Mixed variational principles in space and time for ...Mixed variational principles in space and time...

Acta Mechanica 136, 193-208 (1999) ACTA MECHANICA �9 Springer-Verlag 1999

Mixed variational principles in space and time for elastodynamics analysis

M. B. Quadrelli, Pasadena, California, and S. N. Atluri, Los Angeles, California

(Received February 24, 1997; revised March 24, 1998)

Summary. Nonlinear elastodynamics problems are approached from the point of view of a mixed variational principle. It is shown that four different functionals lead to different formulations of the problem with different independent fields. One of the functionals is specialized to the case of spatial beams undergoing large deformations, and small strains. The extension of these functionals to the case in which kinematic constraints are presented leads to the useful concept of dual constraints, which can be advantage- ously employed in treating problems involving contact, impact, and changing topology.

1 Introduction

This paper deals with the development of primal and mixed variational principles for nonlinear elastodynamics analysis of three-dimensional continua. A previous paper [1] has investi- gated two different methods to compute the tangent stiffness and inertia of a spatial elastic beam, undergoing large deformation. The motivation for that derivation was to formally adopt a variational setting to derive the field equations of motion of constrained multi-flexible bodies. Previously, a consistent variational approach for multi-flexible body dynamics with beams has only been presented by [2]. The first functional for finitely deformed beams, obtained in a consistent fashion from a general three-field mixed variational principle for three- dimensional finite elasticity was proposed in [3] and [4]. [5] and [6] used time finite elements to solve the equations of motion of a rigid body and of multi-rigid bodies. Following these foundations, this paper is organized as follows. First, we overview the basic kinematics of rotations and the kinetics of the problem. Second, we describe primal and mixed variational functionals valid from nonlinear elastodynamics of three-dimensional continua. Additional details of the derivation are omitted and we refer the reader to [1] and [7]. Third, we derive a variational principle for unconstrained linear elastodynamics which correctly accounts for the boundary conditions at the end of a time interval. Finally, we present the analytical steps followed to develop finite elements in space and time for nonlinear elastodynamics problems, and the conclusions of this paper.

2 Kinematics and kinetics

Consider a deformable one-dimensional continuum which is capable of undergoing large displacements and arbitrarily large rotations. To a material element in the undeformed configuration s we assign the triad of basis vectors denoted by Ei. To the same material element,

194 M.B. Quadrelli and S. N. Atluri

but in the deformed configuration Cd, we assign the triad of basis vectors e~, which denotes the basis Ei after a purely rigid body rotation (similarly, S~ and $a denote the undeformed and deformed cross-section). The convected coordinates yi , with i -- 1, 2, 3, denote a curvilinear system associated to a point centered in the material element. The displacement vector u describes the total displacement of the material element the position of which, in C~, is denoted by X. The rotation tensor R describes the rigid body rotation of the material element, and can be parameterized as a function of the finite rotation vector a = 0e, where e is the unit vector fixed in space, and around which the finite rotation of magnitude 0 takes place. A virtual variation of rotation q~e, the angular velocity vector o), and the curvature vector 13 can be expressed, respectively, as q~6 x 1 = 5R. R T, a~ • 1 = R - R T, and 13 x 1 = R,3 " R T, where 1 is the identity matrix, and ('),3 represents the covariant derivative with respect to ya in C~. In addition, for the angular velocity vector m we have the relationship a~ = r ( a ) d, and, for any arbitrary vector field, the associated tangent rotational maps, rou- tinely used during the procedure of consistent linearization, may be derived in a straightfor- ward manner [7]. We have [8] the Euler-Rodrigues formula for the rotation tensor R and the associated tensor F:

R(a) -- 1 + ao(a x 1 ) + a l i a x (a • 1)],

I'(a) = 1 + al (a X 1) @ a2[a x (6g x 1)], (i)

where ao=sO/O, al=(1-cO)/O2, a2=l/O2(l-ao). We adopt the convention that d(.), 5(.), A(,) represent the differential, the virtual variation, and the finite, but small, increment of a generalized coordinate, whereas (')e, (')e, and ( ')a represent the differential, the variation, and the finite increment of a quasi-coordinate. The configuration of the beam is completely known when the internal position of one point P of the cross section S, and the orientation of the cross section itself with respect to a reference triad, are known.

The inertial properties of the motion may be described with a Lagrangean approach, in which the rotational motion is referred to a fixed triad in space. In this paper, a material element labeled with the generalized coordinate vector q = (u, a) T is endowed with a La- grangean density the structure of which we assume to be of the form t; = T(t,q, di) - f l (v, R, U, t), where T is the kinetic energy density and fl is the mixed functional density described in more detail below. We use the usual notation of the polar decomposition theorem, where F, R, U are the deformation gradient tensor, the rotation tensor, and the right stretch tensor, respectively. In our derivation, we make the assumption that the material is linearly elastic, and that the system is scleronomic, i.e., the kinetic energy density is independent of time. Then, our derivation is based on the functional:

El= f ~ f T(q, dl) dAdY 3-Fl (v ,R ,U, t )~dt , (2) )

where dt and dY a denote the time differential, and the differential element of curve along the beam length, of cross-sectional area dA.

2.1 A four-field principle

We denote by S~o(S~o) the portion of the spatial boundary of a material continuum where tractions (dispacements) are prescribed, and by V0 and 0]20 the volume, and boundary, of the undeformed material element. For a general elastic material, a four-field mixed principle [8], [9], involving v, R, U, and t as independent variables may be stated as the stationarity con-

Mixed variational principles 195

dition of the functional:

Fl(v, R, U, t) = f{W~(U) + t T : [(I + Vv) - R . U] - oob" v}dV Vo

- f i. v d A - f N . t . ( v - ~ ) d A , S~o S~O

(3)

where b = f , f being applied body forces per unit mass ; / a re prescribed tractions on S~0 and the prescribed displacements on 8u0, t = tYA~ | az: the 1st Piola-Kirchhoff stress tensor [4]. Here, the symbol ":" represents the double contraction of two 2nd order tensors, i.e., A : B = AiNB ij = trace (A : BT), and V = gs(O/O~ i) is the gradient operator along the vectors g:

associated with the curvilinear coordinates ~. Also, Ws and W~ are the strain and complementary strain energy density per unit undeformed volume. In F1, v must be C O continuous, R orthogonal, U symmetric, and t unsymmetrical, in order to be admissible trial fields. When the condition, 5F1 = 0, is enforced for arbitrary and independent variations: C o continuous 5u, 5R under the constraint 5R. R T = (SR-RT)a, symmetric 8U and un-

symmetric St, the following Euler-Lagrange Equations (ELE) are obtained: CL: OWs/OU = 1 / 2 ( t - R + R a". t r) = ( t . R ) s , CC: ( I + V v ) = R . U , AMB: (t T. U . R T ) ~ = 0, LMB:

V0 �9 t + gob = 0 together with the natural boundary conditions TBC: N . t = t on $~0 and DBC: v = ~ on Suo. Here CL stands for Constitutive Law, CC for Compatibility Condition, LMB for Linear Momentum Balance, AMB for Angular Momentum Balance, TBC for Trac- tion Boundary Condition, and DBC for Displacement Boundary Condition. Here, (.)s and (-)o, represent the symmetrical and unsymmetrical part of a tensor.

2.2 A three-fieM principle

One may derive a complementary variational principle [8] involving only v, R and t. To do this, U must be eliminated from F1 by applying the following contact transformation:

I ( t . R + R r . t r ) : U , (4) + w j r ) =

where r = 1/2(t - R + R T - t T) is the symmetrized Biot-Lur'e stress tensor or Jaumann stress tensor, and W~ the complementary energy density. Making use of this contact transformation is equivalent to adopting the hypothesis that, a priori OW~/Or = U is met. When this contact transformation is substituted into the expression of functional F1, the following Hellinger- Reissner type three-field functional is obtained:

W~, 1 . tT)] + tT -~- QO b r3(v,R,t)=f{- ~ [ ~ ( t . R + R T J : ( I ~Tv)- . v } d V )20

S~ S~o

To be admissible, the independent fields in functional F3 have to satisfy the following requirements: v must be C ~ continuous, R orthogonal, and t unsymmetrical. When the condition r = 0 is imposed for arbitrary and independent 5v, 5R, St, subject only to the additional constraint (R . 5It,T)~ = 0, the following ELE are recovered:

( aw '] : (I + v,,). c c : a . \ - a T / s


\ Or / -t =0 , s a

L B M : V 0 - t + ~ ) 0 b = O ,

together with the natural boundary conditions TBC: N . t i on ,~0 and DBC: v = ~ on S~0.

2.3 The twist and wrench vectors

For a material element of the beam, the Internal Virtual Work (1VW) can be written as [1]

I V W = f [T. (5~ + M . ~~ I dY 3, where 6~ = (5u,3 - qge x (X + u), 3 is the corotational c

variation of the stretch vector h, (5o~ = q~,a is the corotational variation of the curvature vec-

tor ~t (corresponding to a rotation of the cross section parameterized by q~), T are the internal

stress resultants and M the internal moment resultants. The kinetic energy over the beam length/2 may be written as

T = 1/2 f s * . g d V = 1/2 f ( e . u + H . to) dY 3, 12 s

where g and H are, respectively, the linear and angular momentum densities. In terms of the finite rotation vector, we may write that

g = A~(~ + CTd, h = C ~ + rT(a ) I 0 r ( a ) d .

Here, A o is the mass per unit length, ~o0 the material density in C~, C the skew-symmetric

matrix of first moments of inertia per unit length, and I~ is the inertia matrix per unit length of the beam. For simplicity, in the computations we assume the body frame to be a principal

axis frame, hence we take the matrix of first moments of inertia C to be zero. Using the defini-

tion of adjoint of a vector space operator, we may justify the fact that if the intrinsic wrench

(g, H) is conjugate to the intrinsic twist (fa, to), then h = FTH is conjugate to ti [7]. By taking

the variation of the kinetic energy density, we also obtain that

(ST = f [ g . (5~ + H . (5~ dY 3 , L

where ~~ = 6/t3 - tf~ • is the corotational variation of the (absolute) velocity, and 6~ = (sto - q~ x to = to~ is the corotational variation of the angular velocity. Finally, it can be

shown that the External Virtual Work (EVW) may also be written as

E V W f [q. 6u + m - q~] d Y 3 , L

where now q and m denote the vector of external forces and external moments distributed along the length of the beam, respectively. For simplicity, and in order to preserve the symme-

try of the resulting matrices, distributed moments which may result in nonconservative loa- ding are not considered.

We may now collect the increment of the displacement vector and the increment of the

finite rotation vector in the incremental twist vector (also describing quasi-coordinates) A t / = (Au, q~z~) r, and the vector collecting the increments of generalized coordinates in vector

A q = (Au, AeQ r . We may conclude that Atl = X,Sq, where J~ is the nonlinear operator denoted by:

x = ~ r( ,~) " (6)


The connection is now clear with the field of traditional kinematics (whose terminology is commonly adopted in the multibody dynamics literature), and conclude that X is nothing but the Jacobian mapping a vector At/in joint space to a vector Aq in the Cartesian workspace of a mechanical system. In a similar way, one may collect the increment of the linear momentum vector and the material increment of the angular momentum vector in the incremental momentum wrench vector (also describing quasi-momenta) A a = ( Ag, HA) T, and the vector collecting the increments of generalized momenta in vector B p = (Ag, A h ) T. We may conclude that A a = X TAp. The increment of kinetic energy density of the material element now becomes

2. A T = Ag . A u + H a . q)a = Ag . A u + A h . A a (7)

which shows the energetically conjugate pairs in joint space (or twist-wrench notation) and in Cartesian space (or in generalized coordinate notation).

2.4 Contact transformation on kinetic energy

In a completely equivalent way to the static case presented in [1], we may also invoke a contact transformation on the kinetic energy. This is allowed because the inertia matrix AA is always positive definite and invertible. The contact transformation, in this case, involving the Lagrangean density Z: and the Hamiltonian density ~ , may be written as:

: r - + ~ = e . / ~ + h . , ~ . (S)

Using g = A o l . f i and h = F r ( a ) I o F ( a ) . d , and their inversions f i = A 0 - 1 1 - g , and ei = F - l ( a ) Io 1F-r (a) h, we may write that:

1 = g. u + h . d - T = ~ gTAQ-ilg + hTF- i (a) I 0 - i r - T ( a ) h . (9)

We conclude that the contact transformation leads to the (configuration-dependent) Hamilto- nian density:

1 pT..Ad-l(q) . p ~(g, h, u, a) =

and to the dual elastodynamic functional:

A i = f f { g . f l + h . d - ~ ( g , h , u , a ) } d V d t - [t~&+lJ Vo

(10)

f F 3 ( h R , t) d t + B . T . , (11)

where B.T. represent the natural boundary conditions. This form may be called a mixed- mixed form of Hamilton's weak principle. In this paper, we will not deal with 7-{, but with T only, in the inertia contribution of A1. However, the stationarity condition represented by Eq. (11) for a nonlinear beam element can also be subjected to a contact transformation so as to obtain a 5 field stationarity condition equivalent to the 5f ie ld functional AI. The five fields are: g, h, v, R, t. In conclusion, we may synthesize the above derivation as follows. Consider the contact transformation of Eq. (4) and the contact transformation of Eq. (8). Then consider the primal-primal weak pinciple:

f { f S T d Y 3 - (5F1 } dt = [ eb " ~u + Hb "~ 1't~+1" ~ S J I t i " (12) t~ L

Then, from Eq. (12), using Eq. (4), we obtain the primal-dual principle:

f { f 67dY a - 6Fa}dt = [gb " (Su § Hb " ~6JteTlti+llti " ( ] 3 ) t~ L


From Eq. (13), using Eq. (8), by integrating by parts with respect to time, we obtain the dual- dual principle:

f { f [p" 6Cl - q" 615 - 67-l] dY 3 - 6Fa }dt : [pb. 6q - qb. 6p]l~t:+t ti

(14)

where p is the vector of momenta, and q the vector of generalized coordinates. Similarly, we obtain the dual-primal principle:

ti+l f { f [ p ' 6 ~ t - q ' 6 f ~ - O ~ ] d Y a - 6 F 1 } d t = [ p b ' 6 q - q b ' 6 p ] ~ { . t{

(15)

The motivation for introducing these functionals lies in the fact that mixed variational forms of Hamilton's weak principle have been proven [10] to lead to unconditionally stable numerical integration schemes for rigid body dynamics and linear structural dynamics. Despite the redun- dancy of independent variables, there could exist a combined advantage of using complementary energy-based finite elements in space, and mixed finite elements in time, for geometrically (or materially) nonlinear structural elements in three dimensions, leading to time integration schemes for nonlinear structural dynamics with excellent invariant properties.

3 Unconstrained elastodynamics

In this Section, we want to outline the steps of the procedure to derive an unconstrained variational principle for nonlinear elastodynamics of a flexible material element. Previously, use has been made of the concept of the convolution operator [11], [12] in order to outline a variational principle for linear elastodynamics. However, to avoid unnecessary complexity and to outline the problem, let us limit the derivation to introducing a weak functional for linear elastodynamics, i.e., in the case in which the strains are linearly related to the displacements. By unconstrained, we mean that we include the boundary conditions in the space and time variables, as additional requirements to be satisfied a posteriori by the variational principle as in [13] and, furthermore, we relax the restrictions usually included in the stationarity of the action functional, i.e., the requirement that the variations of the generalized coordinates be zero at the time boundaries and the compatibility condition in strong form. Consider an element of a deformable continuum occupying the volume ~v = F0 x [t~,t~+l] in q - t space. Note that holonomic constraints are usually written as a relationship between dependent displacement and rotation variables for points of the body belonging to S.~0. We may postulate an additive decomposition of the functional 12 (Lagrangean density) into kinetic and strain energy densities, as in:

1 = T VV, = ) ~ o v . v - Vr (16)

Now, introduce the action functional $ = f s d4t, where d 4 t = dw stands for the space-time

differential volume element. However, in taking the variation of the action $, which is re- presentative of the internal energy of the continuum element, we must note that, given the unconstrained nature of the principle, the contributions arising from variations due to external surface and body forces, plus the variations of the action occurring at the boundaries of


the slice [ti, ti+l] must be added as well. Consider, then, the functional, called the modified action."

ti+~

ti Vo $oO

S~o 12o

In the first volume integral, the first two terms represent the Lagrangean, and the second term is the compatibili ty equation is written in weak form. The kinetic energy is assumed to be a quadratic form on the generalized velocities, and the strain energy density is a function of the linear strains. The case in which quasicoordinates are present is treated in the next Section. In the third integral, the integrand represents the convective transfer of action at the time boundaries t~ and t~+~. In (17), ~ri represent the momenta at the time boundary. Define now the true trajectory followed by the body in mot ion to be the one that makes 6~91 = 0, under no constraint whatsoever on the independent fields. We must intend the 6(.) operator as a general

isochronous operator, i.e., such that ~t = 0, and such that ~ f f i u i d 4 t = f f~Su~d% We ~cr

assume that only conservative forces are present. Imposing the variation on ~ , we obtain:

( ~ ) 1 : / ( ( ~ - O W s --~r. .[ ~ --Ci,j]-4-O'ij(~Ci,j-- f i (~ i ) dud7 z27

ti+l

+l(i+..,..+j,.,,.,-.,,..+l.,.,..)., ti $~ Suo S"o

1;0 (~8)

where (')Ii,j] represents the skew-symmetric par t of (')i,j. In the linearized theory of elasticity, we assume that the stress tensor is symmetric, i.e., that ~ij = crji, and that the angular momentum balance is satisfied a priori. In general, however, and for finite elasticity problems, or for polar continua, it is necessary to obtain the Angular Momen tum Balance a posteriori, as an Euler-Lagrange equation of a specific functional, In this case, a nonsymmetric stress tensor would have to be introduced a priori. Furthermore, we have that, through application of the divergence theorem (and, by satisfying the traction boundary conditions a priori):

- f ~ i /~ ,5 d~ = - f ( ~ / ~ i ) j dS~ + f ~ . , / ~ d ~ v0 v0 v0

= - f (~j~j) ~ dS + f ~ , / ~ dX? = - f m ~ dX + f ~ . , / ~ dX? 0s l;o OQ Po

= - f T ~ dS - f m ~ dS + f ~ 5 , / ~ d~? 09) 8~o S~o Vo

and, by integrating by parts in time, we also have:

f ST dw = f QoiziSisi dt d[2 = f [cOouiSui]~i *~ - v~ 1r o x. ti WO

~i+l / f poiiiSiti dt d[2. (20) ti


Consequently, we obtain that the stationarity of D1 is equivalent to:

tj+l

Wo

as a general Hu-Washizu type variational principle for unconstrained linearized elastodynamics of a continuum. The Euler-Langrange Equations (ELE) implied by ~1771 = 0 are then: (in WoO

CLS: OW,/&# = c~j,

CCS: U2(u~,j + u~,d = e~,

LMB: ~ij,3 + f~ = L)0ui; (on S~0 x t):

TBC: Ti = 2Pi; (on S~0 x t):

DBC: ui = ~i; (in)20, at t = ti):

DIC: uil~, = uilt~; (on Wo, at t = ti):

VIC: z~lt ' = vlt~; (on W0, at t = t0:

MIC: 7ri[t~ = Q0g~lt~; (on 1;0, at t = ti):

MFC: 7r~[t~+~ = 80t~ilt~+~,

where TBC stands for Traction Boundary Condition, DBC for Displacement Boundary Con- dition, CLS for Constitutive Law in Space, CCS for Compatibil i ty Condition iin Space, LMB for Linear M o m e n t u m Balance, D I C for Displacement Initial Condition, VIC for Velocity Initial Condition, M I C for Momen tum Initial Condition, and M F C for Momen tum Final Condition. This is the weakest possible form for linear elastodynamics in the sense that all constraints on the variations are relaxed. The independent fields are, then, the strains, the stresses, the displacements, the velocities and the momenta . Through a contact t ransforma- tion, we may derive an equivalent Hellinger-Reissner variational principle in which LMB and CCS are met a priori, and therefore obtain the complementary version of 791. Similarly, we may start f rom Eq. (11) and carry out the variation to obtain the equivalent weak form for unconstrained nonlinear elastodynamics [11]. The variation will be analogous to Eq. (18). In the following Section, we consider a primal version of this variational principle, but including the presence of finite rotations.

4 Numerical treatment

Consider now a mechanical system described by the vector of global generalized coordinates x. Equivalently, in local coordinates we have the generalized displacements de, where the sub- script (e) denotes a quantity related to a composit ion of the mechanical system into elements. Since we are analyzing the problem with an incremental approach, we must intend d~ and x as being linearized increments of the respective variables about a reference state: Total


Lagrangean (T.L.) if the reference is the undeformed configuration, Updated Lagrangean (U.L.) if the reference is the current configuration. According to the variational development outlined above, the functionals described next are all functional densities. To the arbitrary material element, we may associate the Lagrangean density L (t,x, • = T(L x , • W,(x) equal to kinetic less potential energy. The potential energy is associated with the strain energy density. We refer the internal forces of the beam to the corotational frame [1]. Hence, the internal forces at node a may be written in global coordinates as:

= r OW~ \ ) 5 i 7 , : (22)

In fact, for a two node beam element, the internal strain energy can be written as W~ = W~(d), where in symbolic form d = d(x) represents the highly nonlinear mapping (because of the rotations) between the local displacements d = (ua ~, 0~ ~) (u = translations, 0 = rotations) of one end of the beam with respect to the other, measured by an observer with respect to a reference point located in the corotational frame, and the global displacements x = (u J , 0~ ~, u J , 069). Defining the internal forces by OW~/Od~ and the local tangent matrix by 02W~/Od~ Odb, w e may obtain the elements of the global tangent matrix as

<b = (~176 \ O~ J" Od~ Odb

\-~bxb( Odb ) + i~= t~if [ O ( Odi ~ (ow ) (2a) :

where rgf is the number of internal forces, and the symbol ":" denotes tensor contraction. More details on the expression of the tangent stiffness will be discussed in a later section. The kinetic energy for a general dynamic system is, in matrix form:

1 T( t ,x ,~ ) = { : ~ - M 2 ( x ) . k + M l ( x ) . • Mo(~ ). (24)

However, we assume that there is no time dependence (i.e., M0(t) = 0) and that M 1 = 0 as well, i.e., we assume the kinetic energy to be a quadratic form of the generalized coordinates. Hence, T(x, • = 1/2 • M2(k) �9 zk. The generalized Lagrangean density becomes

1 1 L(x,k) = ~ k - M 2 ( x ) . ) t - 2 x . K ( x ) . x . (25)

Its partial derivatives are:

OL 0 Ox 0x [~:' g 2 ( x ) ] , k - .7- v (26)

OL 0~- = M2(x) . k , (27)

oxO2L- OOx 0x{k'Oxx0 [k 'M=(x)]} - /Cg (28)

O~L Ok Ok - M2 (x). (30)

O2L _ 2 [ , OxOk Ox " M2(x)] (29)


5 The m i x e d form for e l a s t o d y n a m i e s

Defining the kinetic momenta # at the element level in term of the generalized coordinates x as

OL OT - - M2(x)- • (31)

# = 0 • 0•

we are led to introducing the Hamiltonian density H(#, x) through the contact transformation H(#, x) = #- • - L(x, • which becomes:

1 H(~o,x) = ~ p . M2 -l(x) -@ + ~ x- K ( x ) . x . (32)

The partials of H are:

OH M2_l(x). O ' (33) 0#

1 0 OH _ 7 g + [e- M2-1 (x)] . e , (34) Ox

02H -- M2 -z (x), (35) oeoe

OH 0 0 e 0x - 0x [~o- M2-1 (x)], (36)

02H _ t c g + l 0 { ( a .M2_l(x)]) } 0x0x 2Oxx e ' ~xx[@ . (37)

In explicit form:

1 1 = T(u, a, h, g) + W~(u, a) = 5 g" Ao 11- 6 + ~ h . F -1 (a) I o - tF- r (a ) �9 h + W~(u, a) . (38)

The incremental kinetic energy is given by

03- Ag 07- 1 027- 1 O2T 7 - = T o + A h . ~ + �9 ~ - + 7 Ah �9 ~ - ~ . zlh + 7 A( �9 0~0-~. z~g

OT 1 02 T 1 027-

Oa Oh

07- 027- 02"-27 027- 02 T 02 T 027- (since Ou -- Ou Ou -- Ou Oa -- 06 Oh - Ou Oh - Ou 06 - Oa Og - 0),

where:

02T 07- _ g. Ao- l l ~ = A~-11 06 ~ '

027- = F -~ (a) Io-~ F-T (a) --=OToh h . F l(or)Io-lF-T(a) ~ OhOh

Consequently, the remaining partial derivatives can be obtained as

0 7 - _ 1 h . 0 [F_l(a) io 1F r(a) .h] Oa 2 Oa

1 ------ h . {Lmzfa , Is hi +F-~(a ) IQ ILc-T[a,h]}

2

(39)

(40)

(41)

(42)


02T _ 1 {Lr_ 1 [a, I e - IF -T(a ) . h] + i ,-1 (a) I o- lL~ ~ [a, hi} 7 , (43) Oa Oh 2

02T 1 02 Oa Oa - 2 0 a Oa (h- F I I~- IF-T - h ) ,

1 ~ h - d 6 / " - 1 . ( I e - l F - r . h) 4- 2 (h- F - l i e - l ) �9 d6F - r . h , (44)

where L F 1 and L/.-r are the tangent maps associated with F -1 and F - r . These tangent maps are linear operators which are described in [1].

The incremental strain energy and its partial derivatives with respect to the global coordinates are:

1 W~(u, a) = Wo + A a . . T J + A u . I ' J + ~ A u . K ~ . A u

1 IC~- ~ a + ~ ~ a . ~C L - ~ a , (45) A u .

ow~ 0u - ~ -J ' (46)

Ow~ - f j , (47)

Oa

02W~ - ~ , (48)

0u 0u ]C~

02W~ c92Ws (49) Ou Oa - t C ~ - Oa Ou '

02w~ - ~c~ (50) Oa 0o~ ~a �9

The mixed form, based on Eq. (15), may be written as:

t/+l f { e ' a - x . # - 6H + Q . 6x} dt = [~b. 6X -- X v- @]tti+'. (51) t{

Introducing the vector v = (p, x), we may linearize Eq. (52) about the state v0 as follows:

ti+l f {50. T . vo + 50. T . Av + 5, . Fo + 5Vo" A v } d t = [6v. T . vb]~: § , (52) ti

where, if 16~6 represents the 6x6 identity matrix:

r = ( 016x 6 -16x6 ) 0 ' (53)

Fo = ~ , (54) OH

- SZx + Q

Ko oeoe oeox = = K~ + K2 + K3, (55)

oQ O~ H OQ c~ H + O~x 0 ~ 0 x + 0~ 0 x 0 x


and where (o o) = O @ , H1 OQ +-~x +N

(56)

( -M2-T(x) 0 ) (57) K2 = 0 - K ~ '

and Ka contains the remaining terms. K1 may be set to zero for constant forces or non- follower loads, but it would be different from zero also in the case of actively controlled structural members. To compute the inertia contribution to OH/cg~o 0x and to 02H/Ox 0x, entering in Ka, we must remember the tangent maps of rotation. In particular, the only partition of interest of the inertia matrix is M ~ = FTIeF. Therefore,

O O [h. M 2 ~ ( x ) ] - h = h - 0 [ r _ l . ( i o . H ) ] (58) 0x [0' M~ -1 (x)] - p ~ Oaa Oa

and, denoting by D(h; (-)) the directional derivative of (.) in the prescribed direction (i.e., h):

0{ (0 0-a h- ~ [ h . M2(x)] ~D[h;a, (L'H)]. (59)

We may now introduce a suitable interpolation scheme so that, in terms of the nodal variables /~, the trial functions may be interpolated as v = M �9 tt a n d / , = 1VI./t, and the test functions may be interpolated as 6v = N .p and 6i, = N �9 #. Then Eq. (62) becomes

4~i.1 ti+l

51~. f {N-T.M-t~o+N.r.M.A~}dt+6t~. f {N.Fo+N.Ko.M.AI~}d~ ti ti

= [#,. N . r . M . t, bl~i § (60)

which, for arbitrary variations 5p, reduces to the system of equations:

/ t f ~ { i ' ~ l . T . M + N . K 0 . M } d t ) - A l ~ ti

ti + l

--- IN . T . M . l~b]ttl +' -- f {1N- T - M ./4 o § N - Fo} dr. (62) ti

This system of equations may be solved in an incremental fashion, similar to an updated Lagrangean incrementation procedure, since the end condition at the end of a time step becomes the initial condition of the next time step. See [5] for details on mixed space-time formulations for rigid body dynamics, particularly on the choice of trial and test functions appearing in the incremental functional.

6 Dual algebraic kinematic constraints

In this Section, we explore the extension of the variational methods outlined above in the case in which a material continuum is subjected to a set of algebraic kinematic constraints. We will describe the kinetics of the material continuum by the vector of generalized coordinates q, describing the vector of displacements and, possibly, of finite rotations of the point, by the vector of generalized velocities ~t, and by its energetically conjugate quantity, the vector of kinetic momenta p.


A cont inuum may be subjected to two kinds of structural constraints. Both kinds may be

written as a set of vector equations depending on the generalized coordinates or their derivati-

ves. One kind, material constraints, impose certain restrictions on the constitutive material

law. An example of this is the l imitat ion imposed when, from general three-dimensional elas-

ticity theory, we reduce a problem, for example, to the case of plane strain. In this case, some

o f the (material or spatial) gradients of the independent fields, for instance, some o f the com-

ponents o f strain, are set to zero. These types of constraints are generally expressed in integral

form, i.e., they hold for the whole structural element, We will not consider this type of con-

straint. Instead, we will deal with kinematic (or motion) constraints. These type of constraints

may be written down as sets of algebraic vector equations establishing a relat ionship between

the independent fields q (generalized coordinates) and their time derivatives. Differently than

mater ial constraints, they generally hold pointwise instead of for the whole strncture~ They

may be classified in holonomic, non-holonomic, rkeonomic, and scleronomic. We refer to the

classical l i terature for these definitions. In this report , we will deal with holonomic, possibly

rheonomic constraints. They can be written as g(t , q) = 0 or, after time differentiation:

~5(q, dt) = Og(t, q ) . dl -f Og(t, q) _ .A(t, q ) . (t + a( t , q) = 0 , (63) 0q 0 t

where A(t, q) now denotes the constraint Jacobian. Define now the modified momenta

15 = p 4- ÀT "#, (64)

where the # are independent multipliers, left unspecified for the time being. Then:

,~ = M -1 . (15 - A r �9 p ) . (65)

Substi tut ing this equat ion in the kinetic energy expression, we obta in

1 1 M _ 1 M _ 1 1 T(q, ~1) = ~ 4" M . ~1 = ~ 15" - 15 - 15. �9 ̀A. # 4- ~ p . M -1 , ,AT# = Z(q , 15). (66)

On the other hand, using Eq. (66) in Eq. (64) leads to

# = (ÀM-1AT) 1. [A- M -1 . 15 4- a] = #(% 15). (67)

Therefore, # is really a dependent field. I t can be rewrit ten as # = R + - 15 + D - a, where

R+ = D . ` A . M i

D = (.AM-I.AT) -1 = D T .

(68)

(6~)

Note that R + is the weighted Moore-Penrose inverse (o1" pseudo-inverse) of ,A with weight

equal to M [14].

After some manipulat ion, we may rewrite the kinetic energy as

1 1 T(q, 15 ) = ~ 15. ( M 1.79). 15 + 2 a . D . a . (70)

In the case of fixed constraints, i.e., in the case in which a = 0, the metric in the space of the

modif ied momenta is M -1 - 79. I t can be easily shown that 7 ) is defined by 79 = 1 - .A T �9 R +

and it is a project ion operator . In fact:

792 = 7) . 79 : (1 - .AT. R +) + .AT. R + " .AT. R + _ ,AT. R +

= (1 - M r - R +) + ,A T. D . . A . M -1 - .AT. R + = (1 - .AT. R +) + .AT. R + _ .At. R + = 79.

206

The L a g r a n g e a n dens i ty now becomes

1 1 Z:(q, 15) = T (q , 15) - W ( q ) = ~ 15. S . 15 + ~ a . D . a + W ( q ) ,

where now S = M -1 - 7 ) = S T and

dt = M - ~ . (1 - , A T. a + ) �9 1 5 - ( n + ) . a .

A mod i f i ed L a g r a n g e a n dens i ty can then be def ined as:

M. B. Quadrelli and S. N. Atluri

(71)

(72)

s 15) = s + ~ . # (73)

and the modi f ied H a m i l t o n i a n dens i ty becomes 7~(q, 15) = 15. dl - Z~(q, 15), which us ing the

ident i t ies found above , becomes

1 1 7-~(q, 15) = ~ t 5- S - 15 + ~ a . D . a + W ( q ) - a - / z , (74)

where use has been m a d e of the ident i t ies R + �9 A T �9 R + = R + and R + �9 A T �9 D = D. Cons ide r

n o w the cons t r a in t equa t ion itself. We m a y wri te tha t

4~(q, ~1) = A . 6i + a = 0 = A . [M - 1 - (1 - ,A T. R + ) �9 15 - (R+) T- a] + a

= A . S . 1 5 - . 4 - ( R + ) T . a + a = . 4 . S . ] 5 + ( 1 - A . ( R + ) T . a . (75)

But

A . (R+) T = A . ( D - A - M - l ) T = .A . M - 1 . ,A T . D = D T . D = 1 (76)

hence,

~* (q, 15) = ,4 . S . 15 = 0 . (77)

T h a t this equa t ion is cor rec t m a y be p roven by back-subs t i tu t ion . In fact, using

15 = 79 1. p + 7)-1. ÀT. D . a, we can show tha t

~*(q,15) = À. S . 15 = À. M - ~ �9 79 �9 [79 ~ �9 M . d l + 79-1. AT �9 D . a]

= A . M - 1 . 7 9 . 7 9 - 1 . M . i t + A . M - 1 . 7 9 . 7 9 t . ` A T . D . a

= , A ~t + ( A M - I ` A r ) ' ( ÀM-~,Ar) - 1 a = ,A. ~t + a = 0

as we wan ted to show. Therefore , we have o b t a i n e d a way to rewri te the cons t r a in t equa t ion

in terms of the ad jo in t var iables , i.e., the m o m e n t a , ins tead of the dependen t genera l ized coor-

dinates . W e m a y therefore call r (% 15) the dual constraint equation.

A t this poin t , we mus t po in t ou t the fo l lowing remark . Cons ide r the mod i f i ed H a m i l t o n i a n

with a = 0, for the sake o f s implici ty. Then 7~(q, 15) = 1/215. S . 15 + W ( q ) , wi th

S : {1 - M 1 A T ( A M - 1 A T ) - I , A } M -1 : )/1r 1 : S T " (78)

The m a t r i x W can also be easily p roven to be a p ro jec t ion matr ix . In the non l inea r p r o g r a m m -

ing l i tera ture , it is k n o w n as the oblique projection operator, which one encounte rs in the gra-

dient projection method [15]. The descent d i rec t ion can also be easily shown to be i = - S �9 15.

I f we now cons ider the mechan ica l system under cons ide ra t ion to have n genera l ized coord i -

nates and m cons t ra in ts , then q is (n x 1), M is (n • n ) , ,A is (m • n) , and 15 is (n x 1). Con-

sequent ly , S is (n • n). Now, assume a m o t i o n of the system, which a lways occurs in a con-

f igura t ion space of d imens ion n, and in which the n u m b e r of cons t ra in ts , m, is subject to

change. This p h e n o m e n o n m a y occur dur ing the numer ica l so lu t ion of a con tac t p r o b l e m


such as impact, collision, and in a general situation of changing topology. Therefore, the number of degrees of freedom of the mechanical system, say d = n - m, is subject to change because m changes. However, if ra changes from ral to ra2 (note that ral may be greater, or smaller than m2, provided A never loses rank), S always remains of the same dimension, hence we may conclude that a change in topology does not reduce the dimensionality of the matrix S, This fact may be very useful during mamerical simulation of discontinuous motions of multifle- xible body systems. In [16], a perturbation approach is outlined which is able to treat discontinuous derivatives which occur during phenomena such as impact and collisions of mechanical systems. The technique presented in [16] allows to compute a highly accurate solution without having to interrupt the numerical integration and without having to solve algebraic equations for the velocity jumps. When one uses the Newton-Euler equations, it is always necessary to stop the simulation in order to update, by imposing a momentum balance, the generalized coordinates and their derivatives from immediately before, to immediately after the time instant when the contact occurs. With the new method, which makes use of the concept of the dual constraint, a result similar to the one obtained by [16] can also be achieved. The advantage is the generality of the formulation which can handle general elastodynamic problems. Provi- ded an appropriate nodal collision detection [17] and a constraint update algorithm can be introduced in the computation, the whole sequence of motion may be followed without inter- rupting the simulation, if the independent fields described above (i.e., generalized coordinates and modified momenta) can be incorporated in an implicit integrator, based on Hamilton's weak principle. For instance, introducing Eq. (74) into Eq. (62), and following similar steps to those that lead to Eq. (70), one obtains the discretized mixed formulation associated with the fields r q, and 15. The true momenta p may be recovered a posteriori, through Eq. (65).

7 Conclusions

Linear and nonlinear constrained elastodynamics problems are approached from a mixed variational principle point of view. It is shown that four different functionals lead to four different mixed variational formulations of the problem with different independent fields. This is obtained in terms of the Lagrangean and of the Hamiltonian density, in which the strain energy density and the complementary energy density incorporate the geometric or material nonlinearity. A variational principle is also formulated for the case of linear elastodynamics, but in which all the correct boundary conditions at the end of the time interval are accounted for. One of the functionals is specialized, using finite elements in space and time, to the case of spatial beams undergoing large deformations, and small strains. The natural extension of this work is to incorporate kinematic constraints in weak form into the variational formulation of the problem, generalizing to three-dimensional nonlinear elastodynamics the work presented in [6]. The advantage of this methodology is the potential ability to handle discontinuous phenomena occurring in flexible multibody contact problems, such as impact, collision, and changing topology, in a variational setting.

Acknowledgements

The first author is deeply grateful to Prof. H. Lipkin of the Georgia Institute of Technology for many useful discussions.

208 M.B. Quadrelli and S. N. Atluri: Mixed variational principles

References

[1] Quadrelli, M. B., Atluri, S. N.: Primal and mixed variational principles for dynamics of spatial beams. AIAA J. 34, 2395-2405 (1996).

[2] Cardona, A., Geradin, M.: A beam finite element nonlinear theory with finite rotations. Int. J. Numer. Meth. Eng. 26, 2403-2438 (1988).

[3] Iura, M., Atluri, S. N.: Dynamic analysis of finitely stretched and rotated three-dimensional space- curved beams. Comp. Struct. 29, 875-889 (1988).

[4] Iura, M., Atluri, S. N.: On a consistent theory, and variational formulation, of finitely stretched and rotated 3-D space-curved beams. Comput. Mech. 4, 73-88 (1989).

[5] Borri, M., Mello, F., Atluri, S. N.: Variational approaches for dynamics and time-finite-elements: numerical studies. Comput. Mech. 7, 49-76 (1990).

[6] Borri, M., Mello, F., Atluri, S. N.: Primal and mixed forms of Hamilton's principle for constrained rigid body systems: numerical studies. Comput. Mech. 7, 205 217 (1991).

[7] Quadretli, M. B., Atluri, S. N.: Prima1 and mixed variational principles for dynamics of spatial beams. Int. J. Numer. Meth. Eng. 42, 1071 - I090 (1998).

[8] Atluri, S. N., Cazzani, A.: Rotations in computational solid mechanics. Arch. Comput. Meth. Eng. 2, 49-138 (1995).

[9] Atluri, S. N.: Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitely deformed solids, with application to plates and shells - I: Theory. Comp. Struct. 18, 98-116 (1984).

[10] Borri, M., Bottasso, C., Mantegazza, P.: Basic features of the time finite element approach for dynamics. Meccanica 27, 119-130 (1992).

[11] Hlavacek, I.: Some variational principles for nonlinear elastodynamics. Appl. Mat. 12, 107 117 (1967).

[12] Atluri, S. N.: An assumed stress hybrid finite element model for linear elastodynamic analysis. AIAA J. 11, 1028-1030 (1973).

[13] Chen, G.: Unconstrained variational statements for initial and boundary value problems via the principle of total virtual action. Int. J. Eng. Sc. 28, 875-887 (1990).

[14] Lipkin, H.: Invariant properties of the pseudoinverse in robotics. Proceedings of the 16th Conference on Production Research and Technology, Arizona State University, Tempe, Jan. 8-12, 1990.

[15] Luenberger, D.: Linear and nonlinear programming, 2nd ed. Reading: Addison-Wesley 1984. [16] Ostermeyer, G. P.: Numerical integration of systems with unilateral constraints. Computer Aided

Analysis and Optimization of Mechanical System Dynamics, NATO ASI Series, F9, pp. 415-418. Berlin: Springer 1984.

[17] Belytschko, T., Neal, M. O.: Contact-impact by the pinball algorithm with penalty and Lagrangian methods. Int. J. Numer. Meth. Eng. 31,547-572 (1991).

Authors' address: Dr. M. B. Quadrelli, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, U,S.A.; Prof. S. N. Atluri, Center for Aerospace Research and Education, 7704 Boelter Hall, University of California, Los Angeles, CA-90095-1600, U.S.A.

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