Trefftz direct method

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EISEVIER

Trefftz direct method W. G. Jin Department of Applied Mechanics, Fudan University, Shanghai, PRC &

Y. K. Cheung Department of Civil and Structural Engineering, The University of Hong Kong, Hong Kong

This paperdiscusses a so-calledTrefftz Direct Method (TDM), in which the nonsingular boundary integral equation is used as the starting formulation and complete sets of solution are used as weighting functions. Several relevant characteristicsand problemsof this approximation are also discussed. example to illustrate the basic principle and formulation. The governing equation and boundary condition for a steady potential problem are

1 INTRODUCTION Trefftz method as a boundary solution technique has long been available and applicable to a wide variety of engineering problems. It has been known since the pioneer work of Trefftz in 1926.’ Thereafter, the Trefftz method has been extensively developed in both formulations and applications.2-7 In recent years, the Trefftz method has gained considerable popularity. The important contributions in this field have been made by Quinlan, Jerousek, Herrera & Zienkiewicz.8-‘5 The use of complete solutions of the differential equation in boundary solution methods is an effective approach. It avoids the difficulties associated with integration over singularities in the traditional BEM; causesno disturbance near the boundary; often obtains more accurate results. A common characteristic of these methods is that the Trefftz complete solution is used as the trial function. As a counter part of the Trefftz indirect method, the direct formulation is here again possible and for some problems direct formulation performs in a superior way. In recent years, a Trefftz direct method has been developed and applied systematically to engineering problems. These problems include potential problems, plane elasticity problems, plate bending problems and wave propagation problems.‘6-2’

v2a=o

in R

(14

and a=$

on l?,

(lb) da on r2 q=dn=q where V 2 is Laplacian operator, Cpis the unknown potential function, q = a$/% is the normal derivative on I’, 5 and q are given variables. Trefftz indirect methods In Trefftz indirect method, the unknown potential function @is approximated by the expansion as Q=eNiai=Na

(4

i=l

where ai is the unknown general parameter, the trial function Ni is chosen to satisfy exactly eqn (la) i.e. V2Ni = 0 or

in R

(3)

v2a=o 2 TREFFTZ METHOD FORMULATIONS

In this case, only the boundary conditions (lb) need to be considered. The integral statements equivalent to the boundary conditions (lb) may be written in terms of

For brevity, the potential problem will be used as an 65

66

W. G. Jin, Y. K. Cheung

the generalized Galerkin formulation as

J

W;i(O - &) dI’+

W$(q-q)dI’=O

(4

Jl-2 where W;i and WG are referred to as the weighting functions. Clearly, there are several possible ways to choose Wj*‘s and to determine the unknown coefficients in expression (2). Various Trefftz-type weighted residual methods may be generated from eqn (4) by using different weighting functions such as the collocation method (TCM), Galerkin method (TGM) and least squares method (TLSM) etc. Finally, a set of algebraic equations can be obtained, in matrix form, rl

Ka=f

(5) Obviously, special choices of weighting functions render the final matrix symmetric7 This allows it to be coupled with the standard finite element solutions easily. The unknowns used in Trefftz indirect method are the generalized coefficients a. Trefftz direct method The Trefftz direct method is a technique paired with Trefftz indirect method. This terminology was suggested by Professor 0. C. Zienkiewcz and indicates explicitly the characteristic of this method. The Trefftz direct method is typical of the boundary element methods in which the unknown variables are the physical quantities such as the potential and its normal derivative over the boundary. The TDM is constructed, in general outline, in the following manner:

where Nj is the harmonic function, it may be obtained from the complete system of solution. (c) The trial function (approximation solution) @ and its derivative q within each element employ any suitable interpolation functions, say, the piecewise Lagrange polynomial: - A @e= We@, on re (8) 4e = fi2ek? where the fire and fis are the interpolation functions, and 4, and ij, are the nodal unknowns over each element. Assembling all the elements, one can obtain the interpolation functions for 4 and q over the entire boundary such that a = cii@?‘i on r

where I?tj and iV2i are polynomial expansions, and ii and ii are nodal unknowns on the boundary. Substituting the weighting function (7b), trial function (9) and boundary conditions (lb) into eqn (6), one obtains for the j th equation

J rl

NjCN2i Qi dI’ -

=

Jrl

W*q df’ +

=

Jr2

J

v=w;

=o

in fl

(7a)

which can be chosen by putting Wj* = Nj

(7b)

an

dr

I rl

!!!&r-

(11)

where

W*q dl-’

r $@d, (6) r, an J2 and the boundary is divided into elements. The weighting function is obtained by using the (b) complete sets of homogeneous solution of the differential equation. These Trefftz functions are used as the complementary functions in the present method, in principle, they work for any arbitrary shaped regions and require to be only independent of each other. In potential problems, for example, the weighting functions are assumed a priori to fulfill the Laplacian equation:

an:,NJq

Hx=f

x=

aw* -cpdr+

Jr2

Njq dT’ (10) Jr2 h The final result of the algebraic equation system can be written as

(a) The formulation of TDM is based upon the Green’s formula (reciprocal principle), i.e.

(9)

q = Cfi2i4i

H=

6 0c Jrl

NTfi2 dl? -

E rI;Jtdr J(r2 an >

(12)

?$idrL r2 W dr J Jrl an The features of TDM are briefly summarized as follows: f=

(a) The unknown is the actual physical variables of the problem. (b) The integral equation is regular. (c) The complete set is used as the auxiliary weighting function. (d) The unknown functions are interpolated by shape functions. (e) The final matrix is non-symmetric and full. (f) The values in the interior of the domain are determined in terms of the other procedures, say, the collocation method.

67

Treftz direct method

Surprisingly, although the possibility of using Trefftz function as a wei ting function has been perceived in some literature22’2P etc., its prominence has not attracted sulhcient attention of research work yet. In recent years, the Trefftz direct method has been used practically as the starting point for boundary solution formulation by the authors and others.1”21 Further research will elucidate the advantages of the forms discovered and we expect such developments in the future.

all boundary conditions appear as natural boundary conditions, and the unknown parameters are determined to satisfy them in the best way in an average sense. The potential problem will be used to show how the Trefftz method is constructed by the variational principle. The variational formulation of potential problem can be expressedby a modified principle of potential energy as q@ dr IImp = ; n(V@)2dR J Jr2

3 COMPLETE SOLUTION AND COMPLETE SETS The complete solutions and complete sets are very important for the Trefftz method and also for some boundary procedures in which complete solution sets are used. In the Trefftz methods, one needs to choose the trial and/or weighting functions in such a manner that the differential equations are satisfied throughout the region a priori. Usually they are the complete solutions of the homogeneous differential equations. A criterion of completeness is T-completeness, which can be found in monograph.13 It is interesting to observe that it is possible to develop systemswhich are complete in regions which are, to a large extent, arbitrary. This conclusion is very important for the method’s versatility. The complete systems of solutions have been recently established for a variety of problems, and the question of convergence of some approximations has also been solved by Herrera and other authors. However, the mathematical theory has not yet reached a fairly high degree of development in comparison with the rapid progress in applying the method to engineering. For example, the complete solution of a number of differential equations which are of interest in engineering has not yet been found; the completeness of some known systems of the complete solutions is not yet proved; the criterion of convergence of a new approximating procedure such as the TDM has not yet been established; and so on. In this situation some complete solutions and their associated systems of basic functions will be used without reference to the formal use of their completeness. Numerical experimentation will remain the basic procedure of analyzing the adequacy of the TDM. It is hoped that mathematicians will soon throw more light on these questions and open new frontiers. 4 TREFFI-Z METHOD AND VARIATIONAL PRINCIPLES In 1926, Trefftz presented a new variational method which is formulated as a counterpart to the RayleighRitz variational method. In his variational formulation,

The solution is obtained by the stationary condition of the functional IImp as

-

Jl

(Q - S)b~+$Xj dI’= 0 (14) rl where 6cP is a variation of the potential components from the actual solution. No less general, as used in TDM, we assume W* = S@and V2S@ = 0. In this case, Green’s formula becomes [email protected]@dS1=

JR

Jr

dSQ an dr

Q.-

Introducing eqn (15) into (14), one obtained 5 am .--dr+

Jl-1

-

an

Jr2

J

am an dr - Jrl q-s*

cP.-

r2

dr

q*s@dr=O

or

5 *-dr+ aw* an Jrl -

Jr2

Jr2

aw* an dr - Jrl

fB.-

q- W*dr

q. W*dr=O

Obviously, eqn (16) is the boundary integral equation and is the same as eqn (6) used in Trefftz method. The relationship between the boundary method and variational method has attracted more and more attention of the researchers in recent years.1’*24-26 Professor P. Tong et al., using the potential problem to show how the hybrid displacement method is reduced to the Trefftz method by imposing a priori restriction on the admissible function for the field variables of the hybrid functional and on their variation.25 5 NUMERICAL EXAMPLES Example 1. Circular cavity in an inhite plane The edge of the hole is subject to a uniform normal

68

W. G. Jin, Y. K. Cheung

Table 1. Values of a, at point p(4,O)

Example 3. Wave force

NE

4

6

8

12

Exact

TBlO” TB1lb

-56.25 -56.25

-56.25 -56.25

-56.25 -56.25

-56.25 -56.25

-56.25

Note: ‘constant elements; blinear elements. Table 2. Clamped square plate with unifom distribution load A Method TDM TCM TGM Exact

B

D W/qa4

Mx/qa 2

0.001265 3 0.001265 3 0.001265 2 0.001265

O-022905 0.022 905 0.022904 o-022925

Mn/qa 2

Ww

-0441024 -0.441297 -0440 903 -0441

-0.0513370 -0.051333 4 -0.051300 4 -0.051334 0

The wave diffraction problem of an offshore structure in the form of a vertical cylinder is considered. The vertical cylinder is subjected to a plane incident wave of elevation H. The results of the wave force on the vertical cylinder are listed in Table 5. The present results are given by different elements and compared with the exact solution. For simplicity the note TD102, for example, indicates that the interpolation functions are linear for geometric variables, constant for the potential and quadratic for the wave force respectively. It can be seen that the computed results coincide with the exact solution.

Table 3. Simply supported square plate with uniform load A Method TDM TCM TGM Exact

D W/qa4

B Mx/qa2

Vnlqa

0.004062 34 0.047 8863 O-004062 27 0.047 885 7 0.004062 5 0.047 8876 0.004062 3 0.0479

C ITj/qa2

-0.420 891 0.065 1655 -0.420 467 0.065022 3 -0.420 560 0.064 9730 -0.420 0.064964

Table 4. Uniformly loaded square plate supported at comers Pt Quantity A

B

DW/qa4 Mx/qa2 DW/qa4

TCM

TDM

TGM

Exact

0.025 517 8 O-025555 8 0.025 5039 0.02529 0.111 865 0.111711 0.1119 0.111744 0.0177485 0.0177850 0.0177445 0.01774

6 REMARKS The Trefftz indirect method have been widely applied to the solution of a variety of engineering problems. The Trefftz direct method, however, have received comparatively little attention, although they have certain numerical advantages. The authors hope that this paper will act as a catalyst, sparking interest and further work in this area.

ACKNOWLEDGEMENT

pressure with an intensity of p and the tractions are equal to zero at infinity. Owing to symmetry only one fourth of the boundary is taken into consideration. The results of the stress o;, at point p(4,O) on the boundary are compared with the exact solution.

This research has received financial support from the Research Foundation of The State Education Commission.

Example 2. Square plate

REFERENCES

The analysis of square plates has been carried out for uniform load under different boundary conditions. Owing to symmetry only one eighth of the plate needs to be taken into consideration. The results are presented at the centre A, midside B, and comer C of the plate. Seven quadratic partially discontinuous elements in TDM, 13 coordinate functions in TCM and TGM,

and v = 0.3 are used in this

analysis. The results are compared against the Timoshenko’s solutions, and listed in Tables 2-4. Table 5. Wave force F TDlOO TDlll TD200 TD211 TD222 Exact (k= l;a=d=

12

NE=6

NE=

1.61494 1.38573 1.56979 1.49904 1.63798

1.63422 1.57543 1.62220 1.60373 1.64075

1)

NE=18

NE=24

1.63796 1.61170 1.63256 1.62428 1.64085

1.63924 1,63924 1.63619 1.63153 1.64087 1.64088

1. Trefftz, E. Ein Gegensttick zum Ritz’schen Verfahren. Proc. 2nd Znt. Cong. Appl. Mech., pp. 13l-37, Zurich, 1926. 2. Rafal’son, Z. Kh. A problem arising in the solution of the biharmonic equation. Dokl. Akad. Nauk SSSR, 1949, 64, 8. 3. Birman, M. Sh. Trefftz’s variational method for the equation V4u = f. Dokl. Akad. Nauk SSSR, 1955,101,2. 4. Birman, M. Sh. Variational methods of solving boundary value problems similar to the Trefftz method. Vest. Len. gos. un-ta, Seriya matem mekh. i astr., 1956, 3. 5. Mikhlin, S. G. Variational Methods in Mathematical Physics. Macmillan, 1964. 6. Rektorys, K. Variational Methods in Mathematics, Science and Engineering. D. Reidel, Hingham, Mass, 1977. 7. Zielinski, A. P. & Zienkiewicz, 0. C. Generalized finite element analysis with T-complete boundary solution functions. Znt. J.N.M.E., 1985, 21, 509-28. 8. Quinlan, P. M. The torsion of an irregular polygon. Proc. Roy. Sot., 1964, A282, 208-27. 9. Quinlan, P. M. The edge-function method in elastostatics. In Studies in Numerical Analysis. New York, Academic Press, 1974, p. 10.

Trefftz direct method 10. Jirousek, J. & Leon, N. A powerful finite element for plate bending. Comp. Meth. Appl. Mech. Eng., 1977, 12,77-96. 11. Jirousek, J. & Guex, L. The hybrid-Trefftz finite element model and its application to plate bending. Znt. J. numer. Meth. Engng, 1986, 23, 651-93. 12. Herrera, I. Theory of connectivity: A systematic formulation of Boundary Element Method in Recent Advances in Boundary Element Method. ed. C. A. Brebbia, Pentech Press, London, 1978. 13. Herrera, I. Boundary Method: An Algebraic Theory. Pitman, Boston, 1984. 14. Zienkiewicz, 0. C. The finite element method and boundary solution procedures as general approximation methods for field problems. Proc. World Cong. on FEM in Structure Mechanics, Boumemouth, 1975. 15. Zienkiewicz, 0. C., Kelly, D. W. & Bettess, P. Marriage a la mode - Finite elements and boundary integrals. In Energy Methoak in Finite Element Analysis. ed. R. Glowinski, E. Y. Rodin & 0. C. Zienkiewicz. Wiley, New York. pp. 81-107, 1979. 16. Tang, L. M. & Jin, W. G. An alternative approach to boundary by seriesmethod. Computational Mechanics ‘86, Springer, Tokyo, 1986. 17. Cheung, Y. K., Jin, W. G. & Zienkiewicz, 0. C. Direct solution procedure for solution of harmonic problems using complete non-singular, Trefftz functions. Znt. J. Comm. Appl. Num. Meth., 1989,5, 159-69. 18. Jin, W. G., Cheung, Y. K. & Zienkiewicz, 0. C.

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Application of the Trefftz method in plane elasticity problems. Znt. .Z.numer. Meth. Engng, 1990, 30, 1147-61. 19. Cheung, Y. K., Jin, W. G. & Zienkiewicz, 0. C. Solution of Hehnholtz equation by Trefftz method. Znt. J. numer. Meth. Engng, 1991, 32, 63-78. 20. Jin, W. G., Cheung, Y. K. & Zienkiewicz, 0. C. Trefftz method for Kirchhoff plate bending problems. Znt. J. numer. Meth. Engng, 1991, 36, 765-81. 21. Du, X. L. & Xiong, J. G. Application of boundary element method to wave propagation by using series solution. Earthq. Engng Engng Vibrat., 1988, S(l), (in Chinese). 22. Brebbia, C. A. The Boundary Element Method for Engineering. Pentech Press, London, 1978. 23. Du, Q. H. & Yao, Z. H. Solution of some plate bending problems using the boundary element method. Appl. Math. Modeling, 1984, 8, 15-22. 24. Hu, H. C. A necessary and sufficient condition for the boundary integral equations of harmonic functions. ACTA Mech. Sol. Sinica, 1989, 2, (in Chinese). 25. Tong, P. & Jin, W. G. Derivation of Trefftz method from the hybrid displacement functional. A note on W. G. Jin’s Ph.D. Thesis, 1991. 26. Pian, T. H. H. Thirty-years history of hybrid stress finite element methods. Y. K. Cheung Symposium, 15 Dec. 1994, Hong Kong Proc. and Messages,ed. P. K. K. Lee & L. G. Tham. 27. Timoshenko, S. & Woinowsky-Krieger, S. Theory of Plates and Shells. 2nd edn. McGraw-Hill, New York, 1969.