SUMMARY. This note introduces certain stress smoothing procedures and the so called 'Loubignac-Cantin' iteration for restoration of momentum balance in the ...
COMMUNICATIONS IN APPLIED NUMERICAL METHODS,
Vol. 1, 3-9 (1985)
ITERATIVE SOLUTION OF MIXED PROBLEMS AND THE STRESS RECOVERY PROCEDURES O.C. ZIENKIEWICZ
University of Wales, Swansea, UK LI XI-KUI
Dalian Institute of Technology S. NAKAZAWA
Marc Analysis Research Corporation
SUMMARY This note introduces certain stress smoothing procedures and the so called ‘Loubignac-Cantin’ iteration for restoration of momentum balance in the smoothed stress fields. It is shown that this iteration corresponds precisely to the solution of mixed formulations in which the stresses (or strains) and the displacements are used as primary variables. The procedure has a very wide field of application and promises to add considerable accuracy to f.e.m. results by a small additional expenditure.
INTRODUCTION Displacement formulations of stress analysis problems (or equivalent flow problems) result, as is well known, in discontinuous and inaccurate stress fields. Various techniques of stress ‘averaging’ and ‘smoothing’ have been used since the advent of finite element methods, the most effective of these being based on a ‘projection’ t e c h n i q ~ e l - ~ . In these we proceed as follows: First solve (using the notaton of Reference 4) the equilibrium statement k = K u = f Second, compute stresses (2)
u=DBu
Third, assume a continuous (or at least a more reasonable) stress field U* defined by shape function N* and nodal parameters U*
Fourth, compute parameters O* so that equality between
U
and
U*
is satisfied in the mean sense
i
N*T (U - U*) dR = 0 n
(4)
This yields immediately on substitution of equation (3) U*
=
M-’
I R
074&8025/85/010003-O7$01.00 01985 by John Wiley & Sons, Ltd.
N*T U dR = M-’
(h*T
D B dR)
U
12
Received August 1984
0. c . ZIENKIEWICZ, LI XI-KUI
4
and s.
NAKAZAWA
where
The computation of ( 5 ) is done separately for each component of stress and if M (which has the same structure as a classical 'mass' matrix) is lumped4 the cost of this operation is trivial. Good results are obtainable with both lumped and consistent forms of M', but the consistent form, in general, gives better results. Figure 1 shows such smoothing applied to two elements for different distributions of U. It is immediately evident that the consistent form gives more realistic (statistically equivalent) values whereas the lumped form applied to a smooth solution simply reduces its values.
\' '\
Nodal average IN) or lumped M (I)
\
~0-
(Ll and (NI
Y,
\\,,
v
0
\~~
/A\. \\
I
\\
I iteration w l t h M, (L/Cl)
\\ \
Figure 1. 'Variational' recovery of U using consistent M ( C ) lumped M(L) and iteration LICI. 'Nodal' averaging shown for comparison (N). Linear interpolation Cor U' assumed
To avoid the difficulty of inverting a consistent matrix a simple iteration can be used to obtain the solution of equation ( 5 ) by writing U
dR
-
(M
-
ML)
U"j-1
(7)
This converges very rapidly. In Figure 1 the improvement due to a single iteration is shown. The computed 'smoothed' stress field U* does not, however, satisfy the original equilibrium condition and it is desirable that the equation
I
Bu;b dR = f
R
should be satisfied.
ITERATIVE SOLUTION OF MIXED PROBLEMS
Loubignac er al.".' propose an iteration of the form
U;+l
U1.+ 1
= U;
+ AB;
i
. = D B U , .+ I +
=
1,2.. by smoothing.
= 0 starting value is used)
but these authors use simple nodal averaging to obtain U " . Nakazawa and Nagtegaa18 show that very much improved results can be obtained if the scheme represented by equations (3)-(6) is followed. In particular the algorithm is found highly efficient in problems of plasticity. The results are indeed so good that in many cases a very coarse mesh yields almost exact results. In some problems it is more rational to smooth the strains
rather than the stresses U . Now the sequence of equations (2)-(9) would be replaced by (10)-(14).
including the iteration
(14) In what follows we shall show that the above schemes present a simple iterative solution of a mixed formulation of the problem.
EQUIVALENCE WITH MIXED FORMULATION
In mixed formulation we discretize directly the stresses
U*
and the displacenients
U
as
and start from the differential equation of equilibrium and the constitutive relations written as Lu+b=O
and U
- D L7' U = 0
in which we define the appropriate strain operator L. On discretization we obtain (after Galerkin type weighting of (16) by NT and (17) by N'k'r)
6
o. c. ZIENKIEWICZ, LI XI-KUIand s. NAKAZAWA
where M is the same as in equation (6). Now we see that the above system of equations is identical to the system we solve iteratively by the use of scheme (9) when full convergence is reached. Thus we have proved the equivalence of the two procedures. The mixed formulation we have just obtained is not identical t o that generally used (and derived from the appropriate variational principle). The equation system (18)/(19) is not symmetric - and generally it is convenient to premultiply equation (17) by D to obtain such symmetry. Now the system can be written as
where
-I,
Q = BT N* dfl
and
In the iterative procedure described previously the identity with this symmetric expression is obtained by weighting the stress continuity with N*T D-' and using M as defined in equation (22). This is equivalent to ensuring that the strains caused by U and U* are equal in the mean. Which procedure is superior should be investigated, but obviously no differences arise if D is constant throughout the domain of the problem. The recasting of the mixed form as the independent interpolation of strains E* and displacements U results in the alternative iteration, vis. equation (10-14), which we have described before. Once again the same comment about the use of M rather than M can be made.
7-
10 Elements
20 ELements
B
+4 1 0 Elements
Figure 2. Cantilever beam
-
refinements used
ITERATIVE SOLUTION OF MIXED PROBLEMS
7
CONCLUSIONS ‘Mixed’ rather than ‘irreducible’ formulations have met with much success in the field of finite element treatment of various problems of mechanics.’ In particular the forms in which stresses (or strains) are interpolated independently of the displacements have been used with success in the context of fluid mechanics“’.’’ and visco-elastic flows.I2 However, mixed solution methods have not achieved widespread popularity owing to the large number of solution parameters involved. The applications discussed in References 10 and 11 use discontinuous stress fields to allow for easy elimination of internal stress parameters. Only in the solution of visco-elastic flows, where stress continuity is required by the presence of certain convection terms,12 has their use been realized despite the computational difficulties involved. The iterative procedure outlined here avoids most of the computational difficulties - with convergence being available in a few iterations. Indeed incomplete iteration provides a continuous stress (or strain) field and results which are better than those attainable in pure displacement form and thus the possibility of specifying only a fixed number of iterations is appealing. If the problem is non-linear the iterations can be done simultaneously with those necessitated by the non-linearity and no additional costs are involved. End displacement
10
20
Stress ( a t point B Figure 21
CO
-Number
10
o f elements
10
20
40
10
10
20
40
m
20
CO
20
40
c
20
40
Figure 3. Cantilever beam of Figure 2 . Results of displacement and ‘mixed’ iterative analysis given as relative values to ‘exact’ beam solution. ‘Displacement’ analysis - ‘Mixed’ analysis-3 equilibrium iterations -CL--‘Mixed’ analysis-6 equilibrium iterations (fully converged) -A-(A) with consistent M (10 ML) iterations) (B) with consistent M (3 M, iterations) (C) lumped M
-o---
8
0.
c.
ZIENKIEWICZ. LI XI-KUI
and s .
NAKAZAWA
For the problem of incompressibility this iteration can be combined with another one necessary to impose this additional constraint (Augmented Lagrange Procedure).‘O Indeed the iteration itself bears considerable resemblance to such general processes. We conclude the paper with two numerical examples, both using bilinear elements. The first, that of a cantilever beam of Figure 2, illustrates (probably in an exaggerated manner) the importance of using the consistent matrix in stress smoothing. The stress at point B of the cantilever and the tip deflection are compared with the ‘strength of materials’ solutions in Figure 3 for various numbers of itcration. It will be observed that *Results with three and ten corrective iterations for achieving consistent ‘mass’ matrix give virtually identical results. *All cases show very considerable improvement of stress distribution over the pure displacement solution. *That displacements are very much improved over the pure displacement solution except for the ‘lumped’ M computation where the softening deteriorates the solution. The obvious applicability to practical cases is evident from the example shown in Figure 4.
-
Disp method - - - A - - Mixed solution (with It-itera) Disp method & - 0- -
Figure 4.
L-shaped region analysis: comparison of displacement solution and mixed iteration
REFERENCES 1. J.T. Oden and J . N . Reddy, ‘Note on an approximate method for computing consistent conjugate stresses in elastic finite elements’, Int. j . numer. methods engg., 6 , 55-61 (1973). 2. E. Hinton and J . Campbell, ‘Local and global smoothing of discontinuous finite element functions using 21 least square method, Int. j . numer. methods eng., 8, 461-480 (1974).
ITERATIVE SOLUTION OF MIXED PROBLEMS
9
3 . O.C. Zienkiewicz and S. Nakazawa, 'The penalty function method and its application to the nuincriciil w l u t i o n ol boundary value problems'. Penalty Finite Element Methods in Mechanics. (ed. J . N . Rcddy). Am. Soc. Mcch. Ens. AMD-51. 1982, pp. 157-179. 4. O.C. Zienkiewicz. The Finite Element Method, 3rd edition. McGraw-Hill, 1977. 5 . S. Nakazawa, Y. Owa and O.C. Zienkiewicz. 'On nodal stress recovery in the displacement finite clcmcnt mcthod. I n preparation. 6 . G. Louhignac. G. Cantin and G.Touzot. 'Continuous stress fields in finite element ;tn;ilysis'. AIAR J . . 15. 164.5-47 (1977). 7. G. Cantin, G. Loubignac and G. Touzot, 'An iterative algorithm to build continuous stress and diqkicemcnt solutions'. lnt. j . numer. methods eng., 12. 1493-1506 (1978). 8. S. Nakazawa and J .C. Nagtegaal. 'Mixed finite elements, and iterative solutions for nonlinear structural mcch;inic\'. I n preparation. 9. O.C. Zienkiewicz. R.L. Taylor and J.M.W. Baynham, 'Mixed and irreducible formulation in linitc clcmcnt iinalysih~. Ch. 21, pp. 405-432 of Hybrid and Mixed Finite Element Methods, (ed. S.N. Atluri, R . H . Gallaghcr and O.C. Zienkiewicz). Wiley. Chichester, 1983. 10. R.L. Taylor and O.C. Zienkiewicz, 'Mixed finite element solution of flow problems'. Ch. 1, pp. 1-20. Firiirc Elvrrierrtr irr Fluids. Vol. 4 (ed. R.H.Gallagher et a l . ) , Wiley, Chichester, 1982. 11. C.T. Yang and S.N. Atluri, 'An assumed deviatoric stress-pressure-velocity mixed finite element method lor unsteady, convective. incompressible viscous flow', Int. j . numer. methods eng., 3. 377-398 (1983) and 4. 33-69 (IW4). 12. M. Crochet. 'The flow of a Maxwell fluid around a sphere', pp. 537-458, Fitlife Ekwwnts in Fluids. Vol. 4 (ed. R . H . Gallagher et a l . ) . Wiley. Chichester, 1982. 13. O.C. Zienkiewicz, S. Nakazawa, J.P. Vilotte and S. Toyoshima, 'Constrained problems and their iterative solution', t o be published in Comp. Merh. in Appl. Mech. and Eng.
