Multilevel mesh free method for the torsion problem.

Kozulic, Vedrana ; Gotovac, Blaz ; Colak, Ivo 等

Abstract: This paper presents a mesh free method for solving the torsion problem of prismatic bars. This method applies Fup basis functions, which belong to the class of Rvachev's infinitely derivable finite functions. As it is possible to calculate derivation values of Fup basis functions of high degree in a precise yet simple way, so it is possible efficiently to apply strong formulation procedures. The proposed method represents the problem domain and its boundaries by a set of nodes rather than resorting to the traditional discretization into finite elements. The collocation method was used to form a system of equations in which the differential equation of the problem is satisfied in collocation points of closed domain, while boundary conditions are satisfied exactly at the domain boundary. The proposed mesh free method offers a multilevel approach that achieves the approximate solution of an arbitrary accuracy by hierarchically increasing the number of basis functions in the domain. Values of the main solution function and all the values derived from the main solution are calculated in same points since numerical integration is avoided. The presented numerical model is illustrated by examples of linear and elasto-plastic analyses of prismatic bars subjected to torsion. The propagation of plastic zones in the cross-section is monitored by applying the incremental iterative procedure until its failure.

Key words: mesh free method, basis functions, universality, collocation method, multilevel numerical model, plastic failure

1. Introduction

The method presented in this work belongs to mesh free methods which represent a new approach in the modeling of engineering problems (Liu, 2003). The concept of a mesh free method is in establishing a system of algebraic equations for the whole problem domain without using a predefined mesh. Results obtained by using this new class of numerical methods are more accurate than those obtained by using FEM, as there are no stress discontinuity problems existing on interfaces between the finite elements. In the past few years, there has been an intensive development of mesh free methods for solving of complex processes and problems described by partial differential equations (Atluri, 2005; Griebel & Schweitzer, 2003).

In mesh free methods, the construction of basis functions is the central issue. The domain for field variable approximation (the support domain) should be small in comparison with the entire problem domain (compact support). Satisfaction of the compact condition leads to a banded system matrix that can be handled with good computational efficiency. Therefore, the choice shall be on finite basis functions with a small support, which do not depend upon the type and degree of the boundary-value problem. Basis functions must be infinitely derivable, and what is the most important, their linear combination must give a good approximation of the function from the appurtenant space of the boundary problem solution. In order that the numerical solution is not too complex, it is necessary to calculate simply enough values of basis functions and their derivatives as well as the scalar products of the function with itself, its derivatives and elementary functions. Satisfaction of these requirements ensures both easy implementation of the mesh free method and accuracy of the numerical solutions.

Functions implemented in numerical analyses of this work are the [Fup.sub.n](x,y) basis functions. They belong to the class of finite, infinitely derivable functions (Rvachev & Rvachev, 1971), named Rvachev's basis functions after their authors. The existing knowledge on this class of functions is systemized by Gotovac & Kozulic, 1999. Basis functions are transformed into a numerically applicable form, and first steps for their use in practice realized. These basis functions have properties of good approximation as well as very important property of universality (Rvachev & Rvachev, 1979), which means that the vector space of the n-th dimension is contained within the vector space of the (n+1) dimension. This property makes it possible to add basis functions hierarchically in the domain to the initial base of an approximate solution.

In this work, the collocation method is used to develop the numerical model. The domain of the problem and its boundaries are represented by a set of distributed nodes. The selection of nodes is important to obtain stable and accurate results. The process of node generation can be fully automated by a computer. The numerical model for the linear and nonlinear elasto-plastic analysis of prismatic bars subjected to torsion is developed by applying mesh free method with Fup basis functions. The described algorithm is applied in the computer program that uses the incrementaliterative procedure to monitor the propagation of plastic zones in the cross-section until its failure. It helps analyze bars with different shapes of cross-sections including single and multiplex boundaries. The multilevel process allows a hierarchic increase in number of basis functions in the entire analyzed domain or only in its particular parts. The presented model is illustrated by numerical examples. The results of the analysis are compared with existing exact solutions.

2. Fup basis functions

Fup basis functions belong to the class of Rvachev's basis functions. The simplest among Rvachev's functions is the function up(x), Fig. 1. Function up(x) is a finite function with the support [-1, 1], which is obtained as a solution of non-homogeneous differential-functional equation:

up'(x) = 2 up(2x + 1) - 2 up(2x - 1) (1)

with the norm condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Function up(x) can be expressed in an integral form (Rvachev & Rvachev, 1971):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

It is an even, infinitely derivable function, which is not analytical in any of the points of its support. Expression (3) is not adequate to calculate values of function up(x). Gotovac & Kozulic (1999) offered a numerically more adequate expression for calculating function up(x) values in an arbitrary point x [member of] [1, 0] in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where coefficients [C.sub.jk] are rational numbers determined according to the following expression:

[C.sub.jk] = 1/j! [2.sup.j(j+1)/2] up (-1 + [2.sup.-(k-j)]); j=0,1, ..., k (5)

Expression (x-0, [p.sub.1] ... [p.sub.k]) in Eq. (4) is the difference between the real value of coordinate x and its binary form with k bits, where [p.sub.1] ... [p.sub.k] are the digits 0 or 1 of the binary development of the coordinate x value. Therefore, the accuracy of coordinate x computation, and thus the accuracy of function up(x) in an arbitrary point, depends upon the accuracy of a computer. In binary-rational points the function up(x) values are calculated exactly in the form of a rational number. Those points of the function up(x) support are called characteristic points.

From Eq. (1) it can be concluded that the function up(x) derivatives can be calculated simply from the values of the function itself.

General expression for the derivative of the m-th degree is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [C.sup.2.sub.m+1] = m (m + 1) are the binomial coefficients and [[delta].sub.k] are the coefficients of value [+ or -]1 which determine the sign of each term. They change according to the following recursive formulas:

[[delta].sub.2k-1] = [[delta].sub.k], [[delta].sub.2k] = -[[delta].sub.k], k [member of] N, [[delta].sub.1] = 1 (7)

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Figure 1 shows the function up(x) and its derivatives. It can be observed that the derivatives consist of the function up(x) "compressed" to the interval of length [2.sup.-m+1] and with ordinates "extended" by the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] factor.

A family of [Fup.sub.n](x) functions was developed according to the up(x) function. [Fup.sub.n](x) functions and their derivations retain the properties of up(x) function, but they are more suitable for numerical analyses. Index n denotes the greatest degree of a polynomial which can be expressed accurately in the form of linear combination of basis functions obtained by displacement of function [Fup.sub.n](x) by a characteristic interval [2.sup.-n]. When n = 0 :

[Fup.sub.0](x) = up(x) (8)

Function [Fup.sub.n)(x) values are calculated using linear combination of displaced up(x) functions:

[Fup.sub.n)(x) = [[infinity].summation over (k=0)] [C.sub.k](n) up( x - 1 - k/[2.sup.n] + n + 2/[2.sup.n+1]) (9)

where coefficient [C.sub.0](n) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

and other coefficients are determined as [C.sub.k](n) = [C.sub.0](n) x [C'sub.k](n), where a recursive formula is used for calculation of auxiliary coefficients [C'.sub.k](n) :

[C'.sub.k](n) = 1, when k = 0; i.e. when k > 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Function [Fup.sub.n](x) support is determined according to:

sup p [Fup.sub.n](x) = [-(n + 2) [2.sup.-n-1] ; (n + 2) [2.sup.-n-1]] (12)

Derivatives of the function [Fup.sub.n](x) are also obtained by linear combination of derivatives of displaced functions up(x) according to expression (9). Figure 2 shows function [Fup.sub.2](x) and its first two derivatives.

Basis function for numerical analyses of two dimensional problems is obtained as Cartesian product of functions (9) by each coordinate axis:

[Fup.sub.n](x,y) = [Fup.sub.n](x) x [Fup.sub.n](y) (13)

In solving of the given problem by the collocation method i.e. solving of the partial differential equation of n-th order and satisfying of kinematics and dynamic boundary conditions, values of all partial derivatives of the function [Fup.sub.n](x,y) shall be known, n-th order included. Calculation of all required derivatives of function [Fup.sub.n](x,y) can be written in an algorithm form according to Eq. (13). Fig. 3 gives an axonometric presentation of basis function [Fup.sub.2](x,y) and its partial derivatives.

[FIGURE 3 OMITTED]

3. Application of the mesh free method to the analysis of the torsion problem with Fup basis functions

The first step of mesh free method procedure is the domain representation. The solid body of the structure is represented using set of nodes distributed in the problem domain and its boundary. The density of the nodes depends on the accuracy requirement of the analysis.

An approximate solution of differential equation:

L u(x,y) = f(x,y) (14)

with respective boundary conditions, is sought by the collocation method in the form of linear combination:

[u.sub.N](x,y) = [n.summation over (i=1)][m.summation over (j=1)] [a.sub.ij] x [[phi].sub.ij](x,y) (15)

by solving system of equations of (n x m) x (n x m) dimension:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where L is the differential operator, [[phi].sub.ij] are basis functions, and ([x.sub.1], [y.sub.1]), ..., ([x.sub.n], [y.sub.m]) are collocation points. Function [u.sub.N](x,y) belongs to N-dimensional subspace [X.sub.N] which represents the set of all linear combinations of basis functions {[[phi].sub.ij]: 1 [less than or equal to] i [less than or equal to] n, 1 [less than or equal to] j [less than or equal to] m}. In order to obtain the collocation matrix it is not necessary to perform numerical integration but only to calculate L [[phi].sub.ij]([x.sub.k], [y.sub.l]) images of the basis functions under the operator L.

It is known that functionality of the collocation method depends on the selection of basis functions [[phi].sub.ij] and collocation points ([x.sub.i],[y.sub.j]). Prenter (1989) proved the stability of numerical procedure with spline functions when collocation is performed in so-called natural knots. He developed proofs for existence and uniformity of the solution and error estimate. Since functions [Fup.sub.n](x,y) can be regarded as splines of an infinite degree, it can be shown that for them it is also optimal to perform collocation in natural knots of basis functions, i.e. vertices of basis functions situated in a closed domain such as e.g. for the base in x-direction formed by functions [Fup.sub.2](x) shown in Fig. 4.

[FIGURE 4 OMITTED]

This selection of collocation points provides the simplest numerical procedure, banded collocation matrix is obtained, which is diagonally dominant and thus well conditioned. This selection also implies uniformly distributed nodes set in each coordinate direction.

3.1 Analyses of rectangular domains

The torsion problem is reduced to solving of the Poisson's equation:

[[partial derivative].sup.2][PHI](x,y)/[partial derivative][x.sup.2] + [[partial derivative].sup.2][PHI](x,y)/ [partial derivative][y.sup.2] = -2G[??] (17)

for an isotropic material, with boundary condition:

[PHI]|[GAMMA] = 0 (18)

where [PHI](x,y) is the stress function, G is the shear modulus, while v is the angle of twist per unit length of a bar. Torsion rigidity of the cross-section for v = 1 is obtained as double volume under the surface of stress function [PHI]:

[C.sub.t] = 2 [integral[integral[PHI] dx dy (19)

Approximate solution base is formed on the unit virtual domain defined in the system ([xi], [eta]) according to a scheme shown in Fig. 5.

[FIGURE 5 OMITTED]

Assuming that cross-section of a bar can be contained within one rectangular fragment of a x b dimensions, differential equation of the problem (17) and boundary condition (18) can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[PHI]([xi],[eta]) = 0 for [xi] = 0, [xi] = 1, [eta] = 0, [eta] = 1 (21)

Collocation is performed in ([N.sub.[xi]] + 1) x ([N.sub.[eta]] + 1) equidistant points, while basis functions with the vertex outside the domain are retained so the basis functions set can be complete. Thus, governing equation (20) is satisfied in all collocation points of the domain except in corners; boundary condition (21) is satisfied in all collocation points of the domain sides, while three more conditional equations are satisfied in corners. Boundary conditions are therefore exactly satisfied on the domain boundary and not only discretely in collocation points.

Using the Fup basis functions and a strong formulation, the following collocation equations are obtained:

--equations within the domain [right arrow]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

--equations on the sides [right arrow]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

--equations in corners [right arrow]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

In the aforementioned equations [N.sub.[xi]] and [N.sub.[eta]] denote a number of partitions of a unit domain in directions [xi] and [eta] respectively; i and j are counters of basis functions in [xi] i.e. [eta] directions, while [F.sub.ij] ([xi], [eta]) is the basis function [Fup.sub.2] ([xi], [eta]) of the point (i,j). Depending on the number of partitions [N.sub.[xi]] and [N.sub.[eta]], function [Fup.sub.2] ([xi], [eta]) support is condensed to (4[DELTA][xi] x 4[DELTA][eta]) ; [DELTA][xi] = 1/[N.sub.[xi]], [DELTA][eta] = 1/[N.sub.[eta]]. Partial derivatives values of basis functions in equations (22)-(24) are determined according to the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

Since the function [Fup.sub.2] ([xi], [eta]) is a finite function with the support consisting of 4 x 4 characteristic intervals (see Fig. 2), the solution function value at collocation point (i,j) can be approximated by linear combination in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Values of all other basis functions at point (i,j) are equal to zero. Therefore, a support domain of a point (i,j) is 9. In such a way, banded matrix of the system is obtained. By solving the equation system (22)-(23)-(24), coefficients of linear combination of basis functions [C.sub.ij] are obtained, which can be used to calculate stress function values [PHI] from (26) in any point of the cross-section. Shear stress components [[tau].sub.xz] = [partial derivative][PHI]/[partial derivative]y and [[tau].sub.yz] = -[partial derivative][PHI]/[partial derivative]x are calculated with the same accuracy as the main solution.

3.2 Analyses of curvilinear domains

Surface of the given domain shall be described in a way that mapping matrix and all required partial derivatives of elements of the inverse mapping matrix can be found in each point of the domain. It is important that the surface can be easily and accurately divided into mutually equal partitions in each coordinate direction in order to fulfill the requirement of equidistance of collocation points on the domain.

Parametric form is extremely adequate for description of surfaces and, using the Coons formulation (Yamaguchi, 1988), can be written in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where Q(0,0), Q(0,1), Q(1,0) and Q(1,1) are position vectors at the four corners while Q([xi],0), Q([xi],1), Q(0,[eta]) and Q(1,[eta]) are four boundary curves, see Fig. 6. Changing the parameters [xi] and [eta] in equal steps on the interval [0,1], using Eq. (27), equidistant collocation points within the given domain are obtained.

[FIGURE 6 OMITTED]

Thus, for curvilinear domains, partial differential equation of the torsion problem (17) has the following collocation form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Partial derivatives of elements of the inverse mapping matrix in expressions (29) is determined by derivation of parametric equations of a surface (27), while partial derivatives of basis functions are determined according to Eq. (25).

3.3 Multilevel approach

A possibility of a hierarchic expansion of approximate solution base is established on the property of universality of the vector space formed by basis functions [Fup.sub.2] ([xi], [eta]). When [N.sub.[xi]] x [N.sub.[eta]] partitions on one fragment are selected i.e. ([N.sub.[xi]] + 3) x (N.sub.[eta]] + 3) basis functions mutually displaced by [DELTA][xi] in one and [DELTA][eta] in the other coordinate direction, as shown in Fig. 5, then the selected base is at the "zero level" of approximation. Hierarchic expansion of vector space dimension is obtained by adding of displaced and compressed basis functions. At the first level, functions [Fup.sub.2] ([xi], [eta]) are added, displaced by [DELTA][xi]/2 ; [DELTA][eta]/2 in reference to the functions of zero level, and compressed to a support length (2[DELTA][xi]) x (2[DELTA][eta]). At the second level, added basis functions are displaced by [DELTA][xi]/4 ; [DELTA][eta]/4 in reference to "zero level" with the support length ([DELTA][xi] x [DELTA][eta]) which is 4 1 of the length of basis functions support at zero level. At higher levels of approximation, the base is built by analogy. Fig. 7 shows the distribution of collocation points, in which vertices of basis functions are at the zero, first and second levels of approximation. Compression of the functions to 1/2 of the support from the preceding level is the consequence of basic properties of basis functions (Gotovac & Kozulic, 1999).

[FIGURE 7 OMITTED]

Numerical tests (Kozulic & Gotovac, 2000a) for different densities of collocation points showed that it is sufficient to satisfy the boundary conditions with basis functions of zero level while basis functions of higher levels correct the solution.

Procedure of hierarchic expansion of an approximate solution base is appropriate for computer programming. It can be applied in the entire given domain or only a part of the domain e.g. at concentrated load locations, for singularities such as concave breaks in the edge where stress concentration occur, or in plasticity zones in elastoplastic analyses (Kozulic & Gotovac, 2000b).

4. Elasto-plastic analysis of prismatic bars torsion

The material starts to deform plastically when the resulting shear stress in a point reaches a critical value [[tau].sub.Y]. Then, equation (17) is satisfied in elastic part of the domain while the yielding criterion (Hill, 1985):

[([partial derivative][PHI]/[partial derivative]x).sup.2] + [([partial derivative][PHI]/[partial derivative]y).sup.2] = [[tau].sup.2.sub.Y] (30)

is satisfied in its plastic part. The greatest value of the torsion moment occurs when the entire cross-section is plasticized. It is the limit torsion moment [M.sub.pl]. Elasto-plastic analyses includes determination of the angle of twist . at which the material starts to plastify as well as monitoring of the plastic zones until limit moment [M.sub.pl] is obtained. As common in non-linear numerical analyses, in each iterative step, new stress state is calculated with the assumption that material expansion of behavior is linearly elastic. In this problem, it means that the increase of angle of twist [DELTA]v is applied as load only on the elastic part of the cross-section. In the proposed numerical model, decrease of torsion rigidity of a cross-section for purposes of increase of the plastic zone is obtained in way that the Poisson's differential equation is satisfied only in collocation points that are still elastic within the domain while in plasticized points condition [PHI] = 0 is fulfilled. Similarly, with homogeneous boundary condition, in collocation points on the domain boundary, differential equation is only set in points with elastic behavior while in plasticized ones the condition is [partial derivative][PHI]/[partial derivative]n = 0, where n is the normal on the outer boundary. Since the plasticized zones first occur at the domain boundary and then spread towards the inside of the cross-section, this is a successful numerical simulation of movement of plastic domain boundary in compliance with the membrane analogy.

5. Examples

5.1 Torsion of a prismatic bar with a square cross-section

Torsion of a bar with a square cross-section and made of isotropic material, shown in Fig. 8, is analyzed for v = 1 by mesh free method using the basis functions [Fup.sub.2]([xi],[eta]). An analytic solution for this shape of a cross-section is given by Timoshenko & Goodier, 1951. The effect of hierarchic increase in a number of basis functions is illustrated. Fig. 9 shows the convergence diagrams of numerical solution for torsion rigidity value when number of basis functions increases at zero level only, and when approximate solution base is expanded with basis functions of the first and second levels. It can be observed that with the same total number of basis functions, much better numerical solution is obtained if a hierarchic approach is applied than when all basis functions belong to zero level.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Gradual plasticization of the cross section with the increase in the angle v is given in Fig. 10. Assuming that the bar is not deformed until the yielding limit is exceeded, limit torsion moment according to expression (8[[tau].sub.Y][a.sup.3])/3 for the given values equals [M.sub.pl]=4666.667 kNcm.

Plastic zones first occur at the domain boundary, and then expand towards the inside. At the beginning of non-linear calculation, an initial density of collocation points is selected with 10 partitions in each coordinate direction at zero level. When testing of plasticity criterion determines in which collocation points the yielding has occurred, basis functions are made denser hierarchically according to a scheme given in Fig. 7. Number of basis function is increased only in plastic part of the cross-section while in elastic core initial density at zero level is retained as illustrated in Fig. 10. Thus, movement of the plastic zone boundary is successfully simulated until elastic core completely disappears.

[FIGURE 10 OMITTED]

5.2 Plastic yielding of a bar with a triangular cross-section

Plastic yielding of a bar with a triangular cross-section is analyzed using the conditions of symmetry as shown in Fig. 11.

[FIGURE 11 OMITTED]

Theoretical value of the limit torsion moment is equal to the double volume under the stress function surface for a completely plastic cross-section which is:

[M.sub.pl] = 2[square root of 3]/27 x [[tau].sub.Y] x [a.sup.3] (31)

i.e. for the given values 3103.835 [M.sub.pl] = 3103.835. Table 1 gives a convergence of a numerical solution obtained by presented mesh free method. N denotes total number of basis functions per each coordinate direction, and is obtained by a hierarchic expansion of the approximate solution base until plastic failure is registered.

Fig. 12 gives isolines of stress function [PHI] in the plan and shapes of the stress function [PHI] over the cross-section ranging from elastic to completely plastic state.

[FIGURE 12 OMITTED]

5.3 Cross-section in the form of an eccentric ring

Linear and nonlinear analyses of a bar with a cross-section in the form of an eccentric ring, shown in Fig. 13, are made. An analytic solution exists for this shape of a cross-section (Lurie, 1970).

The real domain of a cross-section is mapped into virtual unit domain using the expression (27) where sides (1) and (2) (see Fig. 14) are described using the parametric equations of a circle; sides (3) and (4) overlap in a real domain.

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

Convergence of torsion rigidity value [C.sub.t] and stress function value [PHI] on the inner boundary [[GAMMA].sub.1] with an increase in the number of collocation points is given in Table 2.

For the purpose of the elasto-plastic analysis, twist angle v increases to the full plastic yielding. Theoretical value of the limit torsion moment is equal to the double volume under the stress function surface for a completely plasticized cross-section. For the yield stress value [[tau].sub.Y] = 14.0, [M.sub.pl] is 37708.746.

Fig. 16 shows surface shapes and isolines of the stress function [PHI] obtained for different load increments, from fully elastic to fully plastic state.

6. Conclusions

Smooth finite functions of Rvachev's class are applied as basis functions in numerical analyses of prismatic bars subjected to torsion. The collocation approach enables efficient, economical and simple procedure.

The proposed mesh free method has very significant advantages: (1) it is very easy to implement because no integration is required; (2) the problem domain and its boundaries are represented by set of nodes without the use of a predefined mesh; (3) an arbitrarily accurate numerical solution is obtained by arbitrary increase in the number of basis functions on the domain; (4) simultaneously, values of the main solution function and all values derived from the main solution are calculated in the same points with the same level of accuracy; (5) all fields derived from the main solution can be expressed by continuous functions on the entire domain; (6) it is possible to increase an accuracy of approximate solution by hierarchic increase of basis function number on the domain, or its parts, without intrusion into the rest of the domain (multilevel approach).

It can be concluded that the presented numerical method efficiently simulates the real non linear behavior. The hierarchic increase in number of basis functions in the model provides a simple way to increase the accuracy of an approximate solution in places where plastic yielding occurs and also accelerates the convergence of incremental-iterative procedure.

The numerical procedure is stable until the cross-section is completely plasticized i.e. until plastic failure occurs. This is the consequence of the fact that numerical integration is avoided so that the criterion of plasticity is tested in the same points for which the values of the solution function are calculated i.e. in collocation points. In comparison with the finite element method based on the weak form, which always records plastic failure before it really happens, the multilevel mesh free method provides more accurate numerical solutions.

7. References

Atluri, S.N. (2005). Methods of Computer Modeling in Engineering & the Sciences, Volume I, Tech Science Press, ISBN 0-9657001-9-4, University of California, Irvine

Gotovac, B. & Kozulic, V. (1999). On a selection of basis functions in numerical analyses of engineering problems. International Journal for Engineering Modelling, Vol. 12, No. 1-4, (1999), pp. 25-41, ISSN 1330 1365

Griebel, M. & Schweitzer, M.A. (Eds.). (2003). Meshfree Methods for Partial Differential Equations, Springer-Verlag, ISBN 3-540-43891-2, Berlin

Hill, R. (1985). The Mathematical Theory of Plasticity, Oxford University Press, ISBN 0-19-856162-8, New York

Kozulic, V. & Gotovac, B. (2000a). Numerical analyses of 2D problems using Fupn(x,y) basis functions. International Journal for Engineering Modelling, Vol. 13, No. 1-2, (2000), pp. 7-18, ISSN 1330 1365

Kozulic, V. & Gotovac, B. (2000b). Hierarchic generation of the solutions of nonlinear problems, CD-Rom Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2000), Onate, E. ; Bugeda, G. & Su rez, B. (Eds.), Book of Abstracts, pp. 283, ISBN 84-89925-69-0, Barcelona, September 2000, CIMNE, Barcelona

Liu, G. R. (2003). Mesh free methods : Moving beyond the finite element method, CRC Press LLC, ISBN 0-8493-1238-8, Boca Raton

Lurie, A. I. (1970). Theory of Elasticity, Nauka, Moskva

Prenter, P. M. (1989). Splines and Variational Methods, John Wiley & Sons, Inc., ISBN 0-471-50402-5, New York

Rvachev, V. L. & Rvachev, V. A. (1971). On a Finite Function. DAN URSR, Ser. A, No. 6, (1971), pp. 705-707.

Rvachev, V. L. & Rvachev, V. A. (1979). Non-classical methods for approximate solution of boundary conditions, Naukova dumka, Kiev

Timoshenko, S. P. & Goodier, J. N. (1951). Theory of Elasticity, McGraw-Hill, New York

Yamaguchi, F. (1988). Curves and Surfaces in Computer Aided Geometric Design,

Springer-Verlag, ISBN 3-540-17449-4, Berlin

This Publication has to be referred as: Kozulic, V.; Gotovac, B. & Colak, I. (2006). Multilevel Mesh Free Method for the Torsion Problem, Chapter 29 in DAAAM International Scientific Book 2006, B. Katalinic (Ed.), Published by DAAAM International, ISBN 3-901509-47-X, ISSN 1726-9687, Vienna, Austria

DOI: 10.2507/daaam.scibook.2006.29

Authors' data: Associate Professor Kozulic V.[edrana]*, Professor Gotovac B.[laz]*, Associate Professor Colak I.[vo]**, *Faculty of Civil Engineering and Architecture, University of Split, Croatia, ** Faculty of Civil Engineering, University of Mostar, Bosnia and Herzegovina, Vedrana.Kozulic@gradst.hr, Blaz.Gotovac@gradst.hr, gf-svemo@tel.net.ba

Table 1. Convergence of numerical solution for triangular
cross-section

 N = 4 N = 10 N = 20 N = 50 Exact

[M.sub.p1] 2888.599 3066.411 3094.227 3102.273 3103.835
[v.sub.p1] 196.102 240.026 7602.955 44788.65 [infinity]

Table 2. Comparison of the results of linear analysis for an
eccentric ring

 [MATHEMATICAL [PHI] -
Number of EXPRESSION [[PHI].sub.
collocation points CANNOT BE exact]
 REPRODUCIBLE [[PHI].sub.
([N.sub.[xi]] +) X ([N.sub.[eta]] +) IN ASCII] exact]

[N.sub.[xi]] = 10, [N.sub.[eta]] = 20 41.387 5.32%
[N.sub.[xi]] = 20, [N.sub.[eta]] = 40 40.279 2.50%
[N.sub.[xi]] = 50, [N.sub.[eta]] = 100 39.649 0.90%
[N.sub.[xi]] = 100, [N.sub.[eta]] = 200 39.445 0.38%
Exact solution 39.297 --

 [MATHEMATICAL
 EXPRESSION
 CANNOT BE
Number of REPRODUCIBLE
collocation points [C.sub.t] IN ASCII]

([N.sub.[xi]] +) X ([N.sub.[eta]] +)

[N.sub.[xi]] = 10, [N.sub.[eta]] = 20 28345.72 2.57%
[N.sub.[xi]] = 20, [N.sub.[eta]] = 40 27976.24 1.24%
[N.sub.[xi]] = 50, [N.sub.[eta]] = 100 27768.75 0.48%
[N.sub.[xi]] = 100, [N.sub.[eta]] = 200 27701.30 0.24%
Exact solution 27634.63 --