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 advantageously 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 investigated 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 threedimensional 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 = 5 R . 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 y a 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, routinely used during the procedure of consistent linearization, may be derived in a straightforward 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
I'(a)
=
1 +
x (a • 1)],
(i)
al(a X 1) @ a2[a x (6g x 1)],
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, incre-
where
ment 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 Lagrangean 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
(3)
S~O
where b = f , f being applied body forces per unit m a s s ; / a r e prescribed tractions on S~0 and
the prescribed displacements on 8u0, t = t Y A ~ | az: the 1st Piola-Kirchhoff stress tensor [4].
Here, the symbol ":" represents the double contraction of two 2nd order tensors, i.e., A : B
= AiNBij = 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 5 R . R T = (SR-RT)a, symmetric 8U and unsymmetric 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 9t + 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 Traction 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:
+ wjr)
= I(t.R+R
r.tr):U,
(4)
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 HellingerReissner type three-field functional is obtained:
r3(v,R,t)=f{-
W~,~ [ ~1 ( t . R + R
T . tT)]J + tT : ( I -~- ~Tv)- QOb . 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:
cc:
( aw ']
a. \-aT/s
: (I + v,,).
196
M.B. Quadrelli and S. N. Atluri
\ Or / -t
=0,
s
LBM:
a
V0-t+~)0b=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 W o r k (1VW) can be written as [1]
I V W = f [T. (5~ + M . ~~ I d Y 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 vector ~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) d Y 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~(~ + C T d ,
h = C~ + rT(a) I0r(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 definition 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~
L
dY 3 ,
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 W o r k (EVW) may also be written as
EVW
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 symmetry of the resulting matrices, distributed moments which may result in nonconservative loading 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)
Mixed variational principles
197
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 ,
ei = F - l ( a ) Io 1F-r(a) h, we may write that:
1
= g. u + h . d - T = ~ gTAQ-ilg +
and their inversions f i = A 0 - 1 1 - g ,
h T F - i ( a ) I 0 - i r - T ( a ) h.
and
(9)
We conclude that the contact transformation leads to the (configuration-dependent) Hamiltonian density:
~(g, h, u, a) = 1 pT..Ad-l(q) . p
(10)
and to the dual elastodynamic functional:
Ai=
f
f{g.fl+h.d-~(g,h,u,a)}dVdt-
f
F 3 ( h R , t) d t + B . T . ,
(11)
[t~&+lJ Vo
where B.T. represent the natural boundary conditions. This form may be called a mixedmixed 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 5field 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{fSTdY
t~
3 - (5F1} dt = [eb " ~u + Hb 1't~+1"
~"~
SJ It i
"
(12)
L
Then, from Eq. (12), using Eq. (4), we obtain the primal-dual principle:
f { f67dY a - 6Fa}dt =
t~
L
[gb " (Su § Hb " ~6JteTlti+llti"
(]3)
198
M.B. Quadrelli and S. N. Atluri
From Eq. (13), using Eq. (8), by integrating by parts with respect to time, we obtain the dualdual principle:
f { f [ p " 6Cl - q" 615 - 67-l] dY 3 - 6Fa } d t : [pb. 6q - qb. 6p]l~t:+t
(14)
ti
where p is the vector of momenta, and q the vector of generalized coordinates. Similarly, we
obtain the dual-primal principle:
f {f[p'6~t-q'6f~-O~]dYa-6F1}dt=[pb'6q-qb'6p]~{
ti+l
.
(15)
t{
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 redundancy 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:
= T
1
VV, = ) ~ o v . v -
Vr
Now, introduce the action functional $ = f s d4t, where
(16)
d4t =
dw stands for the space-time
differential volume element. However, in taking the variation of the action $, which is representative 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
Mixed variational principles
199
the slice [ti, ti+l] must be added as well. Consider, then, the functional, called the modified
action."
ti+~
ti
Vo
$oO
12o
S~o
In the first volume integral, the first two terms represent the Lagrangean, and the second term
is the compatibility 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 m o m e n t a at the time boundary. Define now the
true trajectory followed by the body in motion 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:/((~
z27
-OWs
--~r..[ ~
--Ci,j]-4-O'ij(~Ci,j--
fi(~i) dud7
ti+l
+l(i+..,..+j,.,,.,-.,,..+l.,.,..).,
ti
$~
Suo
S"o
(~8)
1;0
where (')Ii,j] represents the skew-symmetric part 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 M o m e n t u m 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 ~ . , / ~
v0
v0
v0
= - f (~j~j) ~ dS + f ~ , / ~
0s
= - f T~
8~o
l;o
dS - f m ~
S~o
d~
dX? = - f m ~
OQ
dX + f ~ . , / ~ dX?
Po
dS + f ~ 5 , / ~ d~?
Vo
09)
and, by integrating by parts in time, we also have:
~i+l
f S T dw = f
v~
1ro x. ti
QoiziSisidt
d[2 = f
WO
[cOouiSui]~i*~ - f poiiiSiti dt
ti
/
d[2.
(20)
200
M.B. Quadrelli and S. N. Atluri
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,/&#
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 =
VIC:
z~lt ' =
MIC:
7ri[t~ = Q0g~lt~; (on 1;0, at t = ti):
MFC:
7r~[t~+~= 80t~ilt~+~,
=
c~j,
vlt~; (on W0, at t
ti):
= t0:
where TBC stands for Traction Boundary Condition, D B C for Displacement Boundary Condition, CLS for Constitutive Law in Space, CCS for Compatibility 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 M o m e n t u m Initial Condition, and M F C for M o m e n t u m 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 transformation, we m a y derive an equivalent Hellinger-Reissner variational principle in which L M B and
CCS are met a priori, and therefore obtain the complementary version of 791. Similarly, we
may start from 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 subscript (e) denotes a quantity related to a composition 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
Mixed variational principles
201
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~
)
5i7,
(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
and the local tangent matrix
by
Odb, w e may obtain the elements of the global tangent matrix as
OW~/Od~
02W~/Od~
<b (~176
=
\O~ J" Od~Odb
\-~Odb
bxb() + i~=t~i[fO( ~Odi
:
(ow )
(2a)
rgf
where
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,~) ={ :~-M2(x).k+Ml(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) 9zk. The generalized Lagrangean density becomes
L(x,k)
1
= ~
k-M2(x).)t-2
1
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,
oxO2LOOx0x{k'Oxx0 [ k ' M = ( x ) ] }
O2L _ 2 [ ,
OxOk
Ox
" M2(x)]
(27)
-/Cg
(28)
(29)
O~L
Ok Ok - M2 (x).
(30)
202
M.B. Quadrelli and S. N. Atluri
5 T h e m i x e d f o r m 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)- •
#=0•
0•
(31)
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
0#
M2_l(x). O '
(33)
OH _ 7 g + 1 0 [e- M2-1 (x)]. e ,
Ox
02H
oeoe
(34)
-- M2 -z (x),
(35)
OH
0
0e 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 - t F - r ( a ) 9h + W~(u, a). (38)
The incremental kinetic energy is given by
037-=To+Ah.~+
OT
1
Ag 07- 1
0279~ - + 7 Ah 9~ - ~ .
02T
1
1
O2T
zlh + 7 A( 90~0-~. z~g
027Oa Oh
(39)
0702702"-27
02702T
02T
027(since Ou -- Ou Ou -- Ou Oa -- 06 Oh - Ou Oh - Ou 06 - Oa Og - 0),
where:
07- _ g. A o - l l ~ 02T = A~-11
06
~
'
(40)
027- = F -~ (a) Io-~ F-T (a)
-=OToh h . F l(or)Io-lF-T(a) ~ OhOh
(41)
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
(42)
Mixed variational principles
203
02T _ 1 {Lr_ 1[a, I e - I F - T ( a ) . h] + i,-1 (a) I o-lL~ ~ [a, hi} 7 ,
Oa Oh 2
(43)
02T
1 02
Oa Oa - 2 0 a Oa (h- F II~-IF-T - h ) ,
1
~ h - d 6 / " - 1 . ( I e - l F - r . h) 4- 2 (h- F - l i e - l ) 9d6F - 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
A u . IC~- ~ a + ~ ~ a . ~CL - ~ a ,
(45)
ow~
0u - ~ - J '
(46)
Ow~
- f j,
Oa
(47)
02W~
~,
0 u 0u
]C~
(48)
02W~
(49)
c92Ws
Ou Oa - t C ~ - Oa Ou '
02w~ - ~c~
Oa 0o~
~a 9
(50)
The mixed form, based on Eq. (15), may be written as:
t/+l
f { e ' a - x . # - 6 H + Q . 6x} dt = [~b. 6X -- Xv- @]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]~:§ ,
ti
(52)
where, if 16~6 represents the 6x6 identity matrix:
r=( 016x6 -16x6
0 )'
(53)
Fo =
(54)
~
,
OH
- SZx + Q
Ko =
oeoe
O~H
oeox
OQ
0 ~ 0 x + 0~
= K~ + K2 + K 3 ,
oQ
c~ H + O~x
0x0x
(55)
204
M.B. Quadrelli and S. N. Atluri
(o o)
and where
H1 =
O@
OQ +-~x
+N
K2 =
(
-M2-T(x)
0
(56)
,
0 )
-K ~
(57)
'
and Ka contains the remaining terms. K1 may be set to zero for constant forces or nonfollower 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~o0x 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 [0' M~ -1 (x)] - p ~ Oaa
O [h. M 2 ~ ( x ) ] - h = h - 0
0x
and, denoting by
0{ (0
Oa [ r _ l . ( i o . H ) ]
(58)
D(h; (-)) the directional derivative of (.) in the prescribed direction (i.e., h):
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 9tt 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:
/tf~{i'~l.T.M+
N.K0.M}dt)-Al~
ti
ti + l
--- I N . 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 m o m e n t a p.
Mixed variational principles
205
A continuum m a y be subjected to two kinds o f structural constraints. Both kinds may be
written as a set o f vector equations depending on the generalized coordinates or their derivatives. One kind, material constraints, impose certain restrictions on the constitutive material
law. A n example o f this is the limitation imposed when, from general three-dimensional elasticity theory, we reduce a problem, for example, to the case o f plane strain. In this case, some
o f the (material or spatial) gradients o f the independent fields, for instance, some o f the components 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 constraint. Instead, we will deal with kinematic (or motion) constraints. These type of constraints
m a y be written down as sets o f algebraic vector equations establishing a relationship between
the independent fields q (generalized coordinates) and their time derivatives. Differently than
material constraints, they generally hold pointwise instead o f for the whole strncture~ They
m a y be classified in holonomic, non-holonomic, rkeonomic, and scleronomic. We refer to the
classical literature 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 ,
0q
0t
(63)
where A(t, q) now denotes the constraint Jacobian. Define now the modified m o m e n t a
15 = p 4- `AT "#,
(64)
where the # are independent multipliers, left unspecified for the time being. Then:
,~ = M -1 . (15 - A r 9p ) .
(65)
Substituting this equation in the kinetic energy expression, we obtain
T(q, ~1) = ~1 4" M . ~1 = ~1 15" M _ 1 - 15 - 15. M _ 1 9`A. # 4- ~1 p . M - 1 , ,AT# = Z ( q , 15).
(66)
On the other hand, using Eq. (66) in Eq. (64) leads to
# = (`AM-1AT) 1. [A- M - 1 . 15 4- a] = #(% 15).
(67)
Therefore, # is really a dependent field. It can be rewritten as # = R + - 15 + D - a, where
R+ = D.`A.M
i
D = (.AM-I.AT) -1 = D T .
(68)
(6~)
N o t e that R + is the weighted M o o r e - P e n r o s e inverse (o1" pseudo-inverse) of ,A with weight
equal to M [14].
After some manipulation, we m a y rewrite the kinetic energy as
1
1
T(q, 15) = ~ 15. ( M 1.79). 15 + 2 a . D . a .
(70)
In the case o f fixed constraints, i.e., in the case in which a = 0, the metric in the space of the
modified m o m e n t a is M -1 - 79. It can be easily shown that 7) is defined by 79 = 1 - .AT 9R +
and it is a projection operator. In fact:
792 = 7) . 79 : (1 - .AT. R +) + .AT. R + " .AT. R + _ ,AT. R +
= (1 - M r - R +) + ,AT. D . . A . M -1 - .AT. R + = (1 - .AT. R +) + .AT. R + _ .At. R + = 79.
M. B. Quadrelli and S. N. Atluri
206
The L a g r a n g e a n density now becomes
1
1
Z:(q, 15) = T ( q , 15) - W ( q ) = ~ 15. S . 15 + ~ a . D . a + W ( q ) ,
(71)
where n o w S = M -1 - 7) = S T and
(72)
dt = M - ~ . (1 - , A T . a + ) 9 1 5 - ( n + ) . a .
A m o d i f i e d L a g r a n g e a n density can then be defined as:
s
15) = s + ~ . #
(73)
and the modified H a m i l t o n i a n density becomes 7~(q, 15) = 15. dl - Z~(q, 15), which using the
identities f o u n d above, becomes
1
1
7-~(q, 15) = ~ t5- S - 15 + ~ a . D . a + W ( q ) - a - / z ,
(74)
where use has been m a d e of the identities R + 9A T 9 R + = R + and R + 9 A T 9 D = D. C o n s i d e r
n o w the c o n s t r a i n t e q u a t i o n itself. W e m a y write t h a t
4~(q, ~1) = A . 6i + a = 0 = A . [M - 1 - (1 - ,AT. R + ) 915 - ( R + ) T- a] + a
=A.S.15-.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 . ,AT . D = D T . D = 1
(76)
hence,
~* (q, 15) = ,4. S . 15 = 0 .
(77)
T h a t this e q u a t i o n is correct m a y be p r o v e n
15 = 79 1. p + 7)-1. `AT. D . a, we can show t h a t
by
back-substitution.
In
fact,
using
~*(q,15) = `A. S . 15 = `A. M - ~ 979 9 [79 ~ 9 M . d l + 79-1. AT 9 D . a]
t .`AT.D.a
=A.M-1.79.79-1.M.it+A.M-1.79.79
= , A ~t + ( A M - I ` A r ) ' (`AM-~,Ar) - 1 a = ,A. ~t + a = 0
as we w a n t e d to show. Therefore, we have o b t a i n e d a way to rewrite the c o n s t r a i n t e q u a t i o n
in terms of the adjoint variables, i.e., the m o m e n t a , instead of the d e p e n d e n t generalized coordinates. W e m a y therefore call r (% 15) the dual constraint equation.
A t this point, we must p o i n t out the following remark. C o n s i d e r the m o d i f i e d H a m i l t o n i a n
with a = 0, for the sake o f simplicity. T h e n 7~(q, 15) = 1/215. S . 15 + W ( q ) , with
S : {1 - M
1AT(AM-1AT)-I
, A } M -1 :
)/1r
1:
ST "
(78)
The m a t r i x W can also be easily p r o v e n to be a p r o j e c t i o n matrix. In the n o n l i n e a r p r o g r a m m ing literature, it is k n o w n as the oblique projection operator, which one encounters in the gradient projection method [15]. The descent direction can also be easily shown to be i = - S 9 15.
If we n o w consider the m e c h a n i c a l system u n d e r c o n s i d e r a t i o n to have n generalized coordinates and m constraints, then q is (n x 1), M is (n • n),,A is (m • n), a n d 15 is (n x 1). C o n sequently, S is (n • n). N o w , assume a m o t i o n of the system, which always occurs in a conf i g u r a t i o n space of d i m e n s i o n n, and in which the n u m b e r of constraints, m, is subject to
change. This p h e n o m e n o n m a y occur d u r i n g the numerical solution of a contact p r o b l e m
Mixed variational principles
207
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 multiflexible 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. Provided 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 interrupting 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
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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.