvalue. This difficulty can be removed by using an indicator based on two per turba t ion

forces which allows us to compute two eigenvalues (see Fig. 2 (b) and Table 2).

6.2 Plastic buckling

In this section we consider the buckling of a plastic beam with data: 1 = 100 mm, E =

2.105 Mpa, ~Ty =2.102 Mpa , minimal and maximal cross-section radii }:~min = 1 mm,

Rm~x = 1.25 mm and n = 7 for three sorts of variable cross-section. Here the fundamental

branch is Uf(A) = 0 and A0 = 0. Table 3 reports the values of the first load of bifurcat ion

138 M. Jamal et al.

Table 3. Comparison between ANM computation of Eqs. (8), (9) with one force at orders 2, 4, 5 and direct method of Eq. (10). (a) beam with thin edges (b) beam with thinning at the centre (c) conical beam

First bifurcation Thin beam at edges Thin beam at centre Conical beam

ANM order 2 ANM order 4 ANM order 5 direct computation

0.537 136 825 561 52 0.533 214 859 008 79 0.532 973 083 496 09 0.532 867 233 276 37

0.277 996 284 484 86 0.276 517 364 501 95 0.276 451 934 814 45 0.276431 23626709

0.386 179458 618 16 0.385 182411 19385 0.385 148 193 35938 0.385 137481 68945

Table 4. Comparison between ANM computation of Eqs. (8), (9) with two forces at orders 1, 3, 6, 9 and 10 and direct method of Eq. (10)

Second bifurcation Thin beam at edges Thin beam at centre Conical beam

ANM order 1 ANM order 3 ANM order 6 ANM order 9 ANM order 10 direct computation

1.653 557 159 423 8 2.041 945 452 309 2 1.220 943 774 223 3 1.200 410 018 920 9 1.1980500159264 1.193 690767 645 8

1.282 341 805 458 1 0.988 647 795 200 35 0.931453 824 043 27 0.927 723 532 199 86 0.927 664 710 044 86 0.928 162 943 482 40

1.474 842 034 339 9 1.814 706 162 094 3 1.076 030 576 705 9 1.062171251 773 8 1.059981 1072350 1.052 419 535 461 8

obtained by a direct computa t ion of Eq. (10) and by two parameters A N M computa t ion

respectively of Eqs. (8) and (9) with one force at respective orders 2, 4, 5 for three considered

beams: a beam with thin edges, a beam, thin at the centre and a conical cross-section beam

having the radius R1, R2 and R3 so that

4x(1 x ) }{1 - Rmi~ : R m ~ - R2 = - ~ (}{ . . . . - R ~ n ) x ( R m ~ - Rmin)

1 , R3 - Rmin = l

We can see from Table 3 that an expansion of order 2 gives a good estimate for the first value

of bifurcat ion with an accuracy of 0.2%.

However the A N M computa t ion of Eqs. (8) and (9) with one force only allows the compu-

tat ion of the first bifurcat ion value. The A N M computa t ion of Eqs. (8) and (9) with two

forces allows us to get the two first bifurcat ion values. We report in Table 4 the various values

of the second load of bifurcat ion computed by means of a direct computa t ion of Eq. (10) and

by a two parameters A N M computa t ion of Eqs. (8) and (9) until order 10 with two forces for

the three considered beams: a beam with thin edges, a beam thin at the centre and a conical

cross-section beam. Wi th an expansion at order 10 the second point of bifurcat ion is obta ined

with high accuracy.

7 C o n c l u s i o n s

In this paper we propose new bifurcat ion indicators that have been tested in elasticity and

plasticity. The computa t ion of these indicators can be performed by a direct method or by an

asymptot ic numerical computa t ion of Eqs. (8) and (9) with one or two random forces. In the

latter, the indicator is represented in the form of a single or double integro-power series of

one or two appropr ia te parameters. The resolution of a sequence of linear problems permits

Bifurcation indicators 139

to detect the bifurcation points. Moreover, only one stiffness matrix decomposition is needed to achieve this calculation.

The computed series can be improved by using the so-called Pad6 approximants technique [10]. Here, the interesting feature of Pad4 approximants is their ability to correctly represent a function which has poles. This aims at computing the first few values of bifurcation with one force only. Nevertheless, in the case of the expansions with many parameters, as the ones considered in plastic buckling beams, the calculus of Pad6 approximants becomes more com- plicated.

The introduction of two perturbed forces has the advantage of eliminating the first pole, which bounds the radius of convergence of the indicator equation (10). The computation of the indicator equation (8) and (9) with two forces give access to the two first bifurcation values.

Appendix

Let us recall that we are looking for the eigenveetor U and the corresponding eigenvalue A of the bifurcation problem

LAU = L U - AL~U = 0. (A1)

In Sect. 2 we have proposed the determinant equation (6), det (G), as the bifurcation indicator of the eigenvalue problem Eq. (A1). In this Appendix we detail the ANM algorithm to com-. pute the indicators in (8), (9) and for one and two perturbations forces, in order to compute numerically the matrix elements G i'~.

Case o f one perturbation force.

In this case, the bifurcation indicator equation (6), G 11, is the solution of the linear problem equations (8) and (9) with one force perturbation

L A U 1 - A L c ~ / ~ U 1 = Gll f 1 , (A2.1)

(AU 1 - AUo 1, LAUo z} = 0. (A2.2)

To compute this indicator by the ANM, the solutions (AU I, G 1~) of the problem Eq. (A2) are sought in the form of integro-power G011 = 1, ,X0 = 0):

A U 1 = zi21Uo 1 -~- AAU1 1 @ A2AU2 1 -[- . . .

G 11 = G011+ AG111+ A2G211 + . . .

series around the initial solution (AU01 = L - l fl..

(A3.1)

(A3.2)

Injecting the above expansions Eq. (A3) into Eq. (A2), we get a set of linear problems satisfied by the vectors AUp 1 and the scalars @11, at each order p:

Lz~Up I = Lc~AUI_I -~- Gpll f 1 , (A4.1)

(LAUp 1, AU01> = 0 . ( A 4 . 2 )

By combining Eq. (A4.1) and Eq. (A4.2) we can eliminate AUp z from Eq. (A4.2) and get the following problems satisfied by the terms Gp 11 and AUp 1 of the series Eq. (A3) for p = 1 to N,

140 M. Jamal et al,

N being the desired truncature order:

Gp 11 = (L~'AU~-I, AUol> ( f l , Ago 1)

L-_./..~Up 1 L 1 < LGA~UI 1, s = GZ~ Up- 1 ( f l , AGo1}

(AS.i)

f l , ibr p - l t o N . (A5.2)

The right-hand sides of Eq. (A5.1) and Eq. (A5.2) depend only on the t e r m AUpl_I that has been computed at the previous order. The value of Gp 11 is obtained from Eq. (A5.1) and the value of the vectors AUp 1 is obtained by solving numerically Eqs. (A5.2). This is achieved by a classical finite element method. The discretization of the problem Eq. (A5.2) then leads to a discrete problem that can be written in the classical form

[Ke] {Avp} = {Fp}, (A6)

where {Avp} is the nodal unknown of AUv 1, [Ke] is the stiffness elastic matrix obtained by discretizing the operator L which is the same for all these linear problems. The vector {Fp} is the discretized form of the right-hand side of Eq. (A5.2); it depends only on the already com- puted term AUI_I and on the initial vector/AU01. In this manner we obtain the following ANM algorithm:

1) compute the starting point (AUol, G011, /~0), 2) for p = 1 to N, compute Gp n from Eq. (A5.1) and solve the linear problem (A5.2) to get

A C p I .

The solution \ G11 j (A) of the problem Eq. (A3) with ANM is acceptable up to a certain

value of path parameter A, because the radius of convergence of series expansions is generally finite. Cochelin [l 5] proposed a continuation procedure that has been summarized in Sect. 2, but we do not need it in the cases that have been analyzed in this paper.

Case o f two perturbation jbrces:

In this case, the bifurcation indicator equation (6) can be written: G l l G 22 - G12G 21, where G n , G12, G 2~ , G z2, solve the problem Eqs. (8) and (9) with two forces perturbation

L A U 1 - ALoAU 1 = Gll f a + G12 f 2, (AT.l)

(A~U 1 - A]Uo 1, LA~Uo I = O, (A7.2)

(AU 1 - AUo z, LAUo 2 ---- 0, (A7.3)

and

L A U z - ALoAU 2 = C21f 1 + G22f 2 , (A8.1)

(AU 2 - AUo 2, LAUo 1} = 0, (A8.2)

(AU 2 AUo 2, LAUo z} = 0. (A8.3)

To compute this indicator by the ANM, the solutions (AU'~, G "~) m = 1, 2 and i = 1, 2 of the two problem Eqs. (A7) and (A8) are sought in the form of integro-power series around the

Bifurcation indicators 141

initial solutions (AU0 ~ = L l f~ , G0i~ = 5i~, A0 = 0) for m = 1, 2 and i = 1, 2 with respect to A:

AU m = A U o m+AAU1 ~+A2AU2 "~+ . . . for m = 1 , 2 , (A9.1)

G i ~ = G 0 im+AG1 i'~+A2G2 i ~ + . . , for i = l , 2 a n d r n = l , 2 . (A9.2)

Injecting the above expansions Eq. (A9) into Eqs. (A7) (A8), we get a set of linear problems satisfied by the vectors AUp ~ and the scalars @i~, at each order p:

LZ~Up 1 = Lo-,~Ulp_l + Gpn f 1 + Gpe2f 2 , (al0.1)

<~z~Up 1, z~U01} = 0, (A10.2)

<LzSUp 1, AU02> = 0, (A10.3)

and

LAUp 2 = LzAU~_I + Gp21f I 4- GpZ2f 2 , (Al1.1)

(Lz~Up 2, z~Uol} = O, ( A l l . 2 )

<LAUp 2, AUo2) = 0. (All.3)

By combining Eq. (A10.1) and Eqs. (A10.2) and (A10.3), we can eliminate z~Up 1 from Eqs. (A10.2) and (A10.3) and get the following equations satisfied by the terms Gp n, @1~ and the v e c t o r s Z~Up 1 of the series Eq. (A9) for p = 1 to N, where N is the desired truncature order:

Gp11(f1 z~U0 I} + Gpl2{f2, z~U01) 1 : -<L~AUp_I, z~Uol} ,

Cp11(f I , z21Uo 2) d- ap12 (f 2, z~Uo 2> 1 = -<LzZ~U;_I, z~U02} ,

Lz~Upl = Lcrz~Ulp_l + Gpll f l + @12f2 .

(A12.1)

(A12.2)

(A12.3)

In the same manner, from Eq. (A11) we obtain the following problems satisfied by the terms @21, @22 and the vectors AUp 2

Gp21(f 1, z~Uol} + Gp22(Z 2, z~Uo 1) = - ( LaZ~U2 1, z~U01) ,

Gp21{fl, AUo2} + Gp22(f2, z~Uo2} 2 = -<L~UL~, ~uJ>, LAUp 2 = L~/XC~ i + @21fi + G 22f2.

(A13.1)

(ala.2)

(A13.3)

From Eqs. (A12.1) (A12,2) one can see that the right-hand side of Eq. (A12.3) depends only on the term AU 1 1 that has been computed at the previous order. In the same manner, from Eq. (A13), the right-hand side of Eq. (A13.3) depends only on the term AUp2_I computed at the previous order. When the G; im have been obtained, one can deduce the value of the vec- tors AUp ~ by solving the equations (A12.3) or (A13.3). This is achieved by a classical finite element method. The discretization of the problem Eq. (A12.3) or Eq. (A13.3) then leads to an algebraic problem as in Eq. (A6).

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Authors' addresses: M. Jamal, H. Elasmar, B. Braikat, N. Damil, Laboratoire de Calcul Scientifique en M~canique, Universite'Hassan II, Facult6 des Sciences Ben M'sik, B.P. 7955 Sidi Othman Casablanca Maroc; E. Boutyour, M. Potier-Ferry, Laboratoire de Physique et M6canique des Mat6riaux, UMR CNRS n ~ 7554 Universit6 de Meth, ISGMP, Ile du Saulcy 57045 Metz Cedex 01 France and B. Cochelin, Laboratoire de M6canique et d'Acoustique, UPR CNRS n ~ 7051 Ecole Sup6rieure de M6canique de Mar- seille, ESM2, Technopole de Chateau-Gombert, 13451 Marseille, France