2-2-2011.pdf قطر قطر نصف نصف .و، داخلي )isotropic ( .تكاملالكترك وبذلك ،.ال 10 )computational cost ( . للمرة األولىLEFM . تم . مقارنة النتائج مع بحوث مماثلة تم للتحقق. In the recent year, a kind of meshless or meshfree method has been developed based on moving least square (MLS) approximation [1], which does not require a structured mesh or elements. Among these methods, element free Galerkin method (EFGM) [2-5] is particularly attractive, due to its uncomplicatedness, easing of numerical difficulties of mesh entanglement, faster rate of convergence, and a formulation that corresponds to well-established FEM. Because of the MLS properties in the continuity, and the smooth approximation results, EFGM use to extract the data in the discontinuities such as in the crack [6-10], and shear band [11-14]. One of the drawbacks of EFGM is forced imposition of essential boundary conditions rather than the direct method. This is existed because of the MLS does not have the Kronecker delta property, where the shape function associated with a node is not exactly equal to one at this node, and this shape function is not exactly zero at the other nodes in the domain. This drawback is solved by using a variety of the enforced boundary methods [4, 15, 16]. In the MLS approximation, each node has a domain of influence, in which a function of compact support is used as a weight function. The weight function determines the intensity of the effect of a node at various points in its domain of influence. Figure 1 represents discretization using meshless methods: nodes, circular domains of influence. Examples of commonly used test functions are exponential function, and spline functions. Excellent details on systematic ways to construct weigh functions can be found in [15, 17-18]. The nodal influence domain is usually considered having a consolidated form in the shape of a circle or sphere. In the EFGM, accuracy and effectiveness are dependent on the nodal domain of influence and type of the weight function. In this work, non-consolidated (anisotropic) weight function in the elliptic form is introduced to improve the efficiency of the EFGM in some problems. In the using non-consolidated weight functions, the influence domain of each node may vary with direction. As a consequence, the definition of the influence domain based on non-consolidated weight function, improves the numerical efficiency of EFGM. In such case, the influence domain of each node can be determined so that the nodal overlapping decreases. Thus, good results can be achieved with less computational efforts. It has been shown that moving least-square methods and weight functions share many features for the constructing the approximation of the solution. In the MLS technique that presented by [1], the approximation is expressed as the inner product of a vector of the polynomial basis and a vector of the coefficient . 1 (1) where denotes the number of terms in the basis. In two dimensions a complete polynomial basis of order m is given by ,...,,...,,...,,,,,,1, 22 (2) For bi-linear, the basis function is considered as ,,,1, , =4 (3) 11 Also, the linear basis is provided by ,,1, , =3 (4) Vector of unknown parameters that depended on is given as ,...,2,1 (5) These basis functions are not required to be polynomials as shown in Equations 3 and 4. When solving problem involving cracks, a convenient way of capturing /1 stress-singularity in linear-elastic fracture mechanics is calculated by using [19,20] ,,,1, , =4 (6) Or sin2cos,sin2sin,2sin,2cos,,,1, , =7 (7) Where and are polar coordinates with the crack tip as the origin. Equations 6 and 7 represent fully enriched and partially enriched basis functions, respectively. In the equations 1, the coefficient vector, is determined by minimizing a weighted discrete L2 norm, defined as 1 2 (8) Where n is the number of nodes such that ;0 is the weight function associated with node which is non-zero over a limited support called the influence domain of node . Furthermore, it is assumed that is non-increasing for 0 , where . The parameter determines the influence domain of node for weight functions. Equation 8 can also be written as [15] (9) Where ,...,, 21 (10) 1 1 1 2222 1111 (11) 12 00 00 00 2 1 (12) The stationarity of with respect to yields 0 (13) Therefore, (15) 2211 The MLS approximants can be defined as 1 (16) Where the shape function is = 1 (17) The partial derivatives of can be obtained as follows 1 , 11 , 1 ,, (18) In which , . Consider a two dimensional problem. By substituting the approximation in Equation 16 into the associated Galerkin weak form, one can obtain the final form of the discredited system of equations [2,15,20] 0 (19) Where 13 (20) (21) (22) (23) Where and are specified traction vector, body force vector, matrix of elastic constants and vector of Lagrange multipliers, respectively. Matrices and are defined as follows , , , , 0 0 (24) 0 0 (25) 0 0 (26) In Equation (26), are the conventional FEM shape functions which are used for the approximation of Lagrange multipliers on the essentials boundaries [4, 15]. This is because as mention, the MLS shape functions do not possess the Kronecker delta function property. In the process of generation of the system of Equation (19), two domain integrals must be computed. These integrals are involved in the stiffness matrix, Equation (20), and load vector Equation (22). Usually in the EFG method, these two integrals are computed by Gaussian quadrature method [19]. The shape functions are obtained from the weight functions, these functions have influence domain of the node. The domain size is defined by the so called dilatation parameter or smoothing length. It is critical to solution accuracy, stability and plays the role of the element size in the finite element method. The final characteristics of weight functions are its functional forms. The weight function should be continuous and positive in its support. Some commonly used weight functions are 14 • the cubic spline weight function: 1,0 1 2 1 , 3 4 44 3 4 2 1 ,44 3 2 32 32 (27) • The quadratic spline weight function: 1,0 1,3861 432 (28) With (29) Where is the normalized distance between the node and point . The parameter for each node should be chosen large enough that matrix be invertible everywhere in the domain. Also, it should be small enough to retain the local characteristic of the approximation. Equation 29 directly applicable in a circular domain as in the following equation: • Two dimensions, circular domain (30) In elliptic weight functions, each node has a major radius of influence , a minor radius of influence and an angle θ that determines the direction of the maximum radius of the influence ellipse. The influence elliptic domain of a node is determined by three parameters in contrast to one in the circular domain weights. The availability of more controlling parameters for determination of the influence domain of each node helps to increase computational efficiency of EFGM. The normalized distance in the elliptic form (2D) is 22 (31) The use of the circular support may add the number of points in the influence domain of a node ineffectively. In the Figure 2, the influence domain or support of a node that is at the center of the ellipse or circle is shown. There are more sample points such as integration points in the circular support than the elliptic support. 15 The derivatives of the weight functions can be computed using the chain rule. For example, for circular supports: 2 (32) The performance of elliptic domain is studied for Timoshenko beam (Figure 3). The problem is solved for =1000 as a parabolic tangential stress at the free end, = 3 0 × 107, and = 0 3. The beam is considered to be of unit depth and it is in plane stress state. This problem was numerically solved by [4, 15, 20]. The regular node distribution together with the background mesh that is used for numerical integration of the weak form is shown in Figure 4. In each integration cell, 4 × 4 Gauss quadrature is used. A linear basis and cubic spline weight function are used in the MLS approximation. For this problem, circular nodal support of radius 3.5 times from longer nodal spacing is employed ( 5.3 ). Also, elliptical nodal support of radius 3.5 times from longer nodal spacing ( 5.3 ), and smaller nodal spacing ( 5.3 ), in x and y direction respectively are employed. In Table 1, the vertical displacement at the point ( 0) calculated by EFG is compared with the exact solution. This table shows excellent agreement between EFG and the analytical solution, particularly in the results of the elliptic domain. For more clarification, the deflection of the beam is shown in Figure 5. Also, The stresses at the center of the beam (x = L/2, y [−D/2,D/2]) compared with the exact solution are shown in Figure 6. In addition, the distribution of the stresses in the beam is plotted in Figure 7. It is of particular interest that very smooth stresses are obtained by using the elliptic domain-EFGM without any additional treatment as is necessary in FEM. Furthermore, reduction of computational cost is verified. The ratio of the elapsed time for solving the problem with elliptic domain to the circular domain is approximately equal to 0.61 at 1800 field nodes. This is due to less value of in Equation 8 when the elliptic domain is used. This example involves cracked plate under remote tension (Figure 8). Consider an infinite plate in stretching with crack at the center. Due to the symmetry, twofold symmetry (ABCD) is used as a model for this problem. Along ABCD the closed form solution in terms of polar coordinates in a reference frame , centered at the crack tip is 2 cos22 2 cos 2 12 , 2 (33) 2 cos22 2 sin 2 12 , 2 (34) Where is the stress intensity factor, is Poisson’s ratio and is Young’s modulus. The square of 10mm×10mm, =100 mm; = 103 N/mm2, = 0 3 and remote stress of unit N/mm2. Enriched EFGM is used to solve this example where the re-mesh (redistribute of nodes in the crack tip region) is not needed. Geometry of the crack is created as a line as shown in Figure 9 by using the level set method [20]. Additionally, same figure shows the detection of the enriched method for the nodes with the elliptic domains set. More information about the enriched meshless method is found in [9]. After the solution, the error of the displacement norm by using the circular 16 and the elliptic domain is extracted. Table 2 is cleared that the results by using elliptic support is better than using circle support. Anyway, major advantages of the use elliptic support in the cracked plate problem can be demonstrated in the following points: The error of the displacement norm is curtailed, and it is decreased rapidly when the nodes increase or when the area of the elliptic decrease. This gives the problem multi other parameters to control on the numerical solution. Reduction of computational cost is occurred. This is due to less value of the nodes in the local domain (support). The reach path to the final solution is done under a small total number of nodes. This can be illustrated by Figure 10-12 that done at same longer domain size. Figure 10 represents the exact distribution of stress σyy, that is more coincide with Figure 11, which represents the distribution of numerical stress σyy using the elliptic domain. It is a very clear the failure of the distribution of numerical stress using the circle domain for the exact distribution, as in Figure12. Elliptic domain of influence is used in the element free Galerkin method to study the possibility of using this local domain in the computational mechanics. This domain changes the behavior of work of the used weight function in the extract the data from the nodes, because in the elliptic support, each node has three characteristic indications that are major radius, inner radius, and the direction of major local domain . 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Liu, "An Introduction to the Programming of Meshfree Methods", Book, Springer, (2005) [19] B. N. Rao, and A. S. Balu., "Fuzzy Meshfree Method for Fracture Analysis of Cracks"، ASME Pressure Vessels and Piping Division Conference, (2007) PVP2007-26792. [20] V. P. Nguyena, T. Rabczuk, S. Bordas, and M. Duflot, "Meshless Methods: A Review and Computer Implementation Aspects", Math. and Comp. in Simulation 79 (2008) 763–813. 18 : Comparsion of Vertical Displacement End of Beam uy EFGM Error % Nodes uy Exact Elliptic Domain Circular Domain Elliptic Domain Circular Domain 7 x 5 -0.00890 -0.00848 -0.00831 -4.71 -6.26 11 x 5 -0.00890 -0.00877 -0.00868 -1.46 -2.47 15 x 9 -0.00890 -0.00881 -0.00879 -1.01 -1.23 20 x 9 -0.00890 -0.00881 -0.00879 -1.00 -1.23 30 x 15 -0.00890 -0.00883 -0.00882 -0.78 -0.89 60 x 30 -0.00890 -0.00890 -0.00884 0.00 -0.67 : The Error of the Displacement Norm Results Nodes Circular Domain Size Disp. Norm Error Elliptic Domain Size Disp. Norm Error 2.5 x 2.25 0.372 2.5 x 2.00 0.3625 x 5 2.5 0.372 2.5 x 1.50 0.202 2.5 x 2.25 0.103 2.5 x 2.00 0.1027 x 7 2.5 0.115 2.5 x 1.50 0.075 2.5 x 2.25 0.073 2.5 x 2.00 0.04310 x 10 2.5 0.081 2.5 x 1.50 0.028 19 Discretization Using Meshless Methods - Nodes, Circular Domains of Influence. Requirement Representation of Elliptic Domain, and Circular Domain to the Integrations Points. 20 The Timoshenko Beam. 20 x 9 Regular Nodes Distribution with Background Mesh Configuration for the Timoshenko Beam. Deflection Representation of the Beam. 21 Stress Comparison, Shear Stress. Stress Comparison, Normal Stress. 22 Stress Distribution, Shear Stress σxy. Stress Distribution, Normal Stress σxx. 23 Infinite Cracked Plate Under Remote Tension. Creation and Detection of ABCD Crack. 24 Exact Stress Distribution (Deformed Scaled). Numerical (Elliptic Domain Used) - Stress Distribution (Deformed Scaled). 25 Numerical (Circle Domain Used) - Stress Distribution (Deformed Scaled).