INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 43, (1998) INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS G. JELENI C AND M. A. CRISFIELD ; Department of Aeronautics; Imperial College of Science; Technology and Medicine; London; U.K. ABSTRACT The formulation of dynamic procedures for three-dimensional (3-D) beams requires extensive use of the algebra pertaining to the non-linear character of the rotation group in space. The corresponding extraction procedure to obtain the rotations that span a time step has certain limitations, which can have a detrimental eect on the overall stability of a time-integration scheme. The paper describes two algorithms for the dynamics of 3-D beams, which dier in their manifestation of the above limitation. The rst has already been described in the literature and involves the interpolation of iterative rotations, while an alternative formulation, which eliminates the above eect by design, requires interpolation of incremental rotations. Theoretical arguments are backed by numerical results. Similarities between the conventional and new formulation are pointed out and are shown to be big enough to enable easy transformation of the conventional formulation into the new one.? 1998 John Wiley & Sons, Ltd. KEY WORDS: non-linear dynamics; 3-D beams; dissipative algorithms 1. INTRODUCTION The modern setting for dynamics of 3-D beams is based on the use of displacement and rotation variables with respect to the inertial frame and among other works we can mention those of Simo and Vu-Quoc, 1 Cardona and Geradin, 2; 3 Simo et al., 4 Bauchau et al. 5 and Ibrahimbegovic and Al Mikdad. 6 In such a setting the resulting kinetic energy takes an exceedingly simple, quadratic form, while the non-linear relations between global displacement and rotation variables and the adopted strain measures introduce a complex denition of the strain energy, regardless of whether the strains are actually small or not. Consequently, such an approach naturally supports the use of the so-called geometrically exact nite-strain beam theory. 7; 8 The establishment of the relationships between the large overall 3-D rotations and strains appears to be one of the biggest problems in the geometrically exact beam theory, not the least because the representations of 3-D rotations by the direction cosine matrices of the rotated frame with respect to the inertial frame constitute a non-linear manifold rather than a linear space. The (non-additive) update of 3-D rotations has to be performed carefully in order to preserve the nature of the Correspondence to: M. A. Criseld, FEA Chair of Computational Mechanics, Imperial College of Science, Technology and Medicine, Prince Consort Road, London, SW7 2B7, U.K. m.criseld@ic.ac.uk FEA Professor Contract=grant sponsor: Engineering and Physical Sciences Research Council of Great Britain; Contract=grant number: AEST 2504 CCC /98/ $17.50 Received 21 February 1997? 1998 John Wiley & Sons, Ltd. Revised 23 February 1998

2 1194 G. JELENI C AND M. A. CRISFIELD manifold, and in the theory proposed by Simo 8 and Simo and Vu-Quoc 9 the iterative rotational changes are being interpolated along the length of the beam and consequently the associated update of the strain measures is based on the conguration at the last iteration rather than the last converged conguration. On the other hand, in order to solve a dynamic problem, some timeintegration scheme has to be applied, whereby the update of velocities and accelerations is always based on the kinematics at the last converged step and not the one at the last iteration. 1 It will be shown in the paper that the direct combination of these two concepts (as applied in Reference 1) often does not give satisfactory results and can lead to numerical problems. This motivates the development of an alternative approach, which overcomes these discrepancies. The key issue in the new algorithm will be the update of strain measures based on the last converged conguration, which leads to the interpolation of incremental rather than iterative rotational changes. Such an idea is not new Cardona and Geradin regarded it as the most suitable choice 2 and in the energy conserving algorithms for 3-D beams (see References 4 and 5) the interpolation of incremental rotations is mandatory. However, in neither case has the idea been proposed to directly tackle the problem described above. Both the conventional and the new algorithm are presented within a framework of the Newmark s family of time-stepping schemes, 10 4; 11 which has often been reported to produce unreliable results when applied to problems of non-linear dynamics. This problem is not related to the earlier update discrepancy and aects both the conventional and the new algorithm. In this paper we propose a way to apply a modication of the Newmark s algorithm proposed by Hilber et al. 12 (the so-called -method), which often eliminates the unreliability of the original algorithm, to the systems with large 3-D rotations. The conventional formulation 1 can be easily transformed into the new one. This is in sharp contrast with the introduction of an energy conserving formulation, 4; 5 which requires more signicant modications. We compare the new formulation with the conventional one, validate the results on a standard benchmark problem, and compare the two formulations by analysing a new example. 2. DYNAMICS OF 3D BEAMS GENERAL THEORY 2.1. Kinematics of 3D beams The beam model analysed here is basically due to Reissner 7 with the parametrization of rotations as given by Simo 8 and nite element technology as given by Simo and Vu-Quoc. 9 Here we only outline the basic kinematic assumptions. For a more detailed account the reader is referred to Reference 8. The line of centroids of cross-sections of the undeformed beam element is for the sake of simplicity taken to be a straight line which coincides with the x-axis of the inertial Cartesian frame (x; y; z) with g 1, g 2, g 3 as the unit base vectors (Figure 1). The initial position vector of a material particle (x; 0; 0) on the line of centroids is denoted by r 0 (x)=xg 1. The cross-sections of the undeformed beam in the co-ordinate plane x = const. are orthogonal to the line of centroids and their normals coincide with the base vector g 1. The remaining two base vectors, g 2 and g 3, are directed along the principal axes of inertia of the cross-section. The geometric shape of the cross-section is assumed to be arbitrary and constant along the axis of the beam. In the deformed state, the line of centroids is a space curve (Figure 1) dened by the position vector r(x)=xg 1 +u(x) in 3-D ambient space R 3. The Bernoulli hypothesis of the plane

3 INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS 1195 Figure 1. Reference coordinate system, body attached frame and deformed conguration of the beam cross-sections remaining planar after deformation and retaining their shape and area is assumed to hold. The position vector R of an arbitrary material particle (x; y; z) of the deformed beam 0y } may now be written as R(x; y; z)=r(x)+(x){ ; where y and z are the co-ordinates of material particles within a cross-section at x with respect to its centroid (see Figure 1) and is z an element of the Lie group of proper orthogonal transformations SO(3) with well-known properties det = 1 and 1 = t, which denes the relationship between the base vectors of the body attached frame G i (x); i =1;2;3 and the base vectors of the inertial frame g i ; i =1;2;3 via G i (x)=(x)g i ; i=1;2;3. The matrix of components of the linear transformation will be referred to as the rotation matrix. The deformed conguration of the beam at any co-ordinate x between its ends is thus completely dened by (i) the position vector of the deformed line of centroids, and (ii) the rotation matrix. Consequently, we say that R 3 SO(3) is the conguration space of the 3-D beam element. Here we proceed by parametrizing the rotation matrix using the rotational pseudovector = i g i (see References 8 and 13 for more detailed account), whereby = exp = I + sin 1 cos where = e ijk k g i gj t is the skew-symmetric matrix associated with the rotational pseudovector, the repeated indices i, j and k are summational over the dimension of the space, e ijk is the permutational symbol with all the components being zero apart from e 123 = e 231 = e 312 = e 132 = e 321 = e 213 = 1 and i ; i =1;2;3 are the components of the rotational pseudovector with respect to the inertial frame. Note that for any two vectors v; w R 3 the identity v w = vw = ŵv can be established, where here and throughout the paper the superimposed hat denotes the skew-symmetric matrix of the respective vector quantity. We also note that for any v R 3 and any SO(3) we have v = v t. The rotation matrices at congurations k and k + 1 are related via the rotational pseudovector ṽ which rotates the base vectors G i; k into the base vectors G i; k+1 around the axis ṽ= # for the angle # through k+1 = exp ṽ k.

4 1196 G. JELENI C AND M. A. CRISFIELD During the linearization of non-linear equations of motion, we will need the variation of the current conguration (r; ). Variation of the position vector is computed from r(x)=xg 1 +u(x)as r=u, while the variation of the rotation matrix is dened through its directional derivative 9 in the direction of v as = v. Note that r, ṽ and v are all dened with components given with respect to the inertial frame. Such kinematics gives rise to a so-called spatial setting of the problem, which enables easy interpretation of the results Strain measures and constitutive equations In Reissner s beam theory, 7 9 translational and rotational strain measures, S and Z, dened with respect to the body attached frame at x, are related to the conguration (r; ) through the equations { given in Tables 1 and 2 of Reference 9 as S = t 1 } r 0 and Ẑ= t. The prime ( ) in these 0 equations denotes the derivative with respect to the axial co-ordinate x. A convenient expression for the vector of rotational strains at conguration k + 1, which is related to the conguration k and the rotational pseudovector ṽ between the two congurations is14; 15 Z k+1 = Z k + t t kt (ṽ)ṽ where ( 1 T (ṽ)= 1 1 #2 sin # # ) ṽṽt + sin # 1 cos # I + ṽ # # 2 The choice of the so-called material strain measures, dened with respect to the body-attached frame, is motivated by the fact that the constitutive law for an elastic material can then be taken as a linear relation between the (material) stress and stress-couple resultants and the adopted strain measures as N = C N S and M = C M Z, where C N = diag[ea GA 2 GA 3 ] and C M = diag[gj t EI 2 EI 3 ] are constitutive matrices which dene the relations between translational strains and cross-sectional forces, and rotational strains and cross-sectional moments, respectively. Here, E and G denote elastic and shear moduli of material, A is the cross-sectional area, A 2 and A 3 are the shear areas in the directions of principal axes of inertia of the cross-section, J t is the torsional inertial moment of the cross-section and J 2 and J 3 are the cross-sectional inertial moments about the principal axes of inertia of the cross-section Strong form of the equations of motion According to Newton s second law of dynamics, the sum of all the forces and moments acting on a body is equal to the time change of linear and angular momenta acting on the same body. The corresponding dierential equations of motion of a beam are derived in Appendix I of Reference 8 and for any x (0;L) and t 0 read (N) + n = Au (M) + r N + m = (ŴI W + I A) (1) where the superimposed dot denotes the derivative with respect to time, is the mass density, I is the mass moment of inertia tensor, dened as I = diag[j 1 + J 2 J 1 J 2 ], Ŵ = t and A = Ẇ are the angular velocity (skew-symmetric matrix of) and acceleration, the components of which are expressed in the body attached frame, and n and m are specic force and moment loads. At a particular time t, the boundary value problem to be solved is fully dened when partial dierential equations (1) are complemented with the boundary conditions as given

5 INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS 1197 Table I. Essential and natural boundary conditions for dierential equations of motion (1) Essential (kinematic) boundary conditions Natural (static) boundary conditions x = 0 Either r = r 0 or N = F 0 Either = 0 or M = T 0 x = L Either r = r L or N = F L Either = L or M = T L in Table I, where F 0, T 0, F L and T L are the vectors of applied forces and torques at x = 0 and x = L. The dierential equations of motion along with the prescribed boundary conditions and the dened kinematics, strain measures and constitutive equations fully describe the mechanical problem to be solved. We want to solve this problem for the unknown conguration (r; ):[0;L] R 3 SO(3) and in the next section we perform a transformation of the problem into its weak (variational) form, as a rst step toward applying a nite element discretization Weak form of the equations of motion The weak form of the problem dened by dierential equations (1) and the associated boundary conditions (Table I) is obtained by providing vector test functions f :[0;L] R 3 and :[0;L] R 3, which are (i) continuous, (ii) at least once continuously dierentiable and (iii) identically equal to zero at all co-ordinates x with prescribed essential (kinematic) boundary conditions. By taking a dot product of our test functions with dierential equations of motion (1) we obtain f t [(N) + n Au]=0 and t [(M) + r N + m (ŴI W + I A)] = 0 (2) which can be integrated over the domain of the problem (note that (i) implies integrability of the test functions) to give L 0 [f t Au + t (ŴI W + I A)] dx = L 0 {f t [(N) + n]+ t [(M) + r N + m]} dx After performing partial integration of terms consisting of (N) and (M) (note that (ii) implies integrability of the rst derivatives of the adopted test functions) we obtain L 0 (f t N t r N + f t Au)dx+ L 0 L 0 [ t M + t (ŴI W + I A)] dx (f t n + t m)dx (f t N + t M) L 0 = 0 (3) The term (f t N + t M) L 0 has to be complemented with the corresponding predened boundary conditions given in Table I. This means that the stress resultants therein either become zero Note that the minus signs in Table I arise from the applied loads and the stress=stress-couple resultants at x = 0 acting on the cross section from dierent sides. Also note that in each of the four pairs of boundary conditions in Table I only one condition can be satised

6 1198 G. JELENI C AND M. A. CRISFIELD (unloaded ends), or equal to the applied loads as stated in Table I (these two cases dene the so-called natural boundary conditions, see e.g. Reference 16), or are equal to reactions (essential boundary conditions 16 in that case the test functions are zero and the term vanishes). Since the essential and the natural boundary conditions are mutually exclusive in the way pointed out in Table I, equation (3), Table I and (iii) nally give L 0 (f t N + f t Au)dx+ ( L f t n dx + 0 L 0 L 0 [ t M t r N + t (ŴI W + I A)] dx t m dx + f 0 F 0 + f L F L + 0 T 0 + L T L )= 0 (4) Note that in case of essential-only (pure Dirichlet) boundary conditions all of the boundary terms in equation (4) vanish, which accounts for the extra boundary terms in equation (4) that do not exist in equations (2:5) and (2:6) of Reference 1. The equivalence of equation (4), which was derived directly from the strong form (1) and associated boundary conditions (Table I), with no restrictions on f and other than earlier conditions (i) (iii), with the principle of virtual work can be observed by noting that the latter is immediately recognized with f and written as u and v, respectively. The virtual work complements of the stress and stress-couple resultants can then be recognized as t (u + r v) and t v, respectively, which is equivalent to the variations of the adopted strain measures S and Z (see 8; 17; 18 Section 2:2). This equivalence is typical of the geometrically exact beam theories. 3. FINITE ELEMENT FORMULATION The conventional nite element approach for solving the dynamic problem dened by a weak form is based on the following steps: 1. Interpolation of the test functions. In this step the problem becomes partially discretized in space, and we proceed by seeking the solutions of the set of dierential equations with respect to time. 2. Numerical time integration of the velocities and accelerations. In this step the solutions for linear and angular velocities and accelerations at time t n+1 will be assumed to be based on the known kinematics at some time t n t n+1 and the known conguration at time t n+1. In this way, the problem becomes discretized in time. This step introduces some nite-dierence concepts into the method, rather than being a part of a consistent variational nite element methodology. 3. Interpolation of the chosen unknowns (the so-called trial functions of the problem). In this step the problem becomes fully discretized in space and can be viewed as the system of non-linear equations. If the same interpolation is used for the trial and the test functions, such a method is referred to as the Bubnov Galerkin method, a special case of the more general method of weighted residuals. 19; 20 The latter includes the so-called Petrov Galerkin method, 20; 21 where the trial and the test functions are interpolated dierently. 4. Iterative solution of the system of non-linear equations. In this step the non-linear equations are solved through the repetitive solution of the systems of linear equations.

7 INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS Interpolation of test functions In order to solve the weak form of the equations of motion (4) by using the nite element method, the test functions have to be suitably interpolated along the length of the beam. By turning the beam into an N-noded nite element, as depicted in Figure 2, we can dene the polynomial distribution of test functions along the element using f(x)=i i (x)f i ; (x)=i i (x) i (5) where the index i is summational (unless stated otherwise, from now on all the repeated indices will be understood as summational over the nodes of the element) and I i (x); i =1;N are interpolation polynomials of degree N 1 satisfying the well-known conditions I i (x j )=j i ; i; j =1;:::;N and N i=1 I i (x)=1; x [0;L] with j i being the Kronecker s symbol, i.e. j i =1 if i=j and i j =0 otherwise. The arbitrariness of the test functions f(x) and (x) is now entirely contained in their nodal values f i ; i =1;:::;N and i ; i =1;:::;N and the continuous problem (4) is turned into fi t i t g i = 0 (6) where g i is the contribution of all the forces acting on the beam at node i of the beam nite element depicted in Figure 2. Since the nodal values f i ; i =1;:::;N and i ; i =1;:::;N of interpolated test functions in equation (6) are arbitrary, it follows that g i = 0; i =1;:::;N for any t 0. For a specic time t n+1 we thus obtain g i n+1 q i mas;n+1 + q i int; n+1 q i ext; n+1 = 0 (7) where the vectors of inertial, internal and external forces follow from equations (4) (7) as { } L Au n+1 qmas;n+1 i = I i dx (8) 0 n+1 ( [W n+1 I W n+1 + I A n+1 ) L qint;n+1 i = I i [ ]{ } I 0 n+1 0 Nn+1 0 I i r n+1 I i dx (9) I 0 n+1 M n+1 { } { } { } L nn+1 F0 FL qext;n+1 i = I i dx + 1 i + N i (10) 0 m n+1 T 0 T L The vector equation (7) for i = 1;:::;N denes the dynamic equilibrium of the beam nite element at time t n+1. Before proceeding with the numerical integration of velocities and acceleration it is worth generalizing the dynamic equilibrium (7) in the way proposed by Hilber et al. 12 In their -method, the dynamic equilibrium is substituted with 12 g i n+1+ q i mas;n+1 +(1+)q i int;n+1 q i int;n (1 + )q i ext;n+1 + q i ext;n=0 (11) Figure 2. N-noded beam nite element

8 1200 G. JELENI C AND M. A. CRISFIELD where qmas;n+1 i, qi int;n+1 and qi ext;n+1 are nodal vectors of inertial, internal and external forces at time t n+1 and qint;n i and qi ext;n1 are nodal vectors of internal and external forces at time t n. In the present work, qmas;n+1 i, qi int;n+1 and qi ext;n+1 are dened by equations (8) (10) and qi int;n, qi ext;n are also computed from equations (9) and (10), but at time t n. The parameter in the -method introduces a numerical damping into the algorithm, which reduces the inuence of higher frequency modes, and is limited to values between 0 and 1 3. Obviously, for = 0, the original dynamic equilibrium (7) is recovered. Note 1: The -method was originally proposed for the problems of linear dynamics, so in order to apply it to the present problem we have to extend it so as to address both the problem of non-linearity of the governing equations and that of the non-linearity of the conguration space. Cardona and Geradin 3 and Simo et al. 4 applied the method in two dierent ways and in this work we present yet another approach. The fact that equation (11) takes identical form to that originally proposed by Hilber et al. 12 for the problems of linear dynamics originates from our generalization of the strong form of the equations of motion along the lines of the -method. In this way equations (1) are generalized as (1 + )[(N) + n] [(N) + n] =Au (1 + )[(M) + r N + m] [(M) + r N + m] =(ŴI W + I A) It must be noted that an -generalization, in general, and therefore the above equations, in particular produce an algorithmic (imaginary) rather than a real (mechanical) equilibrium. Constructing algorithmic equilibria of the above type (with particular forces at t n+1 and t n being subtracted from one another) is not uncommon for instance, the right-hand sides of equation (1) have in Reference 4 been transformed into (A=t)( u n+1 u n ) and (1=t)( n+1 I W n+1 n I W n ), respectively, which was a necessary step in devising the energy momentum algorithm described therein. Transformation of the above algorithmic strong form along with the boundary conditions at t n and t n+1 into a corresponding algorithmic weak form is performed in the way described in Section 2:4. Because the above equations constitute only an algorithmic and not a mechanical dynamic equilibrium, the resulting weak form represents only an algorithmic and not a mechanical virtual work and it is not necessary to associate any virtual displacement=rotation character to the test functions f and in Section 2:4 (although they can be thought of as algorithmic virtual displacements=rotations at t n+1+ ). The three sucient conditions which the test functions must full (see Section 2:4) in order to make a transition from a strong form to a weak form are, however, fully satised. In devising conserving algorithms for systems with 3-D rotations, such an approach is not only common but also necessary. 4 After interpolating the test functions as shown earlier in this section, the algorithmic nodal equilibrium equation (11) follows. Cardona and Geradin 3 argued that in order to apply the -method to non-linear problems with large rotations one cannot simply subtract q int;n from (1 + )q int;n+1 as we do in equation (11), since the terms at t n and t n+1 are conjugate to two dierent sets of virtual rotations. However, as argued earlier, we do not intend to compare virtual works at t n and t n+1 and believe that in applying the -method to non-linear problems with large rotations our approach is a valid alternative. Indeed, Simo et al. propose yet another extension of the -method to non-linear problems with large rotations (see Remark 5.1 in Reference 4), whereby the internal forces are calculated at the conguration at time t n+1+ and are not obtained by the subtraction typical of our approach

9 INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS 1201 and that of Reference 3. Their method also diers in the treatment of the inertial forces, which are approximated by an algorithmic dierence of momenta over a time step. However, the procedure to construct the weak form in Reference 4 is practically the same as the one we apply here Numerical integration of velocities and accelerations There are many dierent schemes for the numerical integration of velocities and accelerations between times t n and t n+1, and here we limit our attention to those from the Newmark s twoparameter family of algorithms, whereby velocities and acceleration at time t n+1 are dened using the displacements, velocities and accelerations at time t n, the displacements at time t n+1 and the parameters and as 10 u n+1 = ( t ũ + 1 ) ( u n +t 1 ) u n (12) 2 u n+1 = 1 ( ] 12 [ũ t 2 t u n t )u 2 n (13) ũ = u n+1 u n (14) As pointed out in Reference 22, the application of such an integration to angular velocities and accelerations only makes sense if performed in a body attached frame. In this way we obtain W n+1 = ( t + 1 ) ( W n +t 1 ) A n (15) 2 A n+1 = 1 [ ( ) ] 1 t 2 tw n t 2 2 A n (16) where W n, A n, W n+1 and A n+1 are angular velocities and accelerations at times t n and t n+1, the components of which are given in the body attached frame, and is the rotational pseudovector between congurations at times t n and t n+1 with components in the body attached frame. The latter is related to the rotational pseudovector expressed in the inertial frame through = t nṽ = t n+1ṽ (17) Note 2: Cardona and Geradin 2 used Newmark s integration in a dierent way and substituted the angular velocities and accelerations W and A in equations (15) and (16) with and. The constants and in equations (12), (13), (15) and (16) provide the maximum possible accuracy and unconditional stability of the solution of linear systems when they are related to the parameter using the relations 12 (1 )2 = ; = 1 (18) 4 2 Equations (8) (18) for i =1;:::;N dene the system of equations for the unknown conguration r n+1 (x), n+1 (x); x (0;L).

10 1202 G. JELENI C AND M. A. CRISFIELD 3.3. Interpolation of the chosen unknowns The rotational part of the unknown conguration n+1 (x) is an element of the non-linear differential manifold SO(3), so an attempt to interpolate it in a similar way, to Equations (5) would destroy the properties det = 1 and 1 = t of SO(3). Since equations (8) (18) also have to be linearized (see step 4 in the introduction to Section 3) in order to solve the problem, an obvious choice is to interpolate the innitesimal changes of displacements and rotations (i.e. the trial functions of the problem) using u(x)=i i (x)u i ; v(x)=i i (x)v i (19) with I i (x) being the same polynomials as in equation (5). These interpolations do not suer from the above drawback, because they are applied to u R 3 and v R 3. Such an approach has been described in Reference 1 for the ordinary Newmark scheme ( = 0), and here it will be generalised for any value of. From now on, this approach will be referred to as Algorithm 1. Note that in Algorithm 1 we have utilized a standard Bubnov Galerkin approach, 20 in which the same interpolation is used to approximate both the test and the trial functions. Alternatively, we can choose to interpolate the incremental (over a time step) displacements and rotational pseudovectors, i.e. ũ(x)=i i (x)ũ i ; ṽ(x)=i i (x)ṽi (20) Such an algorithm will be referred to as Algorithm 2. The innitesimal changes of the incremental displacements and rotations ũ(x) and ṽ(x) are in Algorithm 2 computed directly from equation (20) as ũ(x)=i i (x)ũ i ; ṽ(x)=ii (x)ṽi (21) Due to equation (14), the interpolation of the innitesimal changes of displacements (the translational trial functions) in Algorithm 2 will be the same as in Algorithm 1 and therefore dened by equation (19) 1. However, this is by no means true for the innitesimal changes of rotations. This can be realized by rst observing that n+1 = exp ṽ n yields n+1 = (exp ṽ) n and hence due to = v and k+1 = exp ṽ k we obtain v = (exp ṽ) exp ṽ t, which eventually leads to with T(ṽ) = 1 #2 ṽṽt + ( 1 T (ṽ) = 1 1 #2 ṽ = T(ṽ)v v = T 1 (ṽ)ṽ # ( ) 2 tan # I 1 #2 ṽṽt 1 ṽ 2 2 ) sin # # ṽṽt + sin # 1 cos # I + ṽ # # 2 (22) (23) See Simo and Vu-Quoc, 1 Cardona and Geradin 2 (who took a material setting, which led to opposite signs in the last terms in equations (23)), Ibrahimbegovic et al. 15 and Criseld 23 (who referred to

11 INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS 1203 v as the spin variable and ṽ as the additive variable) for more detail on the derivation of the above. By inserting equation (22) into equation (21) 2 we then obtain v(x)=t 1 [ṽ(x)] N I i (x)t(ṽi)v i where in order to avoid confusion due to a triple occurrence of the index i, the summation convention has on this occasion been dropped and the summation sign introduced instead. This equation can be written in a more concise form as i=1 v(x)=j i (x)v i (24) which immediately reveals the Petrov Galerkin character of Algorithm 2, in which the rotational test functions (x) have been approximated using interpolation polynomials I i (x) (see equation (5) 2 ), while the rotational trial functions v(x) have been approximated using dierent interpolation via J i (x)=t 1 [ṽ(x)]i i (x)t(ṽi) By introducing an algorithm-dependent interpolation as given in Table II, the approximation of the rotational trial functions and their derivative with respect to axial co-ordinate x for both algorithms is dened by equation (24). We emphasize that the two algorithms basically dier in the choice of interpolation of rotational variables. In both cases, however, we are solving the problem for the unknown conguration (r; ):[0;L] R 3 SO(3). During the process of linearization, the latter leads to the innitesimal changes of displacements and rotations (u; v):[0;l] R 3 R 3, which are interpolated by equations (19) 1 and (24), with the algorithm-dependent shape functions J i (x) as given in Table II. Note 3: The choice of additive interpolation, as in equations (19) (21) and (24), is standard in the nite element approach, but strictly speaking it only makes sense for the interpolation of additive elds, like displacements. Rotations in 3-D space are not additive, so such an interpolation is less sensible. This limitation, which vanishes as the mesh is rened, would seem to aect all isoparametric formulations and has been further investigated (with particular reference to the 18; 24 invariance of approximated strain measures) in separate publications. Note 4: The rotational trial functions in Algorithm 2 can be expressed in terms of nodal unknowns ṽi rather than v i. This would lead to a dierent (vectorial) update of rotational nodal Table II. Shape function J i (x) and its derivative J i (x) in the interpolation of the rotational trial functions and their derivatives in Algorithms 1 and 2 (no summation over the underlined indices) Algorithm 1 Algorithm 2 v(x) =J i (x)v i J i (x) =I i (x)i J i (x)=t 1 [ṽ(x)]i i (x)t(ṽ i) v (x) =J i (x)v i J i (x) =I i (x)i J i (x)= d dx {T 1 [ṽ(x)]}i i (x)t(ṽ i)+t 1 [ṽ(x)]i i (x)t(ṽ i) Underlined indices indicate that there is no summation

12 1204 G. JELENI C AND M. A. CRISFIELD unknowns. 2; 15 In this work we opt for the (spin) rotational nodal unknowns v i in order to provide a unique framework for both algorithms Iterative solution of the system of non-linear equations Linearization of non-linear dynamic equilibrium equations Equation (11) represents the contribution to the algebraic non-linear dynamic equilibrium equation of an isoparametric beam nite element at node i. This equation is iteratively solved using the Newton Raphson method by developing it into a Taylor series around an arbitrary conguration and neglecting the higher-order terms. In this way, we obtain g i n+1+ = g i n+1+; i =1;:::;N (25) The right-hand side of the above equation is dened by equation (11). By dierentiating equations (8) (11) we obtain the left-hand side of equation (25) as [ ][ ]{ } L 0 0 gn+1+ i n+1 0 Nn+1 =(1+) dx 0 I i r n n+1 M n+1 [ L I i ][ ]{ } I 0 n+1 0 Nn+1 +(1+) dx 0 I i r n+1 I i I 0 n+1 M n+1 [ L I i ][ ]{ } I 0 n+1 0 Nn+1 +(1+) dx 0 I i r n+1 I i I 0 n+1 M n+1 { } L Au + I i n+1 dx 0 n+1 ( [W n+1 I W n+1 + [W n+1 I W n+1 + I A n+1 ) { } L 0 + I i dx (1 + )q n+1; ext ; i =1;:::;N 0 n+1 ( [W n+1 I W n+1 + I A n+1 ) (26) which may be written as g i n+1+ = K ij p j ; i=1;:::;n K ij = K ij g1 + Kij g2 + Kij mat + K ij mas + K ij ext (27) where K ij is the ij-contribution to the equivalent tangent stiness matrix and K ij g1, Kij g2, Kij mat, K ij mas and K ij ext are the ij-contribution to the geometric stiness matrix due to the linearization of the translational part of the conguration, the ij-contribution to the geometric stiness matrix due to the linearization of the rotational part of the conguration, the ij-contribution to the material stiness matrix, the ij-contribution to the inertia matrix and the ij-contribution to the load correction matrix. In this work we will limit our attention only to displacement=rotation-independent loads so that K ij ext = 0. These matrices are derived in Appendix A.

13 INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS 1205 The vector p j in equation (27) 1 comprises the innitesimal changes of displacements and rotations, i.e. { } uj p j = (28) v j for both algorithms. Inserting K ij g1, Kij g2, Kij mat and Kmas ij from Appendix A into equations (27) denes the left-hand side of the linearized dynamic equilibrium problem (25) for an arbitrary beam nite element. The right-hand side of equation (25) is computed from the denition (11) Solution procedure In the preceding sections, the denition of the element residual vector (11) has been given and its linearization explained. In the case of higher-order elements with three or more nodes (see Figure 2), the internal degrees of freedom may be condensed out of the linearized equilibrium (25) and resolved at the element level (see e.g. Reference 25). The assembly of the equivalent tangent stiness matrix and the dynamic residual of the structure and the inclusion of kinematic boundary conditions are performed in the standard way (see e.g. Reference 26). In this way, we obtain the following linearised dynamic equilibrium of the structure Kp = g n+1+ (29) where K is the equivalent tangent stiness matrix and g n+1+ the dynamic residual vector of the structure. This equation should be solved for the global unknowns p Update procedure The update of the iterative translations and rotations obtained by solving equation (29) is performed in accordance with the dened kinematics, which states r R 3 and SO(3) at all times. We have to provide, at all the nodes of all the elements in the analysed structure, the quantities that can be interpolated over the elements. In Algorithm 1 these quantities are provided by the solution of equation (29) (see equation (19) and equation (28)). In Algorithm 2, however, these quantities are nodal incremental displacements and nodal incremental rotational pseudovectors over a time step. As argued in Section 3.3, the interpolation of translations is the same in both algorithms due to the vectorial character of the displacements, so the update of the nodal position vectors is always simply r j = r j; old +u j ; j=1;:::;n (30) for all the elements in the structure. The update of the nodal incremental rotational pseudovectors ṽ j =v j ṽj; old; j =1;:::;N (needed in Algorithm 2) has to be done in a way which is compatible with the kinematic condition exp ṽ j SO(3), i.e. exp ṽ j = exp v j exp ṽ j; old ; j=1;:::;n (31) The nodal incremental rotational pseudovector ṽ j can be extracted from the rotation matrix exp ṽ j using Spurrier s algorithm 27 as given in References 9 and 23. Hence, by using equations (30), (31) and (20), we are able to evaluate the interpolated elds and update the conguration at any

14 1206 G. JELENI C AND M. A. CRISFIELD Table III. Distribution of interpolated position vectors and rotations and update of the rotational part of the conguration in the two algorithms Algorithm Position vector Rotation Rotation matrix Algorithm 1 r n+1(x) =I j (x)r n+1;j v(x)=i j (x)v j n+1(x) = exp v(x)n+1; old (x) Algorithm 2 r n+1(x) =I j (x)r n+1;j ṽ(x)=i j (x)ṽ j n+1(x) = exp ṽ(x) n(x) Table IV. Update of translational strain measures in both algorithms Position vector derivative Translational strain Both algorithms r n+1 (x) =I j (x)r n+1;j n+1(x) = t n+1(x)r n+1(x) {100} t Table V. Update of rotational strain measures in the two algorithms (T 1 given by equation (23) 2) Algorithm Rotation derivative Rotational strain Algorithm 1 v (x) =I j (x)v j Z n+1(x) =Z n+1; old (x)+ t n+1; old(x)t t [v(x)]v (x) Algorithm 2 ṽ (x) =I j (x)ṽj Zn+1(x) =Zn(x)+t n(x)t t [ṽ(x)]ṽ (x) co-ordinate x [0;L] for all the elements. However, we only need quantities at the Gauss points. This process is for both algorithms summarized in Table III. The update of strain measures is given in Table IV for the translational strains and in Table V for the rotational strains. These updates provide the necessary information for the denition of the updated internal (elastic) force vector (9). To compute the updated inertial force vector (8), we need to update the translational and angular accelerations and angular velocities. Translational velocities are also needed for future updating. The updates are based on equations (12) (17). The update of angular velocities and accelerations makes use of the incremental rotation which is already computed in Algorithm 2. In Algorithm 1, 9; 23; 27 it has to be recovered from the rotation matrix exp ˆṽ(x)= n+1 (x) t n(x) (32) The update procedure described above completes the denition of the Newton Raphson iterative approach for solving the non-linear dynamic equilibrium equation (11) for all the elements in the structure. The procedure should be repeated until the norm of vectors p and=or g n+1+ in equation (29) falls below a specied tolerance. Note 5: Instead of applying update formulae (15) and (16) directly, they can be used in an alternative way, 1 whereby each of them is rstly written for two consecutive iterations (marked with superscripts (i) and (i + 1), respectively), and then those at iteration i are subtracted from

15 INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS 1207 those at iteration i + 1 (see equations (3.7) and (3.5) in Reference 1) to obtain W (i+1) n+1 = W (i) n+1 + t ( (i+1) (i) ) (33) A (i+1) n+1 = A(i) n t 2 ( (i+1) (i) ) (34) However, for i = 0 the original update formulae have to be applied (see Remark 3.2(3) in Reference 1) Comparative analysis of Algorithms 1 and 2 Due to the additivity of displacements and their iterative changes, the update of all translational quantities (displacements, position vectors of the line of centroids, translational strains, velocities and accelerations) is the same for both algorithms. The additional work in the update procedure of Algorithm 2 requires the computation of the rotation matrix exp ṽ j ; j =1;:::;N for all elements in the structure using equation (31) and the retrieval of the associated incremental rotational pseudovector ṽj using Spurrier s algorithm. 9; 23; 27 However, no additional coding is ever needed, as the same has to be done in Algorithm 1 on the Gauss point level using equation (32). The interpolation of rotations, the update of the rotational part of the conguration and the rotational strain measures is dierent in the two algorithms (see Tables III and V) but again, no new procedure coding is needed for Algorithm 2. However, more substantial changes are needed in the consistent linearization. Computation of K ij g2, Kij mat and K ij mas (see equations (36), (40), (45) 1 and (46) and Table II) requires the evaluation of T (ṽ) and (d=dx)t 1(ṽ) at Gauss points and T(ṽ) at nodes. Note that again no new coding is required for T 1 (ṽ) and T(ṽ) as they are both already used in Algorithm 1 (see equation (45) and Table V). Thus the only procedure that has to be coded in Algorithm 2, which does not already exist in Algorithm 1 is the one to compute 1 the matrix (d=dx)t (ṽ) using equation (46). Both of the algorithms feature extraction of rotational pseudovectors from rotation matrices. Since there is no one-to-one correspondence between the two, we must be aware of the following consequences of this operation, which are closely related to the choice of interpolated rotational variables Magnitude of the extracted rotational pseudovector The update of angular velocities and accelerations makes use of the incremental rotations extracted using Spurrier s algorithm. It is very important to note that not any incremental 9; 23; 27 rotations can be extracted, because the extraction procedure employs the inverse trigonometric functions arccos and=or arcsin to obtain #=2, so the range of # thus extracted will be between 0 and 2 or and. Any real incremental rotations (by real we mean those consistent with the composition of iterative rotations within a time step, regardless of whether an equilibrium has been achieved or not) outside these bounds cannot be correctly retrieved. For example, an incremental rotation ṽt =(9=4){1 00} t will be retrieved as (=4){1 00} t. This means that the update of rotational strain measures in Algorithm 2 (Table V) and the update of angular velocities and accelerations (equations (15) (17)) in both algorithms cannot in these circumstances be performed properly.

16 1208 G. JELENI C AND M. A. CRISFIELD Uniqueness of the extracted rotational pseudovector Further to the discussion in Section 3.5.1, it follows that some incremental rotations, like ṽ t = (=2){1 0 0} t cannot be uniquely extracted, as the possible extracted rotation is either (=2){1 00} t or (3=2){1 00} t. As the extracted rotational pseudovectors are used to update the angular velocities and accelerations, it makes sense to provide a unique extraction, whereby the extracted rotational pseudovector satises ṽ 6. This is provided by choosing such a sign of the associated quaternion q = q 0 + q in Spurrier s algorithm, 9; 23; 27 which makes its scalar component q 0 positive (note that a positive and a negative quaternion are associated with the same rotation matrix). In this way, the singularity of T(ṽ) in equation (23) 1 at # =2n; n N can never take place. However, the latter convenience is lost if alternative nodal rotational unknowns (see Note 4) are used. In the current implementation we also try to minimise the round-o errors in the extraction procedure by comparing the magnitude of the vector part of the quaternion q with the scalar part of the quaternion q 0 and applying either ṽ =2q= q arcsin q if q q 0 or ṽ =2q= q arccos q 0 if q q 0. Note that q = 0 implies ṽ = Internal and=or inertial forces based on the extracted rotation Further to the discussion in Section 3.5.1, it also follows that in Algorithm 1, where the iterative rotations have been interpolated, it is possible that the updates of rotational strains (which is performed using the iterative rotations, which can be of any magnitude) and angular velocities and accelerations (which is performed using the incremental rotations, which are limited according to the discussion in Section 3.5.2) are not necessarily consistent with each other. This problem is additionally emphasized in the update procedure given in Reference 1 (see also Note 5), whereby both the incremental rotation at iteration i and the one at iteration i+1 are utilized. This inconsistency may lead to higher energy equilibria and will be commented on further during the numerical experiments. However, in Algorithm 2 this can never happen, because both the update of rotational strains and the update of angular velocities=accelerations is performed using the incremental rotation (see Table V and equations (15) (17)). This assertion will be assessed against the results of the numerical experiments. 4. EXAMPLES The described algorithms have been coded and mounted in the research version of the nite element system LUSAS. 28 The two examples we numerically analyse will enable us to draw certain conclusions regarding the stability and convergence properties of Algorithms 1 and 2. The two algorithms, which will from now on be denoted as A1 and A2, respectively, will be used with three dierent amounts of numerical damping: (i) = 0 (trapezoidal rule), (ii) = 0 05 (recommended by Hibbitt and Karlsson 29 as the optimal choice), and (iii) = 0 33 (recommended by Reference 12 as the maximum numerical damping of practical interest). The six integration schemes thus obtained, along with the parameters and, which are evaluated using relations (18), are summarized in Table VI. The iterative solution procedure is set up in such a way that, if the equilibrium for a particular time step has not been obtained within a prescribed number of iterations, the time-step length is halved and the iterative procedure for that time step is repeated. Such a reduced time step will then be gradually increased by the factor of 1 2 until the original time-step length is restored.

17 INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS 1209 Table VI. Integration schemes used for numerical analysis Integration scheme Algorithm A1 0 A A2 0 A A A A A A A A A Figure 3. L-shaped cantilever subject to out-of-plane force load Figure 4. Load history of the L-shaped cantilever 4.1. Example 1 Large displacement vibration of an L-shaped frame We rst examine an L-shaped cantilever subject to an out-of-plane force load applied at its elbow. The geometric and material properties of both legs of the structure are the same and are given in Figure 3. The load history is dened by the diagram in Figure 4. This example was originally solved by Simo and Vu-Quoc, 1 who have adopted a time-step length t =0 25 and analysed the motion during the rst 30 units of time using the trapezoidal rule. In order to expose the problems of numerical instability typical of the trapezoidal rule, we analyse the motion during the rst 50 units of time.

18 1210 G. JELENI C AND M. A. CRISFIELD Figure 5. Out-of-plane displacements of the elbow and the tip of the L-shaped cantilever using integration schemes A1 0 (identical to Reference 1 for the rst 30 units of time) and A2 0 Due to the articially exaggerated mass inertia tensor, this example serves as a rigorous test for non-linear dynamics with large rotations. As shown in Figure 4, the load acts only during the rst two units of time and thereafter the cantilever undergoes large-scale free vibrations. Each leg of the frame has been modelled using a single 3-noded (quadratic) element, with 2-point Gaussian integration rule for the internal force vector and the material and geometric stiness matrices and 3-point integration for the inertial force vector and the inertia matrix (see Note I.1 in Appendix I). The dynamic equilibrium at a time step is taken to be satised when both the square root of the sum of the squares of the iterative displacement components in p in equation (29), as a percentage of the square root of the sum of the squares of the total displacement components, and the square root of the sum of the squares of all the residual forces ( g n+1+ in equation (29)), as a percentage of the square root of the sum of the squares of all external forces, including reactions, become less than By employing integration scheme A1 0 (see Table VI), the results of Reference 1 (for the rst 30 units of time) are fully reproduced, as shown in Figure 5. The results obtained by applying integration scheme A2 0 are eectively the same (Figure 5). By analysing the total energy history for the two schemes (see Figure 6), we observe that both of them experience stability problems (in the energy sense) toward the end of the analysed period of time. These stability problems very soon result in a sudden energy growth, at which point it becomes impossible to achieve equilibrium. When some numerical damping is introduced, the results of Algorithms 1 and 2 remain eectively the same, as shown in Figure 7 for integration schemes A and A2 0 33, where maximum numerical damping is utilized. However, there is some disagreement with the results of integration schemes A1 0 and A2 0 in Figure 5. This disagreement is not big and is consistent with the fact that the numerical damping tends to decrease the inuence of the higher and not of the lower frequency modes. The far more important dierence between the (undamped) schemes A1 0 and A2 0 and the numerically damped schemes A and A is seen in comparing the corresponding

19 INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS 1211 Figure 6. Total energy of the L-shaped cantilever using integration schemes A1 0 and A2 0 Figure 7. Out-of-plane displacements of the elbow and the tip of the L-shaped cantilever using integration schemes A and A2 0 33

20 1212 G. JELENI C AND M. A. CRISFIELD Figure 8. Total energy of the L-shaped cantilever using integration schemes A and A energy graphs (Figures 6 and 8). Obviously, the numerical instability is eliminated by introducing numerical damping. This example was run by using the two damped schemes for units of time (total of time steps) without encountering any numerical stability problems. This example both advocates the introduction of the numerical damping into time-stepping algorithms (however, it must not be forgotten that a huge amount of energy may get dissipated in this example the total energy at the end of the analysed period has fallen to less than 10 per cent of its value at t = 2) and justies the adopted -generalization (see Note 1). However, it says nothing of the additional problems caused by the mutually inconsistent update of the rotational strains and the angular velocities=accelerations (see Section 3.5.3), as Algorithms 1 and 2 behave eectively in the same manner. This problem will be exposed in the next example, which features large rigid-body rotations Example 2 Free ight of a exible beam In the second example we analyse the ight of an unrestrained exible beam. The initial con- guration and the geometric and material properties of the beam are given in Figure 9. No static or kinematic boundary conditions are present for this problem, which means that the beam is free to move in 3-D space according to the laws of dynamics and depending on the chosen initial conditions. In the present example, we only dene the initial translational velocities of the beam along the global co-ordinate axes X and Z as depicted in Figure 9. It can be seen that the prescribed translational velocity along the global co-ordinate axis Z will cause the beam to rotate around the axis orthogonal to its centroidal axis and the co-ordinate axis Z, while the prescribed velocity along the global co-ordinate axis X will cause the translation of the beam along the global