Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
d e c * * * *
Series: Monografias em Ciência da Computação
N9 8/82
ON THE SOLVABILITY CF ASY.ÍMETRIC QUASILINEAR FIHITE ELEMENT
APPROXIMATE PROBLEMS IN NONLINEAR INCOMPRESSIBLE ELASTICITY
Vítorxano Ruas
departamento de Informática
PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO OE JANEIRO
RUA MARQUES DE SlO VICENTE, 225 - CEP-22453
RIO DE JANEIRO - BRASIL
PUC/RJ - DEPARTAMENTO DE INFORMÁTICA
Series: Monografias em Ciência da Computação N9 8/82
Editor: Marco Antonio Casanova September 1982
ÜN THE SOLVALIBITY OF ASYMMETRIC QUASILINEAR FINITE ELEMENT
APPROXIMATE PROBLEMS IN NONLINEAR INCOMPRESSIBLE ELASTICITY*
Vitoriano Ruas
* This work has been sponsored in part by FINEP.
ABSTRACT
Tkis paper--4aa4-«--yirtfr a class of siaplicial finite
*".lei»»nts for solving incompressible elasticity problems in
n-dimen$ional space» n-3 or 3,'An asymmetric structure of the
shape functions with respect to the centroid of the simplex ,
renders then particularly stable in the large strain case, in
which the incoapressibility condition is- • nonlinear.
We prove 'that under certain assembling conditions of
the elements, there exists a solution to the corresponding dis
crete problems. Numerical examples illustrate the efficiency of
the method.
KEY-WORDS; Asymmetric, compatible, compression, displacements,
energy, existence, finite elements, incompressible, Mooney-Rivlin
material, nonlinear elasticity, pressure, quasilinear, rubber,
simplex, stability, strain, tetrahedrons, triangles.
RESUMO
Considera-se uma classe de elementos finifos de tipo
simplex para a resolução de problemas relativos a materiais hi -
perelísticos incompressíveis como a borracha. Una estrutura as
simétrica dos elementos com respeito ao baricentro do simplex
tornam a simulação numérica de grandes deformações de corpos de
tais materiais particularmente estável e realista do ponto de
vista físico, especialmente no caso tridimensional onde falham
métodos clássicos,»
-.Prova-se que certas construções de partições do corpo
nesses elementos, conduzem a problemas discretos bem colocados
matematicamente. Exemplos numéricos ilustram a eficiência do mé_
todo em casos de forte compressão.
PALAVRAS-CHAVE;
Assimétricos, borracha, compressão, deformações, desl£
camento, elasticidade não linear, elementos finitos, energia ,
estabilidade, existência, incompressivel, material de Mooney-
-Rivlin partições compatíveis, pressão, quasilinear, simplex ,
tetraedros, triângulos.
C O N T E N T S
Introduction 1
The finite eleaent approximate problem 6
The asymmetric finite elements 11
Stability properties of the asyaaetric eleaents .... 16
2
Existence results in the case of partition t. 26
The case of partition T* and numerical exaaples .... 47
Acknowledgement 55
References 56
-i-
1- INTtODUCTION
In this work ve discuss two finite elements of asyaae-
tric type introduced respectively in [14] and [16], for solving
finite incompressible elasticity probleas.
Let us first give soae notations:
Q being a bounded set of IK, for every open subset D of
Q, we shall denote by ||.{! _ and by | . | _ k n the usual norm
and seai-nora respectively, of the Sobolev space '«' (D) (see
e.g. C 1 D),»,* € m t m 2 0 and 1 S * S -, with W0'K(D) = L^fD) .
Siailarly in the case * = 2 we denote by (.,.) „ the usual inner
product of rfj,2(D) 3 H%(D) and by |.! n~ |.| , n the corres -
ponding nora, while we will represent the nora of fir' (a) = H (U)
b v M-llm n instead of £|.|| „ n . In all cases we shall drop
the subscript D whenever D is Q itself.
Por every space of functions!' defined on D3 V will re -
present the space of vector fields whose n components belong to
V. In the case where V is if'^CD) or }/l'K(D)t we define the norm,
seai-nora and inner product (if A-2) for £, by introducing obvi
ous modifications in the scalar case, and keeping the saae nota
tions.
We shall denote by x.y the euclidian inner product of
two vectors x and y of JR and by |.| the corresponding norm. I
will be either equal to n in the case of vectors of JR , or equal
to n in the case of tensors of JR
Finally for every function or vector field y defined
over a certain set D, we shall denote by y/„ its restriction to
a subset S, S c D.
-2-
Now our problem can be described as follows:
We are given an elastic body represented by a bounded
domain (1 c J? , n = 2t3t with a smooth boundary I*. Keeping fixed
a part To of r with meas(Tü) ? 0, we consider a loading of Q
consisting of body forces f acting on a set r c r, such that .— — * _* —
meaetTç) n T ) = 0 and V u fo^r, having a density f per unit of
measure of r . Although it is physically possible to have r* - 0
we will not consider this case in this paper.
The effect of / and g is to deform ft into an equilibrium
configuration defined by a displacement vector field that we will
denote by M. In this way, the new position of every point * of Si
is given by x + u(x).
Now the fact that every element of Q is measure inva
riant in its deformed statetcan be expressed mathematically by:
(1.1) Jlx + u(x)l = 1 for almost every x e Í1 ,
where Jlv(x)l denotes the Jacobian of a vector field V at point
X.
(1.1) is called the incompressibility condition in finite elas
ticity and we shall often rewrite it as:
(1.2) det(J + ?!*; - 1 a.e. in a ,
where £ is the identity tensor n*n and V represents the gradi -
ent operator.
REMARK: Condition (1.1) is obviously nonlinear but in
the case of small strains, that is to say, when
maxlv u(x) I << 1 ' *v «* ^ •
one cae neglect products of derivatives of u of order higher
then one.(1.1) becomes then the well-known linear incoapreasuti
lity condition arising in infinitesimal elasticity or in fluid
mechanics, namely:
di-o }i(x) = 0 for a.e. x e a. 0
Although there is a rather large range of incoapressi -
ble Materials, in this work we would like to focus our study to
the case of Mooney-Rivlin materials, because they are particu
larly representative of the ciass of Materials for which (1.1)
holds. We note by the way that aaong Mooney-Rivlin Materials rub
ber is a typical case.
For a Mooney-Rivlin Material the elastic energy for a
certain admissible displaceMent vector field £ is given by [13J:
(l.3)2 *rrHM ~ I |j * i u \% - 2 - J/ . 2 dx _| a.£ ds Q or*
for n = 2
(l.3)3 W(Z)= ~ J \l * v v\2dx. - * • |i f \adj(i*y£)\ dxr* -
- I f'Z <*£ " j £-2 d9 foi „ = 3 n~ r*
where adj A denotes the transpose of the Matrix of cofactors of
an nxn Matrix A and t'j and C2 are positive physical constants.
Taking into account (1.2) and the fact that W Must be
finite, it is natural to choose the following set of admissible
displacement vector fields:
X = ÍJJ/E <K '*'»; » U/rn * £ > dett£ * I £'*»» ••«• *"a> 2 V 2 *
with HZ2(n-l), whereas we shall assume that £ tL (Si) and g (H ' (V ) .
- 4 -
Th« problea we want to so lve can now b« s ta ted a* f o i l
Piad ft < I such that
(P) VCjtl s Uitl * £ c J
It i* interesting to not* that J is a noa convex sat and
that it is a subset of the vector space V defined by:
Zs W Z* I1'*!»* . i/#§ -£>
which can be noraed by the seai-nota |.|i K t0 b*"« C O B"
nected [111 ) .
Instead of the ainiasation problea (Pi itself, we «ill
consider the following weak, femulation obtained by dualisation
of (1.2) with the help of a Multiplier p, and by differentiation
of W(u) along £ over V .
(P')
Find i%,p)t Z*Q s o c h t h , t
%lji»q) = 0 * q e Q
where Q = L*(a), with t such that n/K * t/t i 1 , and
(1.4) afj*,£; - (?! J & .ZZ & + C2\ adjfl .+ •&). 0 ft
.CadifJ * Zh * Z2.)- adà I t* •**» ^2"° i« n s 2 ,
(1.5) b'(%,z,q)= Ja iCadif^Vjt^. x E3d£
(1.6) b(z q) = J qld*t(£ * % %) -r 13 dj
(1.7) üfg; = [ X. J> i g + f £. Z d» ' Cl f div t d* J0 If* Jfl
-5-
According to results by Le Tallec [10 3* under reasoni -
ble assumptions, there exists a hydrostatic pressure p, vith
p e L (Q)t associated with every solution u to problem (P), and
in this case (u,p) is a solution to (P * ) .
At this stage ve would like to point out that in practi^
ce, it seen* unwise to use formulation (P') for numerical compu
tations with mixed finite elements, such as those we are going to
treic here. Indeed, there are other mixed formulation of (P) much
more suitable for such a purpose and in this respect we refer to
[ 6!, for instance. However, for the sake of clearness, we prefer
to consider ' (P*) in this work, as it appears to be the most n« -
tural formulation of all.
Bearing in mind that our mixed finite element methods
apply to other mixed formulations of (P) as well, we shall from
now on, consider that we are actually going to approximate pro -
blem (P'). For this purpose we will define two finite dimensional
spaces j£. and Q, aimed at approximating V and Q, associated with
two n-siaplicial finite elements for n=2 and n-3, respectively ,
which have an asymmetric structure with respct to the centroid
of the simplex. The three-dimensional element can be viewed as a
certain generalization of the two-dimensional one and it should
be mentioned that the latter was first introduced in [14],whereas
both were discussed in [16] for linear prcblens, arising in He -
chanics of incompressible media.
An outline of the paper is as follows:
In Section 2 we define in a general way a discrete ana
logue (Pi) of (P'), based on finite element approxinations. In
Section 3 we briefly recall the asymmetric elements and we
describe the corresponding problem (P/), in connection with two
kinds of partitions of ft. In Section. 4 we consider some basic
properties of both elements toot justify s priori their adequacy
.for the numerical solution of problem if). In Section 5 we con -
sidcr in deteil the vell-posednes* of (F/) an the case of one of the
types of partition considered, in Section 3. Finally in Section 6
we discuss the sane question for the other type of partition in
a particular case, and we give corresponding numerical results.
2. THE FIIITE BLKMBMT APFIOXIMATB PK01LEM
Henceforth» except where otherwise specified, in this
paper we consider ft to be a domain of m , 1«f,3, having n poly
hedral boundary Y. For the case n=3 we also assume that F*nF0 is
a set of special polygonal lines.
He are given a family (t.). of partitions of"0 into
n-simpiices. satisfying the classical assembling rules for the
finite element method. Some additional compatibility conditions
for (T^)JL related to our asymmetric element» will be specified in
Section 3. We also assume that r* and To can be viewed a» the
union of faces of elements of T. and that (*»)* i» regular in tlm
following sense:
Denoting by h^ the diameter of the circumscribed sphere
and by pj. the diameter of the inscribed sphere of element X» K * T.
and setting
h - mas hv and p - min pr ,
* *h * * **
there exists a strictly positive constant o such that ph > a Vh.
With each partition t* we associate the finite dimensional
•paces Q^ and %., approximations of Q and £(resp. Q •»« t )*•»-
psctively. We sssune that CCC^# whereas in general a similar, in-
-7-
elusion will sot hold for F.. Lot [.[be the aom of Cfc induced
by L (Ü), and il> Ij, a h (.reap. { | fl ft) ho obtaiaed by tuMM -
tioa over the eleaeats XCT. of the squares of the || . || --noras
(resp. |.| y-seaniooras). In particular we «ill us* the ^-dis
crete nora for 7ft defined by:
(2.i) \\VJ = « !•»!*, I h *eT* * *
How ia the discrete aaalegue of (P'), we weaken the re
quirement that the approximation £i*Zt- e i t B C solution u to pro-
blea (P) satisfy exactly (1.1), in the following way:
The incoaprcssibility condition is to be satisfied only
at those points of Í1 to which we attach the degrees of f reedoa of
Qfr. This is equivalent to require that u, belong to an approxi -
nation X. of X defined by:
Xh * { 2ft / Hft € 4k ' bh(^h'%} = ° *1h€ V
where b, is a suitable approxiaatioo of b given by (1.6).
A natural way of defining b, is to set
<2'2> bh fZh ' «A; = r
E h (Zh > <*h> Ktrh
where b% corresponds to an approxiaaticn of the integral of (1.6),
restricted to eleaent X, whose quadrature points are those asso
ciated with the degrees of freedom of Q,. We consider two possi
bilities of perforaing this numerical quadrature, according to
the way of defining the elements of t,,
To be more specific, if the domain 0 is a polygon or a
polyhedron, the eleacnts of the pa:tition T, are as prescribed
above. Notice that in this case we have:
em*» %} Every X « T . i s the r e c i p r o c a l iaage of the
«. aa l X(see Figaro 2.1) by an a f f i a e traaafometioa 4_r X • X.
Ia this casa we defina th a approx iaat ioa of
J qh ZdMtll * I j^) - n dx to b e :
(2.3) VJVV =J, "V V # c** ¥* V-13/^ mmta
where ix.)* 2 is tha sat of points used to defiaa 4gA»
aad the »'. a ara the weights of the nvaarieal quadrature forasla.
On the other hand, if Q has a carved boundary aad
F, is conforaing it aay be interesting to partition it
into carved elements defined in the classical way, na -*
•ely :
cat* ii) Every Jf « T. is the reciprocal iaage of X by
a bijective isoparaaetric traasforaation /C :* •*• X.This
aaaas that^-J 1%) - la*(£},..., aj^l. i*era <*J e PJ lííí n, A
P being a space of shape f u n c t i o n s def ined over K, such
that vh= vh,x*flx c P,¥ vh. « Kft end * X c Tfc .
In'this case the approximation of J q£d*t(l + V fy) - I ] dx
i s given by:
where{£*} J . J i s the set of points of X whose reciprocal iaages
througbft~ are the point* of X to which wa attach the degrees of
freedoR of Q-,
Mow both To and r* are approximated by the onion of carved
faces or edges of aleaents of t..
- 9 -
5,ro,o; S2(l,0)
The r e f e r e n c e e lement K f o r n=2
Figure 2 .1
How, tak ing i n t o c o n s i d e r a t i o n ( 2 . 2 ) , we can v e r i f y
• t h a t i n both cases i) and ii) we have :
* vh € Xh >
( 2 . 5 ) det(I + V *-h)/xK = 2 * í , M Í S » u i ^ J l e ^ 3
Indeed, in ease i) this it trivial provided meas (K)
in nonzero for all K e T, . One the other hand, from
the well-known formula of Calculus [3 ] we have:
(2.6) J(& - J<Z) J(A) where | - A(x) and £ • A(x) - %(&)
Thus we see that (2.5) also holds for case ii) by set
ting %(x) - Z^^ * * *nd A V*X * and t*kin8 i n t o ac"
count th« identity J (A) - J(A ) .
- 1 0 -
BEMASK í I f th.* f , , s are the p o i n t s of a quadrature for
aula tha t i n t e g r a t e s e x a c t l y f u n c t i o n s of form 'Í£fc^
over K ¥ £fc e P, then l i k e in [ 1 5 ] we can draw the
f o l l o w i n g c o n c l u s i o n :
If (2.5) holds and t « , = 3 , we have meas(K) = mgas(K)
¥K e TT , if being the defomed state of K induced by g . . 0
Mow we further set
(2.7) bjfat„qj , ^ . l S h
and we define the discrete nixed fornulation of problea (P) to be:
that
sL%> **h « 4 4-
[h%>%)=° **h €Qh
According to Í 9 1, the existence of a solution to problea (P*J
is directly dependent on the validity of a nonlinear discrete
Brezzi-type compartibility condition between the spaces JV and Q-..
However now this condition oust be expressed in terms of the vector
field iu itself. Since \u is supposed to ninimize the energy V in
soae sense» the following result proved in [10], Theorem 4.1 is
of crucial importance:
The problem
(2.8) find u, c X, to minimize Mty) over X, has a solution .
Row, let Ui, be a local minimum of W. Let also ([. |J be the norm of 9
)U and j. | be the norm of Qh induced respectively by £ and I (11).
The nonlinear compatibility condition can be stated as follows :
-11-
There exist? 6, > 0 such that
" • " ^ ii «íik ' »» '«*' * « * « « *
According to [103, Theoreraa 4.3, if condition (2.9) is
fulfilled, there exists a unique pressure p. e Q. such that
~/j •» Pft *s * solution to ^ft^ *
3 - THE ASYMMETRIC FINITE ELEMENTS
We first define Q, to be the space of functions q, that
are constant over each elenent of T, , and we clearly have Q,c Q.
For convenience we consider the degrees of freedom of Q, to be
functional values at the centroid G of the elements. V, in turn
consists of functions whose restriction to each simplex K e T,
belongs to a space P defined as follows :
Let S. denote the vertices of a simplex
K c T, , i = 1,2,...,n+l. We first assign to K a priviliedged
face, say the face opposite to verter ^n+i* that will be called
K K
the base B of K, and let F. be the face opposite to vertex
S. , i = 1,2,...,n. The F. s will be called the lateral faces of
K :
Let X. denote the area coordinate of K associated with
vertex S., i = 1,2,...,n*l and S „ denote the centroid of B
Now we define P to be the ftt*2J-dimensional space
spanned by the functions A. = i=l ,2, . . . ,n+l and $, where :
n (3.1) * = E XAT,
j,k=l ó k
i<k
•1-2-
One can easily verify that the set of degrees of
tn*2 freedom íõvív^j » where a . is the value of the function at point
St.. is P
given by:
S., is P -unisolvent and that the associated oasis functions are
<3.2)
Pi = Ai —=• • t = J, 2, ... ,n n-2
< pn*l = An+J
* - 2n A
In Figure 3 . 1 . we i l l u s t r a t e the s o - d e f i n e d asymmetric
f i n i t e e lements where o r e p r e s e n t s degrees of freedom for V.
and x represents those for Q,.
Mote that the following inclusions hold : P. <= P c P
where Pj, denotes the space of polynomials of degree less or
equal to k defined over K .
«3
The asymmetric quasilinear elements
Figure 3.1.
As remarked in [14] and [15], the elements associated
with P must be used in connection with partitions of ft into a
n-eimplices constructed in a special way, which are called com -
patible partitions. Let us briefly recall two kinds of s'<ch parti
tions given in [16 ] for both elements :
-13-
1 Partition x, : In the two-dimensional case we first construct a
partition of Q into artibrary convex quadrilaterals (like in the
case of the bilinear Q. element). Next, every quadrilateral is
subdivided into two triangles by an arbitrarily chosen diagonal.
Those diagonals will be the only bases of the elements of the
so-generated triangulation.
In the three-dimensional case we first construct a
partition of Q into arbitrary convex hexahedrons having quadrila_
teral faces. Now we refer to figure 3.2b where we show a classi
cal subdivision of a hexahedron into 5 tetrahedrons. We next
take an arbitrary point in the interior of each central tetrahe
dron ABCD, say point E, and we join it to A,B,C and 0, so that
each hexahedron becomes the union of 8 tetrahedrons. These form
partition T, if we assign its bases to be the faces of tetrahe
drons ABCD.
-14-
n=3
Bases: Faces of
ABCD
Figure 3.2b 1
An.Illustration of compatible partition T.
Figure 3.2
Partition T. : Ve first construct an arbitrary partition T. of Q
into n-sinplices K. Then we subdivide each K ex, into n+1 sin -
plices having a common vertex situated in A . 2
This subpartition of t. becomes the compatible partition T. if
we define its bases to be the faces of t.. Note that the interior
point of the simplex KIT. can be arbitrary, although in this
work we will choose it to be the cancroid Gfsee figure 3,3).
An illustration of compatible partition tjj Figure 3.3
-15-
With the above considerations, we define the degrees of
freedon of V, to be the functional values at the vertices and at
the cuntroid of the bases of a compatible partition T, of ft ,
except the values at those nodes lying on FQ, where a function
vif 7» vanishes necessarily.
With the above definition of V, we can say that Vj,c C. (Õ.)
if n=2, but if n=Z this inclusion does not hold and therefore we
have a nonconforming element. Nevertheless, for n=3t a function
of V, is necessarily continuous along the bases of the partition.
Let us now examine the particular case of problem (?U
for the spaces V, and Q, defined above:
k We have m=l, Uj = 1, and the quadrature point at- is
centrpid of K in ear,a i), and the image of the centroid of K
through transformation/!> 7 in case ii).
It is then possible to verify, using arguments to be de-
velopped in Section 4, that in both oases i) and ii)f the
so-obtained numerical quadrature formula integrates exactly
det(I+Vuh) over K, that is to say:
bK(*h * <V = 1 qhldet(l + iKh* ' 2 ^ d&
and
*bk(uh,qh)
This means that, at least when fl-Bt s v K, we have
b,= b and bS Mb', for Vh and Q, defined in this section.
-16-
REMARK : Strict.y speaking, if the union of the -eleaent K over
T^ is different of ft, we should redefine problem ( i)
by replacing a. end L by approximate functional» a, and
L^ chat take into account integration over 8, rather
than over ft.D
4. STABILITY PROPERTIES OF THE ASYMMETRIC ELEMENTS
In this section ve intend to justify our proposal of
the elements of asymmetric type for the numerical solution of
problem (P') from the point of view of the simulation of (1.1).
First of all let us briefly recall some a priori argu -
nents already considered in [14 ] and £15 ].
If a vector field of an approximation space .£» - of £ i*
such that each component restricted to an element K of x. is a
polynominal of T. , its Jacobian is a polynomial of ^-/-L-I) o v e r
K. This implies that one must satisfy constraint (1.1) in a re -
latively large number of points of K in order to simulate the
incompressibility phenomenon in a meaningful way. Note that this
question becomes particularly critical in the three-dimensional
case, where nunerieal instabilities are frequently observed
whenever the number of these point constraints per element i'.
taken small, specially under compression loads.
However, the total number of constraints to be satisfied
in the discrete problem associated with (P') - that is precisely
dim Q-. - must not exceed the total number of displacement degrees
of freedom» i.e. din £t, otherwise condition (2.9) fails to hold
(see e.g. I 9 ]). This fact is usually expressed numerically by
requiring that the following aeymptptia ratio:
-17-
e = tim dif9% h+0 dim lh
be strictly less than on* (actually in practice 9 should not be
too cloaa to on*).
On th* other hand, from a mathematical point of view.it
is not appropriate to choose a space Q, satisfying continuity
requirements at points si tua tec' on the interface of the elements.
This fact prevents oat fron reducing the diaension of Q^ signi -
ficantly like in the case of linear problens solved with the
so-called Taylor-Hood elenents [ 7 ] .
Let us also add that K. should he preferably conforaing.
Indeed, even if condition (1.1) is properly satisfied elementwisc,
the nonconforaity nay lead to a aeaningless representation of the
incoapressibility phenomenon at, the global level, unless one can
prcve that the resulting interpenetrations of neighboring defomed
elements cancel each other or are negligible.
Summing up all the above considerations, we can say that,
except for a vary few cases, one cannot expect to approximate
problem &'} by using standard spaces £<L and Q.t such
as those that work well for fluid problems or for linear iacoa -
pressible elasticity. Therefore, a solution that seems ressoaa -
ble, is to construct JV by means of spaces of special polyaoaism
of degree k, for which the Jacobian is ai maximal degree sigai -
ficaticaatly lesa than n(k-l). As we show hereafter this is pre
cisely th* case of F .
Th*Q99m 4.1. If £ - (•oJ»"'*vni d s f i a' d o v a r K *• • o c h eh*c
vi * *a *** t n* B 'f£ * fcOs^ *• * poly»o»i«l of
*l.
-18-
Proof: According to C3.2), each component v. c«o he written.«si
i i* where the ct , • and the 6 * ere scalers end * is the quadratic function given (3.1). We here :
(•.l) Jts+ &£)! =
*n + B2 H tej «J**»' £, '• •*i»*fl2£n
eM*a» £ J 2 n
2 2 n
where constant a'., is the *. - derivative of the linear part of
xi+vi<*>'
Mow we expand the above determinant into a SUB of 2
deterainants whose j - t h , column is either (<*iJ*e2J M"*'°nJ^ OT
3d 1 2 n T 3~ fi * P ...,B ) . As one can easily see, the only determinants
i of this expansion that do not vanish identically are those having
•t most one column with linear entries =*• £ , and the results
follows. q.e.d.
An immediate consequence of Theorem 4.1, is the fact that it
suffices to satisfy (1.1) at the centroid G of K to have incom
pressible elements in the following weak sense:
The measure of Jf in deformed state induced by g c £ is
invariant.
Indeed, if v« denote by A the deformed state induced by
j£ of any sbbset A of K, K e t. , according to a well-known nume
rical quadrature formula, we haves
-19-
nb -\K Jix + MHISOU = \K Jtx * a(x)l d£ = JIG * u(G)l wmas(K) = mas (V.
This shows that, the space Q. defined in section 3 is
• proper choice for these asymmetric elements.
1IMAMS :
i) The relation above holds in the isoparametric case
too, if G is replaced by the image of G under/t^ .
ii) Dsing the same arguments as in [ 15 ], ve can con -
elude that for both n=2 and n=Z, we have * « 1/2 ,
Z i
in case partition x. is used. In the case of T, ,
the same value of 0 applies for n-Zt while *• - 4/9
for n=3.
Hi) In the two-dimensional case, the standard Q. * P0
element has the same properties as the quasi-linear
asymmetric element, as far as the degree of the
Jacobian and & are concerned. It can actually give
satisfactory numerical results as shown by many
examples in £10 J. However, in the three-dimensional
case, the property of Theorem 4.1 no longer holds
for the Qj element.
iv) Another generalisation to the case n~Z of the two-
dimensional asymmetric element satisfying the pro
perty of Theorem 4.1, was presented in [1* 1. This
element has the advantage of being conforming, but
the value of the asymptotic ratio is rather high ,
namely 0 - 4/5 or 0 * 8/11 in the soit favorable
case of partitions. This explains the introduction
of the present nonconforming generalisation.
-20-
Kuw, having proved that th« iacompressibility can be
properIT treated for each element, we would like to assert that
the same is true for Q .
More precisely, letting A denote any subset of fi, set
ting A- = A n K3 K € T, and defining
A = U Ar with Ar- u(Ar) , X* rh ~
where u/i e £ , we would like to verify that
meaa(K) = m*as(K) f I t t. ^ meaeíü) = m*c»(Q) t
or yet that
metis (H) - I uveas fJÜ
Actually letting Q be the deformed state of fi induced
by ji to be defined hereafter, we will prove that :
(4.2) meas(ü) - Z meaB(K) Jfc rh
In the two-dimensional case i t w i l l be convenient to
set ? - fi. Indeed i f SCg + {<f£>2 * 0 * g e 0 , ( 3 . 2 ) i s trivially
sa t i s f i ed s ince V, is conforming and therefore the elements in
deformed s tate do not interpenetrate . However even under the
above assumption, th i s i s not necessar i ly the case of a noncon
forming Y.. That i s why for n=Z ve w i l l s e t 6 = U X , where t
denotes the deformed s t a t s of K induced by the vector f i e ld *&
that interpolates u at the vertices of the elements of T.. In this way 0
can be viewed as a certain interpolation of ft at the points 5,5 being
a vertex of an element of t^. In so doing we can prove that (4.2)
is exactly satisfied for some kind of partitions, whereas in the general
-21-
2 case it is satisfied up to an 0(h } tern.
Before giving the proofs, let us say that, whenever
the above Jacobian is negative for sone x e ft, vc Bust define
A for A c 0 , not as the union of the A's, but with modifica
tions taking into account the interpenetrations of the elements
in deformed state that occur in the general case. This can be
achieved by assigning a subtractive meaning to the sets A such
that Jlx + u(x)l < 0 * x i A, A = JC. In so doing, all the
above assertions for the so-defined Q would be true, and in par
ticular (4,2) with or without a pertubation term. Bearing this
in mind, for the sake of simplicity, we shall assume in this
section that U c V, is such that :
(4.3) Jlx * u(£)l 2 0 for a.e. x e Q
(4.4) JIGK * uCGK)l - 1 * K £ th , where G& is the centroid
of X.
Mow we note that (x * vu) ,„ is nothing else than the
the linear part of (x+u) ,„ .Therefore, since T-*- vanishes at r — — /K ' ix .
•7 vertex S - , J = 1,2,...,n and recalling (3.1) we have :
JlSn+l * *(Sn+in = Jl* * * &&n ¥ « e K .
Since meas(K) = Jlx * nu(x}ldx , assumption (4.3)
implies that meets(K) i 0, which in this case means that the K's
are oriented in the same way as the K's, or yet that the X's do
not interpenetrsnte.
Let us now consider the particular case n=Z. We fur -
ther define A to be the deformed state induced by *g of every
subset A of fi. Notice thkt we are actually defining Q -8.
-22-
Vc first need the following lemma proved in [16 ] :
Lemma 4.1 : Let K be • tetrahedron and n_ denote the outer unit — — — — ^ "K
normal vector with respect to 3K, the.boundary of K. Let • be a
Teeter field defined over K such that v = £ v * with £ e J? and
f be given by (3.1). He then have :
^ diV * d£ = J |^ t. Sf A
Nov ve note that since "«is conforming «e clearly have:
Actually ve can prove that, under a reasonable assump
tion the above equality also holds if the K's are replaced by
the £'s.
Theorem 4.2 : If x, is a compatible partition of C that has no *
base on V ve have :
voi(á) = r vol(K)
REMARK : It is interesting to note that partition -r. satisfies
the assumptions of this theorem.
Proof : a partition satisfying the assumptions of the theorem
can be viewed as a subpartition of a first partition X. of Q, con
sisting of hexahedrons having triangular faces. Each hexahedron
H of Kh generates two tetrahedrons of T^, say Kj snd K^, having
a common base lying in the interior of Ht and lateral faces
coinciding with the faces of the hexahedron (see Figure 4.1).
-23-
Since u is continuous over 8, the common basis of K-
and #,, we have:
Vol(H) = voKKj) + Vol (KJ
Now we want to prove that we actually have
Vol(H) = voliB) ¥ H e Kh
which will yield the result we are looking for, since
Vol(il) = E Vol(.H) .
For this purpose we introduce a new variable x with
the help of the following affine transformation over each K :
x •*• x = x + tiu(x) r+* *S* *s «w «sr
In t h i s way H can be regarded as the deformed state o K
obtained by the a p p l i c a t i o n of the displacement vec tor f i e l d i|>
defined by:
$(x) = i>(x)
3 where <l> = B •, with 3 = (u), - [ Z (u) .] / 3 , (u) . being the
t-2 value of u at S., i = 1,2...,5.
If we denote by \.(x) the area coordinates of K, we have necessa-
rily *--(*) " ^s(&)* which means that jji - S $ where
j < k
Now we have :
r Vol(K) - , Jl'x + i(x)1 dx
K
where J represents the Jacobian with respect to the new variable x.
-24-
Expanding the integrand above, ve have
3 vo lit) =vol(K) + f div Idx + f C I LJ(it) +J($)l<3x
where jj>» is the vector field obtained by replacing the £-th com-
ponent of £ by x~ and div represents the divergence .-. operator
associated with x .
Since each Jacobian of the second integrand above has
at least two columns of form £ •, they vanish identically.
On the other hand, according to Lemma .4.1 we have :
Í div * (x) dx~\ f *. n-K di
However, since irjt is conforming, B coincides for both Jt, and K,
together with $/_, whereas n = -n /B, «„««««, a / J { / f l - - f t / ^ / f l
Therefore we have :
vol(H) - Vol(Kj) + vol(K2) s voKKj) + vol(K2) s vol(H).
q . e . d .
A hexahedron of partition X, figure 4.1
Now for the general case we have:
•23-
Tkeorem 4.3: For any compatible family (T,J, of partitions of Si
we have:
vol(n) - l void) \ < C h2 \u\ Kç T, 2,<
where C is a constant independent of h.
Proof : According to Theorem 4-2, all we have to do is proving
that
I# Ivol(K) - vol(K)J Jta,
s C h* \u\ 2,~
where T* - {K/K e x , , meaa (BK n T*) t 0}
By a direct computation of the increments of volume of
K over its faces, due to the quadratic component J| $ of u, we
get:
Vol(K) - vol(K) dx
According to Lemma 4.1 we get
vol(K) - vo 0 [ ' '
l(K) = - 2 Z \, i>(x) . J» ' dx
Now, F being a lateral face of element Kt we define
the set fin as follows:
Let E be the edge of F belonging to the basis of te -•» ~
trahedron K and let L„ be the plane surface delimited by E and E.
A„ is defined to be the solid delimited by F, F and L_ as ilus-
trated in Figure 4.2 below.
-26-
E
1/41 h
A pertubation of P due to the quadratic components of u .
Figure 4.2
Using classical arguments, if ft,>>, i» regular we
can estimate:
4 Vol(L ) s C k |g| ¥ F 2t-
Now not ing that
if ' ' <áar
we have :
void) - vol(K) £ 6 C h4 \u\
-2 Since card T * S C h the r e s u l t f o l l o w s .
A q . e . d ,
5. EXISTENCE RESULTS IN THE CASE OF PARTITION t£
Let us now prove that under suitable assumptions on
j^, the compatibility condition (2.9) is satisfied for any par -
tition TV . We treat sepaxstréJty oases ii and ii), and the latter
only for the two-dimensional asymmetric element. For partition
T, in some particular cases» the existence
problem (P^) will he examined in section 6,
T, in some particular cases» the existence of a solution to
_?7-
For the sake of simplicity ve vill work with the li
near manifold vf of £,, defined to he x + J£, . We also define
the following subset of V\ :
*f = {u? / uf - * c Xu) n "h ~h ~ n
In both cases i) and itJ we shall prove the validity
of (2.9) under the following basic assumption on u, .
ASSUMPTION A) Let ww? denote the piecewise linear interpolate ~n
x * B *• x of u, def ined in S e c t i o n 3 . The t r i a n g u l a t i o n T, of Q, = JT «, ft^»
defined to be: + ••
x^ = {K / K = J[ u* (K) , K e T p ,
i s such tha t there e x i s t s a c o n s t a n t a. > 0 for which
we have:
- area(K) 2 areafx; & a areaix; Y K e \ \ . D a ft
x Notice that Assumption A) implies that J(Uu,) > 0 a.e.
in il . It also implies that i-, belongs to a regular family of
'2 partitions ÍI^}L » whenever \i* belongs to a bounded subset of
W1'" (ilh) * h.
Indeed, in this case if we set :
h = mar { hv = diameter of K)
and
p - min ípv = diameter of the inscribed circle in K)
-28-
we have ph. i o -pfc, where a is given by -y~ —* with IT ^
27 = max |u.| , 0 being the constant such that pbT i c > 0 h ~* 1,-
•Pi. e ^Tfc fc * •• on* c*n *a*ily verify.
We now consider <?a«* ij ;
In this case both Q = 0, and 0 = 8, are polygons. Thus,
since HM, defines an affine transformation over each triangle K
onto X, we can define a space 7* over 8. associated with T, in
2 ' the saae way as V. is associated with T, , and 7, will have the
saae structure as V, . ~n
Also we define Q. to be the space.of pressures analo
"2 gous to Ç, for triangulatiõn T. .
* ";
Let us f i r s t cons ider the subspace Q? of tho<se pressures
that are constant over Kt K, be ing a s implex of T . . According to
C 4 ] leama C23 i f 7 = {v / v e B*<n), % = 0 on f„ 5 r 0 } , * ^ « «fy 5 v « £ such that
(5.1) (dlv g , qp , â Bo Ú2I2 -
(5.2) | í | 2 í 5 íC.líJj 0 i -
where Bo > 0 and CQ are independent of q,-
Lemma 5.1. There e x i s t constants So > 0 and C0 such that with
every q9 e Q? we can a s s o c i a t e a u,c 7, that s a t i s f i e s :
( 5 . 3 ) g^fS-J = 0 for a l l v e r t i c e s 5 of a n a c r o s i a p l e x AL1 K^t
the KJ'B being the simplices of a nscros implex K c i-., where 2
T. i s the f i r s t p a r t i t i o n of 8 upon which T^ i s constructed.
(5.4) (dtvfo $ # 0 ^ f c « íol*tl0,£
•29-
Proof : Let V e V satisfy (5.1) and (5.2). We associate with V a
vector field u, c V, such that w,,' satisfies ^ Í e T, :
iS-fá; = 0 if S is a vertex of Í. i = 2,2...,n+I
J . £ da B .
fcrV-l — where B. is the base of K. and Áf. is its mid-point. (5.3) ;s X x x
thus fulfilled.
Using Lemma 4.1, letting i, be the partition of 0 into
macrosimplices K, K e T,,, we obtain:
J * div ú, da: - div V dx V- K e T.
X
This yields :
(div ik , q\) = (diV h «i > *
which in turn gives (5.4), taking into account (5.1).
In order to prove (5.5) we first use the Trace Theorem and we
get: A *
bh\iat. S * * ' V c(Ki} * c'(Kx} l U H i . i . ,
which a c c o r d i n g t o Aesumption A) y i e l d s :
\wh\ , s c (n, C.h) \z\ - with c < -
Thus using (5.2) we get (5.5) with C0 - C0C q . e . d ,
Let now i- = me as (K.), lsiin+1. Without loss of gene X X
rality we can assume that 4. 2 4„ £... 2 'n*l
-3-0-
Proof : Let q\ - q\ + q\ where qjj e $° and <?£ < 5J .
i» «
tfe first construct « vector field £, € J£, satisfying
(5.3) in the followings way :
É '*
£.= â «t every vertex or aid-point of the bases of X ,
« ti . If C is the coaaon vertex of K. , i=l,2,..,,n+2, we
define *l(G) to be of the form:
Zh(G) = * A »i
where the mis are the oriented edges G S. of the fc'-e, as indi-i. * *
cated in Figure 5.1 below, and the y. e are given scalers de -
K' pending on the q. e only (see (5.6)).
X € T.
Macrolenents of partitions r, and x, for ns2
Figure 5.1
First of all for n=2 we set
K .. K . K _ K V2 - q 2 and Y 3 - Q3
Now using Assumption A) one can easily estiajats :
-31-
Let 0. be the subspace of Ç, generated by the set of
orthogonal functions l»,»,...^ j.7 1/ such that suppin..) c X
i - 2>...,n+l- and
w - z
<
* - 7
J _ "1 n2 ~ i0+ 4
if x ' t A'.
if x e K2 v K2
i|7 = 0 if x t X.
< n 3 •j if x c X „
—1L- if I £ Í, 4,, ~ 3
r A
K
n = 3
2 if í e Í.
-4,
-77" " £ K2 0 if x e X, u £„
~ 5 4
<
n» - 0
-4
if x e if u K 2
-2— if x c X 7
n, - 1 if x c £ "3
r
< K
i f k * ki •*• * j
*7>1 ' 3 4
As one can easily verify we have n . 1 1?? ¥ a\ <: <2, , i- 2,
and «, = *?©<
Let now q, be any function of Q- . We can write
(3.6) <,' = £ E' 4 nj
where the a",'s are given scalars
"n
Lemma 5. 2 : If Anaumpl.inn A) holds, for every q-, c Q, t h e r e
e x i s t s £ . < V, s a t i s f y i n g ( 5 . 3 ) t o g e t h e r w i t h
( 5 . 7 ) p - , ^ k A l 0 > Q
for som»? 3 , > 0 i n d e p e n d e n t of q-.
-32-
(5.8) \ih\ *C(uhl |,Jl 1*Q 0,0
Now dropping the superscript K we get after siaple calcutaltions:
2 -3
1 2 !*fcL Í ~ ^ ' " í L l ' *(3|,*31^ I * " 78^ "*2 ' "«3*
InJ2 - + <i|n,|* . * 2-t± tq\* qi)
Since /ssumpt-tort A J ianlies that *, St a 4. vc have
0,A a
How we prove that
(5.9) <dU'zh, « í v s * * ' i « i i ; , 5 , * * > < > .
A straighforward calculation gives :
jw. Ih^h*** ui+ V T7? i3 *1 + V*. 2 ,
Thus we have:
low as one can easily check, for any a > 0 we have
2
*1A2
v%7 • V ! < 4(A1*l2)(Â1**3)
which yields
' diV íh qh d* * A2(q2 * qZ]
This in turn implies Í5.9) with a' - a4/2 .
In the case n=3 we set
Y2 * q2 Y3 ~ <i * qK3 •nd A = q¥4~ «3
-3 3-
Like in the case n—2 it is s trai;>htf orward to derive
the estiaate (5.8), and dropping again the superscripts A vi
also get:
4 &Z
1 " 0,K i^2 i l 0,K *4 * ' q
S i n c e Assumption A) i m p l i e s t h a t * . ' > a" 4 , , we h a v e
e\ m \ I,.1!2 • 2 meas(K) , 2 ^ 2 ^ . 2 , (3.10) jqj * (q2 + q3 + tq4) O ,K a
On the other hand simple calculations yield:
mi * V 2
Thus we have :
7 ' * 1 *T 1 2 7 4 P
mean (A) 0,K Z 3
or yet
.-,' ' 1, N , 2 ^ 2 ^ _ 2 , meas(K) (dtv zh> q}) i (q£ * q3 * Zq.^ g
U , K
,á This, together with (5.10), imply again (5.9) with a ' -Co/2;'
Finally we proceed like in C 16 ] Theorem 4.2, namely we
s e t Tit, ~ ®lÀh * 2,fr * where u, is defined in Lemma 5.1 and 8 > 0.
From (5.4), (5.5), (5.6) and (5.9) it is clear that for 6 suffi
ciently small there exists 8, > 0 such that (5.7) holds together
with (5.3) q.e.d.
-34-
Hon we further prove;
Lemma 5.3 : With every q, c Q, we can associate g^ e V^
that satisfies
(5.11) £t(SJ - £ , for every vertex S of a supertrian-
gle K, K e rB
(5-12) — m p 2 B*! «*' wher* B^ is a strictly positivo parameter independent
of qh.
Proof : Using an identity encountered in [ ,page 1083
we obtain :
(5.13) bh(Uuh , 2ft, qh) = Jfi qh dlv Jfc 4 "
where fajt) ~ Zh(&)
On the other hand, fro* Assumption A) i t i s straight -
forward to establish the existence of a constant CÍQ,^) such
that :
ll£fcll* C iüê%h) l í f c l^ í
whereas
Now, if g. is che field defined in Lem»a 5.2 we have
(5.11) and (5.12) with 6ft = a3'* C~* »h > 0. q.e.d.
As a final preparatory result we have :
Lgmma 6,4 : Under Aasumption A)$ for any g^ e fy sa
tisfying (5.11) we have:
-T5-
Pvoof : Taking into account the definitions of :' anJ . it" w* prove that
e T,
«re have the Lenaa. In order to prove the above equality we rewrite :
\KadjT l u l . l i h ^ = lim I \K [_J(x + nj^ + £ * + 6 vh) -
where + is given by (3.1) and £ is a linear combination of w., i=I,Z,...,n
2 and w +«* j<£ being the value of j£. at node S. of K e T. (see Figure 3.1)
Passing to element K using the affine transformation and notations
already e'.countered in Section * we get;
'JC «í?T 7 H? . 7 £, dr = lim ^ L LJ(x * B | + 6 £, ./
(5.
9->0 v 'X
-hx + % in dx
Expanding the right hand side above and taking the limit one gets :
14) f adf y4 . %^dx= f dCv^to* T (-!>*+*[ d-ibi^di
where av.
•tot. 3
9v. _ i l
* 3x. f t
, ?>$
~T —-—
3
Zv.
3*,
3u
%r.
*/
, v.-ft',, .. •, v J
Now, according to (5.11) we can write V, as the sum of two compo -
nents, namely v, - a X - + i#. Then if we expand the above determinants
into finis of two determinants corresponding to these components of £,, the
one associated with b\ is readily seen to vanish identically . Thus we have:
-36-
A - ftlüll li **»»J 111 „ w -«Vi ~ •* • ' -*• "" — — -*-j c . . «her* c.. •= ** aí, aí. aí. aí. ** **
Row we notice that :
i* *' aí- i- ai. aí. L aí.
0. a.
• j as
iSJ K 3 xi 9 xi 'í '*
- n ax. . But since •=*• = t «— X, , we have
axk y = i a«k
í *4 - í mea9(K) t n ax.
Taking into account the eleaentary identity I A. 5 *~^-.-,»
finally g«t :
- ax f já s-?zi (K) —22-K * * ***
(5.15). j, d^. Taking the above relation to (5.15), j, d^. dx is readily
seen to vanish.
The result then follows taking into account (5.13) and
<5.1«) q.e.d.
How, as an ianediate consequence of Leaaae 5.2,5.3 and
5.4 wa have :
Th*or*n S.J : If g^ satisfies Assuaption A) for any a > 0, (2.9)
holds in eat* i). n
Let us now turn to oa»e H). This we do for n-i only.
-37-
In this case £, will be the union of triangles with one
parabolic edge, such that its boundary T, coincides with T at
least at the nodes of those triangles that have a parabolic edge
(base) on T.. Let also Tj be the portion of V, consisting of n h n
ouch parabolic edges that have its three nodes on TQ.
Now instead of fivxurnptir-n A) we make a stronger one, na
mely
ASSUMPTION B) J(u IA 7 j 0 almost everywhere in Í22 D
Taking into account (4.5), the above assumption implies
Ar.sumyt'L-on A). Moreover, it allows us to say that u, is a bisec
tion between Q, and ft, - u, (Q J. In this case Q, is a domain that h n ~h , h
has the same structure AS si,, in the sense that it can also be
viewed as the union of isoparametric elements X,where K^u.fK), K e T,.
~2 . . ~ Let then T, be the triangulation of Q, consisting of the
K's . Similarly, let T, be the set of curved superelements
3 2 ~ K ~ u K. upon which T, is constructed, and let T, be the
i=l partition of 0.. into curved superelements K vheie K = Ui(K) . Kei, r n r ~n h (see Figure 5 . 2 ) .
K C T . n areafK^J =avea(K.) , i- 1,2,1
S u p e r t r i a n g l e s o f p a r t i t i o n s \- and T ^ F i g u r e 5 . 2
-38-
For simplicity we consider the case where ¥ K c T,tarea(KJ=
- <zrea(Kç} - area(K„) , although the more general case can be trea -
ted without major difficulties.
Ni if /T. = Jlu^fK), Assumption B) , hence A), implies'
- aread) 2 3 area (K.) > a area ft) , 1 < i i 3 .
~ 1 Since areaUC.) = area (K.) = -= area (K) ¥ K e T* .
t 1 0 n
Let us now define the following spaces of functions defined
over .1, :
We equip V, and Q, with the norms || • || and |* | g iven
r e s p e c t i v e l y by - || j ^ | | = | ^ i _ , y ^ j ^ and \qh\= \qh\ „ , % * %
(Since y, = 0 on r 0 , = ?Q , Ü • |[ i s a c t u a l l y a norm) . n n fa Let us also denote by x the new variable u, (x)
More generally, for every function / defined over ÍI. we
denote by / the function defined over 0. such that
In order to prove that (2.9) holds, we use the following
theorem given by Le Tallec :
Tneprem 5.2 ; [10 , Theor. 4.5] : Under Assumption B)J(2,9) is
equivalent to :
"3BL > 0 such that
-39-
f % div 2h d ~
(5.16) Sup » , \ |;fcI J£?fc« \
where diw represents the divergence operator with respect to the
x variable . .-
The above result states th't if suffices to prove the
linear discrete compatibility condition between spaces 7, and Q^
to have existence of a solution to (Pp.} in the isoparametric case.
Now, in order to prove (5.16) for the asymmetric trian
gle we eive the following lemmas :
Lemma 5.5 : Let Q? be the subspace of Q, of those functions
that are constant over , K* ¥ K e T, . Then for every q-, e Q£
there exists a vector field w, e V, such that :
(5.17) j^ qh dZv wh d x 2 0O 1^1
'°h
(5.18) |!^|| s ^ \qh\
where Bo and C, are strictly positive constants inde -
pendent of q-, .
o re
Proof ' According to [ 4 ], Lemma C2, for a given q, c Q , the 1 ~
exists £ c # Cfi, with y •= £ on r0, such that
' S i ,
-40-
Now we construct a vector field w, e V-, associated ~h —n
with V in the following way :
— ~2 For each triangle X « T, , we define two perpendi -
•+K •*•](
cular axes x_ and x, oriented in such a way that they corres -•+ -*•
pond to rotations of the reference cartesian axes x? and x„ of
an angle $
Dropping the supercript X for simplicity, we determine
$ in such a way that the straight line passing through nodes S,}
and 5, of K forms an angle of H/4 with both x, and x. .
Let x. be the variable with respect to axis J
*j > * * é « < •
Clearly x- and x. will coincide for any pair of elements of T,
that have a base 5 as a common edge. Let the local numbering of
the vertices of each element respect the usual permutation con -
vention (in this way, S„ and S, interchange within each element
of such a pair, as shown in Figure 5.3). Now for each X £ T, ,
let & be the curved abcissa along B with origin in S„ and n(&)
denote the outer unit normal vector along o with respect to K
We also denote by n.(&) the component of r\ with respect to x. .
- 4 W
Figure 5.3
Let W = (wltu„) , ü - w, ,v and w and tf be given by ~ 1 6 •** "n/*. a 4
w~ - Wj oom • + w„ s in +
w4 - w- «in + + w, COB +
Mow we check that w« can uniquely define w, and w. < (and
consequently w) in the fol lowing way :
The values of w_ and w. at the v e r t i c e s of X are
given by
tf* C5.J = vA(S.) = 0 a t 4 V i = J, 2, S
The value of ü - and w. at node 5 . are such that
jL vj n^UJcU ' J Vj rijUJdA j = Z,4
where V, i s the component of £ with respect to * . , j = Z, 4 .
-42-
Since u, e Kj»»ve c a n compute the coordinates x_ and
*^ in terms of the reference coordinates x. and x,(see Figure 2.1)
used for defining P over X, in the following way :
*4 = i44- 2(* 'i* * J/l», *, * ç í Í, + 1 I £B • *l
where Çj = ** - JC, and Ç * = x* - *. i = 2,3,4 and
i í . , i . . ;;_ - *, cos i * xn 8in +
Using the above relations we make a change ct variables in
the integral I w . n . U ) d* » Ô = 3,4, namely from i to i ,
where A is the abcissa along the edge S of Í with origin in £,
(see Figure 2.1).
d x d x
Since we have n,(A) = — - r r - «nd n.(6) - - «. , for
a vector field £ defined over B, whose components with respect
to x, are /. , j - 3,4, we have for the X- -component :
* s2 82 L a£, d* aifl df J .
where /.Co J * f4U) . Since - r ~ - = f-JJ t - 5 we have : * * dh 2
-43-
whereas an entirely analogous relation holds for the x. - com -
ponent.
Now since w . .Z = 2 w ,($) & (V2 - £) , we have :
(5.19) f w3 n3 U) *" = | ÍÇ* - çjj w3(S4) B
and ana logous ly
(5.20) T w^aJd* = 4 ííj - «J > W / V •
Since by construction U* - ?f |= |çj - çf |= y^ length(i)#i,
w can be defined uniquely.
Furthermore, proceeding in the same way for every ele
ment we can define a vector field g. e 7, such that":
j u, . n(i) d& =? £.n(«)d4 for every base S of X e T.
B~ 8
This yields :
f lh &v £/, d l = f £ A <# v £ á* * ?, c «J , «ft °*
and consequently (5.17) holds.
On the other hand we have
Kti•, * K Kex* 'K
But
J. l** I* di - *J C^J | IXP4 I2 4 , * « 3,4 ,
where j?4ix) ^(x) , p4(xi = 4 Sfa t * - | £ / J f C/jj (*)!
Now, according to Assumption B) and standard estimates we have :
-44-
\z h\2 4 * <? *'l»»li'.-~2 PX
where p denotes the diameter of the inscribed circle in X
Now if area(K) a area(K) we clearly have :
~ „ ' „ 2ai>ea(K) „ 2a area(K) PK * p* * = * 3fc, **IK*I
J.-
If area(K) i area(K) we use Assumption B) together with geone -
tirical arguments sketched in self-explanatory Figure 5.4 (we omit
details for the sake of conciseness). It is then possible to
prove that p„ is greater than the diameter of the inscribed cir-
cle in a triangle K', defined to be the hoaotetical reduction of
K with ratio 1/2 .
Hence we have in this case :
Tangent to Í at S2 and S-
Tangency points
2!j&|8tn9
Triangles K and K when area(K) S area(K) Figure 5 .4
-45-
Th i s gives ; 4
4'lüjfcl 4
L 2
«here c i s the constant of regularity of { T J . (see
Section 2) .
On the other hand, by construction , (5.19), (5.20)and
the Trace Theorea we have :
L Md* is *
l»/V *
Therefore
/2 ' 3 PX
B ' V II Ell ~ iKfcl *C 2*f l l * i
i i y * * *~2i*/ iiE»j s * c<V *"I'*i* si -2,« *?ft 2,» 2,0^
which proves (5 .23) with Z?fc = C (Qh) j« fc |J m /h. * q.e.d.
Let us now construct a vector field s. e 7, associated
with the subspáce 5ft of $ft, iucb that Qh = 3j © 3ft. Like space
Q, of case Í.J, Qh is spanned by a set of orthogonal basis V V
functions ívô * Y,}*? , ** defined in an entirely analogous way
(for •** ~ •*« - * 3)• N o w w e prove «
Lemma 5.6 ; Let'4, b a function of 5L whose coaponents with
K K K K
respect to Y« and y- are respectively ÇJJ and £* » * « T-Under
Assumption B), the vector field {^ e £ft that vanishes at all
the vertices of T« and who3e value at the coaaon vettex G of
K.t 1,2,Z, K. c K is givan by (refer to Figure 5.5).
- 4 6 -
s a t i s f i e s :
~1 (5.21) | | « f c | | * « S ^ l í í l , CC^ } < -
(5.22) J_ qJh dZv zh 4x 2 e j k j l vith Bj > 0
Proof : (5.21) is a trivial consequence of the definition of £^.
On the other hand a straightforward computation gives:
where 4 . - area(K.) i = lt2t3 .
Assuaing again that the local nushering of the nodes
of K is such that A- 2 &„ 2 A_ we have :
if we just have A- 2 a area(K)/3 > 0
Thus we can write :
A
which y i e lds (5 .27) with 1 ; = | o q.e .d,
Supere lament X Figure 5,5
-47-
Now defining üft - 9 «^ +zh, from (5,22), (5,23) ,
(5.26) and (5.27) ve have (5.18) just like in Lemma 5.2, for a
sufficiently small 0. Hence, as an immediate consequence of
Theorem 5.2 and Lemmas 5.5 and 5.6 ve have :
Theorem 5.3 : Under Assumption Bl the compatibility condition
(5.16) holds for ease ii). 0
REMARK : Assumption B) and A) with a > 0 express in particular
the fact that the area delimited by the base of the triangle in
deformed states B and B do not account for the whole of
area(K) -area(K). This fact was crucial for the assertion of the
existence results in both oases i) and ii). . 0
6. THE CASE OF PARTITION xl AND NUMERICAL EXAMPLES
h
Let us finally consider the existence of a solution to
problem (Pi) when one uses a partition of type T, for the special
case described below ;
Let a be a domain that can be viewed as the images of
a rectangle Q with boundary T, through a mapping A : g * v (x) .
Here ti is an element of a reference vector space V, such that det (L * t S^S^ * °« •••• in &• Zu is «efioed in the same way
2 as V. in Section 3, for a compatible partition T, of {I into
equal triangles illustrated in Fig. 6.1. T. is constructed upon
• first partition Xi of 0 into rectangles by means of a uniform
M x S grid, in such a way that the edges of T, over which
£. c £. is necessarily linear, are the edges parallel t« the
reference axes £- and £
-48-
Reference rectangle 5 aad partition T.
Figure 6.1
We assume that the fixed portion To of r over which
v, € V, vanishes, is the union of edges of rectangles" of Xt» If
we define r0 to be the iaage through A of r0 it is clear that TQ
consists of polygonal lines (eventually disjoint), just like
r = A(T)
Now we define T, to be the partition of ft into isopa -
~ '1 rametric elements K that are the image of K through A3 ¥ Kt x-, .
Similarly we define Xt to be the partition of ft into elements
that are the images through A of rectangles of Xt •
Notice that the union of a pair of elements K and X'
of i. that are the images of two triangles of T\ contained in a
given rectangle of Xh * i» * quadrilateral (with four straight
edges). Therefore every element of x^ i» • quadrilateral (aee
Figure 6.2).
- 4 9 -
S*
Elements of Xh and T.
Figure 6 . 2
Nov, according to C 10 2, TLeorem 4.5, it suffice
prove (5.16) to «•sere the existence of • solution to fPtJ
to
assuming of course that Jfu.J > 0 a.e. in fi.
Let us denote the quadrilaterals of Xt D7 v *
í - 1,2,..., Af x JP , vhere R. - A(R.). R. are the rectangles of
X, that we number in a systematic way along the columns, row by
row, as indicated in Fig. 6.1
Let n - in^H^n,^} i - 2 b* tn' D«»*» of t n e «P«« o f
pressures Q. associated with T. , in such a way that suppí^) c R^
suppfn^^^g) c Rj t 1 & i & M x M, with :
\(£> = *
nt.r2; = -J
t where K_. and X. are the curved triangles into which R^ is sub -
divided.
Let also v s ív.^}^ b« the usual basis of j^, where
h is the number of free nodes of T^ . Each y^ is associated
with a degree of fTeed a of £ft which are assigned to two diffe -
rent blocks. The first one corresponds to the 2MxN components of
a field of Y.h t n â t •rt associated with the nodes lying in the
-50-
iaterior of R. « Xt > while the remaining degrees of freedom are
assigned to the second block. How we number the degress of freedom
of j£, *tt such • way that those i» the first block carry the nur
her from one to 2MxR and those in the second block the numbers
fro» 2M*N*1 to 2L
Fin*i W , l«t Bfc be the (2L ) * '1**M) matrix whose
entry at the i -th row and j-th coins» is given by
j n. div M.dx.
According to C 9 3, Lemma 5.1, the existence of Í. > 0
such that (5.16) holds is equivalent to the rank of B, being
equal to dim Q, = ZM x M.
In order to examine this rank condition, it"is conve
nient to split B, into four rectangular matrices, according to
the pattern below:
" ^ « ^ J =
• I -
2
H*M
ms+i
ZM*n
i
<
*H
. . . . 2M*S 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1 1
'
<
<
21
-51-
First ve notice that all the terms of B, vanish» since the
basis functions of £, associated with nodes lying in the interior
of the quadrilaterals have zero flux along its boundary.
Secondly, recalling (5.19) and (5.20), we can say that
the entries af B, in the positions 3 = 2i-l ór 3 = 2% , 1 S i s M * Nt
+ a 2 are given by expressions of the form - ^^xp ~ aV^^ » k = 1»2 *
I I where (x-,x_J t t - 2,3 are the coordinates of the vertices of
the curved diagonal of B. .Since those vertices are necessarily
distinct, at least one of the above terms of B, is nonzero.
2 Finally we notice that matrix 3, has exactly the sane
entries as the matrix studied by Le Tallec for the Q- * PQ ele -
ment associated with a partition of SI into quadrilaterals, like
With the above considerations it is easy to conclude 2
that the rank of B, is 2M x N, provided the rank of B, is H x N .
Therefore the condition of existence and uniqueness of p. such
that (%, , p,) is a solution to (Pi) becomes the same as in the
case of the Q~ x P0, at least for domains defined as above. That
is why we refer to the work of Le Tallec C 9 for the proper
answer to this question in various situations depending on the
shape of Tg.
Nevertheless, with the purpose of givi.^ a brief
illustration of his results we mention here the following case :
If To is contained in • set that is the in*<i through
A of two non disjoint edges of ft, then the »>>*ve axis't a and
uniqueness result is guaranteed. If on the oth> hand To . does
not fall in this category this can only be asserted under some
restrictive condition.
-52-
Finally, we would like to give a short account of the
three-dimensional case, when one uses a partition of type T, .
First ve note that, as far as linear problems related
to incompressible media ar.e concerned, we can prove that, under
identical assumptions on V and on the first hexahedral mesh upon
which T, is constructed, the existence and convergence results
••.hat hold for the tetrahedral element considered in this paper ,
are the same that apply to the mixed element defined as follows:
V-, is the space associated with the classical isopa -
rametric trilinear functions (Q-) defined on the hexahedral mesh,
and Qi is the space of constant functions (P$) over each hexahe
dron.
Corresponding proofs will be given in a forthcoming
paper on the Stokes problem. Note that in the case of the
Q. x p0 element, the existence and convergence analysis of two -
-dimensional linear problems for a rectangular ft, due to
Pit-dranta [8], have later been shown to apply in an analogues
way to the case where SI is a parallelepiped [12], Por this reason
one can expect that the existence results given above for our
nonlinear problem, can be easily extended to the case of a
parallelepipedal domain, if one uses a partition x, constructed
upon a partition of fl also consisting of parallelepipedal elements.
In [15] one can find numerical results related to the
two-dimensional element treated in this paper. Lat us now illus
trate the superiority of the three-dimensional one, compared to
(*) classical methods . Indeed, for n=Z, the appropriate numerical
-Comparison has actually been made with standard elements such
as the Q1 * PQ element.
-53-
solution of (Pi becomes a critical issue, as the Jacobian is a
high order polynomial for standard elements. We have taken a
compression test-problem, which is precisely one of the most
difficult cases to simulate correctly from a numerical point of
view. As one will see,particularly stable and realistic results
are obtained.
In our test-problem we take £2 to be a cube having a
fixed face To. We bring the face opposite To closer to it
parallelely to itself of a certain percentage of the edge length
I of R. Due to symmetry only the eight of the cube shown ir. Fi -
gure 6.3 is taken into account in the computations.
The TV partition is obtained in the following way:
We first subdivide ft into 27 equal cubes. Hext the
slices of cubes adjacent the faces x. = 0., i = 2,2,3, are sub -
divided into three equal slimmer slices, parallel to these faces.
This yields a mesh consisting of 125 parallelepipeds. Finally
each parallelepiped is subdivided into eight tetrahedrons,in the
way shown in Pigure 3.2b.
We show in Figure 6.4 the boundary of the eighth of
the cube in deformed state induced by a compression of 40%. " It
is interesting to notice that this deformed configuration
corresponds to what one can expect to obtain,by performing a si
milar experience with a rubber cube.
-54-
Initial configuration of the cube Figure 6.3
Deformed configuration ox 1/8 of the cube induced by a com
pression of ÜOX Figure 6.4
-55-
ACKMOWLEDGEMENT :
The numerical results given in this paper were obtai
ned by combining the author's finite element methods, with an
algorithm of augmented lagrangian type due to Glovinski and Le
Tallec (see e.g. [ Í ]) for solving the nonlinear problem (?).
The author wishes to thank Dr. Le Tallec for having supplied
him with FORTRAN programs corresponding to this algorithm.
-56-
R E F E R E N C E S
1 I Adams, R.A., Soholev Spaces, Academic Press, New York,
1975.
2 3 Breszi, F., On the existence, uniqueness and approxi
nation of saddle point problems arising from Lagrange
multipliers, RAIRO Analyse Numerique,8-R2, pp. 129-151,
1974.
3 ] Cartan, H., Calcul Différentiel.Herrmann-Collection Mé-
tbodes, Paris, 1971.
4 1 Oebongnie, J.P., Sur la formulation de Herrmann pour
l'itude de solides incompressibles, Journal de Mecanique,
Vol.17, n9 4, pp. 531-557, 1978.
5 ] Girault V., ft Raviart, P.A., Finite Element Approxima -
tion of the Navier - Stokes Equations, Lecture notes in
Mathematics, Springer Verlkg, Berlin, 1979.
6 ] Glowinski, R., Le Tallec, P. ft Ruas, V., - Approximate
solution of nonlinear problems in incompressible finite
elasticity, in: Nonlinear Finite Element Analysis in
Structural Mechanics, edited by W. Uunderlich, E. Stein
and K-J. Barthe, Springer Verlag, Berlin, 1981.
7 ] Glowinski, R. ft Pironneau, 0., On a mixed finite element
approximation of the Stokes problem (I), Humeriache Ma -
th ema tile 33, pp. 397-424, 1979.
8 3 Johnson C. ft PitkHranta, J., Analysis of Some Mixed Fi -
nite Element Methods Relate J to Reduced Integration ,
Department of Computer Sciences of the Chalmers Univ. of
Technology and the Univ. of GSteborg, Research Report
80.02 R, 1980.
9 1 Le Tallec, P., numerical Analysis of Equilibrium Problems
in Incompressible Nonlinear Elasticity, Thesis, TICOM ,
The University of Texas at Austin, 1980.
-57-
[10] Le Tallec, P., Les Problemes. d'Eguiljbre d'un Corps
Hyperelastigue Incompressible en Grandes Deformations ,
These de Doctorat es Sciences» Université Pierre et
Marie Curie, Paris, September 1981.
Cll] Lions, J.L. A Magenes E., Prohlemes aux Limites Won
Homogenes et Applications, Dunod, Paris, 1968.
[12] PitkMranda, J., On a mixed finite element method for the 3
Stokes problem in IR , to appear in RAIRO - Analyse Nu-
merique.
[13] Rivlin, R.S., Large elastic deformations of isotropic
•aterial, Philosophy Transactions Research Society ,
1948.
[14] Ruas, V.» Sur 1'Application de Quelques Methodes d'Ele
ments Finis a la Resolution d'un Probieme d'Elasticity
Incompressible Non Linéaire, INRIA, Rapport de Recher -
che n° 24, Rocquencourt, 1980.
[15] Ruas, V. A class of asymmetric finite element methods
for solving finite incompressible elasticity problems ,
Computer Methods in Applied Mechanics and Engineering ,
27, pp. 319-343, 1981.
[16] Ruas, V., Methodes d'elements finis quasilineaires en
déplacement pour 1'etude de milieux incompressible* ,to
appear in RAIRO Analyse KumSrique.