Evaluation of simple bifurcation points and post ... - Science Direct

Computer methods in applied mechanics and engineering ELSEVIER

Comput. Methods Appl. Mech. Engrg. 175 (1999) 137-156 www.elsevier.com/locate/cma

Evaluation of simple bifurcation points and post-critical path in large finite rotation problems Alberto Cardona*, Alfredo Huespe ~ Centro International de M~todos Computacionales en hzgenier(a CIMEC, INTEC, Universidad Nacional del Litoral/Conicet Giiemes 3450, 3000 Santa Fe, Argentina Received 31 December 1997; revised 28 July 1998

Abstract

The main aspects of an algorithm for the exact evaluation of single bifurcation points along the nonlinear equilibrium trajectory of flexible mechanisms with large 3-D rotations are discussed. A first version of this algorithm was presented elsewhere [1]. In this work we extend its range of application to problems involving large finite rotations. Several flexible mechanisms and structures presenting bifurcation points are analyzed. Solutions of perfect bifurcation cases are compared to solutions of the literature, which were computed by geometric perturbation. @ 1999 Elsevier Science S.A. All rights reserved.

I. Introduction

The stability analysis of nonlinear structures and mechanisms involves the solution of large systems of nonlinear algebraic equations for varying values of a control parameter, which is, in most cases, associated with a load amplitude. In structural mechanics, this problem is usually referred to as that of tracing the equilibrium path of the system. Instabilities are associated with singular points on the equilibrium p a t h - - a l s o called critical p o i n t s - - w h i c h can be broadly classified into limit and bifurcation points. Standard methods of solution of systems of nonlinear algebraic equations--i.e. Newton's m e t h o d - - a r e not able to traverse singularities in the solution path. In order to analyze these systems, continuation methods have been successfully used in structural analysis as well as in many other fields of engineering and science. Basically, they consist in the enlargement of the set of unknowns of the problem while adding constraints that raise the singularity. Much interest in the structural analysis field is focussed on the exact calculation of limit and bifurcation points and, in the latter case, switching to secondary equilibrium branches and tracing the full equilibrium manifold. In a previous paper, we have discussed different implementations of continuation methods, suitable for treating nonlinear structural and mechanism applications [2]. In this paper, we present methods that allow us to compute critical points accurately on the equilibrium path, while keeping the basic framework of the continuation methods implemented previously. Several authors have treated this problem in a general context, giving particular attention to the analysis of bifurcation. We can mention the works of Allman [3], Wriggers et al. [4], Ofiate et al. [5] and Eriksson [6-8] within the engineering literature. The subject has received much attention in the numerical analysis literature: the works of Moore et al. [9], Decker and Keller [10,11], Fink et al. [12] and Rheinboldt et al. [13-15] should be mentioned with many others.

* Corresponding author. Professor UNL and researcher from Conicet. Researcher from Conicet. 0045-7825/99/$ see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S0045-7825(98)00365-X

A. Cardona, A. Huespe / Comput. Methods AppI. Mech. Engrg. 175 (1999) 137-15;6

138

Nevertheless, the large 3-D rotations case has not been particularly treated in the literature. A feasible algorithm for analyzing bifurcation in large 3D rotations situations--and specially when large precritical rotations are developed--raises some remarkable issues that we want to emphasize in the present work, within the general framework of the finite element method. We have considered the problem of lateral buckling of thin beams and arches as a test problem of analysis. This problem has deserved special attention in the structural analysis field (see e.g. Reissner [16] and Lee et al. [17]). It is a particularly challenging problem whose solution can be envisaged by the algorithms we have developed. The examples to be shown all exhibit this phenomenon. However, the algorithms are quite general and are not restricted at all to this kind of structural behavior. The paper lay-out is as follows. In Section 2 we do a short review of the treatment given to large three-dimensional rotations. Section 3 presents the general formulation of flexible multibody systems, especially the equations for a rigid body element. We stress aspects concerning the symmetry of the tangent stiffness matrix, assumption which has been used intensively in the proposed algorithm for computing critical points. In Section 4 we briefly mention some aspects concerning algorithms for nonlinear path following (i.e. continuation methods). The method we used has been described elsewhere [2]. Here, we restrict the analysis to the methods used to compute the critical points accurately. Aspects concerning the evaluation of the null eigenvector at the critical point in case of large finite rotations are discussed in Section 5. This section is essential to the algorithm. We discuss how the null eigenvector should be updated when changing the reference configuration, in order to be able to evaluate the path tangents at the critical point correctly. Finally, Section 6 describes the computation of the path tangents at the critical points, while Section 7 shows several numerical applications which illustrate the power of the approach.

2. P a r a m e t r i z a t i o n o f finite r o t a t i o n s

We adopt the rotation vector q¢ in order to parametrize finite rotations [18]: q t : q'n

(1)

with n E ~ a unit vector pointing the axis of rotation and ~ the angle of rotation around it. The rotation matrix is thus evaluated as 1 - c~2 os(~) R = exp(tlk) = I + sin(~) tlk+

~tak

(2)

where ~r is the skew-symmetric matrix associated to the axial vector qt through the identity

@.)=qtx(.)

(3)

Relations between rates R and qt are next detailed. The angular velocity o~ and its associated skew-symmetric matrix ~ are related to R in the form R -- ,Z,R

(4)

Material angular velocities are defined: g) = RVdoR = R T R

(5)

The axial vector 12 is a material object, which transforms vectors in the material setting. Meanwhile, transforms vectors in the spatial setting. Material and spatial angular velocities are related by the rotation matrix: oJ = RI2

(6)

The following identity between qt and 12 holds [18] 12 = T(qt) ~

(7)

A. Cardona,A. Huespe / Comput.MethodsAppl.Mech.Engrg.175(1999)137-156

139

where T is the nonlinear operator: q~

T(!It)-

sin(~)

7~ I

+(1

sin(qt) )nnX

~

-2

1

2

sin(~-) ~

~

_

) q¢

(8)

Relations for variations are obtained directly taking the rates as admissible rotation variations: 5R = R 5 0

(9)

with

(Io)

5 0 = T(q') 5 ~

and where 5 0 is the material rotation variation.

3. Formulation of flexible muitibody systems The Lagrange equations of constrained flexible multibody systems are stated as [19] d 5~o.cE_. 0o~f

dt ~1 Oq : Q . k CI~(q ) = 0

(11)

Here, ~/' is the system Lagrangian; q and ~/are the vectors of generalized coordinates and of their rates; Q is the vector of constraint forces associated to the constraints ~(q), which are given by Q = (kA - p@)r ~-q = (kA

-

pCI))T~q~)

(12)

where an augmented Lagrangian approach is used, with A the Lagrange multipliers vector, p a penalty factor and k a scale factor to balance the equations. Since the constraints are introduced as generalized internal forces, the implementation into a finite element code is straightforward. We are interested in treating quasi-static motion problems. Therefore, we assume that the kinetic energy of the analyzed system is null: changes in the system Lagrangian are produced by changes in the total potential energy T" due to deformation and changes of external forces (i.e. 2{ : - 7 / ) . The generalized coordinates vector is formed by positions and rotation parameters at the nodes of the model

q={~}

(13)

being x the vector of positions and q¢ the vector of rotation parameters. When formulating the principle of virtual power, the rotations increment arises naturally as one of the primary variables in the quasi-static problem. For this reason, the relation (10) plays a main role in the formulation. From (11) the equilibrium equation results g-[

kq~(q)J--[oJ

(14)

In order to solve the nonlinear system (14) by way of a Newton method, we need to evaluate the (stiffness) matrix, which can be written:

~qg = [~qqo'/' :] where B = [Tq~Ot

+

I-]3BB T-~n] _ -kB T

7qq)~ c ...].

+

[~_ut(tg(1)i-k/~i)Lq~i :] 0

(15)

140

A. Cardona, A. Huespe / Comput. Methods Appl. Mech. Engrg. 175 (1999) 137-1.56

REMARK. We remark that to evaluate the exact stiffness matrix we need the second derivatives of the restrictions with respect to the coordinates q. As pointed out in [2], inclusion of this term is particularly important when using continuation methods. If this term is neglected, the convergence rate is severely affected. An updated Lagrangian formulation is used: in order to evaluate equilibrium at the load level (k + 1), we consider as reference configuration that defined by the generalized coordinates q k = (x k, tltk) at the previous step. The equilibrium solution at the step (k + 1) determines the coordinate increments Aq k = (Ax k, At/¢~). The coordinates updating is written: k-I =X k

~k k

x + exp( ~ k + i ) = exp( tlkk) exp(A t ] / )

(16)

where we remark that the rotation parameters updating is not additive: ~.~+1 ~ q p

+ A~rf

(17)

3. I. The rigid bo@ element In order to illustrate the form of the equilibrium equations we are dealing with, we briefly describe those of a rigid body element. We will emphasize the computation of terms arising from the second derivatives of constraints, i.e. the third term on the right-hand-side of Eq. (15). We define a rigid body element formed by two nodes. At each node we have six degrees of freedom: three positions and three rotations. Rotation parameters at both nodes are equated by using a Boolean assignment:

~,,:~

(18)

thus the rotations at node B are eliminated from the set of degrees of freedom of the element. Therefore, three restrictions are used to impose the fact that the distance between nodes is constant:

~ = e l f ( x , - x A --RAg )

i = 1,2,3

(19)

where xa, RA are the position and rotation at the first node, x 8 the position at the second node, and X is the relative position of node B with respect to node A at the initial configuration. Here, in the context of the rigid body formulation, e, is the ith vector of the canonical basis in ~3. The generalized coordinates of the element result: q=

~ XR

(2o)

The constraints gradients matrix B of the element is computed next: B =Vq4~ = [Vq•,

Vq4): Vq@3]

(21)

with vq ~ =

f - r T--ei ~Rre, 1 L

ei

i = l, 2, 3

(22)

J

The second derivative of constraints Vqq ~ is evaluated by iinearizing Eq. (22) giving ~ A • A(V~,) =

- ~

.

(ATTXRTei + T'rx ARTe,)

= --5~A" (ATVXR~ei + TTX(R~Q) T Aq~) The Frechet derivative of T v is given by AT T = DT v • A~, expression which results [18]:

(23)

A. Cardona, A. Huespe / Comput. Methods Appl. Mech. Engrg. 175 (1999) 137-156

141

AT v = DT T • Allt - q t 1 (cos gt

sin~\

I ( 3 -sin~ -~

+ ~

-

qz/2

~

)(n T A t / t ) / +

-~1 (1 _

sinqZ'~ ~JtA

~ T n + n AqtTI

)

- cos qz - 2 (n T Aqt)[nn T]

/

7q; )(n

At/¢)ti +~- \

qz/2

After replacing equation (24) into (23), we get the contribution of the second derivative of constraint ~ to the stiffness matrix:

gVtA -(p~, - kA,) A(V~) = gqca"-x~'' ~,~, Aqta

(25)

The full expression of the symmetric matrix k'c'~ is given in Appendix A. Symmetry of the tangent stiffness matrix is verified for all kinds of elements of the flexible mechanism model. It is lost only in case of considering non-conservative loading.

4. Nonlinear path following methods We consider now the problem of finding the set of points {(q, p) E R" × R}, such that g(q, p) = 0

(26)

where g : E"+ ~--~ E" is a nonlinear algebraic function. This equation can be seen as an implicit function representing the response of rather general discrete nonlinear physical systems, in which p is a parameter controlling loads and q the set of dependent variables. In the context of flexible multibody systems discussed in Section 3, these points characterize the equilibrium states of the discrete system. This system of equations may be highly nonlinear. When describing mechanical structures, the high degree of nonlinearity is produced mainly by the geometrical description of motion. The kinematics of large rotations magnifies such phenomenon: the form of the equilibrium equations has been shown in Subsection 3.1 for the rigid body element. The set of solution points (q, p) of the nonlinear equation (26) is a one-dimensional manifold in E"+ ~ that can be described by the map g(s) = ~(q(s), p(s)) = 0 ; s C:_~

(27)

where the new parameter s is usually given the meaning of an arc-length. Continuation methods are usually used for finding the map (27). We discussed elsewhere [2] the particular implementation we use to treat flexible mechanisms problems, without considering singular points. In this work, we restrict the analysis to the computation of singular points along the nonlinear solution path. Inherent to the nonlinear response, singular points are commonly detected. At the points (q0, pO) where Vqg is singular (we consider only cases with nullity one, i.e. N(Vqg °) = span(~b~)), two different possibilities may occur:

(1) Vt~g if_ R(Vqg): these points are called turning or limit points. (2) V g E R(Vqg): this condition is also characterized by '/'zT Vag = 0, being ,;b,T the null vector of (Vqg) T. These points are called bifurcation points. For engineering purposes, full knowledge of the structural response requires the correct and exact evaluation of critical points. For example, the maximum loads that can be supported by a structure are normally associated with limit points. Also in some cases, bifurcation points mark the separation between stable and unstable behavior.

A. Cardona.A. HuespeI Comput.MethodsAppl.Mech.Engrg.175(1999)137-156

142

4.1. Detection of singularpoints along the equilibriumpath Let us suppose we are tracing the equilibrium curve along the fundamental path, and a change of sign of det(Vqg) is detected at point (q,, p,,). Let us call Vqg =K and Vpg =-gex,. We assume K symmetric, a consistent assumption with the form of the equations presented in Section 3. Then, we can determine exactly a limit point by solving the following augmented system of nonlinear equations (LPS):

fg(q'P)] F(q,~l,p)= i K~b, ~ : 0

(28)

L ~(~,) J where function t~: ~ " - - + ~ is /(4,, ) --114,,112 - 1. The latter system of equations may be used to determine a solution only in the case of a limit point [9]. At a bifurcation point, the system (28) is singular. Therefore, to eliminate the singularity, the following system (BPS) is solved instead, which is still augmented by adding one further equation and a new unknown y:

rg(q, p)+ yei] F(q'~bi'P'Y)-_jI

efd~,-I

tf = O

(29)

l -gL,4,, J We adopt now g(~b~ ) = e,T~b. -- 1, and ei is the ith vector of the canonical base in ~ . Component i is selected as that corresponding to the component of ~b~ with maximum absolute value. The solution process at a singular point is started as if it corresponds to a limit point. If a bifurcation point is found instead, the system of equations to be solved is switched from Eq. (28) to Eq. (29) (see Fig. 1). Such situation is detected by evaluating:

Ig~)