Spectral Formulation for Geometrically Exact Beams:A Motion Interpolation Based Approach∗

Shilei Han and Olivier A. Bauchau†

Department of Aerospace Engineering, University of MarylandCollege Park, Maryland 20742

Abstract

This paper proposes a novel spectral formulation for geometrically exact beams based onmotion interpolation schemes. Motion interpolation schemes based on matrix, quaternion, andgeodesic metrics yields simple expressions for the sectional strains and linearized strain-motionrelationships at the mesh nodes. Consequently, the expressions for the nodal forces and tangentstiffness matrices of spectral elements are simplified dramatically. Furthermore, the motionformalism is used to describe the kinematics of the problem, leading to equations of motionthat present low-order nonlinearities. Spectral elements based on Gauss-Lobatto and Gaussquadrature rules are investigated. For both cases, the combination of the spectral formulationwith the motion formalism leads to geometrically exact beam elements that are much simplerto implement than their counterparts based conventional finite element interpolation schemesusing a classical description of kinematics. A global parameterization-free generalized-α schemeis used to integrate the equations in time. Numerical examples demonstrate the accuracy ofthe proposed formulation. The elements based on Gauss-Lobatto quadrature rules are shownto suffer from axial and shear locking, while those based on Gauss rules are locking free. Theconvergence rate of (N + 1)-node spectral elements based on Gauss quadrature rules is about2N − 0.5 to 2N for all three interpolation schemes. As the number of elements increases, theproposed formulation becomes more accurate than its conventional counterpart.

1 Introduction

This paper is concerned with the formulation of geometrically exact beam models, i.e., Cosseratcurved material lines with distributed sectional stiffness. Geometrically exact beam models weredeveloped by Reissner [1] based on the rigid cross-section assumption. Simo [2, 3] generalized theformulation to beams undergoing large motion. When dealing with beams presenting complexsectional geometries and made of anisotropic laminated composite materials, sectional in- and out-of-plane warping have been shown [4–6] to alter stress distributions and sectional stiffness propertiessignificantly and hence, the rigid-section assumption is no longer valid. For these problems, Hodgesand his coauthors [5,7,8] have shown that the three-dimensional nonlinear problem decomposes into

∗AIAA Journal, to appear.

1

a nonlinear, one-dimensional analysis along the reference line and a linear, two-dimensional analy-sis over the cross-section. Their approach is based on variational asymptotic methods and on thedecomposition of the rotation tensor. More recently, Bauchau and Han [9–11] have proposed a re-duction procedure based on the Hamiltonian formalism that brings the three-dimensional, nonlinearelasticity problem to geometrically exact beam problems.

Because geometrically exact beams are Cosserat material lines, their kinematic description in-volves both displacement and rotation fields and it is customary to treat these two fields indepen-dently, following the footsteps of Simo [2] and of numerous other researchers [12–15]. The key to therigorous description of geometrically exact beams is the treatment of finite rotation, a topic thathas been the subject of intensive investigation [16–19]. Indeed, rotation tensors form the specialorthogonal group SO(3), in contrast with the displacement field that forms an Euclidean space. Asunderlined by numerous authors [20–22], the traditional interpolation techniques of finite elementmethods cannot by used to interpolate the rotation tensor. Although suitable techniques have beendeveloped based on the vectorial parameterization of rotation, the relationship between the straincomponents and rotation vectors is highly nonlinear, leading to complex equations of motion.

To avoid the complexity introduced by finite rotations, intrinsic formulations has been pro-posed by Hegemier and Nair [23] and Hodges [24, 25]. In intrinsic formulations, the unknownsare the sectional strains and velocities: displacement and rotation variables are eliminated. Theresulting equations of motion exhibit low-order nonlinearities and space-time conservative schemescan be developed easily. With this formulation, however, the assembly of the beam elements ismore complex because displacement and rotation variables do not show up explicitly. Zupan andSaje [26, 27] developed a novel beam element based on the interpolation of the strain field, whichis integrated to yield the displacement and rotation fields. The element is free of locking becauseshear and axial strains are interpolated directly. The strain based formulation, however, requiresmore computational effort than that based on displacement and rotation fields.

In recent years, a new approach to the description of the kinematics of beams has been developedby Borri and Bottasso [28], McRobie and Lasenby [29], Merlini and Morandini [30], Sander [31],Sonneville et al. [32], and Demoures et al. [33]. In this approach, called the motion formalism, thedisplacement and rotation fields are treated as a unit that forms the special Euclidean group SE(3).This unified treatment of the displacement and rotation fields leads to simple governing equationspresenting low-order algebraic nonlinearities and simplifies time integration for dynamic problems.As was the case for the interpolation of rotation, the interpolation of motion is a thorny issue thatmust be treated carefully, as discussed by numerous authors [22,30,34,35].

In a recent paper [35], the authors have shown that the interpolation of rotation and motionfields can be recast as a minimization problem: the interpolated field minimizes the weighted sumof the distances between the motions at a set of mesh nodes and the interpolated field. Many exist-ing interpolation schemes can be obtained as solutions of minimization problems based on variousdefinition of distance metrics. In general, these schemes produce complicated expressions for thecurvatures at the Gauss points and the evaluation of the elastic forces and stiffness matrix of beamelements becomes arduous. To alleviate this problem, many authors [15, 36, 37] simplify the ex-pressions for the curvatures, leading to more compliant beam elements that although incompatible,converge under mesh refinement.

These interpolation schemes, however, yield very simple, closed-form expressions for curvaturesat the mesh nodes. The spectral element formulations proposed in this paper take advantage thisfact. In spectral formulations, the nodes are located at the Gauss-Lobatto-Legendre (GLL) points.

2

Two types of spectral elements are investigated, depending on the quadrature rule used to integrateover the element: (1) elements based on Gauss-Lobatto quadrature rules and (2) elements basedon Gauss quadrature rules. Gauss-Lobatto quadrature rules lead to very simple expressions forforce vectors and tangent stiffness matrices, but the resulting beam elements suffer from axial andshear locking. For the Gauss quadrature rules based elements, curvatures at the Gauss points areobtained from the interpolation of curvatures at the GLL points. The Gauss quadrature rules basedelements are locking free.

2 Preliminary of dual number and rigid-body motion

In this paper, vectors and matrices are indicated by notation • and •, respectively, while unit vectorsare indicated by notation •. Notation • and axial(•) are linear maps between a vector of size 3× 1and a matrix of size 3× 3, defined as

a =

0 −a3 a2a3 0 −a1−a2 a1 0

, axial(A) =1

2

A32 − A23

A13 − A31

A21 − A12

(1)

Calligraphic letters indicate dual entities of dimensions 3: A = A + εA represents a dual matrix

with primal part A ∈ R3×3 and dual part A ∈ R3×3.Dual numbers were first introduced in the nineteenth century by Clifford [38]. Applications of

dual number algebra to kinematics have been presented in refs. [39–44]. More recent presentationsfocusing on computational issues include those of Borri et al. [45], Bauchau and Choi [46], Pennestrıand Stefanelli [47], Condurache and Burlacu [48], or Han and Bauchau [49]. Typically, dual numbersare written as a = a+ εa, where a and a are referred to as the primal and dual parts, respectively,and parameter ε is such that εn = 0 for n ≥ 2. The zero and identity dual numbers are 0 = 0 + ε0and 1 = 1 + ε0, respectively.

A rigid-body motion is defined as the transformation that brings inertial frame F = [O, I =(ı1, ı2, ı3)] to material frame Fb = [B,B = (b1, b2, b3)]. Rigid-body motion can be represented bymotion tensor R = R+ε rR, where rotation tensor R brings inertial basis I to material basis B, and

vector r is the relative position vector of reference point B with respect to the origin, O. Considera given motion field R (t), where t represents time. The dual velocity resolved in the material frame

is defined asv = R T R = ω + εv, (2)

where notation ˙(·) indicates a derivative with respect to time. The angular and linear velocitiesexpressed in the material basis are denoted as ω = axial(RT R) and v = RT r, respectively. Thedual velocity can be expressed as a dual vector of size 3× 1, v = ω + εv.

The definitions of the virtual and incremental motion vectors are similar to that of the dualvelocity vector, eq. (2),

δu = R T δR = δψ + εRT δr, (3a)

∆u = R T∆R = ∆ψ + εRT∆r, (3b)

3

where RT δr and δψ = axial(RT δR) represent the virtual displacement and rotation vectors, respec-

tivel; RT∆r and ∆ψ = axial(RT∆R) represent the incremental displacement and rotation vectors,respectively.

Chasles’ theorem [50] states that the most general motion of a rigid body consists of a translationalong a line followed by a rotation about the same line. Hence, a general motion is characterizedby its Chasles’ line of Plucker coordinates n and the magnitudes of the rotation and intrinsicdisplacement, combined into dual number d = d + εd. The motion tensor can be expressed interms of dual vector q = q(d )n , of size of size 3× 1, as

R (q) = I +sin d

qq +

1− cos dq2

q q . (4)

Different choices of scalar dual function q(d ) lead to different motion parameter vectors. Thefollowing choices [46,51] will be used in this paper:

q = q(d )n , q(d ) =

sin(d ), linearparameterization

sin(d /2), Euler −Rodriguesparameterizationd , Cartesianparameterization

(5)

The linear motion parameter vector corresponds to q = axial(R ), the Euler-Rodrigues motion

parameter vector is the vector part of the unit dual quaternions, and the Cartesian motion parametervector corresponds to q = axial[log(R )].

Derivatives of the motion parameter vector are related to the dual velocity vector by

q ′ = T −1(q)v . (6)

For the linear, Euler-Rodrigues, and Cartesian motion parameter vectors, the inverse of tangenttensor reduces to

T −1(q) =

tr(R )I

3− R T , linear motion vector

±√

1− qT q I3

+ q , Euler-Rodrigues motion vector

I3

+ q/2 + a2q q , Cartesian motion vector

(7)

where a2 = [1 − d /(2 tan(d /2))]/d 2. The sign of the first term for Euler-Rodrigues parameter isdetermined by sign of the rotation angle d: choose the positive or negative sign if d ∈ [−π, π] ord ∈ [−2π,−π] ∪ [π, 2π], respectively. Choosing the positive sign is always correct in the presentpaper because the relative rotation angles inside one element are always in the range of [−π, π].

Tangent tensor T enjoys the following remarkable properties

R (q) = T (−q)T −1(q) = T −1(q)T (−q). (8)

Let R12

denote the relative motion between R1(q

1) and R

2(q

2), i.e., R

12= R T

1R

2, and let q

12

denote the motion parameter vector associated with R12

. Taking a variation of this relationship

yields T (q12

)δq12

= δu2 − R T

12δu1 and introducing identity (8) then yields

δq12

= −T −1(−q12

)δu1 + T −1(q12

)δu2. (9)

4

3 Kinematics of the problem

Figure 1 depicts an initially curved and twisted beam with a cross-section of arbitrary shape. Thevolume of the beam is generated by sliding the cross-section along the reference line of the beam,which is defined by an arbitrary curve in space denoted C. Curvilinear coordinate s defines thearc-length of C. Point B is located at the intersection of the reference line with the plane of thecross-section.

Referenceconfiguration

Deformedconfiguration

OI

A B

B

B0

Bi2_

i1_

i3_

b01_

b02_b03

_ b1_

b2_

b3_

R (s)=

R 0(s)=

Figure 1: Configurations of a geometrically exact beam.

Frame F0 =[B,B0 = (b1, b2, b3)

]defines the cross-section in the reference configuration. The

plane of the cross-section is determined by two mutually orthogonal unit vectors, b2 and b3. Thereference point and orientation of the cross-section change as it slides along curve C and hence,frame F0 is a function of arc-length coordinate s. The motion tensor that brings frame FI to F0 is

R0(s) = R

0+ εr0R0

, (10)

where r0 is the position vector of material point B and rotation tensor R(s) brings basis I to basisB0. The beam’s dual curvature vector resolved in the material frame in its initial configuration is

k 0 = R T

0R ′

0= k0 + εt0, (11)

where notation (·)′ indicates a derivative with respect to coordinate s; t0 = RT

0r′0 and k0 =

axial(R0R′

0) are the tangent vector and curvatures resolved in the material basis, respectively.

The curvature vector is expressed as a dual vector of size 3× 1, k0

= k0 + εt0.

In the deformed configuration, the cross-section is defined by frame F =[B,B = (B1, B2, B3)

].

The motion tensor that bring frame FI to F is

R (s) = R + εrR, (12)

where r is the position vector of material point B after deformation and rotation tensor R(s) bringsbasis I to basis B. The curvature vector of the beam after deformation is

k = R TR ′ = k + εt, (13)

5

where t = RT r′ and k = axial(RTR′). The curvature in the deformed configuration can be expressedby a dual vector of size 3×1, k = k0+εt0. The sectional strain measures [10] of the beam are definedas the differences between the curvature vectors in the deformed and reference configurations

e = k − k0. (14)

Taking an increment of eq. (11) and a spatial derivative of kinematic compatibility equation (3a),leads to ∆k = ∆R TR ′ + R T∆R ′ and ∆u = R ′T∆R + R T∆R ′, respectively. Subtracting these two

equations and using eqs. (11) and (3a) then yields ∆k − ∆u′

= k ∆u − ∆uk . Elementary vectoridentities lead to

∆k = ∆u ′ + k ∆u, (15a)

∆v = ∆u + v∆u, (15b)

v ′ = k + vk , (15c)

where eqs. (15b) and (15c) are obtained in a similar manner. These equations are called “com-patibility equations” because they result from the continuity of the motion tensor with respect tospatial, temporal, or incremental variables. The compatibility equations (15b) are referred to thetransitivity equations in analytical dynamics [52]. Compatibility equation (15c), also referred to asthe “transpositional relationships,” are at the heart of the intrinsic beam formulation developed byHodges [25].

4 Interpolation of rigid-body motion

Consider a beam element with a non-dimensional coordinate ξ ∈ [−1, 1] along its axis; the endpoints of the beam are located at ξ = ±1. In the spectral formulation, the nodes are located atthe GLL points, ξGLLk , k = 0, . . . , N . The N Gauss points are denoted as ξGi , i = 0, . . . , N − 1.The configuration of the beam is defined by the nodal values of the motion tensors, denoted asRk, k = 0, . . . , N . The interpolated motion tensor at point ξ is denoted as R , as indicated in

fig. 2. The relative motion tensor from ξ to ξGLLk and from ξGLLj to ξGLLk are R TRk

and R T

jRk,

j, k = 0, . . . , N , respectively. The motion parameter vectors associated with relative motion tensors,R TR

kand R T

jRk, are denoted as q

kand q

jk, respectively; the linear, Euler-Rodrigues, and Cartesian

motion parameter vectors will be used in this work. Let arrays ∆uT = ∆uT0 , . . . ,∆uTN and

∆kT

= ∆k T0, . . . ,∆k T

N store the nodal incremental motion and generalized curvature vectors,

respectively.

ξ0 = -1 ξN = 1Rjk qjk_

Rk qk_

R(ξ)=

R0= Rj= Rk= RN=

Figure 2: Interpolation of rigid-body motion in a spectral element, GLL points: (), Gauss points:(4).

6

As discussed by Han and Bauchau [35], interpolation of motion can be recast to a minimizationproblem

R (ξ) = arg minR T R =I , det(R )=1

N∑k=0

`k(ξ)dist2(R ,Rk), (16)

where `k(ξ) are Lagrange’s polynomials based on the GLL points, as discussed in appendix 10.1and notation dist(·, ·) indicates the metric or distance measure between two rigid-body motions.Different metrics leads to different interpolation schemes. Most of the existing interpolation schemesfor rotation and motion can be derived from minimization problem (16) equipped with three differentmetrics: the matrix metric distm = ‖R

1− R

2‖F , the quaternion metric distq = ‖e1 − e2‖, and the

geodesic metric distg = ‖axial[log(R T

1R

2)]‖. Solutions of minimization problem (16) also satisfy

the following equationN∑k=0

`k(ξ)qk

= 0 . (17)

For the matrix, quaternion, and geodesic metric based interpolation schemes, relative motion pa-rameter vectors q

kare the linear, Euler-Rodrigues, or Cartesian motion parameter vectors associated

with relative motion tensors R TRk, respectively. Explicit solutions of minimization problem (16)

and eq. (17) can be found when matrix or quaternion metric based interpolation is used [35]. Forthe matrix based interpolation, you need to perform the polar decomposition, which is iterative innature. I think it is only the quaternion based that has a closed form solution Iterative techniquesmust be used to solve the minimization problem when the geodesic based scheme is used.

The corresponding curvature vector resulting from the interpolation can be found as

k (ξ) =

[N∑k=0

`k(ξ)T −1(−qk)

]−1 N∑k=0

`′k(ξ)qk. (18)

The term in bracket can be simplified to∑N

k=0 `k(ξ)T −1(−qk) =

∑Nk=0 `k(ξ)

√1− qT

kqk

I3

when qk

are the Euler-Rodrigues parameter vectors in quaternion metric based interpolation. Equation (18)is the curvature-motion relationship resulting from the motion interpolation. Linearization of cur-vatures (18) requires linearization of the inverse of tangent tensor T −1(−q

k), leading to complex

tangent stiffness matrix in finite element formulation.At the nodes (the GLL points), the term in bracket of eq. (18) vanishes and the curvatures have

very simple expressions

kj

=1

Jj

N∑k=0

dj,kqjk, (19)

where dj,k denote derivatives of Lagrange’s polynomial of order k at the nodes, see eq. (52), andJj is the Jacobian associated with the transformation from the arc-length coordinate s to the non-dimensional coordinate ξ. The curvature field over one element is constructed by interpolating thenodal curvatures. This approach was pioneered by Bathe and Dvorkin [53]: the nodes, or GLLpoints, are used as tying points. The following assumed curvature field is introduced

k (ξ) =N−1∑i=0

¯i(ξ)

[j=N∑j=0

`j(ξGj )k

j

], (20)

7

where ¯i(ξ), i = 0, 1, . . . , N −1 are Lagrange’s polynomials defined by eq. (51a), and k

jare the cur-

vatures at the GLL points defined in eq. (19). Clearly, the assumed curvature fields are polynomialsof degree N − 1. Because the Gauss points are located at the zeros of the Legendre polynomial ofdegree PN+1(ξ), the assumed curvatures equal the true curvatures at the Gauss points if the truecurvature (18) are polynomials of degree N . Because quantities dj,k and `j(ξ

Gj ) are numerical values,

the linearization of assumed curvature field only requires the linearization of curvature expressionsat the nodes, k

j.

In view of identity (9), the linearization of curvature expressions at the nodes yields

∆k = B ∆u, (21)

where strain interpolation matrix B is composed of (N + 1) × (N + 1) sub-matrices of size 3 × 3.Notation [·]jk indicates the sub-matrix of size 3× 3 at location (j, k), j, k = 0, . . . , N ,

[B]jk

=

kj −

1

Jj

N∑i=0

dj,i T −1(qji

), for k = j,

1

Jjdj,kT −1(q

jk), for k 6= j,

(22)

where the inverse of tangent tensor T −1(qjk

) is defined in eq. (7).

5 Recasting dual entities to vector/matrix form

The dual number formalism used in the previous sections is ideally suited to the treatment ofkinematics problems. Dynamics, however, involve both kinematic quantities, such as velocities orcurvature, and force quantities, such as elastic or inertial forces. While Brodsky and Shoham [54,55]have shown that dual numbers can be used to solve dynamics problems by introducing the conceptof dual inertia operator, their approach is cumbersome. In this paper, dual entities are expandedinto matrix/vector form composed of their primal and dual components using the following changeof notation

q = q + εq =⇒ Q =

q

q

, (23a)

v = ω + εv =⇒ V =

vω

, (23b)

k = k + εt =⇒ K =

tk

, (23c)

δu = δψ + εRT δr =⇒ δU =

RT δrδψ

, (23d)

∆u = ∆ψ + εRT∆r =⇒ ∆U =

RT∆r

∆ψ

, (23e)

where q and q are the primal and dual parts of vector q . Vectors Q, V , etc. now become vectorsof size 6× 1, replacing their dual counterparts of size 3× 1. The sectional strain measures defined

8

in eq. (14) are recast asE = K −K0, (24)

where KT0 = tT0 , kT0 recast from k0.

Similarly, dual matrices v and k , of size 3×3 are recast as matrices of size 6×6 in the followingmanner

v = ω + εv =⇒ V =

[ω v0 ω

], (25a)

k = k + εt =⇒ K =

[k t

0 k

]. (25b)

The inverse tangent tensor T −1(q) defined in eq. (7) is recast as a matrix of size 6×6 in the followingmanner

T −1(q) =⇒ T −1(Q) =

[primal[T −1(q)] dual[T −1(q)]

0 primal[T −1(q)]

]. (26)

Equation (21) becomes∆K = B∆U , (27)

where ∆UT

= ∆UT0 , . . . ,∆UTN and ∆KT

= ∆KT0 , . . . ,∆KTN; matrix B is composed of (N + 1)×(N + 1) sub-matrices of size 6 × 6. Each sub-matrix [B]jk indicates the sub-matrix of size 6 × 6at location (j, k), j, k = 0, . . . , N , is constructed form the primal and dual part of dual sub-matrix[B]jk in the following manner

[B]jk =

[primal([B]jk) dual([B]jk)

0 primal([B]jk)

]. (28)

6 Governing equations

The governing equations of the problem will be derived from Hamilton’s principle. Inertial effectsdue to sectional warping can be ignored for beams undergoing low frequency motion, i.e., frequencieswhose associated wave lengths are much longer than the dimensions of the cross-section [56, 57].After integration over the cross-section of the beam, the kinetic energy can be found as

K =1

2

∫VTMV ds =

1

2

∫VTP ds, (29)

where array P = MV stores the components of the momentum vector resolved in the materialframe. The sectional mass matrix, M, is defined as

M =

[mI

3mqTc

mqc %B

], (30)

where m is the sectional mass per unit span, vector qc

is the position vector of the sectional centerof mass with respect to reference point B, and tensor %

B, of size 3×3, is the sectional mass moment

9

of inertia per unit span computed with respect to point B. Taking a variation of the kinetic energyexpressed by eq. (29) gives δK =

∫δVTMV ds. Using compatibility relationships (15b) and

integrating by parts then leads to

δK = −∫δUT

(P − VTP

)ds. (31)

Sectional warping leads to strain components of the same order as those due to rigid-sectionmotion and hence, warping effects must be taken into account when evaluating the strain energy.Hodges et al. [5, 7, 8] have shown that the three-dimensional beam problem can be decomposedinto a linear, two-dimensional analysis over the cross-section, and a nonlinear, one-dimensionalanalysis along the beam’s span. Those authors used the variational asymptotic method to reachthis conclusion. More recently, the same conclusion was reached by Bauchau and Han [9, 10] usingthe Hamiltonian formalism. A byproduct of the two-dimensional sectional analysis is the sectionalstiffness matrix, D, of size 6 × 6, which takes into account the warping effects due to geometriccomplexity and material heterogeneity of the cross-section. The strain energy can be found as

V =1

2

∫ETDE ds =

1

2

∫ETF ds, (32)

where vector F = DE , of size 6 × 1, stores the sectional stress resultants resolved in the materialframe and vector E , of size 6 × 1, stores the components of the sectional deformation measuresresolved in the material frame as defined by eq. (24).

Taking a variation of the strain energy expressed by eq. (32) gives δV =∫δETDE ds. Using

the compatibility relationships of eq. (15a), variation of the strain energy becomes

δV =1

2

∫ (δU ′TF + δUT KTF

)ds = −

∫δUT

(F ′ − KTF

)ds, (33)

where the second equality results from integrating by parts.For the problem at hand, the virtual work done by the externally applied forces is expressed as

δW =

∫δUTL ds, (34)

where vector LT = nT mT, of size 6× 1, stores the components of the externally applied force, n,and moment vector, m, per unit span of the beam, respectively, resolved in the material frame.

The principle of virtual work states that δV −δK−δW = 0, introducing eqs. (31), (33) and (34)leads to

∫[δUT (P −VTP+KTF−L)+δU ′TF ] ds = 0 and the weak form of the governing equations

of motion then result from integration by parts∫δUT

[P − VTP − F ′ + KTF − L

]ds = 0. (35)

Because the virtual motion vector is arbitrary, the strong form of the governing equations becomes(P − VTP

)−(F ′ − KTF

)= L, (36)

where the first and second terms on the left-hand side are the contributions of inertial and elasticforces, respectively.

10

7 Finite element formulation

The virtual motion and velocity vectors are interpolated within the elements as

δU =N∑k=0

`k(ξ)δUk = L(ξ) δU , (37a)

V =N∑k=0

`k(ξ)Vk = L(ξ) V , (37b)

where `k(ξ) are Lagrange’s polynomials defined by eq. (51b), L(ξ) = [`0I6, . . . , `NI6] stacks all

the shape functions, δUT

= δUT

0 , . . . , δUT

N and VT

= VT

0 , . . . , VT

N store the nodal values formotion increments and velocities, respectively. Introducing the interpolation into the weak formof the governing equations (35) and using quadrature rules will yield the discretized governingequations

M˙V − G + F = L, (38)

where the mass matrix M , gyroscopic force G, elastic force F , and external force L are defined as

M =

∫ 1

−1J(ξ)LT (ξ)ML(ξ) dξ,

G =

∫ 1

−1J(ξ)LT (ξ)VT (ξ)MV(ξ) dξ,

F =

∫ 1

−1

[L′T (ξ)DE(ξ) + J(ξ)LT (ξ)KT (ξ)DE(ξ)

]dξ,

L =

∫ 1

−1J(ξ)LT (ξ)L(ξ) dξ,

(39)

where J(ξ) = ds/dξ is the Jacobian associated with the coordinate transformation.A linearization of gyroscopic force G yields

∆G =

∫ 1

−1J(ξ)LT (ξ)

[VT (ξ)M+ (M V(ξ))U

]∆V = G∆V , (40)

where notation (•)U , a linear map between a vector of size 6×1 and a matrix of size 6×6, is definedas KTQ = (Q)UK. Similarly, eqs. (27) are used to linearize the elastic forces, leading to

∆F =

∫ 1

−1

L′T (ξ)DL(ξ) + J(ξ)LT (ξ)

[KT (ξ)D + (DK(ξ))U

]L(ξ)

B∆U = K ∆U . (41)

Finally, the linearized governing equations are found as

M∆˙V −G∆V +K ∆U = r, (42)

where r is the residual.

11

To evaluate the integration in eqs. (39), (40), and (41), two types of quadrature rules areconsidered: the reduced Gauss-Lobatto and Gauss rules. The force vectors and matrices based onGauss-Lobatto quadrature rule (54a) are found as

G = diag(VTk )M V ,

F =[N + diag(Jkw

GLLk KTk )

]diag(D) Ek,

M = diag(JkwGLLk M),

G = diag(VTk )M + diag(JkwGLLk DEk),

K =[[N + diag(Jkw

GLLk KTk )

]diag(D) + diag(Jkw

GLLk [DEk)U ]

]B.

Derivative matrix N is defined as N = [dTdiag(wk)] ⊗ I6, where matrix d is defined by eq. (52),

notation ⊗ indicates a Kronecker product, wGLLk are the weights at the GLL points, and vector

ET

= ET0 , . . . , ETN stacks strain components at the GLL points. The external force vector is

LT

= J0w0LT (ξ0), . . . , JNwNLT (ξN). Because the nodes and quadrature points are collocated,the mass matrix becomes diagonal and the expression for the elastic forces is simpler than thatresulting from Gaussian quadrature in conventional finite element formulations.

The force vectors and matrices based on Gauss quadrature rule (54b) are found as

G =N−1∑i=0

J(ξGi )wGi LT (ξGi )VT (ξGi )MV(ξGi ),

F =N−1∑i=0

[wGi L

′T (ξGi )D E(ξGi ) + J(ξGi )wGi LT (ξGi )KT (ξGi )D E(ξGi )

],

L =N−1∑i=0

J(ξGi )wGi LT (ξGi )L(ξGi )

M =N−1∑i=0

J(ξGi )wGi LT (ξGi )ML(ξGi ),

G =N−1∑i=0

J(ξi)wGi L

T (ξGi )[VT (ξGi )M+ (MV(ξGi ))U

]L(ξGi ),

K =N−1∑i=0

[wGi L

′T (ξGi )D + J(ξGi )wGi LT (ξGi )

[KT (ξGi )D + (DE(ξGi ))U

]]L(ξGi )B.

Therein, wGk are the weights at the Gauss points. The curvature and strain components at the Gausspoints are evaluated through interpolations of the nodal quantities, i.e., K(ξGi ) =

∑Nk=0 `k(ξ

Gi )Kk

and E(ξGi ) =∑N

k=0 `k(ξGi )(Kk −K0k).

8 Time integration scheme

The global parameterization-free generalized-α scheme developed by Arnold and Bruls [58,59] willbe used to integrate the equations of motion in time. One time step extends from time ti to time tf ,

12

and h = tf − ti is the time step size; subscripts (·)i and (·)f indicate values of quantities evaluatedat time ti and tf , respectively. If motion tensors R

iand R

fdenote the motion tensors of node k at

times ti and tf , respectively, the incremental motion tensor becomes R = R T

iRf. Motion parameter

vector qk

is associated with incremental motion tensor R and is recast to vector Qk, of size 6× 1.

These incremental motion vectors at all nodes are collected into a single array, QT

= QT0, . . . ,QT

N.

The generalized-α scheme involves the following equations,

Q = hV i + h2[(1

2− β)Ai + βAf

], (43a)

Vf = V i + h[(1− γ)Ai + γAf

]. (43b)

These equations are expressed in terms of nodal algorithmic accelerations Ai and Af that are relatedto the actual nodal accelerations of the system through the following recurrence relationship,

(1− αm)Af + αmAi = (1− αf ) ˙Vf + αf˙V i, A0 =

˙V0, (44)

where αm = (2ρ∞ − 1)/(ρ∞ + 1), αf = ρ∞/(ρ∞ + 1), γ = 1/2 + αf − αm, and β = (γ + 1/2)2/4.Spectral radius ρ∞ ∈ [0, 1] are tuned to achieve the desired numerical dissipation, see refs. [58, 59].Spectral radius ρ∞ = 0 is used for all the dynamics problems in section (9). Using identity (9) andlinearizing of eqs. (43a), (43b), and (44) leads to

∆˙Vf = β′T

−1∆Uf , (45a)

∆Vf = γ′T−1

∆Uf , (45b)

where matrix T = diag[T (Qk)], β′ = (1− αm)/[h2β(1− αf )], and γ′ = γ/(hβ).

Introducing eqs. (43), (44), (45a), and (45b) into the governing equations (42) at time tf and

linearizing with respect to ∆Uf yields[β′M T

−1− γ′G T

−1+K

]∆Uf = r. (46)

The final step of the scheme is the kinematic update that evaluates the motion of node k at timetf as R

f= R

iR (∆uk).

9 Numerical examples

To validate the proposed approach, a set of numerical examples will be presented. In all cases,the sectional mass and stiffness matrices of the beam are computed using SectionBuilder, a finiteelement based tool for the analysis of cross-sections of beams of arbitrary configuration made ofanisotropic materials [9–11]. Reference solutions will be provided by Dymore 4, a finite elementbased, flexible multibody system analysis tool that uses the classical description of kinematics, i.e.,the displacement and rotation fields are treated separately. Predictions based on Gauss-Lobattoand Gauss quadrature rules are compared in the first examples; for the remaining examples, onlythe predictions based on Gauss quadrature rules are presented.

13

9.1 Cantilevered beam with a 45-degree bend

B

P45o

100 in

O

1 in

i1_

i2_

i3_

b1_

b3_

b3_

b2_

b2_

B1 in

Figure 3: Cantilevered beam with a 45-degree bend.

In this example, the response of the 45-degree bend cantilevered beam shown in fig. 3 is inves-tigated. The beam is cantilevered at point O and subjected to a static tip load, P = 600 lb, actingalong unit vector ı3. The initial curvature of the beam about unit vector ı3 is k3 = −0.01 in−1. Thecross-section of the beam is 1×1 in2. The beam is made of isotropic material with the followingproperties: Young’s modulus E = 107 psi, and Poisson’s ratios ν = 0.0. The non-vanishing entriesof sectional stiffness matrix D predicted by SectionBuilder are D11 = 1.00 107 lb, D22 = D33 = 4.17106 lb, D44 = 7.03 105 lb·in2, D55 = D66 = 8.33 105 lb·in2, and D16 = −8.33 103 lb·in.

No. of elements u1 (in) u2 (in) u3 (in)

Proposed, Gauss-Lobatto 8 (3-node element) -13.670 -23.712 53.421Proposed, Gauss 8 (3-node element) -13.731 -23.818 53.607

Ibrahimbegovic [13] 8 (3-node element) -13.729 -23.814 53.605Bathe and Bolourchi [60] 8 -13.4 -23.5 53.4

Table 1: Displacement components of the free tip.

Table 1 lists the present predictions based on the Gauss-Lobatto and Gauss quadrature rules,those of Ibrahimbegovic [13], and those of Bathe and Bolourchi [60]. Figure 4 and 5 show theerror measure, ‖u − ur‖/‖ur‖, for the tip displacement of the beam versus the number of spectralelements on a logarithmic plot, for the Gauss-Lobatto and Gauss quadrature rules, respectively.The reference solution, ur, is obtained using 128 spectral element of order N = 6. The figuresshows the error measures for spectral elements with N = 2, 3, 4, and 5 (corresponding to 3, 4, 5,and 6 nodes in each element). The predictions based on Gauss-Lobatto rules are quite inaccuratecompared with those based on Gauss rules. For the Gauss rules based approach, the elements basedon quaternion and geodesic metric interpolations are slightly more accurate than those based on thematrix metric interpolation. For the three interpolation schemes, the convergence rate is between2N−0.5 to 2N . For 3-node elements, the quaternion and geodesic metric based interpolations havethe same order of accuracy as the 3-node elements based on classical kinematics implemented inDymore 4. For 4-node elements, the proposed approach become more accurate than those based onclassical kinematics when the number of elements is larger than 8.

For the solutions of 8 4-node elements, figures 6 and 7 show the axial and shear strains, re-spectively, obtained from the interpolated and assumed strain fields defined by eqs. (18) and 20,respectively, over the root element of the beam. The interpolated strain field behaves as a poly-nomial of degree 3 and presents 3 zeros. The interpolated and assumed strain fields are nearlyidentical at the Gauss points. The beam elements based Gauss-Lobatto quadrature rules suffer

14

2 4 8 1610-15

10-10

10-5

100

Number of Elements

Err

or

in T

ip D

isp

.

N=2

N=3

N=4

N=5

Figure 4: Tip displacement error versus numberof elements and degree of the Lagrange poly-nomial for elements based on Gauss-Lobattoquadrature rules. Matrix metric: (), quater-nion metric: (), geodesic metric: (×).

2 4 8 1610-15

10-10

10-5

100

Number of Elements

Err

or in

Tip

Dis

p. N=3

N=4

N=5

N=6

Figure 5: Tip displacement error versus numberof elements and degree of the Lagrange polyno-mial for elements based on Gauss quadraturerules. Matrix metric: (), quaternion metric:(), geodesic metric: (×), Dymore 4 : ().

from locking because the axial and shear strains interpolated at the GLL quadrature points arevery inaccurate and hence, the strain energy associated with axial and shear deformation is grosslyoverestimated. The assumed strains is a polynomial of degree 2, which is consistent with the motionfield interpolated via polynomials of order 3. Consequently, beam elements based on assumed strainand Gauss quadrature rules are locking free.

9.2 Post-buckling of a circular arch

Figure 8 depicts a circular arch of radius R = 100, hinged at one end and clamped at another,subjected to a vertical force, P , applied at mid-span. The post-buckling behavior of this structureis investigated. The beam’s sectional stiffness matrix is D = diag(108, 108, 108, 106, 106, 106, 106).This example was first described by Crisfield [61], who provides all input data in non-dimensionalform.

The mesh consists of 12 4-node elements and the arc-length method of Crisfield [61] is used totrace the buckling and post buckling behavior of the structure. Figure 9 shows the applied load asa function of the magnitude of the displacement vector of the mid-span point; the configurationsof the arch for P = 897.9 and - 17.8, labeled as curves 1 and 2, respectively, are shown in fig. 10.For the present approach, all the three metrics provide the same critical buckling load Pcr = 897.9,to four significant digits. This compares favorably with the buckling load Pcr = 897 reported byDaDeppo and Schmidt [62].

9.3 Stability of a rotating shaft

A flexible shaft of length L = 6.0 m is supported at its two ends by revolute and cylindrical joints,as depicted in fig. 11. A rigid disk is attached to the shaft at its mid-span point M. Initially, thedisk’s center of mass is located a distance d = 0.05 m above the axis of the shaft.

15

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-250

-200

-150

-100

-50

0

50

100

150

200

250A

xia

lst

rain

(με)

Nondimensional coordinate ξ

Figure 6: Axial strain ε1 in the root ele-ment. Interpolated strain (eq. (18)): dashedline; assumed-strain (eq. 20): solid line. Gausspoints: (×), Gauss-Lobbato points: ().

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-80

-60

-40

-20

0

20

40

60

80

Sh

ear

stra

in (με)

Nondimensional coordinate ξ

Figure 7: Shear strain γ12 in the root ele-ment. Interpolated strain (eq. (18)): dashedline; assumed-strain (eq. 20): solid line. Gausspoints: (×), Gauss-Lobbato points: ().

O

φ=145

i1_

i2_

R

P

Figure 8: Configuration of a clamped-hinged circular arch.

The cross-section is a circular tube with inner and outer radii rI = 46 and rO = 50 mm,respectively. The wall consist of four, 1 mm thick plies of graphite/epoxy material with the followingmaterial properties: mass density ρ = 1.6×103 kg·m−3, longitudinal modulus EL = 181 GPa,transverse modulus ET = 10.3 GPa, shearing modulus GLT = 4.17 GPa, and Poisson’s ratios νLT= 0.28, νTN = 0.33. The stacking sequence of the lay-up is [0, 45, 45, 0]; 0 fibers are alignedwith the axis of the beam and a positive ply angle indicates a right-hand rotation about the localouter normal to the wall. The shaft’s sectional stiffness properties are

D105

=

1254 0.0 0.0 0.1565 0.0 0.00.0 162.3 0.0 0.0 −0.06481 0.00.0 0.0 162.3 0.0 0.0 −0.06481

0.1565 0.0 0.0 0.7474 0.0 0.00.0 −0.06481 0.0 0.0 1.449 0.00.0 0.0 −0.06481 0.0 0.0 1.449

.

16

0 30 60 90 120 150 180-100

0

200

300

400

500

600

700

800

900

Loa

d

1

2

Norm of mid-span displacement

Figure 9: Load-displacement curve, matrix met-ric: (), quaternion metric: (), geodesic metric:(×).

-120 -80 -40 0 40 80 120 160-80

-40

0

40

80

120

Coordinate along i1

Coo

rdin

ate

alon

g i 2

2

1

Figure 10: Buckling modes at the critical points,matrix metric: (), quaternion metric: (),geodesic metric: (×).

The units of stiffness matrix components Dij are as follows: N·m−1 for i, j = 1, 2, 3; N·m fori, j = 4, 5, 6; and N for all other components.

The mass per unit span is m = 1.930 kg/m, the moments of inertia per unit span are m22 = m33

= 2.227 g·m and the resulting polar moment of inertia per unit span is m11 = m22 + m33 = 4.455g·m. The mid-span circular disk is of mass md = 40.0 kg, radius rd = 0.24 m, and thickness td =0.05 m. Its inertial tensor computed with respect to the center of mass is diagonal, diag(2.0, 1.0,1.0) kg·m2. The acceleration of gravity is 9.8 m/s2.

At the initial time, the shaft is at rest and deformed under the effect of the gravity loads actingin the vertical direction, as indicated in the figure. The angular velocity at root R of shaft isprescribed as

Ω(t) =

0.8ω[1− cos(πt/0.5)]/2, 0 ≤ t ≤ 0.5,

0.8ω, 0.5 < t ≤ 1.0,

0.8ω + 0.4ω1− cos[π(t− 1.0)/0.25]/2, 1.0 < t ≤ 1.25,

1.2ω, t > 1.25,

(47)

where ω = 54 rad/s is close to the first natural frequency of the shaft in bending (ω1 = 56.7 rad/s).As the shaft accelerates, it passes through the first natural bending frequency of the system

and the operation goes from sub- to super-critical. As predicted by linear rotor dynamics theory,the shaft becomes unstable when operating at the critical speed. In the present example, themagnitudes of lateral oscillations and corresponding internal forces rise as the shaft is acceleratedthrough the critical speed.

The response was simulated for a total time of 2.5 s with a constant time step ∆t = 5 ms. In thepresent approach, 8 four-node elements are used to model the shaft. Figure 12 shows the trajectoryof the shaft’s mid-span point as viewed by a rotating observer in sectional basis B. Because theshaft operates above its critical speed for time t > 1.25 s, it becomes self-centering, explainingthe circular trajectory of its geometric center seen by a rotating observer. Figure 13 shows the

17

Cylindricaljoint

Revolutejoint

i1_

i2_

i3_

b1_

b3_

b2_Ω

Groundd

R

T

L

Ground

M

g

Cross-section

Figure 11: Configuration of the rotating shaft

phase plot of shaft’s mid-span for the displacement and velocity components along unit vectors b3.Here again, the predictions for the interpolation schemes based on matrix, quaternion, and geodesicmetrics are nearly identical and are in good agreement with the predictions of Dymore 4.

Figures 14 and 15 show the component of axial and transverse shear forces, Fj, twist andbending moments, Mj, along unit vector bj, j = 1, 2, 3, at the root of the beam. Here again,excellent agreement is observed between the present solutions and the predictions of Dymore 4.

9.4 The four-bar mechanism

Figure 16 depicts a flexible four bar mechanism. Bar 1, 2, and 3 are of lengths 0.12, 0.24 and 0.12m, respectively. The axes of rotation of three revolute joints at points A, B, and D are normal tothe plane of the mechanism. The axis of rotation of the revolute joint at point C is rotated by +5

about unit vector ı2 indicated in fig. 16 to simulate an initial defect in the mechanism.The cross-sections of bar 1, 2, 3 are hollow square boxes with outer widths, 16, 16 and 8 mm, and

inner widths, 9.6, 9.6 and 4.8 mm, respectively. Each box is composed of two plies of graphite/epoxymaterial of equal thickness. The stacking sequence is identical for all the three lay-ups: [30,−30];0 fibers are aligned with the axis of the beam and a positive ply angle indicates a right-handrotation about the local outer normal to the wall.

In the reference configuration, the bars of this mechanism intersect each other at 90 angles.The angular velocity at point A of bar 1 is prescribed as

Ω(t) =

0.3[t− cos (πt)] rad/s, 0 ≤ t ≤ 6.0 s;

0.6 rad/s, t ≥ 6.0 s.(48)

18

-0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

u2 [m]

u3

[m]

Figure 12: Trajectory of shaft’s mid-spanviewed by a rotating observer, matrix metric:(), quaternion metric: (), geodesic metric:(×), Dymore 4 : ().

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2-15

-10

-5

0

5

10

15

v 3 [m/s]

u3 [m]

Figure 13: The phase plot of shaft’s mid-spanfor components along unit vectors b3, matrixmetric: (), quaternion metric: (), geodesicmetric: (×), Dymore 4 : ().

0 0.5 1.0 1.5 2.0 2.5-30

-25

-20

-15

-10

-5

0

5

10

15

20

Time t [s]

F [

KN

]

F3

F2

F1

Figure 14: Axial and shear forces at the beam’sroot A, matrix metric: (), quaternion metric:(), geodesic metric: (×), Dymore 4 : ().

-30

-20

-10

0

10

20

30

40

0 0.5 1.0 1.5 2.0 2.5Time t [s]

M [

KN

m]

Figure 15: Torsion and bending moments at thebeam’s root A, matrix metric: (), quaternionmetric: (), geodesic metric: (×), Dymore 4 :().

19

0.24 mBar 2

Bar

1 Bar 3

A

B C

D0.

12 m

Revolute jointsBeams

Ω = 0.6 rad/s

i2_

i1_

Bar 1&2 section Bar 3 section

8 mm16 mm

16 m

m

8 m

m

b3_

b2_

b3_

b2_

Misaligedaxis of rotation

Figure 16: Configuration of the four-bar mechanism.

The sectional stiffness matrices evaluated by SectionBuilder for the three cross-sections are

D1

103=D

2

103=

10660 0 0 −8.676 0 0

0 2532 0 0 5.013 00 0 2532 0 0 5.013

−8.676 0 0 0.2462 0 00 5.013 0 0 0.3091 00 0 5.013 0 0 0.3091

,

D3

102=

2665 0 0 −1.085 0 0

0 633.0 0 0 0.6266 00 0 633.0 0 0 0.6266

−1.085 0 0 0.01539 0 00 0.6266 0 0 0.01932 00 0 0.6266 0 0 0.01932

.

The units of stiffness matrix components Dij are as follows: N·m−1 for i, j = 1, 2, 3; N·m fori, j = 4, 5, 6; and N for all other components. Stretching/twisting and shearing/bending couplingterms stem from material anisotropy. The sectional mass properties for bars 1 and 2 are m00 =1.9968 kg/m and m22 = 42.598 mg·m and the corresponding quantities for bar 3 are m00 = 0.4992kg/m and m22 = 2.6624 mg·m.

Bar 1, 2, and 3 are modeled with 2, 4, and 2 four-node elements , respectively. For this problem,the simulation was run for 36 s, starting from initial conditions at rest and using time steps ofconstant size ∆t = 9 ms. The predictions for the third revolute of the mechanism are shown infigs. 17, 18, 19, and 20. The present results for interpolations based on matrix, quaternion, andgeodesic metrics agree well with predictions of Dymore4.

10 Summary and conclusion

A new solution procedure was proposed for geometrically exact beams within the framework of themotion formalism. It combines the spectral method with motion interpolation schemes. Motioninterpolation schemes based on matrix, quaternion, and geodesic metrics yield simple expressionsfor the sectional strains and linearized strain-motion relationships at the nodes. Beam elementsbased on Gauss-Lobatto and Gauss quadrature rules were investigated. Gauss-Lobatto rules only

20

0 π/2 π 3π/2 2π-5

-4

-3

-2

-1

0

1

2

3

4

Driving angle θA

Dis

pla

cem

ent

at C

, u3

[mm

]

Figure 17: Displacement of point C along ı3,matrix metric: (), quaternion metric: (),geodesic metric: (×), Dymore 4 : ().

-0.10

-0.05

0

0.05

0.10

0.15

0 π/2 π 3π/2 2πDriving angle θA

Vel

ocit

y at

C, v

3 [m

m/s

]

Figure 18: Velocity of point C along ı1, matrixmetric: (), quaternion metric: (), geodesicmetric: (×), Dymore 4 : ().

requires summation over the nodes of the elements. Gauss quadrature rules requires curvatures atthe Gauss points that are evaluated through an interpolation of the nodal curvatures. In both cases,the expressions for the internal forces and tangent stiffness matrices are simplified. Consequently,the proposed spectral element formulation is much easier to implement than its conventional coun-terpart. The global parameterization-free generalized-α scheme was used to integrate the motionfield in time.

Numerical examples have demonstrated the Gauss-Lobatto rules based elements suffer from axialand shear locking, while the Gauss rules based elements are locking free. The convergence rate ofa (N + 1)-node spectral element based on Gauss rules is about 2N − 0.5 to 2N for all the threeinterpolation schemes. For the static problem investigated, the matrix metric based interpolationis less accurate than the quaternion and geodesic metric based interpolation. As the number ofelements increase, the Gauss rules base formulation becomes more accurate than the conventionalbeam element in Dymore 4. For all test cases, excellent agreement is observed between the variousmotion interpolation strategies and the prediction of classical formulations. The proposed methodscan be generalized easily to geometrically exact shell formulations.

Appendix

10.1 Legendre polynomial, Gauss-Legendre Quadrature and Lagrangeinterpolation

Spectral methods are based on orthogonal polynomials originating from the solution of eigenvalueproblems for ordinary differential equations, a class of problems known as “Sturm-Liouville prob-lems.” Legendre’s polynomials, denoted as Pk(ξ), ξ ∈ [−1, 1], are the solutions of the following

21

-250

-200

-150

-100

-50

0

50

100

0 π/2 π 3π/2 2πDriving angle θA

Sh

ear

forc

e at

M, Q

3 [N

]

Figure 19: Shear force at the mid-span of bar 2along b3, matrix metric: (), quaternion metric:(), geodesic metric: (×), Dymore 4 : ().

-20

-15

-10

-5

0

5

0 π/2 π 3π/2 2πDriving angle θA

Ben

d.m

om. a

t M

, M2

[N.m

]

Figure 20: Bending moment at the mid-span ofbar 2 along b2, matrix metric: (), quaternionmetric: (), geodesic metric: (×), Dymore 4 :().

Sturm-Liouville problem,

(1− ξ2)d2Pkdξ2

− 2ξdPkdξ

+ k(k + 1)Pk = 0, (49)

with boundary conditions Pk(−1) = (−1)k and Pk(1) = 1. The few lowest-order polynomialsare P0(ξ) = 1, P1(ξ) = ξ, P2(ξ) = (3ξ2 − 1)/2, P3(ξ) = (5ξ3 − 3ξ)/2. Alternatively, Legendre’spolynomials are generated by the following recurrence relationship

kPk+1(ξ) = (2k − 1)ξPk−1(ξ)− (k − 1)Pk−2(ξ), k ≥ 2. (50)

The set of polynomials of degree less or equal to N forms a vector space of dimension N + 1,denoted as PN . Clearly, the set of Legendre’s polynomials up to the N th degree, P0, . . . , PN,forms an orthogonal basis of PN .

The Gauss quadrature points are denoted as [−1 < ξG0 , ξG1 , . . . , ξ

GN−1 < 1]. Points ξGk , i =

0, 1, . . . , N − 1, are the real zeros of polynomial PN+1(ξ). The Gauss-Lobatto-Legendre (GLL)points are denoted as [ξGLL0 = −1, ξGLL1 , . . . , ξGLLN = 1]. Points ξGLLk , k = 1, . . . , N − 1, are the realzeros of polynomial P ′N+1(ξ) of degree N − 1, where notation (·)′ indicates a derivative with respectto ξ. Lagrange’s polynomials based on the Gauss and GLL points are

¯i(ξ) =

N−1∏j=0, j 6=k

ξ − ξGkξGLLk − ξGj

, 0 ≤ j ≤ N − 1, (51a)

`k(ξ) =N∏

j=0, j 6=k

ξ − ξGLLk

ξGLLk − ξGLLj

, 0 ≤ j ≤ N, (51b)

22

respectively. Lagrange’s polynomials satisfy identity `k(ξGLLj ) = δkj and ¯

i(ξGj ) = δij , where

δ denotes the Kronecker delta. Furthermore, the condition of partition of unity is satisfied i.e.,∑N−1i=0

¯i(ξ) = 1 and

∑Nk=0 `k(ξ) = 1.

At the GLL quadrature points, the derivatives of Lagrange’s polynomials `k(ξ) with respect toξ become

dj,k =d`kdξ|ξ=ξGLL

j=

−N(N + 1)/4, j = k = 0,

0, j = k ∈ [1, 2, . . . , N − 1],

N(N + 1)/4, j = k = N,

PN(ξGLLj )/[(ξGLLj − ξGLLk )PN(ξGLLk )], j 6= k.

(52)

These derivatives are stored in matrix d, of size (N + 1) × (N + 1), such that entry (j, k) of thismatrix are [d]jk = dj,k.

The following identities holds for any polynomial p(ξ) ∈ P2N−1 [63, 64]∫ 1

−1p(ξ) dξ =

N∑k=0

p(ξGLLk )wGLLk =N−1∑i=0

p(ξGi )wGi , (53)

where wGLLk and wGi are the weights of Gauss-Lobatto and Gauss quadrature rules, respectively.For polynomials p(ξ) of degree higher than 2N , reduced quadrature rules are used and quadratureformula (53) still provides a good approximation∫ 1

−1p(ξ) dξ ≈

N∑k=0

p(ξGLLk )wGLLk , reduced Gauss-Lobatto rule (54a)

∫ 1

−1p(ξ) dξ ≈

N−1∑i=0

p(ξGi )wGi . reduced Gauss rule (54b)

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[3] J. C. Simo and L. Vu-Quoc. A three-dimensional finite strain rod model. Part II: Computationalaspects. Computer Methods in Applied Mechanics and Engineering, 58(1):79–116, 1986.

[4] V. Giavotto, M. Borri, P. Mantegazza, G. Ghiringhelli, V. Carmaschi, G. C. Maffioli, andF. Mussi. Anisotropic beam theory and applications. Computers & Structures, 16(1-4):403–413, 1983.

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23

[6] E. Carrera, G. Gaetano, and M. Petrolo. Beam Structures: Classical and Advanced Theories.John Wiley & Sons, New York, 2011.

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