Application of Multi-Region Trefftz Method to Elasticity
J. Sladek1,V.Sladek1,V.Kompis2, R. Van Keer3
Abstract: This paper presents an application of a direct where unknowns are fictitious parameters. That is the main Trefftz method with domain decomposition to the two- reason, why in this paper a direct formulation is used inspite dimensional elasticity problem. Trefftz functions are substi- of the more frequently occurred indirect formulation in the lit- tuted into Betti’s reciprocity theorem to derive the boundary erature [Leitao (1998); Zielinski and Herrera (1987); Herrera integral equations for each subdomain. The values of displace- (1995)]. If the number of Trefftz functions used as test func- ments and tractions on subdomain interfaces are tailored by tions is high the final system of algebraic equations can be ill- continuity and equilibrium conditions, respectively. Since Tr- posed. To overcome this difficulty the whole domain is subdi- efftz functions are regular, much less requirements are put on vided into smaller subregions. For the same purpose the spe- numerical integration than in the traditional boundary integral cial Trefftz finite element formulations are frequently used by method. Then, the method can be utilized to analyse also very many researchers [Jirousek and Leon (1977); Jirousek (1978); narrow domains. Linear elements are used for modelling of Jirousek (1987); Freitas and Ji (1996); Freitas (1998); Jirousek the boundary geometry and approximation of boundary quan- and Zielinski (1993); Kompis and Bury (1999)]. On interfaces tities. Numerical results for a rectangular plate with varying of subregions displacement continuity and equilibrium of trac- aspect ratio and cantilever beam are presented. tions have to be satisfied. Adding such restriction conditions at nodal points to the discretized integral equations, we create keyword: plynomial Trefftz functions, direct formulation, a complete set of algebraic equations for the unknown values linear approximation, boundary integral equation at nodal points. Boundary integral formulations combined with Trefftz func- 1 Introduction tions as test functions can be used for the computation of boundary unknowns. An integral representation of displace- The effort to predict more effectively the response of the com- ments at internal points is not available in such a formula- plex problems of continuum mechanics motivates the authors tion. If such quantities are required, the integral representation to seek for formulations in which the governing equations in- with the fundamental (singular) solution has to be used [Balas, side the approximated domain are identically satisfied. Such Sladek and Sladek (1989)]. analytically derived functions are called Trefftz functions [Tr- Numerical tests were carried out on a rectangular plate with efftz (1926)] which can be found in the form of polynomials, varying side aspect ratio and a cantilever beam. Legendre, harmonic, Bessel, Hankel, singular Kupradze func- tions, etc. [Zielinski (1995)]. The main characteristic of the Trefftz method is the use of trial 2 Boundary integral equations functions that satisfy the governing differential equations in a For linear isotropic problems of elasticity, the governing equa- domain. The method can be classified into the indirect and tions are force equilibrium equations expressed in displace- the direct formulations [Jin, Cheung and Zienkiewicz (1990); ments, the Lame-Navier equations in domain
Kita, Kamiya and Iao (1999)]. In the indirect formulation
the trial function is given as a superposition of Trefftz func-
λ · · = ; ; µ u ; µu X (1) tions. Then, the unknown parameters in the trial function are j ji i jj i determined so that boundary conditions have to be satisfied. In the direct formulation the Trefftz functions are considered where Xi is a body force vector. Recall that in two-dimensional as a weighting (test) function in Betti’s reciprocity theorem to problems under plane stress conditions special care should be derive boundary integral equations. Boundary values are ap- paid to the association of the coefficients with the material pa- proximated by standard polynomials within elements on dis- rameters. In general, cretized boundary. In this approach, unknown values have ´ ν ν
a physical interpretation in contrast to the indirect approach, 2µ ; plane stress
; ν = ; λ = ν ν 1 · 1 2 ν ; otherwise 1 Institute of Construction and Architecture, Slovak Academy of Sciences, 84220 Bratislava, Slovakia. 2 Faculty of Mech. Engn., University of Zilina, 01026 Zilina, Slovakia. where ν is Poisson’s ratio and µ is the shear modulus associ-
3
´ · µ Dept. Mathematical Analysis, University of Gent, B-9000 Gent, Belgium. ated with the Young modulus as E = 2µ 1 µ . 2 Copyright c 2000 Tech Science Press CMES, vol.1, no.4, pp.1-8, 2000
An equivalent to the Lame-Navier equation is the force equi- It makes the numerical integrations much easier in the Tre- librium in terms of stresses fftz boundary integral equation formulation than in the conven- tional formulation based on singular boundary integral equa-
σ = : ij; j Xi 0 (2) tions. On the other hand, the numerical stability is more ques-
tionable in the former case because of a higher condition num-
Γ = Γ [ Γ µ Γ Γ Γ ´ Let u and t denote parts of the boundary u t ber of the discretized BIE. with prescribed displacements and tractions, respectively. The boundary integral equation (8) can be used only for the Traction vector ti is defined as a scalar product of stress tensor computation of unknown boundary quantities. If we are in- σ ij and outward normal vector n j. The weak formulation of terested in displacement and stress values in the interior of
eq. (2) can be expressed in weighted residual form the body, the Somigliana identity and integral representation Z
Z for stresses, respectively, has to be utilized for the evalu-
´σ µ Ω · ´ µ Γ W1i ij; j Xi d W2i ui ui d ation. Fundamental solutions corresponding to the Lame- Ω Γu
Z Navier governing equation (1), with the Dirac delta function
´ µ Γ = ; ´ µ W1i ti ti d 0 (3) distribution of body forces, are denoted here as Uij x y . Γ
t Then, the governing equations can be written as
where prescribed quantities are denoted by overbar and ui, ti
λ · ´ µ· ´ µ=δ δ ´ µ ; ; are trial functions. Weight functions are chosen in the follow- µ U jm; ji x y µUim jj x y im x y (9) ing form
with the fundamental solutions in two dimensions being given
£ £
; = ; = as
W1i ui W2i ti (4)
¨ ©
£ £ 1
´ µ= ν µ δ · ;
where u and t are the displacements and tractions corre- ´ ;
i i Uij r 4 3 lnr ij r;ir j
π ´ νµ sponding to the weighting field. 8 µ 1
1 ∂r
´ ; µ= νµδ · ] ´
Applying the Gauss-Green formula to the domain integral in [ ;
Tij x y 1 2 ij 2r;ir j
π ´ νµ ∂
eq. (3), we have 4 µ 1 r n
©
´ νµ´ µ : ;
1 2 r; n r n (10)
Z Z
Z i j j i
£ £ £
´σ µ Ω = Γ Γ ui ij; j Xi d ui ti d ti ui d
Ω Γ Γ The Somigliana identity has the form [Balas, Sladek and
Z Z £
£ Sladek (1989)
σ Ω Ω : · u d u X d (5) ij; j i i i
Γ Ω
Z Z
µ= ´ µ ´ µ Γ ´ µ ´ ; µ Γ ui ´y t j x Uij x y d x u j x Tij x y d x Substituting eq. (5) into (3), with taking into account the pre- Γ Γ
scribed boundary conditions, one obtains Z
´ µ ´ µ Ω :
· Xj x Uij x y d x (11)
Z Z Z
Z Ω
£ £ £ £
Γ · σ Ω = Ω : u t dΓ t u d u d X u d (6) i i i i ij; j i i i Γ Γ Ω Ω If an interior point y is lying very close to the boundary, the second integral on the r.h.s. of eq. (11) is nearly singular. The If the weighting field is selected as a Trefftz function the ho- accurate evaluation of such integral requires special attention. mogeneous governing equation is satisfied It is well known that the integral representation (11) can be
σ£ regularized ( see e.g. [Balas, Sladek and Sladek (1989)] ) as = 0(7)
ij; j
Z Z
¼ ¼
Ω ´ µ= ´ζ µ· ´ µ ´ µ Γ [ ´ µ ´ζ µ] ¢ in Ω ,whereΩ . ui y ui t j x Uij x y d x u j x u j Γ Γ
Finally, one obtains the boundary integral equation Z
; µ Γ · ´ µ ´ µ Ω ;
Tij´x y d x Xj x Uij x y d x (12)
Z Z
Z Ω
£ £ £
Γ Γ = Ω : ui ti d ti ui d Xiui d (8) Γ Γ Ω Γ where ζ ¾ can be selected as the nearest boundary point to Ω The boundary integral equation (8) relates the boundary dis- y ¾ . placements and traction vectors. Then, the unprescribed quan- Similarly, one can derive a regularized integral representation tities can be computed from that equation. In contrast to the for stresses [Balas, Sladek and Sladek (1989)]. Note that in the conventional boundary integral equations the free term is miss- integral representation (12) the strong singularity of the kernel Γ
ing here. The governing equation for the fundamental solution Tij (as y approaches ζ ¾ ) is smoothed by the vanishing factor
µ ´ζ µ contains the Dirac delta function which causes a singular be- u j ´x u j in the numerical integration. A more advanced haviour of the fundamental solution. Here, Trefftz functions cancellation of divergent terms is discussed elsewhere [Sladek are nonsingular and all the integrals exist in a regular sense. and Sladek (1998)].
Application of multi-region Trefftz method to elasticity 3
; ;::: The Trefftz formulation of the boundary integral equations for b = 1 2 ,whereNq and Ne is the number of boundary supplemented with integral representations based on singular and domain elements, respectively. The symbol Jq denotes fundamental solutions can be utilized for the numerical analy- the Jacobian for transformation of Cartesian coordinates into
sis of the whole domain. isoparametric ones. A matrix form of eq. (16) is given by
= ; 3 Trefftz functions for 2-d elasticity problems Tu Ut F (17) The Trefftz function is a homogeneous solution of the partial differential equation (1). We try to find the Trefftz function in where the definition of the matrices directly follows from eq. a polynomial form. Then, for displacements one can write (16). The number of rows in eq. (17) is equal to the num- ber of the Trefftz functions. Since the maximum number of s s the columns is twice the number of the boundary nodes, the
£ n m
u1 = ∑ ∑ anmx1x2 number of the Trefftz functions must be equal to or larger than = n=0 m 0 s s twice the number of the nodes. In this paper we have selected
£ n m
= such a number of Trefftz functions that we obtained a square
u2 ∑ ∑ bnmx1x2 (13) = n=0 m 0 matrix. where 2s is the order of polynomial. Substituting polynomi- Kita, Kamiya and Iio (1999) have analysed the conditioning of matrices in the Trefftz boundary integral equation method als (13) into the governing equation (1) (Xi = 0), one obtains a system of algebraic equations for the computation of the un- for potential problems. They considered a narrow cut of thick- known parameters anm and bnm. For such purposes symbolic walled cylinder with a high aspect ratio of the arc length over computations by MATHEMATICA software has been utilized. the difference of radii. From numerical results it follows that Explicit expressions of some of the Trefftz functions are given the condition number is dependent on the number of the func- in Appendix. tions rather than on the aspect ratio. Provided that the num- ber of Trefftz functions is restricted, the accuracy is reason- 4 Numerical implementation able even if the objects with long and narrow profiles are anal- ysed. Therefore, for solving the large-scaled problems accu- In the direct BIE Trefftz formulation the boundary is dis- rately, the domain decomposition is required. For simplicity cretized by conforming elements. Boundary quantities are ap- we will consider a domain divided into two subdomains. The proximated within the elements: subdomains are refered to as Ω1 and Ω2, respectively, with n boundaries Γ1 Γ2 and Γl (interface). Then, eq. (17) on the
a aq Ω
µ= ´ξµ ; gi ´x ∑ N gi (14) subdomains α can be written as
a=1
aq α α α α α =
T u U t F (18)
; g where gi is the nodal value of a physical quantity gi ¾fui ti on element q at the node with local number a.Thevaluen
α ; denotes the number of nodes on a element. For a linear ap- for = 1 2. where the superscript is related to the subdomain.
a
´ξµ proximation, n = 2 and the shape functions N have the On the interface boundary both the displacements and traction following form vector are unknown. The system of integral equations (18) is
1 not sufficient to get a unique solution. Therefore, they have
ξµ= : ´ ξµ ; N ´ 0 5 1 to be supplemented by additional equations which satisfy dis-
2
ξµ= : ´ · ξµ ξ ¾< ; >: N ´ 0 5 1 1 1 (15) placement continuity and force equilibrium on the interface: For the evaluation of the domain integral in eq. (8) we can use
1 2
µ= ´ µ ; an analytical method if the domain and body force distribu- ui ´x ui x
1 2
µ= ´ µ ; tion are simple. Otherwise a numerical method has to be used. ti ´x ti x (19) Standard isoparametric elements are very convenient [Balas, Sladek and Sladek (1989)]. Making use of the approxima- Γ for x ¾ l . Then, the integral equations (18), supplemented tion formula (14) in the BIE (8), we obtain a system of alge- aq aq with the tailored conditions (19), give a complete system for a braic equations for nodal displacements ui and tractions ti , large-scaled boundary value problem.
respectively,
Z i
Nq 2 h
´ µ £´ µ
£ b aq b aq a q 5 Numerical examples
ξµ ´ξµ ´ξµ ξ = ∑ ∑ u ´ t t u N J d
Γ i i i i = q=1 a 1 q Two numerical examples will be presented here to show the
Ne Z
´ µ
£ b a ae e accuracy of the Trefftz method in comparison with the con-
ξ ; ξ µ ´ξ ; ξ µ ξ ξ ∑ u ´ 1 2 N 1 2 Xi J d 1d 2 (16) Γ i ventional boundary integral equation method.
a=1 e 4 Copyright c 2000 Tech Science Press CMES, vol.1, no.4, pp.1-8, 2000
x 2 x2
s22 D P B
A B C x1 L/3 L/3 L
h=1. Figure 2 : Cantilever beam with a parabolic shear end load
a 5.2 Example 2. Cantilever beam The behaviour of the present Trefftz BIE is also studied in the cantilever problem (Fig. 2) for which the following exact so- x1 lution is given by Timoshenko and Goodier (1970). The dis-
placements in the beam are
=
Figure 1 : A rectangular plate with varying aspect ratio a h
2
ν µ ν σ P´1 D 2
under a uniaxial uniform load 22 = 1
´ µ· ´ µ u1 = x2 3x1 2L x1 νx2 x2 D 6EI 2 1
2
ν µ ν
P´1 2 3 2
´ µ· ´ µ´ = µ · u2 = x1 3L x1 ν L x1 x2 D 2 6EI 1 ν 5.1 Example 1. Effect of the aspect ratio in a rectangular 4 · 2 νD x1 ; (20) domain 4 4
3 = A drawback of the conventional boundary element method where I = D 12 is that it is not convenient for analysis of domains with 8
< E plane strain
high aspect ratios and a special regularization approach is re-
´ · ν µ E = 1 2 E
quired [Sladek and Sladek (1998); Sladek, Sladek and Tanaka : 2 plane stress
· ν µ ´1 (1993)]. When the distance between the elements on close boundary faces is smaller than the size of elements, the ac- The stresses corresponding to eq. (20) are curacy fails due to an inaccurate evaluation of nearly singu-
σ P ´ µ´ = µ ; = L x x D 2 lar integrals if conventional BEM formulation and/or standard 11 I 1 2
numerical quadratures are employed. To investigate the de- ; σ22 = 0 pendence of the accuracy on the aspect ratio in the presented
Px2
´ µ : Trefftz method, we have analysed a rectangular domain with a σ12 = x2 D (21) 2I varying aspect ratio (Fig. 1). The plate is subjected to a uni- In the numerical analysis the following geometry and material
form load in x2 direction. The following material constants are
= = ν = :
parameters were considered: L = 3, D 1, E 1, 0 3,
ν = : considered: Young modulus E = 1andPoissonratio 0 3.
P = 1. The beam domain is divided into three subdomains Due to two axial symmetries only one quarter of the plate is of the same size with 12 linear elements on each subdomain modelled. The analysed domain is discretized by 8 linear el- boundary. ements. The height of the plate is kept constant and the plate
Numerical results obtained by the present Trefftz BIE method
= = = : width is varying from a =h 1toa h 0 001. Numerical re- sults are given in Tab. 1. One can observe essentially higher are compared with the analytical solutions at three points A, stability of numerical computations by the presented method B, C (Fig. 2). The displacements at these points are given in than by the conventional BEM formulation combined with Tab. 2. using regular Gauss quadrature for numerical integration. A A quite good agreement of results can be observed. A relative quadratic approximation in the BEM is used on 4 elements (8 error for the maximum bending stresses σ11 at the clamped σ nodes). When increasing the number of elements in the BEM, side is less than 1% (exact value: 11 = 18 and the numerical
σ : the accuracy of the results is expected to be higher. value 11 = 18 16). Application of multi-region Trefftz method to elasticity 5
Table 1 : Numerical results for a rectangular plate under a uniform tension Trefftz method BEM Exact σA B σ σA B σ σA B a =h 22 u2 Err. %Err.u % 22 u2 Err. %Err.u % 22 u2 1. 1. 0.91 0. 0. 0.9993 0.9108 0.07 0.088 1. 0.91 0.5 0.9992 0.9097 0.08 0.033 0.9995 0.9104 0.05 0.044 1. 0.91 0.1 0.9989 0.9091 0.11 0.099 1.114 0.8695 11.4 4.45 1. 0.91 0.05 0.9956 0.9099 0.44 0.011 0.583 0.817 41.7 10.22 1. 0.91 0.01 1.05 0.9241 5.0 1.549 1.526 0.6372 52.6 29.97 1. 0.91 0.005 1.019 0.948 1.9 4.175 2.079 0.865 107.9 4.94 1. 0.91 0.001 1.52 0.769 52.0 15.49 2.484 9.587 148.4 953.5 1. 0.91
Table 2 : Displacements at the cantilever beam A A B B C C u1 u2 u1 u2 u1 u2 Analytical solution 10.50 14.25 16.80 43.95 18.90 82.05 Trefftz BIE 10.61 13.98 17.03 44.22 19.21 82.41 Error [%] 1.047 1.894 1.369 0.614 1.640 0.438
6 Conclusions Herrera, I. (1995): Trefftz-Herrera domain decomposition. Advances in Engn. Software, vol. 24, pp. 43–56. The present Trefftz BIE method is a promissing computational alternative to more popular finite element or boundary element Jin, G.; Cheung, Y. K.; Zienkiewicz, O. C. (1990): Appli- methods. A multi-domain Trefftz BIE formulation is keeping cation of the Trefftz method in plane elasticity problems. Int. the simplicity of the FEM and the boundary character of the J. Num. Meth. Engn., vol. 30, pp. 1147–1161. BEM. The number of subdomains in the presented method can be much less than in the conventional FEM due to higher mod- Jirousek, J. (1978): Basis for development of large finite elling accuracy. As compared with the conventional BEM for- elements locally satisfying all field equations. Comp. Meth. mulation, the treatment of singularities is avoided. Therefore, Mech. Engn., vol. 14, pp. 65–92. the numerical integration is much easier. Without any special effort the method is also convenient to analyse very narrow Jirousek, J. (1987): Hybrid-Trefftz plate bending elements objects due to the regular integrals in the formulation. with p-method compabilities. int. J. Num. Meth. Engn.,vol. A drawback of the Trefftz BIE formulation is the necessity 24, pp. 1367–1393. to use a multi-domain approach because of the decrease of the condition number of the discretized BIE with increasing Jirousek, J.; Leon, N. A. (1977): A powerful finite element number of Trefftz functions. Consequently, this leads to an for plate bending. Comp. Meth. Appl. Mech. Engn., vol. 12, increase of the number of unknowns on the subdomain inter- pp. 77–96. faces also in physically homogeneous problems. Of course, Jirousek, J.; Zielinski, A. P. (1993): Dual hybrid Trefftz in the case of piecewise nonhomogeneous problems this effect element formulation based on independent boundary traction disappears because of the necessity of domain subdivision in time. Int. J. Num. Meth. Engn., vol. 36, pp. 2955–2980. such problems. Kita, E.; Kamiya, N.; Iio, T. (1999): Application of a di- References rect Trefftz method with domain decomposition to 2d potential Balas, J.; Sladek, J.; Sladek, V. (1989): Stress Analysis by problems. Engn. Analysis Bound. Elem., vol. 23, pp. 539–548. Boundary Element Method. Elsevier, Amsterdam. Kompis, V.; Bury, J. (1999): Hybrid Trefftz finite element Freitas, J. A. T. (1998): Formulation of elastostatic hybrid formulations based on the fundamental solution. In Mang, H.; Trefftz stress elements. Comp. Meth. Appl. Mech. Engn.,vol. Rammerstorfer, E.(Eds): Discretization Methods in Structural 153, pp. 127–151. Mechanics, pp. 181–187. Kluwer, Doordrecht.
Freitas, J. A. T.; Ji, Z. Y. (1996): Hybrid Trefftz finite ele- Leitao, V. M. A. (1998): Applications of multi-region ment formulation for simulation of singular stress fields. Int. Trefftz-collocation to fracture mechanics. Engn. Analysis J. Num. Meth. Engn., vol. 39, pp. 281–308. Bound. Elem., vol. 22, pp. 251–256.
6 Copyright c 2000 Tech Science Press CMES, vol.1, no.4, pp.1-8, 2000
´ µ
£ 2 ;
Sladek, V.; Sladek, J. (1998): Some computational as- u1 = x1 (29)
´ µ
pects associated with singular kernels. In Sladek, V.; Sladek, £ 2 ; u = 0 (30)
J.(Eds): Singular Integrals in Boundary Element Methods. 2
´ µ
£ 2 e ; CMP, Southampton. = t1 2 n1 (31)
eν 1
Sladek, V.; Sladek, J.; Tanaka, M. (1993): Nonsingular bem eν
´ µ
£ 2 e ; formulations for thin-walled structures and elastostatic crack t2 = 2 n2 (32)
eν 1
problems. Acta Mechanica, vol. 99, pp. 173–190.
´ µ
£ 3
= ; u1 = x2 2 (33)
Timoshenko,P.S.;Goodier,J.N. Theory of Elas-
´ µ
(1970): £ 3
= ;
ticity. McGraw-Hill, New York. u2 = x1 2 (34)
´ µ
£ 3 e ; t1 = n2 (35)
Trefftz, E. (1926): Ein gegenstuck zum ritzschen verfahre. In e
· νµ 2 ´1
Proc. 2nd Int. Congress on Applied Mechanics, pp. 131–137.
´ µ
£ 3 e ;
Zurich. t2 = n1 (36) e
· νµ
2 ´1
´ µ
£ 4
; Zielinski, A. P. (1995): On trial functions applied in the = u1 0 (37)
generalized Trefftz method. Advances in Engn. Software,vol.
´ µ
£ 4 ;
24, pp. 147–155. u2 = 1 (38)
´ µ
£ 4 ;
Zielinski, A. P.; Herrera, I. (1987): Trefftz method: fitting t1 = 0 (39)
´ µ
£ 4 ;
boundary conditions. Int. J. Num. Meth. Engn., vol. 24, pp. = t2 0 (40)
871–891. ´ µ
£ 5 ;
u1 = 0 (41)
´ µ
£ 5 ; Appendix A: u2 = x2 (42)
eν
´ µ
In this Appendix, we collect the explicit expressions for sev- £ 5 e ;
t1 = n1 (43)
´ µ £´ µ £ 2
eν n n 1
eral Trefftz functions ui and ti in terms of polynomials
´ µ
£ 5 e ; with respect to cartesian coordinates of a field point in a global t2 = 2 n2 (44)
eν 1
coordinate system. First of all, let us introduce two parameters
´ µ
£ 6
= = ;
e and eν u x2 2 (45)
´
´ µ
£ 6
= ; E plane stress u2 = x1 2 (46)
= E
´ µ
e (22) £ 6 ; plane strain = ν2 t1 0 (47)
1
´ µ
£ 6 ; t = ´ 2 0 (48)
ν plane stress
´ µ
£ 7 ; ν =
e u x x (49) ν = (23) 1 2
plane strain
´ µ
£ 7 2
ν
e
´ · νµ = ; 1 = u2 1 x2 4 (50)
which allow to write the Trefftz functions in a compact form e
´ · ν µ
´ µ
£ 7 e 2
; σ =
for both the plane stress and plane strain problems. Recall that 11 x2 (51)
e
· νµ 2 ´1
the Hooke law takes the form
´ µ
£ 7 e
; σ = x1 (52)
νe 12
e
· νµ
σ e 1 2 ´1 ; ε · ε ε = ´ · µ : = δ ;
ij ij ij kk ij ui ; j u j i
e e
· ν ν
´ µ
1 1 2 £ 7 e
; σ =
22 x2 (53)
e
´ · νµ
´ µ
£ 2 1
The tractions t n , corresponding to the Trefftz displacements
¡ i
´ µ
£ 8 1
´ µ £´ µ
£ n n 2 2 2
e
· ν ; u , can be obtained by contraction of stresses σ by the u1 = 2x2 x2 x2 (54)
i ij e ν 1
normal vector n j as
´ µ
£ 8 ;
u2 = 0 (55)
£´ µ
£ n
µ=σ
t ´n n j (24)
´ µ
i ij £ 8 2e
; σ = 11 2 x1 (56)
eν 1
Trefftz functions:
´ µ
σ£ 8 2e = ;
´ µ
£ 1 12 2 x2 (57)
e ; = ν
u1 1 (25) 1
´ µ e
£ 1 ν
´ µ
£ 8 2e ;
= σ ;
u 0 (26) = 2 22 2 x1 (58)
eν
1
´ µ
£ 1 ; t = 0 (27)
eν
1 ·
´ µ
£ 9 1 2
= ;
´ µ
£ 1 u1 x2 (59)
e
´ν µ t2 = 0 (28) 2 1
Application of multi-region Trefftz method to elasticity 7
£´ µ
´ µ
9 £ 14 e
= ;
; σ = u2 x1x2 (60) 12 x1x2 (87)
eν
1 ·
e
ν ¡
´ µ
£ e
´ µ
σ 9 £ 14 e 2 2 2 = ;
e
= ν x (61) σ ;
11 2 1 22 x2 x1 2x1 (88)
e
ν
e
· ν µ 1 2 ´1
eν
£´ µ
´ µ
£ 9 e 15 3 ;
σ = ; = u x x (89) 12 2 x2 (62) 1 1 2
eν
1
¡
´ µ
£ 15 1 4 2 2 2 2 4
e e
e
; ν ν ·
ν =
´ µ
£ 9 e u2 x2 3 x1x2 3x1x2 x1 (90)
e
· νµ
σ ´ ; 22 = 2 x2 (63) 2 1
eν
1
¡
´ µ
£ 15 ex2 2 2 2
e e
; ´ µ = ν · ν £ σ
10 11 2 x2 x2 3 x1 (91) ;
= 2
e
· ν µ
u1 0 (64) ´1
¡
¡
£´ µ
´ µ
10 2 2 2 £ 15 ex1 2 2
e
= = · ν ;
e
ν · ; u 2x x x 2 (65) σ =
2 1 2 2 12 2 3 x2 x1 (92)
e
· ν µ ´1
eν
´ µ
£ e ¡
10
´ µ
£ 15 ex2
;
σ = x (66) 2 2
2 e
ν · ; 11 σ = x 3x (93) eν
1 · 22 2 2 1
e
· ν µ
´1
´ µ
£ 10 e
£´ µ
; σ =
12 x1 (67) 16 3
= ;
eν
1 · u1 x2x1 (94)
´ µ ¡
£ 10 e
´ µ
£ 16 1
;
σ = x 4 4 2 2 4 4
e (68) e
= ν · ν 22 2 u x x 12x x x 3x ; (95)
eν 2 2 1 2 1 1
· 2
1 e
· νµ
4 ´1
¡
´ µ
£ 11 x1 ¡
2 2 2
£´ µ
e ex
= · ν u 6x x x ; (69) 16 2 2 2
2 1 1 e
; ν ·
1 σ =
6 11 2 x2 3x1 (96)
e
· ν µ
´1
e
· ν
£´ µ ¡
11 1 3
´ µ
£ 16 ex1
;
u = x (70) 2 2 2
e
= ν 2 2 σ ;
6 12 2 3x2 x1 2x1 (97)
e
· ν µ
´1
¡
´ µ
£ 11 e
¡
2 2 2
e
; ν · σ =
´ µ
11 x2 2x2 x1 (71) £ 16 ex2 2 2 2
e
ν ; σ =
e
· νµ
2 ´1 22 2 x2 3 x1 6x1 (98)
e
· ν µ
´1
´ µ
£ 11 e
¡ ;
σ = x x (72)
´ µ
12 1 2 £ 17 1 4 4 2 2 4 4
e
· ν
e e
; ν · ν ·
1 u1 = x2 3x2 12x1x2 x1 x1 (99)
e
¡ ´ · νµ
4 1
´ µ
£ 11 e 2 2
e
; · ν
σ = x x (73)
´ µ 22 2 1 £ 17
e 3
· νµ
2 ´1 ;
u2 = x1x2 (100)
¡
£´ µ ¡
12 x2 2 2 2
´ µ
£ 17 ex
e 1
· ν ; u = 2x 3 x 3x (74) 2 2 2
σ e ν · ;
1 2 1 1 = 3 x 6x x (101)
e
ν µ
´ 11 2 2 2 1
3 1 e
· ν µ
´1
¡
e
· ν
´ µ
£ ex
´ µ
£ 12 1 3 17 2 2 2 2
e
; ν · σ = ;
u2 = x1 (75) 12 2 x2 2x2 3x1 (102)
e
e ´ · ν µ
ν µ
3 ´ 1 1
¡
´ µ
£ 17 ex1 2 2
´ µ
£ 12 2e
e
= · ν
σ σ 3x x ; (103) ;
= x1x2 (76) 22 2 2 1
e
· ν µ 11 2 ´
eν 1
1
¡
¡
´ µ
£ e
´ µ 12 2 2 £ 18 1 4 2 2 2 2 4
σ e ; = · ν
e e
= · ν ν
12 2 x2 x1 (77) u1 x2 3 x1x2 3x1x2 x1 ; (104)
e
ν
e
· νµ 1 2 ´1
eν
£´ µ
´ µ
£ 12 2e 18 3 ;
σ = ; = u x2x (105)
22 2 x1x2 (78) 2 1
e
ν
¡
1
´ µ £ 18 ex1 2 2
σ e = · ν ;
eν 3x x (106)
· 2 1 ´ µ
£ 13 1 11 2
e
· ν µ
3 ´1 ;
u1 = x2 (79)
e
ν µ ¡
3 ´ 1
´ µ
£ 18 ex2 2 2
e
· ν ; σ = ¡
x2 3 x1 (107)
´ µ
£ 13 x1 12 2
e
· ν µ 2 2 2 ´
e 1
; ν ·
u2 = 3 x2 3x2 2x1 (80)
e
ν µ ¡
3 ´ 1
´ µ
£ 18 ex1 2 2 2
e e
ν ν σ = 22 2 3 x2 2 x1 x1 ; (108)
e
ν e
´ · ν µ
´ µ
σ£ 13 2e 1 ;
= x1x2 (81) ¡
11
£´ µ eν x
1 19 2 4 4 2 2 4 4
e e
ν · ν · ;
u = x 3x 20x x 5 x 5x (109) ¡
1 2 2 1 2 1 1
´ µ
£ 13 e 2 2 20
e
ν · ; σ =
12 2 x2 x1 (82)
¡
e
e
ν
· ν
´ µ
1 £ 19 1 4 4
;
u2 = x1 5x2 x1 (110)
´ µ
σ£ 13 2e 20 ;
= x x (83) ¡
22 2 1 2
£´ µ
eν e
1 19 2 2 2
e
ν · ; σ =
11 x1x2 x2 2x2 x1 (111)
e
´ · ν µ
eν
· 1
´ µ
£ 14 1 3
= ; ¡
u x (84)
´ µ
1 1 £ 19 e 4 4 2 2 4
e e
ν · ν ; 6 σ =
12 x2 2x2 6x1x2 x1 (112)
¡
e
´ · ν µ
´ µ
£ 14 x2 2 2 2 4 1
e
ν · ;
u = x x 6x (85) ¡
2 2 2 1
´ µ
6 £ 19 e 2 2
e
· ν σ =
x x x x ; (113) ¡
22 1 2 2 1
´ µ
£ e
e
· ν µ
14 2 2 ´1
e
ν · ; σ =
11 x2 x1 (86)
e
· νµ 2 ´1
8 Copyright c 2000 Tech Science Press CMES, vol.1, no.4, pp.1-8, 2000
¡
¡
e
ν £´ µ
· 2ex
´ µ
£ 20 1 4 4 24 2 4 4 2 2 4
e e
ν · ν : σ =
;
= x 2x 10x x 5 x (138)
u1 x1 5x2 x1 (114) 22 2 2 2 1 2 1
e
´ · ν µ
e
ν µ
10´ 1 5 1
¡
´ µ
£ 20 x2 4 2 2 2 2 4
e e
; ν ν ·
u2 = x2 5 x2x1 5x2x1 5x1 (115) Other higher order Trefftz functions can be obtained analo-
e
ν µ
5 ´ 1
¡ gously.
´ µ
£ 20 e 4 4 2 2 4
e e
= ν · ν σ ;
11 2 2 x2 x2 6 x1x2 x1 (116)
e
ν µ
2 ´1
¡
´ µ £ 20 2e 2 2
σ e ; ν · 12 = 2 x1x2 x2 x1 (117)
eν
1
¡
´ µ
£ 20 e 4 2 2 4 4
e e
ν · ν ; σ =
22 2 x2 6x1x2 x1 2x1 (118)
e
ν µ
2 ´1
¡
e
· ν
´ µ
£ 21 1 4 4
; u = x x 5x (119)
1 20 2 2 1
¡
´ µ
£ 21 x1 4 4 2 2 4 4
e e
ν · ν ; u = 5 x 5x 20x x x 3x (120)
2 20 2 2 1 2 1 1
¡
´ µ
£ 21 e 2 2
e
ν · ; σ =
11 x1x2 x2 x1 (121)
e
· νµ
´1
¡
´ µ
£ 21 e 4 2 2 4 4
e e
ν · ν ; σ =
12 x2 6x1x2 x1 2x1 (122)
e
· νµ
4 ´1
¡
´ µ
£ 21 e 2 2 2
e
; ν σ = 22 x1x2 x2 x1 2x1 (123)
eν
1 ·
´ µ
£ 22 1 6 2 4 6
· ; u = 2x 15x x x (124)
1 15 2 1 2 1
¡
´ µ
£ 22 4x1x2 4 2 2 2 2 4
e e
; ν ν ·
u2 = 3 x2 5 x1x2 5x1x2 3x1 (125)
e
· νµ
15´1
¡
´ µ
£ 22 2ex1 4 4 2 2 4 e
σ e = ν · ν
11 2 10 x2 5x2 10 x1x2 x1 ; (126)
e
· νµ
5 ´1
¡
´ µ
£ 22 2ex2 4 4 2 2 4
e e
ν · ν ; σ =
12 2 2 x2 x2 10 x1x2 5x1 (127)
e
· νµ
5 ´1
¡
´ µ
£ 22 2ex1 4 2 2 4 4
e e
σ ; ν · ν
22 = 2 5 x2 10x1x2 x1 2x1 (128)
e
· νµ
5 ´1
´ µ
£ 23 1 6 4 2 6
· ; u = x 15x x 2x (129)
1 15 2 1 2 1
¡
´ µ
£ 23 2x1x2 4 4 2 2 4
e
ν · ;
u2 = 3 x2 3x2 20x1x2 9x1 (130)
e
· νµ
15´1
¡
´ µ
£ 23 2ex1 4 2 2 4 4
e e
= ν · ν σ ;
11 2 5 x2 10x1x2 x1 2x1 (131)
e
· νµ
5 ´1
¡
´ µ
£ 23 2ex2 4 2 2 4 4
e e
σ ; ν · ν
12 = 2 x2 10x1x2 5 x1 10x1 (132)
e
· νµ
5 ´1
´ µ
£ 23 2ex1 4 2 2 2 2
e
ν σ =
22 2 5x2 10 x1x2 20x1x2
e
· νµ 5 ´1
4 4 ¡
e
; ν ·
·2 x1 3x1 (133)
¡
´ µ
£ 24 2x1x2 4 4 2 2 4 4
e e
ν · ν · ;
u1 = x2 9x2 20x1x2 3 x1 3x1 (134)
e
· νµ
15´1
´ µ
£ 24 1 6 2 4 6
· ; u = 2x 15x x x (135)
2 15 2 1 2 1
´ µ
£ 24 2ex2 4 4 2 2
e e
ν · ν σ =
11 2 2 x2 3x2 10 x1x2
e
· νµ 5 ´1
2 2 4 ¡
· ;
20x1x2 5x1 (136)
¡
´ µ
£ 24 2ex1 4 4 2 2 4 e
σ e ν · ν ;
12 = 2 5 x2 10x2 10x1x2 x1 (137)
e
· νµ 5 ´1