TRACKING NONLINEAR EQUILIBRIUM PATHS BY A HOMOTOPY METHOD

LAYNE T. WATSON, SIEGFRIED M. HOLZER and MARK C. HANSEN Virginia Polytechnic Institute and State University, Blacksburg. VA 24060. U.S.A.

(Received 10 March 1983; received for publication 13 April 1983)

Key words and phrases: Homotopy algorithm. nonlinear equilibrium path. structural mechanics, curve tracking, Chow-Yorke algorithm.

INTRODUCTION

IN STRUCTURAL mechanics, the Newton-Raphson method, with various modifications, is widely used to trace nonlinear prebuckling paths [17]. However, the method tends to break down in the vicinity of limit points where the tangent stiffness matrix becomes singular. To overcome this problem, and to trace equilibrium paths through limit points into the postbuckling range. several methods have been proposed. The most commonly used methods are described and evaluated by Ramm [18]. Among them, the modified Riks/‘Wempner method appears to be particularly effective.

The modified Riks/Wempner method is described and applied in conjunction with the finite element method in [18]. [19], [20]. It differs from the other methods described by Ramm [18] in that the loading parameters and the generalized displacements are dependent variables. The independent variable, which controls the progress along the equilibrium path, is an approximation to the arc length. Specifically, the length of the tangent vector to the cwrenf equilibrium point is taken as a measure of the arc length increment. Iteration from the tangent back to the equilibrium path can proceed along a “normal plane” [18] or a “sphere” [19]. In its initial form, the Riks/‘Wempner method [21], [22] destroyed the symmetry and bandedness of the stiffness matrix. To eliminate the problem, a modification was introduced [18], [19] at the cost of restoring the singularity at the limit point. It is justified by the argument that it is highly unlikely to obtain the singular point exactly.

There are two other methods for tracking curves through “singular points” which are applicable to structural mechanics-Rheinboldt’s algorithm [12] and Watson’s algorithm [2], [3]. These are frequently referred to as “homotopy methods”, since they view the problem as tracking some homotopy curve. They are both based on arc length as the independent variable. Tracking homotopy curves by solving a differential equation dates back at least as far as Davidenko [23], and using arc length in the context of tracking bifurcation curves has been popularized by Keller [24], [25]. The theoretical foundations, as well as the implemen- tation details, of Rheinboldt’s, Keller’s and Watson’s algorithms are quite different. If a large number of points on the curve and very accurate values for limit points are desired, the homotopy method proposed here guarantees nonsingularity and is superior to the others. Rheinboldt’s algorithm is preferable for just a few points, but nonsingularity cannot be guaranteed (as with the modified Riks/‘Wempner method). Keller’s work on bifurcation [24] is related but has not been applied to large finite element models.

The homotopy method proposed here has no inherent numerical instability near a limit

1271

1272 L. T. WATSOV. S. !A. HOLZER and .M. C. HAMEN

point, in fact, a limit point is no different (as far as the method is concerned) from any other point on the load-displacement curve. The method bears a superficial resemblance to standard continuation, imbedding, or incremental methods but is fundamentally different in two important respects: (1) aside from bifurcation points, there are no “singular points” along the curve, so turning points (e.g. limit points) pose no special difficulties whatsoever; (2) the imbedding or load parameter k is a dependent variable along the curve. In the algorithm, L does not necessarily change monotonically, and part of the power of the algorithm derives from this ability of A to both increase and decrease along the curve.

Theoretical background and results pertaining to the homotopy algorithm are in [2) and [3], and computer code is described in [4] and [5]. The method has been successfully applied to nonlinear complementarity problems [6], nonlinear two-point boundary value problems [3], fluid dynamics [7-91, highly nonlinear elastica problems [lo] and optimal structural design

[Ill.

HOMOTOPY METHOD

In the notation of Almroth and Felippa [l], the equilibrium equation has the form

F(q, A, 6) = 0 (1)

where q is the generalized displacement vector, 1 is the scalar load parameter, & is the load distribution vector and the vector function F is the force imbalance. q, 0, and F are n- dimensional vectors, where n is the number of degrees of freedom, Note that F is viewed as a function of q, A, and 0. The theoretical basis for the homotopy algorithm is the following fact from differential geometry [2].

Fact. Suppose that the n X (2n + 1) Jacobian matrix of F

DF = [F, FA FQ] (2)

has full rank on F-‘(O) = {(q, A, e)iF(q, I., 0) = 0,O < A < =}. Then for almost all a the Jacobian matrix of

p(q, A) = F(q, A, e) (3)

(I’ is just F with 0 fixed) also has full rank on F;-‘(O) = {(q, A)jF(q, A) = 0, 0 < i. < x}. “Almost all @ means for all Q except those in a set of measure zero. Alternatively, if 0

were picked at random, the result holds with probability one. In practical terms, the conclusion is virtually always true. The full rank of the n X (n + 1) Jacobian matrix

Di’ = [&F;.] (4)

at points on the load-displacement equilibrium curve I has two important consequences [2]: (1) I is a smooth curve with a continuously varying tangent, and I does not intersect itself nor other zero curves of P (in the (B + l)-dimensional q, A space). (2) the matrix DE has a one dimensional kernel (null space), which is crucial to the implementation of the algorithm.

Since I is smooth, it may be parameterized by arc length s. Thus q and A are functions of s along T, and equation (1) becomes

I’(&), i(s)) = 0 (5)

Tracking nonlinear equihbrium paths by a homoropy method 1173

Thus

and

-&q(s). A(s)) = [F,(q(s). 4s)) w?(s). j.(s))1 = 0

dq x [.I = 1 dA

dr

(6)

(7)

dqT d,I since T is parametrized by arc length -q = q(s), A = A(s)- and ds z

[ I IS the unit tangent

vector to T pointing in the direction of increasing arc length. T is the trajectory of the initial value problem equations (6) and (7) with initial conditions

q(0) = qo, W) = 0. (8)

Tracking I- simply amounts to solving equations (6)-(8). Recall that the Jacobian matrix [F’, r’,] in equation (6) has full rank everywhere along T, so in this formulation a limit point is oblivious. The true problem is to efficiently and accurately solve the initial value problem equations (6)-(g) for q(s), A(s).

Note that the derivative 5 $ [ I

is only indirectly specified by equations (6)-(7). and any

ordinary differential equation algorithm requires the problem to be in the form

Y’ = f(s, Y) (9)

dqT dA - -

[ 1 ds ds 1s calculated by finding the one-dimensional kernel of the matrix [F;, Fj.] and then

using equation (7) and the continuity of the derivative. This kernel calculation is done via Householder reflections (see [2] for the details), and the work involved is comparable to solving a linear system of equations.

There are basically two approaches to tracking the curve T defined parametrically by q = q(s), L = A(s). The first is to use a low order ODE method (e .g. Euler’s method) to predict the next point and then use Newton’s method with an (n -L 1) x (n + 1) Jacobian matrix (formed by augmenting P, with a row and column to give a full rank matrix) to iterate back to T. This is simple, accurate (there is no drift from T), and has no difficulty with limit points. The disadvantages are that an appropriate step As for Euler’s method is difficult to determine, there is no guarantee that the augmented matrix wilt have full rank, and iterating back to T exactly is very expensive if Euler’s method is forced to take small steps to maintain convergence. See Rheinboldt [12] for an excellent discussion of and argument for this approach.

The second approach is to solve equations (6)-(g) by a high order ODE method (e.g. a sophisticated variable step, variable order Adams method as in [13]) and use Newton‘s method only rarely to verify the accuracy of the ODE solver. Using an ODE subroutine package, this

1771 L. T. WATSOX, S. M. HOLZER and M. C. HANSEY

approach is simple, accurate enough for most engineering applications. and very efficient because Adams’ methods are efficient. The step size selection and order choice are done automatically, relieving the user of two difficult and burdensome chores. Furthermore, a good implementation. as in [13], will be stable and robust. Drawbacks are that the solutions do drift (although slightly), the startup cost is high, and on rare occasions the method simply “loses” the curve and wanders off in the wrong direction.

Both approaches breeze through limit points with no difficulty and neither approach is clearly superior. At the moment there is considerable computational evidence in favor of the second approach [2]-[ 111, and it is the second approach that is advocated here.

APPLICATION

The homotopy method was applied in the analysis of a space truss whose stability charac- teristics under multiple independent load combinations are described in [14]. A selection of results is presented. First, the mathematical model of space trusses, which is valid for large rotations [15], is expressed in the form of equation (5).

MATHElMATICAL MODEL

The axial deformation of the truss element in Fig. 1 is

e=L-Lo (10)

where LO and L are the element lengths in the initial and deformed states, respectively. It follows from Fig. 1 that

(11)

Fig. 1.

Tracking nonlinear equilibrium paths by a homotopy method 1275

(12)

Lj = Loj + A, (13)

AX0 = Xb -X,;A=U,,-Un;AX=AX,,fA (14)

where X,, Xb are the initial coordinate vectors at the CI, b-end of the element; U,, U, are the global displacement vectors at the a, b-end of the element; and the vectors AX,,, AX coincide with the initial, deformed state of the element. The strain energy of the element is

n=:yl)e”;yo=~ (15)

where y. is the extensional stiffness of the element. The principle of virtual work yields the equation of equilibrium

F(q,A,&) = -@+ ,$Ri= 0 (16)

where

ani R;=-;k=l,2 ,..., n.

@k (17)

The superscript i identifies the m elements of the assemblage. If Newton’s method is used to verify the accuracy of the ODE solver, the generalized force vector of element i. R’, is computed by the code number method [16] from the global force vector of element i. which can be expressed as

pi = P;

[ I .-p~=p~= Pi ’ [ 1 g =[c;jp’ I

where

L. ,j=++ Yf)e’ L

(18)

(19)

cj, i = 1, 2, 3, are the direction cosines and pi is the axial force of element i in the deformed state.

By equations (3), (4): (16), and (17), the n x (n -t 1) Jacobian matrix becomes

DE=[K--Q] (20)

where

(21)

KA is the tangent stiffness matrix of element i expressed relative to the generalized displacements of the assemblage. It is computed by the code number method from the global stiffness matrix

1276

of element i

L. T. WATSON. S. M. HOLZER and M. C. HANSEN

-.

K’ = [

K’ -jp

_Ki Ki I where

Ri =

[ 1 aw ah$bk

= Y

CIC?

k - (1 - c;>

y’F.

CIC3

czc3

1 & - (1 - c$)

(22)

(23)

(24)

RESULTS

To test the ability of the ODE solver to track with precision highly nonlinear equilibrium paths, the reticulated spherical cap in Fig. 2 was subjected to a variety of loads. The most

t2

IO 8

--------a 3

II 13

0 FREE JOINT

5 0 FIXED JOINT

E G 0” 2 i

25cm I 250-n j

43.3cm 43.3cm

Fig. 2.

Tracking nonlinear equilibrium paths by a homoropy method 1277

revealing result was obtained by applying a load at joint 1 in the direction of global l-axis and tracking the equilibrium path through nine successive zero load configurations in a single run. The projections of the path onto the x - ql, 44, and q6 planes are shown in Figs 3-5, where the loading parameter is nondimensionalized as

(2%

and the joint displacements, qk, which have units of cm, are numbered in the sequence of the global 1, 2, 3-axes from joint 1 to joint 7; i.e. q1 and ql are the deflections in the direction of the global l-axis at joints 1 and 2, respectively, and q6 is the deflection in the global j-axis at joint 2. The zero load configurations are labeled u-i in the sequence of occurrence. i.e. with increasing arc length. Figure 6 shows qualitatively the positions that the elements connecting joints 1, 2 and 8 assume at the zero load states.

Figures 2 and 6 indicate that the zero load configurations are symmetrical relative to the support plane and that a, c, g, and i are rigid-body states. This is reflected with high precision in Fig. 5. Specifically, the curve passes twice through the largest negative coordinate on the qb-axis, which corresponds to the zero load states d andf, and four times through the origin

.-

Tracking nonlinear equilibrium paths by a homotopy method 1279

8 8 T

8 d (D

Fig. 5.

1280 L. T. WATSOX, S. M. HOLZER and >I. C. HAMEN

of the I- q6 plane, which corresponds to the rigid-body states n, c. g, and i. The last pass occurs after the curve has been tracked over an arc length of approximately 60 units. This remarkable accuracy was achieved solely with the ODE solver, i.e. without a single correction by Newton’s method.

The Jacobian matrix in equation (4) has full rank along the equilibrium curve r for almost all e, but for some e this may not be true, in which case T may bifurcate or even have an endpoint. For these special 0, none of our theory is strictly applicable and the performance of the ODE solver in the vicinity of a bifurcation point cannot be predicted. The ODE solver may go through a bifurcation point satisfactorily (i.e. points on T beyond the bifurcation point

LINE OF SYMMETRY

Fig. 6.

are computed accurately), but the branch it comes out on and the effort required to pass the bifurcation point are largely unpredictable. Precisely what happens depends on round-off error, the ODE tolerance used, the shape of F, and the order and step length vvith which the ODE solver is operating when it reaches the bifurcation point. A successful run through a known bifurcation point, the first critical point on the fundamental equilibrium path [14], is depicted in Fig. 7. The load distribution consists of equal loads in the direction of the global l-axis applied at joints 2-7. q7 represents the deflection in the direction of the global l-axis at joint 3. The second path in Fig. 7 corresponds to a 0.1% imperfection of the load at joint 2.

Tracking nonlinear equilibrium paths by a homotopy method 1251

iT_ 00 0.20 o.lio 5.60 0.80 I.CO 1.20

q?

Fig. 7.

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