Computational Mechanics 22 (1999) 450±462 Ó Springer-Verlag 1999
Mathematical aspects of the general hybrid-mixed
finite element methods and singular-value principle
W. Xue, S. N. Atluri
450
Summary In this paper the general hybrid-mixed ®nite
element methods are investigated systematically in a
framework of multi-®eld variational equations. The commonly accepted concept ``saddle point problem'' is argued
in this paper. The existence, uniqueness, convergence, and
stability properties of the solutions are proved undertaking the assumptions of Ker*-ellipticity and nested BBconditions. The relation between discrete BB-condition
and smallest singular value, and a so-called singular value
principle are proposed for the practical applications using
hybrid-mixed ®nite element methods.
1
Introduction
F. Brezzi [1] considered the variational statement, which is
related to the mixed F.E.Ms. for plate bending problem
and Stokes ¯ow, of the following form:
a…u; v† ‡ b…v; q† ˆ hf ; vi 8 v 2 V;
1:1†
b…u; p† ˆ hg; pi
8p2P ;
where V and P are Hilbert spaces, a…; † and b…; † are
continuous bilinear forms, hf ; i and hg; i are linear
functionals determined by the given functions f and g
respectively.
L.-A. Ying and S.N. Atluri [2] presented a variational
statement, which is summarised from the hybrid-mixed
®nite element method for solving Stokes ¯ow, of the following type:
8
< a…r; s† ÿ b…s; q† ˆ hf ; si
b…r; v† ÿ c…v; q† ˆ hg; vi
:
c…u; p† ˆ hh; pi
8 s 2 T;
8 v 2 V,
8p2P ,
1:2†
where T, V, P are Hilbert spaces, a…; †; b…; † and c…; †
are continuous bilinear forms, hf ; i; hg; i and hh; i are
Communicated by S. N. Atluri, 20 September 1998
W. Xue
Department of Mathematics,
Hong Kong Baptist University, Hong Kong
S. N. Atluri
Center for Aerospace Research Education
7704 Boelter Hall, UCLA, Los Angeles, CA 90095-1600
Correspondence to: W. Xue
The work of the author was supported by Research Grants
Council of Hong Kong.
linear functionals determined by the given functions f , g
and h respectively.
For a given bilinear form b…; † : V P ! R, we de®ne
the associated linear operator B : V ! P and its dual
operator B : P ! V , where the P and V are dual spaces
of P and V respectively, by
hBv; pi ˆ hv; B pi ˆ b…v; p† 8 …v; p† 2 V P :
1:3†
Then the variational Eqs. (1.1) and (1.2) may be rewritten
as
Au ‡ B q ˆ f
Bu ˆ g
in V ;
in P ;
1:1†
in T ;
in V ;
in P :
1:2†
and
8
< Ar ÿ B u ˆ f
Br ÿ C q ˆ g
:
Cu ˆ h
Undertaking the general hybrid-mixed ®nite element
methods, W.M. Xue, L.A. Karlovitz and S.N. Atluri [3, 4]
considered a general form of variational statement which
includes more independent unknown functional variables
and arises from various hybrid-mixed F.E.Ms. based on
the modi®ed Hellinger-Reissner principle. The more general statement may be summarised as the following multi®eld variational equation [5, 6]: ®nd …u1 ; u2 ; . . . ; un † in
Hilbert space V1 V2 Vn , such that
2
A
6B
6 1
6
6
6
6
6
6
6
4
32
ÿB1
0
ÿB2
B2
0
ÿB3
Bnÿ2
0
2
3
f1
6 f 7
6 2 7
6 7
6 f3 7
6 7
ˆ6 7
67
6 7
6 7
45
fn
u1
3
76 u 7
76 2 7
76 7
76 u3 7
76 7
76 7
76 7
76 7
76 7
ÿBnÿ1 54 5
Bnÿ1
2 V 3
0
un
1
6V 7
6 27
6 7
6 V3 7
6 7
in 6 7 :
67
6 7
6 7
45
Vn
1:4†
Variational Eqs. …1:1† and …1:2† are special cases of (1.4)
for n ˆ 2 and n ˆ 3 respectively. More examples of variational Eq. (1.4) for hybrid-mixed FEMs in linear elasticity
are shown in Table B.1 of Appendix B. In this paper the
mathematical theory of the general hybrid-mixed ®nite
element methods will be studied systematically in the
above framework of multi-®eld variational equation. In
Section 2, it is proved that the variational Eq. (1.4) is
equivalent to a minimization problem with simultaneous
equality constrains, which was commonly stated as inequality relations in literatures. Thus the commonly
adopted statement ``saddle point problem'' is argued. The
existence, uniqueness, convergence, and stability properties of the solutions of (1.4) are discussed in Section 3. In
Section 4, the simpli®ed hybrid-mixed ®nite element
methods are shown to be non-standard in viewpoint of the
framework (1.4), and the mathematical foundation of the
methods should be studied individually. Moreover the
simpli®ed hybrid-displacement model [14] is shown to be
unstable. The relations between discrete BB-condition,
rank condition, and smallest singular value are investigated in Section 5. Based on the theoretical results in
Section 5, it is preferable to construct a hybrid-mixed ®nite
element model such that the smallest singular value of the
cross-energy matrix is as large as possible. This suggestion
is called singular value principle, and is applied to a typical
example in Section 6.
2
Equivalence of variational equations
and minimization problems
We use Vi to denote ith Hilbert space provided its element
vi 2 Vi and the norm k:kVi : Let a…; † and bi …; † be
continuous bilinear forms de®ned on product space
V1 V1 and Vi Vi‡1 …i ˆ 1; 2; . . . ; n ÿ 1† respectively.
For a given bilinear form bi …; † : Vi Vi‡1 ! R,
we de®ned the following associated linear operator
Bi : Vi ! Vi‡1
and its dual operator Bi : Vi‡1 ! Vi by
hBi vi ; vi‡1 i ˆ hvi ; Bi vi‡1 i ˆ bi …vi ; vi‡1 †
8 …vi ; vi‡1 † 2 Vi Vi‡1 :
2:1†
1
L…u† ˆ a…u1 ; u1 † ÿ hf1 ; u1 i
2
nÿ1
X
ÿ1†i ‰bi …ui ; ui‡1 † ÿ hfi‡1 ; ui‡1 iŠ :
‡
2:3†
iˆ1
Proof. (see Appendix A)
In [3, 4], also see Tables B.1 and B.2 in Appendix B, quite
enough examples were carried out, based on the special
cases of Theorem 2.1, which is related to the potential
energy principle, complementary energy principle, Hellinger-Reissner principle, Hu-Washizu principle, as well as
various modi®ed variational principles for general hybridmixed F.E.Ms. of ®rst version and second version de®ned
in [7]. Therefore the Theorem 2.1 may be also considered
as the general mathematical modelling of variational
principles in computational mechanics. In the following,
we just take the example of Hu-Washizu principle as the
application of Theorem 2.1.
The governing equations and boundary conditions of
linear elasticity are well known as follows: ®nd …r; ; u† 2
T E W, such that
aijkl kl ÿ rij ˆ 0 in X ;
ij ÿ u…i;j† ˆ 0 in X;
i
at Su ;
ui ˆ u
rij;j ‡ fi ˆ 0
rij nj ti ˆ ti
in X;
at St :
2.4a†
2.4b†
2.4c†
If the material of elastic body is isotropic, the coef®cients
of elasticity aijkl have the following properties of symmetry
and ellipticity [8]:
aijkl ˆ aijlk ˆ aklij ;
aijkl kl ij aij ij
8 ij :
2:5†
By taken the concept of weak solution of P.D.E., the weak
form of (2.4) will lead to the following three-®eld variational equation:
8
8 e 2 E;
< a…; e† ÿ l…e; r† ˆ 0
The following theorem discusses the equivalence of multil…; s† ÿ p…s; u† ˆ hg; si 8 s 2 T;
2:6†
®eld variational Eq. (1.4) and the minimization statement : p…r; w† ˆ hf ; wi
8w2W ;
(2.2) with simultaneous constrains. The later, in fact, is
where the bilinear forms and linear functionals are de®ned
known as the modi®ed variational principle which is
by
considered as the original motivation of hybrid-mixed
Z
F.E.Ms. in engineering.
a…; e† ˆ
aijkl kl eij dX 8 ; e 2 E;
ZX
Theorem 2.1 Suppose that the bilinear form a…; † is
symmetric and positive de®nite. Then the variational
ij sij dX 8 2 E; s 2 T;
l…; s† ˆ
Eq. (1.4) is equivalent to the following constrained miniX
Z
Z
mization problem: ®nd u ˆ …u1 ; u2 ; . . . ; un † in Hilbert
sij w…i;j† dX ÿ
sij nj wi dS 8s 2 T; w 2 W;
p…s; w† ˆ
space V1 V2 ; . . . ; Vn , such that
X
Su
Z
8
< L…u† L…v1 ; u2 ; . . . ; un † 8 v1 2 V1 ;
sij nj ai dS 8 s 2 T;
hg; si ˆ
Su
L…u† ˆ L…u1 ; . . . ; uiÿ1 ; vi ; 8 vi 2 Vi ;
Z
Z
:
i ˆ 1; 2; 3; . . . ; n ; …2:2† hf ; wi ˆ
ui‡1 ; . . . ; un †
ti wi dS 8 w 2 W :
fi wi dX ‡
where the functional L…† is given by the formula
X
St
451
(i) The multi-®eld variational problem (1.4) has a unique
The properties in (2.5) guarantee the symmetric and
solution u ˆ …u1 ; u2 ; . . . ; un † in product space
positive de®nite properties required in Theorem 2.1. Thus
V ˆ V1 V2 Vn ,
the functional (2.3) in equivalent minimization problem
(ii) There exists a constant C > 0, such that
(2.2) with n ˆ 3 becomes that
L…; r; u†
ˆ 12a…; † ÿ ‰l…; r† ÿ hg; riŠ ‡ ‰p…r; u† ÿ hf ; uiŠ ;
i.e.
452
Z Su
St
This is exactly the same as that employed in Hu-Washizu
principle.
A commonly adopted argument in literatures is that the
multi-®eld variational principle for hybrid-mixed F.E.M. is
a saddle point problem [1, 9, 10, 2, 3]. But the authors
would emphasise that the statement (2.2) is indeed a
minimization problem (w.r.t. u1 ) with simultaneous
equality constrains (w.r.t. u2 ; u3 ; . . . ; un ). Due to the mechanical consideration, the equilibrium state of the elastic
body, which is described by equilibrium equation
rij;j …† ‡ fi ˆ 0, reaches the minimum of strain energy,
while the other relations in (2.4), i.e. constitutive law
rij ˆ aijkl kl and compatibility equation ij ˆ u…i;j† , are
equality constrains. The statement in this paper coincides
the mechanical consideration. The argument ``saddle point
problem'' is probably a misleading to the natural mechanical motivation.
3
Existence, uniqueness, and stability conditions
The following theorem discusses the existence, uniqueness
and stability conditions of general variational Eq. (1.4),
which may be considered as an extension of the theory
developed by Brezzi, Babuska, Ying and Atluri, etc.
[1, 11, 2].
Theorem 3.1 (existence, uniqueness, and stability) Suppose that the following Ker -ellipticity condition
8 v1 2 Ker …B1 † ;
3:1†
3:4†
where f ˆ …f1 ; f2 ; . . . ; fn † and the norms are de®ned by
1
L…; r; u† ˆ
aijkl kl ij ‡ rij …u…i;j† ÿ ij † ÿ fi ui dX
X 2
Z
Z
ti ui dS : …2:7†
rij nj
ui ÿ ui † dX ÿ
‡
a…v1 ; v1 † akv1 k2v1
kuk Ckf k ;
kuk ˆ
n
X
iˆ1
kui kVi ;
kf k ˆ
n
X
iˆ1
kfi kV :
i
3:5†
Therefore the mapping c : f ! u is an isomorphism from
V ˆ V1 V2 Vn onto V ˆ V1 V2 Vn .
Proof (see Appendix A)
As a well known result [8], the ellipticity condition (3.1)
3
is provided in …H01 …X†† . We must notice that the Ker ellipticity condition (3.1) is proposed in a subset
3
Ker …B1 † V1 …H 1 …X†† , in which the boundary condition may not be involved. Thus the following theorem,
which is an extension of the result in [8], is essentially
important.
Theorem 3.2 (ellipticity condition) Suppose that X is a
regular bounded domain in R3 , space V is a weakly closed
3
subspace in …H 1 …X†† , and there is no rigid body motion.
Then there exists a positive number a such that the following ellipticity condition is satis®ed:
Z
X
aijkl kl …v†ij …v† dX akvkH 1 …X† 8v 2 V :
3:6†
Proof (see Appendix A)
The elliptic conditions and nested BB-conditions for
various hybrid-mixed FEMs in linear elasticity are explicitly expressed in Table B.3 of Appendix B. Among these
conditions in (3.1) and (3.2), the ellipticity condition (3.1)
is provided in Theorem 3.2, and therefore the positive
de®nite property of a…; † required in Theorem 2.1 is also
satis®ed. The BB-conditions in (3.2) were extensively discussed in literatures [1, 11, 9, 2, 3].
4
On the simplified hybrid-mixed method
bk …vk ; vk‡1 †
One may see from Table B.1 of Appendix B that there are
sup
bkvk‡1 kVk‡1
too many independent variables in some hybrid-mixed
kvk kVk
8vk 2Vk
vk 6ˆ0
models. It has been frequently suggested by the engineers
8 vk‡1 2 Ker …Bk‡1 †; k ˆ 1; 2; . . . ; n ÿ 1
3:2† to combine some independent variables. Actually the ®rst
hybrid ®nite element method, i.e. the assumed-stress hybrid method introduced by Pian [17], can be explained as
are satis®ed. The notation Ker …† is de®ned by
the simpli®cation of the second version of modi®ed
complementary energy principle. We have already put this
Ker …Bk †
result in the Table B.1 of Appendix B. But most of the
ˆ fvk 2 Vk ; bk …vk ; vk‡1 † ˆ 0 8vk‡1 2 Ker …Bk‡1 †g;
simpli®cations of the hybrid-mixed methods summarised
k ˆ 1; 2; . . . ; n ÿ 1 ;
3:3† in the general variational Eq. (1.4) may not be expressed in
the standard form of (1.4). Then the mathematical founand Ker …Bn † is de®ned by Ker …Bn † ˆ Vn . Then we have dation of these methods should be studied individually. In
this section we discuss only two sample problems. One of
the following results
and the nested BB-conditions
these examples, the simpli®ed hybrid-displacement model
is shown to be unstable.
As the ®rst example let us consider the modi®ed form
(2nd version) of the Hu-Washizu principle. The functional
employed, as shown in Table B.2 of Appendix B, has ®ve
independent variables:
MHW2…; r; u; Tq ; u~q †
i
X Z h1
ÿ
aijkl kl ij ‡ rij u…i;j† ÿ ij ÿ fi ui dX
ˆ
Xm 2
m
Z
Z
rij nj
ui ÿ ui † dS
ti ui dS ‡
ÿ
Z
‡
Stm
qm
S um
Tiq …~
uiq ÿ ui † dS ;
4:1†
where is the strain ®eld, r the stress ®eld, u the displacement ®eld, Tq the traction de®ned on the inter-eleÿ
~q the
ment boundary q‡
m and qm independently, and u
displacement uniquely de®ned on the inter-element
boundary qm . If we assume an a priori relation:
Tiq ˆ rij nj ;
then the functional (4.1) becomes
SHW2…; r; u; u~q †
i
X Z h1
ÿ
aijkl kl ij ‡ rij u…i;j† ÿ ij ÿ fi ui dX
ˆ
Xm 2
m
Z
Z
rij nj
ui ÿ ui † dS
ti ui dS ‡
ÿ
Stm
Sum
rij nj …~
uiq ÿ ui † dS :
Z
‡
qm
4:2†
One may derive the associated variational equations of
(4.2):
8
a…; e† ÿ l…e; r† ˆ 0
>
>
<
^ u
~q † ˆ hg; si
l…; s† ÿ r…s; u† ÿ d…s;
> r…r; v† ˆ hf ; vi
>
:^
d…r; ~vq † ˆ 0
8
8
8
8
e 2 E,
s 2 T,
v 2 V,
~
v~q 2 V…q†
,
4:3†
where a…; †; l…; †; hf ; i and hg; i are the same as in
~ † are de®ned by
Section 2; r…; † and d…;
r…r; v† ˆ
XZ
m
^ v~q † ˆ
d…r;
XZ
m
Z
Xm
qm
rij v…i;j† dX ÿ
rij nj ~viq dS :
oXm =Stm
rij nj vi dS ;
questioned and discussed in [15, 16]. In this section we
will analyze this model and see that this is not a stable
model since the ellipticity of a…; † is destroyed.
According to the theory introduced in Sections 2 and 3
of this paper, the second kind hybrid displacement model
based on the modi®ed potential energy principle should
employ the following variational principle (also see Tables
B.1 and B.2 of Appendix B). Find the displacement ®eld
u 2 V0 , traction Tq 2 T…q† de®ned on the inter-element
ÿ
boundary q‡
m and qm independently, and the displacement
~
u~q 2 V…q† uniquely de®ned on the inter-element boundary
qm , to minimize the following functional:
XZ 1
~q † ˆ
aijkl kl …u†ij …u† ÿ fi ui dX
MPE2…u; Tq ; u
Xm 2
m
Z
Z
ÿ ti ui dS ‡ Tiq …~
uiq ÿ ui † dS :
Stm
qm
4:4†
The associated variational equations are given by
8
< a…u; v† ÿ c…v; Tq † ˆ hf ; vi 8 v 2 V0 ,
~q † ˆ 0
8 tq 2 T…q†,
c…u; tq † ÿ d…tq ; u
4:5†
:
~
d…Tq ; ~vp † ˆ 0
8 v~q 2 V…q†
.
This method was ®rst introduced by Tong [14], and has
solid mathematical foundation provided that the shape
functions are well selected to satisfy the BB-conditions in
Table B.3 of Appendix B.
In order to reduce the number of independent variables
in (4.4), Tong [14] also proposed a so-called simpli®ed
hybrid-displacement model by choosing the space T…q† in
the following fashion:
T…q† ˆ ftq ˆ rij …v†nj ;
where v 2 V0 g :
Then the functional (4.4) and the variational Eq. (4.5) are
reduced as follows:
L…u; u~q †
XZ
ˆ
m
Z
ÿ
Stm
1
aijkl kl …u†ij …u† ÿ fi ui dX
Xm 2
Z
ti ui dS ‡
rij …u†nj …~
uiq ÿ ui † dS
qm
and
(
^ u
~q † ˆ hf ; vi
a^…u; v† ‡ b…v;
^
b…u; ~vq † ˆ 0
8 v 2 V0 ,
~
8 v~q 2 V…q†
,
4:6†
4:7†
where the < f ; > is the same as in Section 2, the bilinear
^ † are derived as
forms a^…; † and b…;
a^…u; v† ˆ
X Z
aijkl kl …u†ij …v† dX
Xm
The variational statement (4.1) is not standard in the
m
Z
viewpoint of general form (1.4). Following the method
ÿ …ui rij …v†nj ‡ vi rij …u†nj † dS …4:8†
shown in Appendix A, one may prove without essential
qm
dif®culty that the above variational problem (4.3) has solid
mathematical foundation provided that certain ellipticity and
condition and BB-conditions are satis®ed.
XZ
^
rij …u†nj v~iq dS :
Now let us consider the simpli®ed hybrid displacement b…u; v~q † ˆ
qm
m
method ®rstly proposed by P. Tong [14]. This model was
453
454
One should notice that the bilinear form a^…; † is not
guaranteed to be elliptic. Then the equivalence between the
variational statement (4.7) and the variational principle of
functional (4.6) is violated. Since the functional (4.6) is
used in practical deductions to formulate the ®nite element method, we should examine the functional (4.6) directly. In fact the functional (4.6) has even no
minimization property. To verify this let us assume that u
and u~q are the solutions of the physical elasticity problem,
i.e. u is smooth enough and satis®es the following governing equations and boundary conditions:
P…b ‡ db† ÿ P…b† ˆ
X
m
1
db
H ÿ P db :
2
T
Again it is not guaranteed that the difference 12 H ÿ P is
always non-negative. Thus the mathematical foundation
for the simpli®ed hybrid-displacement model is violated.
5
Discrete BB-condition, rank condition
and singular value principle
As is well known, the ®nite element method is an approach
8
to ®nd the approximate solution uh ˆ fuh1 ; uh2 ; . . . ; uhn gT 2
u†
ˆ
u
in
X,
>
ij
i;j†
Vh of variational Eq. (1.4) in a ®nite dimensional subspace
>
>
>
rij …u† ˆ aijkl kj …u† in X,
>
Vh ˆ V1h V2h Vnh V instead of the in®nite di>
<
rij;j …u† ‡ fi ˆ 0
in X,
mensional space V ˆ V1 V2 Vn : The discrete
form of (1.4) is written as the following fashion.
i
at Su ,
ui ˆ u
>
>
>
>
32 3
2
at St ,
r …u†nj ˆ ti
>
>
u1
A ÿB1
: ij
~iq
ui ˆ u
on qm .
7
6B
7
6
0
ÿB2
76 u2 7
6 1
7
6
7
6
Then one may derive that
76 u3 7
6
B2
0
ÿB3
76 7
6
76 7
6
L…u ‡ du; u~iq † ÿ L…u; u~iq †
76 7
6
Z
76 7
6
X
1
4
Bnÿ2
0
ÿBnÿ1 54 5
rij …du†ij …du† dX
ˆ
Xm 2
Bnÿ1
0
un
m
Z
2 3
2 3
V1
f1
rij …du†nj dui dS :
4:9†
ÿ
6 f 7
6V 7
qm
6 2 7
6 27
6 7
6 7
6 f3 7
6 V3 7
Since the displacement ®eld in hybrid displacement model
7
7
6
‡
in 6
5:1†
ˆ6 7
is not continuous at the inter-element boundary qm and
67 :
7
7
6
6
ÿ
‡
ÿ
qm , the variation dui may be different at qm and qm . Thus
6 7
6 7
45
45
the summation
Z
X
fn
Vn h
rij …du†nj dui dS
qm
By use of the standard terminology and technology in ®m
nite element analysis [13, 9], we have the following conmay be non-zero. Then the term (4.9) is not necessarily
vergence theorem for the general hybrid-mixed ®nite
always non-negative. Therefore it can not be justi®ed to element methods.
formulate the ®nite element method based on the functional (4.6).
Theorem 5.1 In addition to the assumptions (3.1) and
Now we check the ®nite element formulation, which is (3.2) in Theorem 3.1, we assume that the following ellipbased on the minimization of functional (4.6), given in Eq. ticity condition
(17) of [14]. The ®nal formula is shown below.
a…vh1 ; vh1 † akvh1 k2V1 8vh1 2 Kerh …B1 † ;
5:2†
X 1
T
T
T
^
^
b Hb ÿ b Pb ÿ b Pb
Pˆ
2
and the discrete BB-conditions
m
bk …vhk ; vhk‡1 †
1 ÿ ^bT F
2 ‡ bT Gg ÿ ST0 g ;
ÿ bT F
sup
bkvhk‡1 kVk‡1
h
k
kv
h
h
8v 2V
k Vk
^ and G are matrices. vkh 6ˆ0 k
where b; ~b; F1 ; F2 ; g are vectors, H; P; P,
k
One may derive that
8vhk‡1 2 Kerh …Bk †; k ˆ 1; 2; . . . ; n ÿ 1
5:3†
P…b ‡ db† ÿ P…b†
are satis®ed. The notation Kerh …† is de®ned by
X 1
T
T
T
T
db Hdb ÿ db Pdb ‡ db …Hb ÿ …P ÿ P †b
ˆ
2
Kerh …Bk †
m
ˆ fvhk 2 Vkh ; bk …vhk ; vhk‡1 † ˆ 0 8vhk‡1 2 Kerh …Bk‡1 †g;
^ T ^b ÿ F1 ‡ Gg† :
ÿP
k ˆ 1; 2; . . . ; n ÿ 1 ;
5:4†
Since the terms of order …db† are set to ®nd the solution for and Kerh …Bn † is de®ned by Kerh …Bn † ˆ Vnh . Then we have
b, we have
the following results:
The Theorem 5.3 suggests that, from the viewpoint of
(i) The discrete multi-®eld variational problem (5.1) has
unique solution uh ˆ …uh1 ; uh2 ; . . . ; uhn † in the product space computational stability, it is preferable to construct a
hybrid-mixed ®nite element model such that the
Vh ˆ V1h V2h Vnh ,
smallest singular value of the matrix B is as large as
(ii) There exists a constant C > 0, such that
possible. We refer this suggestion as the singular value
ku ÿ uh k C disfu; Vh g ;
5:5† principle in the construction of hybrid-mixed ®nite element models.
where the norm is de®ned in (3.5), and the distance
Example 6.1 (Rank condition of hybrid stress moddisf; g is de®ned by
el) Referring to Table B.3 in Appendix B, the discrete BBn
X
condition for hybrid stress model of second version is
disfu; Vh g ˆ
inf kuk ÿ vhk kVk :
5:6† given by
h
h
8v 2V
kˆ1
k
k
(
P R
~iq
m qm sij nj v
Sup
dS
bk~
vq kV…q†
~
kskT
For the simplicity we discuss one single discrete
8s 2 T0
BB-condition in (5.3). Suppose Vm and Pn are ®nite
dimensional subspaces of Hilbert spaces V and P respectively, and
where the space T0 is de®ned by
~
8 v~q 2 V…q†
;
5:10†
T0 ˆ sij 2 H 1 …Xm †;
The BB-condition may be written as an inf-sup condition: sij;j ˆ 0 in Xm ; sij nj ˆ 0 on Stm ; 8 m :
dim Vm ˆ m;
inf sup
8p2Pn 8v2Vn
dim Pn ˆ n :
b…v; p†
b>0 :
kvkV kpkP
5:7†
Since the operator B de®ned in (1.3) or (2.1) with ®nite
dimensional subspaces Vm and Pn is a mapping from
Vm Pn to R, there must be a n m matrix representation
of the operator B, which is denoted still by B for the
simplicity.
The following two theorems was ®rst proved in [2, 3, 4].
In this section we will apply these two theoretical results to
some practical problems.
Theorem 5.2 The distance BB-condition (5.7) holds, if
and only if the following rank condition holds.
rank…B† ˆ n :
(
Since the condition rank…B† ˆ n implies n m, the rank
condition may be written in the following fashion:
rank…B† ˆ n m :
5:8†
Notice that
XZ
rij nj v~iq dS
qm
m
ˆ
XZ ÿ
sij;j v~i ‡ sij v~…i;j† dX
m
ˆ
Xm
m
Z
Z
ÿ
XZ
Xm
sij nj v~iq dS ÿ
Sum
Z
sij v~…i;j† dX ÿ
Sum
sij nj v~iq dS
Stm
sij nj v~iq dS
8s 2 T0 ;
where the function v~ is an extension of v~q in Sobolev space
H 1 …X†. We assume that v~iq ˆ 0 on Sum . Then we get that
XZ
m
qm
rij nj v~iq dS ˆ
XZ
m
Xm
sij v~…i;j† dX 8s 2 T0 :
According to the trace theorem, we know that
k~
vq kV…q†
a k~
vkV~ ;
~
Moreover the original rank condition proposed by engineers in n m: Thus the expression (5.8) is refereed as the where the space V~ is de®ned by
rank condition for convenience.
V~ ˆ v~i 2 H 1 …Xm † \ C0 …X†; v~i jqm ˆ v~iq 8 m :
Theorem 5.3 Suppose that the rank condition (5.8) holds.
Then the optimal constant b in (5.7) is the smallest sinThen the BB-condition (5.10) becomes that
gular value of the matrix B, i.e.
P R
b…v; p†
inf sup
ˆ min…li † ;
8p2Pn 8v2Vm kvkV kpkP
5:9†
Sup
8s 2 T0
v† dX
m Xm sij …~
kskT
bk~
vkV~
8 v~ 2 V~ :
5:11†
5:12†
The above global BB-condition is dif®cult to be investiwhere li is the ith singular value of the rectangular matrix gated before practical computation. Thus we study the
B, and is determined by the ith non-zero eigenvalue of the following necessary condition:
symmetric matrix BT B, i.e.
XZ
li ˆ ‰ki …BT B†Š1=2
i ˆ 1; 2; . . . ; r ;
where r is the rank of matrix B.
Sup
8s 2 T0 m
Xm
~ ij …~
sij …~
v† dX 0 8 v~ 2 V;
v† 6ˆ 0 :
5:13†
455
To satisfy the above condition it suf®ces to get
Z
Sup
8s 2 T0 …Xm †
Xm
where
~ m †; ij …~
sij …~
v† dX 0 8 v~ 2 V…X
v† 6ˆ 0 :
Ker…D†
n
XZ
ˆ tq 2 T…q†;
m
qm
~
tiq v~iq dS ˆ 08~
vq 2 V…q†
o
:
In fact, instead of the above condition, the so-called local
BB-condition:
For the sake of convenience, the boundary traction tiq ,
R
which is not necessary to be reciprocated at qm , is given by
v† dX
Xm sij …~
a stress ®eld, i.e.
~
Sup
bk~
vk
8~
v 2 V…X
†
456
m†
V…X
kskT…Xm †
8s2T0 …Xm †
m
5:14†
T…q† ˆ ftiq ˆ sij nj at qm ; s 2 T0 g ;
where the space T0 is the same as in (5.10). We assume that
X
is often used.
ktq k2T…q† ˆ
ktq k2 12
:
As a conclusion the local BB-condition (5.14),
H …qm †
m
which is well known in engineering as the condition
Then we have
for zero-energy free model, is only suf®cient to the
X
X
necessary condition (5.13) of the discrete BB-condition
2
2
1
kt
k
ˆ
ks
n
k
jnj j ksij k2 12
q
ij
j
T…q†
(5.10).
H 2 …qm †
H …qm †
m
m
From Theorem 5.2 the condition (5.14) is equivalent to
X
jnj j k ksij k2H 1 …Xm † …trace theorem†
the following rank condition:
rank…Bm † ˆ dim…V~q †m ÿ r dim…s†m ;
m
5:15†
m
where dim…†m represents the number of the parameters of
shape function in element level, r is the number of rigid
body motions of one single element, and the matrix Bm is i.e.
determined by
Z
Xm
sij …~
v† dX ˆ
hBTm b; ai
X
k
ksij k2H 1 …Xm † …jnj j 1† ;
ˆ a ksk2T ;
2
ktq k akskT :
;
Following the same procedure shown in Example 6.1, the
BB-conditions (5.16) may be reformed as
P R
where b is the parameter vector of assumed stress model, a
v† dX
m Xm sij ij …~
bk~
vkV~
is the parameter vector of displacement model assumed in Sup
kskT
element Xm . If the inequality relation in (5.15) is strictly 8s2T0 P R
satis®ed, i.e.
m X ij …v†sij dX
Sup
rank…Bm † ˆ dim…V~ q †m ÿ r ˆ dim…s†m ;
8v2T0
this model is called a least-order hybrid-stress model. The
theory of symmetry groups was applied by Rubinstein,
Punch and Atluri [18] to develop the least-order, stable,
invariant hybrid-stress models. In Section 6, we will investigate some least-order hybrid-stress models of 8-node
cubic elements based on the view point of singular value
principle mentioned above.
m
kvkV
bkskT
8~
v 2 V~ ;
5.17a†
8s 2 Ker…D† ;
5.17b†
where the space V~ is de®ned by (5.11), V0 and Ker(D) are
de®ned by
V0 ˆ fvi 2 H 1 …Xm † \ H 0 …X†; viq jSum ˆ 0g ;
and
XZ
Ker…D† ˆ s 2 T0 ;
~
sij ij …~
v† dX ˆ 0 8~
v2V
Example 6.2 (Rank conditions of hybrid displacement
Xm
m
model) Referring to Table B.3 in Appendix B, the discrete
BB-conditions for hybrid displacement model of second respectively. The ®rst BB-condition (5.17a) is the same as
version are given by
(5.12). Thus the associated rank condition is
P R
Sup
8tq
~iq
m qm tiq v
ktq kT…q†
P R
Sup
8v2V0
dS
m qm
rank…Dm † ˆ dim…~
vq †m ÿ r dim…sq †m ;
bk~
vq kV…q†
~
8~
vq 2 V…q†
;
5.16a†
vi tiq dS
kvkV
bktq kT…q†
8tq 2 Ker…D† ;
5.16b†
5:18†
where the matrix Dm is given by
Z
Xm
sij …tq †ij …~
v† dX ˆ bT Dm a :
The second BB-condition is essentially a global condition,
since the space Ker(D) has meaning only in the global
sense, i.e. the orthogonality property in the de®nition of
Ker(D)
XZ
Xm
m
L…u1 ‡ v1 ; u2 ; . . . ; un † ÿ L…u†
sij ij …~
v† dX ˆ 0
is concerned on whole domain X. According to Theorem
5.2 the global BB-condition (5.17b) is satis®ed if and only if
rank…C† ˆ dim…s† dim…v† ÿ r ;
5:19†
where s 2 Ker…D† and C is given by
XZ
m
Xm
ˆ 12a…v1 ; v1 † ‡ ‰a…u1 ; v1 † ÿ b1 …v1 ; u2 † ÿ hf1 ; v1 iŠ
ˆ 12a…v1 ; v1 †
0;
(positive definite) ;
L…u1 ; . . . ; uiÿ1 ; ui ‡ vi ; ui‡1 ; . . . un † ÿ L…u†
ˆ …ÿ1†i‡1 ‰biÿ1 …uiÿ1 ; vi † ÿ bi …vi ; ui‡1 † ÿ hfi ; vi iŠ
sij ij …v† dX ˆ qT Cb :
ˆ 0;
Let us denote that
i ˆ 2; 3 . . . ; n ÿ 1† ;
457
and
Ne ÿ the number of elements in the whole domain X;
Np ÿ the number of nodes in the whole domain X;
Ke ÿ the number of nodes per element.
L…u1 ; . . . ; unÿ1 ; un ‡ vn † ÿ L…u†
ˆ …ÿ1†n‡1 ‰bnÿ1 …unÿ1 ; vn † ÿ hfn ; vn iŠ ˆ 0 :
The above results indicate that u is the solution of constrained minimization problem (2.2).
Assume, on the other hand, that u is the solution of
Ne …dim…v†m ÿ r† dim…s† :
5:20† problem (2.2). Notice that the last two formula of above
differences are equality deductions. Then it remains to
Notice that the dimension of space T is Ne dim…s†m , and
prove the ®rst equation in (1.4). Let > 0 be an ®xed
the dimension of Ker(D) satis®es
arbitrary constant. The inequality
Then the inequality in (5.19) becomes
dim…s† dim…Ker…D††
~ ÿ r†
dim…T0 † ÿ …dim…V†
Np
r
:
dim…~vq †m ‡
ˆ Ne dim…s†m ÿ
Ne Ke
Ne
Thus one may propose the following suf®cient condition
for rank condition (5.20):
Np
r
dim…v†m ÿ r dim…s†m ÿ
dim…~
vq †m ‡
:
Ne Ke
Ne
Since we usually have
(
r
Ne
Np
Ne
>1 :
Therefore the rank condition should be
5:21†
In engineering the following rank condition is suggested:
dim…v†m ÿ r dim…s†m :
leads to
a…u1 ; v1 † ÿ b1 …v1 ; u2 † ÿ hf1 ; v1 i
ÿ 2 a…v1 ; v1 † 8 v1 2 V1 :
Since may be arbitrary small and a…v1 ; v1 † is bounded,
the above relation becomes
a…u1 ; v1 † ÿ b1 …v1 ; u2 † ÿ hf1 ; v1 i 0 8 v1 2 V1 :
This inequality is indeed an equality, since the left hand
side is linear to v1 . The proof is completed.
1;
dim…v†m ÿ r dim…s†m ÿ K1e dim…~
vq †m :
L…u1 ‡ v1 ; u2 ; . . . un † L…u† 8 v1 2 V1
5:22†
Lemma 1 (Girault and Raviart [9]) Let b…; † be continuous bilinear form de®ned on product space W P and
the operators B and B are de®ned by (1.3). Then there
exists constant b > 0 such that the following three properties are equivalent:
(A) The bilinear form b…; † satis®es the BB-condition
[1, 11, 12], i.e.
b…w; p†
We may see this rank condition is suf®cient to lead (5.22),
sup
bkpkP 8 p 2 P :
A.1†
8w2W kwkW
but is not necessary.
w6ˆ0
On the other hand the condition (5.22) is also necessary,
if the variables v and s should be eliminated in element
(B) The dual operator B is an isomorphism from P onto
level as usually done in practical computation.
(Ker(B))0 , and
The reader is referred to the related work presented in
kB pkW bkpkP 8 p 2 P :
A.2†
Cazzani and Lovadina (1997); Tchonkova and Sture
(1997), and Felippa (1996).
(C) The operator B is an isomorphism from (Ker(B))?
onto P , and
Appendix A
Proof of Theorem 2.1 Assume that u is the solution of
problem (1.4). Let us investigate the differences
kBwkP bkwkW
8 w 2 …Ker…B††? ;
0
A.3†
where b > 0, and the spaces (Ker(B)) and (Ker(B))?
are de®ned by
Ker…B††? ˆ fw 2 W; …w; w0 † ˆ 0 8 w0 2 Ker…B†g ;
2
6B
6 1
6
6
0
Ker…B†† ˆ ff 2 W ; hf ; w0 i ˆ 0 8 w0 2 Ker…B†g ; 6
6
6
(A.4b† 6
6
4
where …; † denotes the inner product in Hilbert space W,
and hf ; i denotes the continuous linear operator in W .
A.4a†
458
Proof of Theorem 3.1 First we consider the case n ˆ 1.
This is exactly the result of well-known Lax-Milgram
theorem [13]. Then we prove Theorem 3.1 by mathematical deduction. Suppose that the result holds for a certain
integer n. Let us investigate the case of n ‡ 1. In addition
to nested BB-condition (3.2), the BB-condition of bn …; † is
also satis®ed, i.e.
bn …vn ; vn‡1 †
bkvn‡1 k
kvn kVn
8vn 2Vn
32
ÿB1
A
0
ÿB2
B2
0
ÿB3
Bnÿ2
0
2
Bnÿ1
2
3
fn
vn 6ˆ0
f~nÿ1 ˆ fnÿ1 ‡ Bnÿ1 …un †? :
Bnÿ1 unÿ1 ÿ Bn un‡1 ˆ fn in Vn
b†
Bn un ˆ fn‡1 in Vn‡1
a†
A.5†
3
u2 7
7
7
u3 7
7
7
7
7
7
unÿ1 5
un †0
7
7
7
7
7
7 :
7
7
7
5
A.8†
Ker…Bn †
where f~nÿ1 is given
Notice that the last two equations in (1.4) for the case of
n ‡ 1 should be
0
V1
6 V
6
2
6
6 V3
6
in 6
6 6
6
4 Vnÿ1
f1
6 f 7
6 2 7
6
7
6 f3 7
6
7
ˆ6
7
6 7
6
7
6~ 7
4 f nÿ1 5
sup
8 vn‡1 Ker …Bn‡1 † ˆ Vn‡1 ;
76
76
76
76
76
76
76
76
76
ÿBnÿ1 54
3
u1
A:9†
The ®rst n-2 equations in (1.4) keep unchanged in (A.8).
The nth equations in (1.4) is replaced by (A.7) due to the
motivation explained previously in (A.7). The (n ÿ 1)th
equation in (1.4) is originally that
Bnÿ2 unÿ2 ÿ Bnÿ1 un ˆ fnÿ1
in
Vnÿ1
:
According to the BB-condition of bilinear form bn …; † and Notice that un ˆ …un †? ‡ …un †0 . The above equation
Lemma 1(C), Bn is an isomorphism from (Ker…Bn ††? onto becomes
Vn‡1
. Thus there exists a unique …un †? 2 …Ker…Bn ††? for
Bnÿ2 unÿ2 ÿ Bnÿ1 …un †0 ˆ fnÿ1 ‡ Bnÿ1 …un †? in Vnÿ1
:
, such that
given fn‡1 2 Vn‡1
Bn …un †? ˆ fn‡1
in Vn‡1
:
Thus the …n ÿ 1†th equation in (A.8) is properly derived.
de®ned in (A.9) is known,
We also notice that fnÿ1
Let …un †0 be arbitrary element in Ker…Bn † and denote
since fnÿ1 is given and …un †? is uniquely determined
un †? ‡ …un †0 ˆ un , clearly un 2 Vn . Then we have
by given functional fn‡1 . Therefore the solution
u1 ; u2 ; . . . ; unÿ1 ; …un †0 † of the n-®eld variational Eq. (A.8)
Bn un ˆ Bn …un †? ˆ fn‡1 in Vn‡1 :
is, according to the assumption of mathematical deducThis means Eqs. (A.5)(b) is satis®ed already. It remains to tion, uniquely determined. Consequently the element un
®nd an appropriate …un †0 2 Ker…Bn † such that the ®rst n is uniquely determined by un ˆ …un †? ‡ …un †0 . Thus the
solution …u1 ; u2 ; . . . ; un ; un‡1 † is uniquely for the case
equations in (1.4) are satis®ed also. Notice that the unknown variable un‡1 is involved only in nth equation of n ‡ 1.
We also notice that the last equation of (A.8) is satis®ed
(1.4) for the case of n ‡ 1, i.e. Eq. (A.5)(a). We rewrite
in
subspace
Ker…Bn †† . Therefore the BB-conditions must
(A.5)(a) as
be nested as de®ned in (3.2).
Bn un‡1 ˆ Bnÿ1 unÿ1 ÿ fn in Vn
A.6†
Now we prove the estimate (3.3). Due to the assumption
of mathematical deduction and (A.9), one may easily
Due to the BB-condition of bilinear form bn …; † and
known that
Lemma 1(B), Bn is an isomorphism from Vn‡1 onto
nÿ1
X
(Ker…Bn ††0 . Thus the Eq. (A.6) has an unique solution
0
kui kVi ‡ k…un †0 kVn
un‡1 , iff Bnÿ1 unÿ1 ÿ fn belongs to the subspace (Ker…Bn †† ,
iˆ1
i.e. hBnÿ1 unÿ1 ÿ fn ; w0 i ˆ 0 for all w0 2 …Ker…Bn ††. This
X
n
leads to
Bnÿ1 unÿ1 ˆ fn
in …Ker…Bn †† :
A.7†
Thus the Eq. (A.6) may be replaced by (A.7). Then it
suf®ces to solve the reduced problem: ®nd …u1 ; u2 ; . . . ;
unÿ1 ; …un †0 † 2 V1 V2 Vnÿ1 ; Ker…Bn †, such that
c
iˆ1
kfi kV ‡ kBnÿ1 kk…un †? kVn
i
:
From Lemma 1(C), we have
k…un †? kVn bÿ1 kBn …un †? kV ˆ bÿ1 kfn‡1 kV n‡1
n‡1
:
Thus we get that
n
X
iˆ1
kui kVi We recall the Korn's inequality [8]
nÿ1
X
Z
Z
kui kVi ‡ k…un †? kVn ‡ k…un †0 kVn
iˆ1
n
X
c
X
kfi kV ‡ …ckBnÿ1 k ‡ 1†bÿ1 kfn‡1 kV iˆ1
n
X
C1
n‡1
i
kfi kV ;
A:10†
i
ij …v†ij …v† dX ‡
8 v 2 …H 1 …X††3 ;
X
vi vi dX Kkvk2H 1 …X†
A:14†
which may be written as
E…v† Kkvk2H 1 …X† ÿ kvk2L2 …X† 8 v 2 …H 1 …X††3 : …A:15†
According to the Korn's inequality (A.15), the inequality
where C1 ˆ maxfc; …ckBnÿ1 k ‡ 1†b g. Moreover we know (A.13) is true, if
from Lemma 1(B) and (A.6) that
E…v† C 8 v 2 V; kvk 2 ˆ 1 :
iˆ1
ÿ1
L …X†
kun‡1 kVn‡1 bÿ1 kBn un‡1 kVn
ÿ1
Suppose the above argument is not true, i.e. there exists a
sequence of vn 2 V and kVn kL2 …X† ˆ 1, such that
ÿ1
ˆ b kBnÿ1 unÿ1 ÿ fn kVn b
ÿ
kBn‡1 kkunÿ1 kVnÿ1 ‡ kfn kVn :
A:11†
The above two inequalities (A.10) and (A.11) lead immediately to the estimate:
n‡1
X
iˆ1
kui kVi n‡1
X
iˆ1
kfi kVn :
E…vn † ! 0 as n ! ‡1 :
According to the Korn's inequality (A.15) and ellipticity
property (A.12), we have
kvn k2H 1 …X†
Z
ÿ1
K
a
aijkl kl …v†ij …v† dX ‡ 1 < ‡1 :
ÿ1
X
Thus the sequence fvn g is a bounded sequence in
H 1 …X††3 . Since the bounded subset of a Hilbert space
must be a weakly compact subset and the space V is
weakly closed, there exists a subsequence of fvn g, simply
Proof of Theorem 3.2 According to the ellipticity property denoted by fvn g again, which converges weakly to v 2 V.
of the elastic coef®cients aijkl [8], i.e.
It is known (see Lions [8], pp. 116) that
That means the inequality kuk kf k is satis®ed for the
case n ‡ 1. The proof is completed.
aijkl kl ij aij ij ;
A:12†
lim inf E…vn † E…v † :
A:13†
This leads to E…v † ˆ 0, since E…vn † ! 0. Therefore v is a
rigid body motion. It is a contradiction to the assumption
of this theorem. The proof is completed.
it is suf®cient to prove that
E…v† Ckvk2H 1 …X†
8v2V ;
where the notation E(v) is de®ned by
Z
E…v† ˆ
X
Appendix B
(The notations in this appendix may be found in references [7, 3].)
ij …v†ij …v† dX :
Table B.1. Multi-®eld variational equations
V.P.
Continuous form
Modi®ed form (1st version)
H.W.
a…; e† ÿ l…e; r† ˆ 0
8e
a…; e† ÿ l…e; r† ˆ 0
8e
a…; e† ÿ l…e; r† ˆ 0
8e
l…; s† ÿ p…r; u† ˆ hg; si
8s
8s
8v
l…; s† ÿ p…s; u† ˆ hg; si
~ q † ˆ hf ; vi
p…r; v† ÿ c…v; T
8s
p…r; v† ˆ hf ; vi
l…; s† ÿ p…r; u† ˆ hg; ri
~ q † ˆ hf ; vi
p…r; v† ÿ c…v; T
c…u; ~t q † ˆ 0
H.R.
a…r; s† ÿ b…s; u† ˆ 0
8s
b…r; v† ˆ hf ; vi
8v
a…r; s† ÿ b…s; u† ˆ 0
~ q † ˆ hf ; vi
b…r; v† ÿ c…v; T
c…u; ~t q † ˆ 0
P.E.
C.E.
a…u; v† ˆ hf ; vi 8v
a…r; s† ˆ hg; si
8s
~ q † ˆ hf ; vi
a…u; v† ÿ c…v; T
~
c…u; t q † ˆ 0
Modi®ed form (2nd version)
8v
8~t q
8s
8v
8~t q
8v
8~tq
~q † ˆ 0
c…u; tq † ÿ d…tq ; u
d…Tq ; ~vq † ˆ 0
a…r; s† ÿ b…s; u† ˆ 0
~ q † ˆ hf ; vi
b…r; v† ÿ c…v; T
8v
8tq
8~vq
8s
c…u; ~tq † ÿ d…tq ; u~q † ˆ 0
8v
8~t q
d…Tq ; ~vq † ˆ 0
8~vq
a…u; v† ÿ c…v; Tq † ˆ hf ; vi
~q † ˆ 0
c…u; tq † ÿ d…tq ; u
d…Tq ; ~vq † ˆ 0
8v
8tq
8~vq
a…r; s† ÿ d…s; up † ˆ hg; si
8s
~q † ˆ hg; si
a…r; s† ÿ d…s; u
8s
~q† ˆ 0
d…r; vq † ÿ c…vq ; T
c…uq ; ~t q † ˆ 0
8v
8~t q
d…r; ~vq †
8~vq
459
Table B.2. Variational principles
V.P.
Continuous form
Modi®ed form (1st version)
Modi®ed form (2nd version)
H.W.
HW …; r; u†
Z
ˆ ‰12 aijkl kl ij
~q†
MHW1…; r; u; T
Z
X
ˆ
‰12aijkl kl ij
~q †
MHW2…; r; u; Tq u
XZ
‰12 aijkl kl ij
ˆ
X
‡ rij …u…i;j† ÿ ij †f i ui Š dX
Z
ÿ t i ui dS
S
Zt
rij nj
ui ÿ ui † dS:
‡
460
Xm
m
S tm
Sum
Z
ui ÿ ui † dS ÿ
Su
qm
Xm
m
‡ rij …u…i;j† ÿ ij †f i ui Š dX
Z
Z
ti ui dS ‡
rij nj
ÿ
‡ rij …u…i;j† ÿ ij †f i ui Š dX
Z
t i ui dS
ÿ
S
Z tm
rij nj
ui ÿ ui † dS
‡
~
ui T iq dS :
Z
‡
H.R.
~q†
MHR1…r; u; T
XZ
‰ÿ12Aijkl rkl rij ‡ rij u…i;j†
ˆ
HR…r; u†
Z
ˆ ‰ÿ 12 Aijkl rkl rij
X
‡ rij u…i;j† ÿ f i ui Š dX
Z
ÿ t i ui dS
S
Zt
rij nj
ui ÿ ui † dS.
‡
Xm
m
P.E.
PE…u†
ˆ
Sum
ÿ
Z
‰12 aijkl kl …u†ij …u†
Z
ÿ f i ui Š dX ÿ ti ui dS:
X
Xm
m
Z
Z
qm
~ q†
MPE1…u; T
XZ
ˆ
Xm
m
Z
t i ui dS
ÿ f i ui Š dX ÿ
Stm
Z
rij nj
ui ÿ ui † dS
‡
Z
~ iq dS :
ui T
‡
ÿ
Stm
t i ui dS ÿ
Z
CE…r†
ˆ
Z
‰ÿ12Aijkl rkl rij ‡ rij Š dX
Z
i dS:
rij nj u
‡
X
qm
Z
~ iq dS :
ui T
ÿ
Z
~q†
MCE1…r; uq ; T
XZ
ÿ 12 Aijkl rkl rij dX
ˆ
m
Su
qm
Tiq
uiq ÿ ui dS :
Xm
m
‡
C.E.
Sum
~q †
MPE2…u; Tq ; u
XZ
‰12 aijkl kl …u†ij …u† ÿ f i ui Š dX
ˆ
‰12 aijkl kl …u†ij …u† ÿ f i ui Š dX
Z
St
qm
Tiq
uiq ÿ ui dS :
~q †
MHR2…r; u; Tq u
XZ
‰ÿ 12 Aijkl rkl rij ‡ rij u…i;j†
ˆ
ti ui dS
ÿ f i ui Š dX ÿ
Stm
Z
rij nj
ui ÿ ui † dS
‡
Su
Sum
m
Z
Z
‡
Sum
qm
V.P.: Variational Principle
H.W.: Hu-Washizu Principle
qm
t i ui dS
Tiq
uiq ÿ ui † dS :
~q †
MCE …r; u
XZ
ˆ
Xm
‡
Stm
Xm
ÿ 12 Aijkl rkl rij dX
Z
‡
i dS
rij nj u
~iq ÿ rij nj † dS :
uiq …T
Z
‡
Sum
qm
i dS
rij nj u
iq dS :
rij nj u
The bilinear forms and linear functionals in Table B.1 are
de®ned as follows.
H.R.: Hellinger-Reissner Principle, in which the only
constrain is
a…u; v† ˆ
rij ˆ aijkl kl
a…; e† ˆ
XZ
m
Xm
m
Xm
XZ
aijkl kl …u†ij …v†dX 8 u; v 2 V ;
aijkl kl eij dX 8 ; e 2 E ;
XZ
( rij ˆ aijkl kl
Aijkl rkl sij dX 8 r; s 2 T ;
a…r; s† ˆ
Xm
m
ij ˆ u…i;j†
XZ
i on Su
ui ˆ u
sij v…i;j† dX 8 s 2 T; v 2 V ;
b…s; v† ˆ
Xm
m
C.E.: Complementary Energy principle, in which the conZ
X
strains are
vi tiq dS 8 v 2 V; tq 2 T…q† ;
c…v; tq † ˆ
8
qm
m
< ij ˆ Aijkl rkl
XZ
rij;j ‡ f i ˆ 0
~
tiq v~iq dS 8 tq 2 T…q†; v~q 2 V…q†
;
d…tq ; v~q † ˆ
:
rij nj ˆ t i on St
qm
m
P.E.: Potential Energy principle, in which the constrains are
C.E.
P.E.
H.R.
H.W.
V.P.
X
Z
Z
X
X
Aijkl skl sij dX akskT 8s 2 Ker…B†
aijkl kl …v†ij …v†dX akvkV 8v 2 V0
Aijkl skl sij dX akskT 8s 2 Ker…B†
R
sij v…i;j† dX
bkvkV 8v 2 V0
sup X
kskT
8s
Z
Continuous form
Z
aijkl kl ij dX akkE 8 2 Ker…L†
X
R
sij v…i;j† dX
bkvkV 8v 2 V0
sup X
kskT
8s
R
eij sij dX
sup X
bkskT 8s 2 Ker…P†
kek
8e
E
Table B.3. Elliptic conditions and BB-conditions
8s
sup
~
bk~t q kT…q†
t q 2 T…q†
~ 8~
bkvkV 8v 2 Ker…C†
~
bk~t q kT…q†
t q 2 T…q†
~ 8~
sij v…i;j† dX
kskT
m Xm
kvkV
m qm
vi~t iq dS
Aijkl skl sij dX akskT 8s 2 Ker…D†
P R
Xm
vi~t iq dS
kvkV
m qm
P R
8v2V0
sup
m
Xm
aijkl kl …v†ij …v†dX akvkV 8v 2 Ker…C†
bkvkV 8v 2 Ker…C†
~
bk~t q kT…q†
t q 2 T…q†
~ 8~
sij v…i;j† dX
kskT
m Xm
P R
XZ
8v2V0
sup
m
vi~t iq dS
kvkV
m qm
P R
XZ
8s
sup
8v2V0
bkskT 8s 2 Ker? …P†
Aijkl skl sij dX akskT 8s 2 Ker…B†
P R
Xm
eij sij dX
kekE
m Xm
bkvkV 8v 2 Ker…C†
~
bk~t q kT…q†
t q 2 T…q†
~ 8~
sij v…i;j† dX
kskT
m Xm
P R
vi~t iq dS
kvkV
m qm
P R
sup
m
Xm
P R
XZ
8s
sup
8s
sup
8v2V0
sup
m
Modi®ed form (1st version)
XZ
aijkl kl ij dX akkE 8 2 Ker…L†
Xm
8tq
sup
m
viq dS
m qm tiq ~
bktq kT…q† 8tq 2 Ker…D†
ktq kT…q†
viq dS
m qm tiq ~
~
bk~vq kV…q†
8~vq 2 V…q†
~
Aijkl skl sij dX akskT 8s 2 Ker…D†
P R
Xm
m
kvkV
~
bk~vq kV…q†
8~vq 2 V…q†
~
aijkl kl …v†ij …v†dX akvkV 8v 2 Ker…C†
bkvkV 8v 2 Ker …C†
bktq kT…q† 8tq 2 Ker…D†
~
bk~vq kV…q†
8~vq 2 V…q†
~
sij v…i;j† dX
kskT
ktq kT…q†
P R
m q vi tiq dS
XZ
8v2V0
m
kvkV
m Xm
P R
sup
8tq
sup
m
viq dS
m qm tiq ~
ktq kT…q†
P R
m q vi tiq dS
XZ
8s
bkskT 8s 2 Ker …P†
bkvkV 8v 2 Ker …C†
Aijkl skl sij dX akskT 8s 2 Ker…B†
P R
sup
sup
Xm
P R
8v2V0
8tq
sup
m
eij sij dX
kekE
m Xm
P R
sup
8s
XZ
8s
bktq kT…q† 8tq 2 Ker…D†
~
bk~vq kV…q†
8~vq 2 V…q†
~
sij v…i;j† dX
kskT
m Xm
kvkV
m
ktq kT…q†
P R
m q vi tiq dS
viq dS
m qm tiq ~
P R
sup
sup
Xm
P R
8v2V0
8tq
sup
m
Modi®ed form (2nd version)
XZ
aijkl kl ij dX akkE 8 2 Ker…L†
461
462
XZ
[7] Atluri SN (1975) On hybrid ®nite-element models in solid
mechanics, in Advances in Computer Methods for Partial
qm
m
Differential Equations (Vichnevetsky, Ed.) AICA, Rutgers
Z
X
University, 346±356
ij sij dX 8 2 E; s 2 T ;
l…; s† ˆ
[8] Duvant G, Lions JL (1976) Inequalities in mechanics and
Xm
m
physics, Springer-Verlag, Berlin Heidelberg, New York.
Z
[9] Girault V, Raviart PA (1979) Finite element approximation
X Z
of the navier-stokes equation, Lecture Notes in Mathematics,
sij v…ij† dX ÿ
sij nj vi dS
p…s; v† ˆ
No. 749, Spring-Verlag.
Xm
S um
m
[10] Brezzi F, Fortin M (1991) Mixed and hybrid ®nite element
8 s 2 T; v 2 V ;
methods, Springer-Verlag.
Z
XZ
[11] Babuska I, (1971) Error bounds for ®nite element method,
t i vi dS
8v2V ;
fi vi dX ‡
hf ; vi ˆ
Num. Math. V. 16, 322±333
X
S
m
u
m
m
[12]
Ladyzhenskaya OA (1969) The mathematical theory of visXZ
cous incompressible ¯ow, Gordon and Breach, New York.
sij nj ui dS 8 s 2 T :
hg; si ˆ
[13] Ciarlet PG (1978) The ®nite element method for elliptic
Sum
m
problems, North-Holland.
[14] Tong, P. (1970) Int. J. Num. Meth. in Engng., Vol. 2, 73±83
[15] Cris®eld MA (1978) Further comments on the simpli®ed
Reference
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d…s; v~q † ˆ
~
sij nj v~iq dS 8 s 2 T; v~q 2 V…q†
;