A 4 CST quadrilateral element for incompressible materials and nearly incompressible materials

A 4 CST Q U A D R I L A T E R A L E L E M E N T FOR INCOMPRESSIBLE MATERIALS AND NEARLY INCOMPRESSIBLE MATERIALS M. BERCOVIER (i) . E. LIVNE (1) ABSTRACt - We introduce a quadrilateral element for plane strain or axisymmetric analysis of incompressible or nearly incompressible materials. This quadrilateral is made of four constant strain triangles. The divergence is taken as constant over the quadrilaterals; Lagrange multipliers are avoided by using a penalty function approach. Numerical examples are given. Such an element can also be used for the solution of Navier-Stokes equations.

Introduction. The rapid development of the finite element method in fluid dynamics has led to a better understanding as to how to deal with the incompressibility constraint div u = 0 . It is clear that the same question arises for incompressible and nearly incompressible materials. In this paper we introduce a simple quadrilateral element for plane strain or axisymmetric analysis of such materials. In constructing finite elements approximations for this type of problem, one has to be careful that the condition div u = 0 does not introduce a number of constraints larger than the total number of degrees of freedom, (cf. NAGTEDnAL, PAR~:S and RICE [7]). This concern has led to the introduction of approximate operators divh ( ) , ( the is defined by some mean). Detailed study of this kind of operators for triangular elements as well as the related error estimates for STOKES' problem are given by CROUZEIX and RAVIA~T [ 4 ] . This approach is actually the basis of The successful construction given by NAYLOR [8] and used by ZIENKIEWmZ and GOOBOLE [ I l l , for 8 nodes isoparametric elements. These authors use the > technique, (a concept introduced by ZIENIOEWmZ, TAYLOR and Too [10] for plates and Received June 28, 1977. (1) Graduate School of Applied Sciertces, The Hebrew University Jerusalem,

M. BERCOVIER- E. LIVNE: A 4 CST quadrilateral shells elements), for the computation of the divergence on one element O. It can be shown that the use of a 2 • gaussian quadrature rule to evaluate fdivZe dx is equivalent to defining divh u as a bilinear polynomial on Q, (cf. Q BERCOVIER and ENCELMAN [3]). We start by introducing the Lagrange multiplier related to the linear constraint (i. e., the hydrostatic pressure). This gives us the mixed variational formulation due to HERRMaNN [5]. It is well known that even then constant strain elements are not appropriate9 To weaken the condition div u = 0 , we define it as a mean over several elements (or by duality we take the hydrostatic pressure as constant over several elements). In fact we use the 4 CST quadrilateral element of the preisoparametric days together with a constant pressure (4 CST-P). in the case of nearly incompressible materials we can eliminate the pressure term at element level, a procedure that is equivalent to penalisation cum reduced integration. We study the approximate solution of some prob!ems with a known closed form solution. In this paper we consider the isotropic case only.

i. Approximation of Incompressible Materials by Nearly Incompressible Ones. Let fills this that this We

S2 be connected domain of R 2, I" its boundary. Suppose that a body domain. To avoid considering rigid body displacements, we suppose body is fixed a part of the boundary, say/'~. define: H~(S2)={O I r

2 (S2),~~ e L 2 (S2), i = 1 , 2, q~trl = 0 } oxl

and V = H I . (.Q) XH~I (.f2) we suppose that the body is made of homogeneous isotropic material. Let E be the modulous of Young and v Poisson's coefficient of the material. Given a displacement field u = (u~, u2), we introduce the (linear) tensor of deformations eq (u), i = 1 , 2 ; j = l , 2 . By Hooke's law we can write down the stress tensor thus: ail (u) = 2 G (eli (u) + v / ( 1 - - 2 v ) div u)

(1.1)

t

cr12(u) = 4G e12 (u)

element [or incompressible and nearly incompressible materials where G = E / ( 2 ( l + v ) ) is the shear modulus. In the axisymmetric case we have: O'kk( u ) = 2 G (e~k (u) + v / ( 1 - 2 v ) div u)

k=r,z,O

(1.1') tO-rz=4G erz (U) the bilinear form on V x V : a (u,

v) = f JEll (U) O'i1(V)-~E22 (V)'-Jf-E12(U) el2 (V)] dx

can be decomposed in a (u, v) = a ( u , v) +v/(1 --2v) b (u, v) where

(o, v)=2Gf

EE. (u) el, (v)+E= (u) E. ( v ) + 2e,2 (u)

el2

(V)]

dx

g~

and b (u, v ) = 2 G . f (div u-div v) dx ~2

Let f be a field of body forces and T a traction on r~= Crrl. We are looking for a solution of (1.2)

~ 1-2v ~min r 21-a(v'v)+-b(v'v)-v

f

f.vdx--.

[

T.vdr.

1"2

In the case of incompressible materials (v=0.5), our problem becomes:

(1.3)

rain v ~ g and div v=O

--la(v,v)--ff.vdx--fT'vd

when v is close to 0.5, v / ( 1 - 2 v ) is actually a penalty term on the constraint div u = 0 . By a classic duality argument, we introduce the hydrostatic pressure,

M. BECOVIER- E. LIVNE: A 4 CST quadrilateral

let

Then (1.3) is equivalent to find (u, p)eV• (1.4)

satisfying:

~(u,v)+Gf pdivvdx-f f.vdx+f T.vdV

f

(1.5)

q div u dx=O

a

for all (v, q)eV• Note that (1.5) is just another way of saying that div u = 0 . manner (1.2) can be written as find (u~, pv)eV•

In the same

satisfying:

(1.6)

-(1-2v)fp~qdx+fqdiv

uv~O

(1.7) for all

(v, q)eVXW. (1.6), (1.7) is a mixed variational principle due to HERRMANN. (1.4), (1.5) is nothing but Stokes stationary problem. (Note that in fluid mechanics one uses the unknown - - p instead of p). Let ]l" 1] be the Sobolev norm on V defined by

if,,H

( ut=ld=) ''=

Jt\ax, ] + \ax~ / j

element for incompressible and nearly incompressible materials

Then we have the following classic lemma of fluid mechanics: LEMMA i: There exists k > 0 such that

( 1.8)

sup

f

it. div v dx

~

>_k

(f

dx

~2

for all p e W. One step in the demonstration of this lemma relies on the fact that for all p e W one can construct a u c V such that (1.9)

div u = p. Inequality (1.8) enables us to establish (cf. [2]) the following.

THEOREM 1. Let (u,p) be the solution of (1.4), (1.5) and (u~,p~) the solution of (1.6), (1.7). Then: (1.10)

[tu--u~i[ + [p--p.lL, ta)~C (1--2v)

where C is a constant depending on ~2, J and 7' only. (1.10) clearly expresses that incompressible materials can be approximated by nearly incompressible ones. In order to apply this remark to the finite element method we have to be careful to check that Lemma 1 or at least 0.9) is still satisfied. Now by the equivalence of the penalty approach (1.2) and the mixed variational formulation (1.6), (1.7), we can conclude that one can use a penalty formulation in the finite element method provided it is with an operator divh (.) that satisfies an inequality ol type (1.8). Intuitively we see that such an inequality will be satisfied if the number of linear constraints induce by divh (-) is small enough in order to solve (1.9). Hence the success of the reduced integration technique. We now apply these comments to the construction of a simple element.

2. The 4 CST-P Element. Let Q be a convex quadrilateral, ai, az, a3, a4 its vertices and ak and arbitrary interior point (cf. figure 1). We subdivide P into four triangles TI, T2, T3, T4 with common vertex ak. To the five nodes aa, a2, a3, a4, ak we

M. BERCOVIER E, LIVNE: A. 4 CST quadrilateral

i0

-

associate the linear shape functions related to the triangles to which they belong (i. e., for ak we 'have a four-sided ). In other words we assume that the displacement field is linear over each triangle. We next take hydrostatic pressure p as constant over Q. (Hence the name 4 CST-P). On each triangle we have the standard construction:

- f ar~(u, u)=uir Ai u';

i=1, 2, 3,4.

where u i is the corresponding vector of the 6 nodal displacements on Ti. A~ is a non-negative symmetric matrix. Next we write:

I" G./p, div u d x = p Bir u i,

i=1,2,3,4.

Tt

Setting: qr = (ul (1), u2 (1) . . . . . u2 (4), u (k), u2 (k), p) after assembling the four triangles, we obtain that (1.6), (1.7) is equivalent Kq=F where:

( K~ ivLr K = \"'""'-'"'-i"'"'"'""'""--'"']vL i v(I

(2.1)

Zv) a2

a3

04

01

Fig. 1 4 CST-P Element.

element /or incompressible and nearly incompressible materials

It

K~ being the result of the assembling of the (6 • 6) matrices Ai, (i= 1, 2, 5, 4), and L of the (1 • 6) matrices B, to B,. REMARK: L involves only the degrees of freedom on nodes al to a4. It can be directly obtained by computing over the boundary aQ the integral of the normal displacement un. We eliminate the degrees of freedom ul (k), u2 (k) at element level. The hydrostatic pressure p being constant on Q, when v+0.5, it can also be condensed out after the assembly of (2.1). Actually this procedure gives the same symmetric non-negative matrix that we would have obtained by applying directly (1.2) and the above remark. A similar idea is to use a 4 nodes isoparametric quadrilateral with a constant pressure (or dually a constant ~ divergence ~). This was done by HUGHES, TAYLORand LEVY [6] for the study of incompressible flows. It is interesting to note that the 4 CST-P construction correspond to the optimal nine-point first order formula for the Laplacian. Finally, let us point out the remarkable study of NAGTEDAALet al. where a 4 CST quadrilateral with non-constant pressure is used to construct a divergence free approximation. The only drawback of this last procedure is that it does not apply to the axisymmetric case.

5. Numerical

Examples.

We start by two examples related to Stokes stationary equations in the bidimensional case. We next study the approximation of Boussinecq's problem for v=0.5 by a nearly incompressible material. EXAMPLE 1: Couette's Problem. A typical exampJe in fluid mechanics, it can also be viewed as a plane strain displacement problem. Figure 2 gives the details of tthe case as well as its drawn displacement solution. The closed form solution is here: u, (x~, xz) =0.25 xz (xz- 1) us (x,, x9 = 0. The error: e = max lu?~

-u,

....

'l/u, .... 'l

on all nodes is given in Table 1. We see that a parabolic displacement field is approximated up to (1--2v) by the 4 CST-P element.

12

M. BECOVlER - E. LIVNE: A 4

CST quadrilateral

r"

/

d

0

/

0 I!

(D ~O II

II

d~ II

0

element /or incompressible and nearly incompressible materials

13

TABLE I

(o,~)

v

e = 1 - - 2v

e

m a x [u?l

0.499

0.002

4.10 -3 .

1.3.10 -4

6.49999

0.00002

3.6.10 -5

1.3.10 -6

u~ = U 2 : 0

UI=U2=O

U1 = U2 =0

U1 =0

U2=O

i (o,o)

___,~ "r 1 h = 1/2~

U~ =U2 =0 Fig. 3 U n i t s q u a r e f o r analytical m o d e l .

(~,o)

14

M. BERCOVIER E. LIVNE: A 4 CST quadrilateral -

EXAMPLE 2. Although this example lacks any physical interpretation it is an interesting one, for the solution of this case by triangular linear finite elements fails completely as v tendes to 0.5. Let $2 be the unit square (figure 3). Imposing all displacements to be zero on the boundary and taking the body forces field: ~fl (X, y) -- 12 Xz (X-- 1)2 (2y-- 1) + 2 (y-- 1) (12Xz - 12X + 2) fr

)

(l~ (x, y) = -- h (Y, x) the analytical solution of (1.4), (1.5) is (with G = I ) : u~ ( x , y ) = - - x 2 (x-- 1)2 y ( y - - l ) (2y-- 1)

nT

l

I.,12(x, y) .~. -- 1.,ll (y, X) p~O.

We divide s in equal squares of size h = 1/28, and by symmetry we compute our solution on ] 0, 1/2 [2 only. Figure 4 gives the error:

e=(

z

~1,2

( z

. .... 1, -,

u o

,2

I)/

z

[u:xaotl2),

2

nodos

for various v. There is a slight deterioration from v=0.499, (e--1.195"10-3), to --0.49999, (e=1.25"103,). This can be attributed to round off errors as we proceeded in our program by elimination of the pressure term instead of direct penalisation. EXAMPLE 3: Point Load on a Hall Space. In cylindrical coordinates, Boussinecq's problem has a known closed form solution (cf. TIMOSHENKO and GOODIER [9]). To study the case v--0.5 and to avoid infinite domains we limit ourselves ro a cylinder of meridian section O A B C (figure 5. a) on A B and BC we impose the displacements of the known solution at v = 0.5. Stresses are recovered on each triangle, and then averaged at the center of the quadrilateral. Stresses at nodes are obtained by averaging the preceeding results over the elements to which a node belongs. Table 2 gives us the displacements along diagonal OB. Results are surprisingly good. Tables 5.1 to 3.4 give us the stresses at the quadrilateral centers I, II, III, IV and at nodes V to IX. Thus by using a smoothing procedure we gee results that seem to be as good as those of NAYLORfor the 8 nodes element.

element ]or incompressible and nearly incompressible materials Relative

error

15

I2

O~ tT~ 0"~

Ob t3~

c5

o~ II e"

O~ .,.3-

6

0'~ .q-