2.2 Weighting function

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1 Annual Report (23) Kawahara Lab. On Shape Function of Element-Free Galerkin Method for Flow Analysis Daigo NAKAI and Mutsuto KAWAHARA Department of Civil Engineering, Chuo University, Kasuga 3 27, Bunkyo ku, Tokyo, JAPAN, E mail: kawa@civil.chuo u.ac.jp Abstract The mesh-free methods, which does not require time-consuming mesh generations for modeling the computational domain, are expected to become a key technology in the next generation of computation methods. The mesh-free method is easy to employ the CAD data because it needs only nodal data and definition of the domain. The mesh-free method is possible to realize seamless system from modeling to computation, thus it is expected as an useful method. The element-free Galerkin method (EFGM) proposed by Belytschko et al []. is one of the practical mesh-free methods. Purpose of this study is to investigate the basic characteristics of the EFGM. To do this, the diffusion equation is analyzed. Introduction In recent years, the performance of computer is improved and the technology of the numerical analysis is highly progressed. Therefore it is possible to simulate the large-scale numerical simulation. The finite element method (FEM) has been successfully applied to solve a wide variety of engineering problems. On the other hand, it is clear that the mesh generation is a most time-consuming process for the analysis. For this reason, it can be said that the preparation of computational data is preventing the technology of simulation from popularization. Recently, study of the mesh-free method which does not need elementnodal data for input data is studied in various fields of engineering. Among the mesh-free methods proposed, the element-free Galerkin method (EFGM) has been considered as the most realistic and practical scheme[2]. The EFGM does not need the element-nodal data, however, to make a interpolation function, the EFGM needs a lot of calculation time because searching the nodes are necessary at every integration point and as the characteristics of the EFGM, making shape function becomes complicated. Therefore, it prevents the EFGM from popularization. It is very important to reduce the calculation time for the EFGM. In this research, for reducing the calculation time, effective scheme for searching the node is presented. As a numerical example, two-dimensional diffusion equation is analyzed. 2 Element-Free Galerkin Method(EFGM) 2. Moving Least Square Method(MLSM) As for the FEM, state value of arbitrary integration points in the domain is interpolated using the shape function based on the state value at node included in an element. On the other hand, as for the EFGM, state value of arbitrary integration points in the domain is solved by a kind of Least Square Method based on the state value of the list of nodal points which exists in the circle of which the center is the integration points. The radius of circle is called domain of influence. Introducing a weighting function which has an influence according to the distance between integration point and every node in the domain of influence to the Moving Least Square Method(MLSM), the method is said as a kind of least square method. In short, weighting value is changed according to the distance between integration point and 232

2 nodes in the domain of influence, then a smoothly approximate curved line is given. For the reason of this characteristic, an approximated value is changed according to the movement of location, this approximate method is called as the MLSM. The characteristics of the MLSM are described as follows referring to Fig., Approximated Function Approximate curved line does not pass the nodes. Compared with the shape function applied to the FEM, differentiated value of node is continuous at all sections. x(sampling point) Fig.. Image of MLSM Not only in the EFGM but also in the mesh-free method, the MLSM is a representative method, in which the unknown function φ(x) h is approximated as follows, φ h (x) ={p (x)} T {a (x)}, () where p(x) is linear basis, a(x) is coefficients, which depends on the position x. The coefficient a(x) is obtained minimizing the performance function J, and m is the numbers of terms in the basis. J = N N w (x x i ) {φ h i (x) φ i} 2 = w (x x i ) {{P (x)} T {a (x)} φ i } 2, (2) i i where N is the number of nodes in the neighborhood of x, and φ i is the nodal state value of φ at x = x i. The weighting function w(x x i ) is the function of distance between integration point x and node x i. The stationary condition of the performance function J in Eq.(2) with respect to a(x) is written as follows, J =, (3) a(x) which leads to the following linear relation between a(x) and φ, where A(x) and B(x) are defined as follows, A(x) = a(x) =A (x)b(x)φ i, (4) N w(x x i )p(x i )p T (x i ), (5) i [B(x)] = [w(x x )p(x ),w(x x 2 )p(x 2 ),...,w(x x N )p(x N )], (6) {φ N } T = {φ,φ 2,...,φ N }, (7) in which A(x) and B(x) are m m, m N matrix as follows, [A mm ] = [D mn ][W NN ][D mn ] T, (8) [B mn ] = [D mn ][W NN ], (9) [D mn ] = [{p(x ) m }, {p(x 2 ) m },...,{p(x N ) m }], () [W NN ] = diag[w(r ),w(r 2 ),...,w(r N )]. () Then the interpolation function φ h (x) is transformed into the following form, φ h (x) ={p(x)} T [A(x)] [B(x)]{φ}. (2) where [A(x)], [B(x)] are functions of space coordinates x, where [A mm ] is symmetric matrix. The partial derivatives of φ h (x) can be obtained as follows, φ h (x) x =( {p(x)}t x [A mm ] [B mn ]{p(x)} T [A mm] x [B mn ]{p(x)} T [A mm ] [B mn] ){φ N }. (3) x 233

3 2.2 Weighting function It is very important to select a weighting function w(x x i )=w(r i )in the MLSM. The choice of the weighting function affects on the resulting interpolation function φ h (x). There is a large range for selecting the weighting function. The basic characteristics are as follows, Number of weighting function is positive. Weighting function is defined as the function of distance between the two points. The Quartic Spline Function as shown in Fig.2 is applied in this study for the weighting function w(r i ) w(r i )=. 6.( r i ) 2 8.( r i ) 3 3.( r i ) 4, (4) d m d m d m where this weighting function is satisfied the following conditions, w(r) Quartic Spline Function r/dm Fig.2. Weighting Function w(d m )=, (5) w (d m )=, (6) w (d m )=, (7) w () =, (8) w() =. (9) in which r i, d m is the distance between integration point x i and node x, radius of the domain of influence supported by the weighting function. By reason of these conditions, in case that integration point is moved to the next integration point, the new node is not employed to the interpolation of the next integration point. Then, C continuity is satisfied by these conditions of the weighting function. The weighting function is effective at the integration point x only. If the integration point is changed, the other domain of influence should be taken. The new function is defined by re-evaluation of the weighting function. Continuing this process in every integration point x, a(x) becomes a function of x. Thus, even if linear basis is employed, a smoothly approximated function is obtained. The relation of distance and weighting value is expressed in Fig.3. r Weight Function w(r) r evaluation point candidate node Fig.3. Domain of Influence 2.3 Numerical integration As for FEM, integration of domain is carried out over every element. Contrary to this, in case of EFGM, technique is required, because the EFGM has no element. As for EFGM, the domain is divided into latticed domain called background cell as shown in Fig.4. This integration point in the cell becomes a unit of integration. Usually, the Gauss integration is applied to integration in all cells. The number of integration points is 4 at inner cell, and 6 at boundary cell in this paper. node integration point Fig.4. Background Cell 234

4 3 Comparison of Shape Function To clarify the basic characteristics of the MLSM interpolation, the sinusoidal curve is interpolated by the EFGM and the FEM. The total number of node is 2. One cycle of sinusoidal curve is interpolated. The differentiated value of sinusoidal curve becomes cosine curve, which is illustrated in Figs.5 and 6. sin(efgm) cos(efgm) sin(fem) cos(fem) Fig.5. EFGM Fig.6. FEM From these figures, it is clear that even if linear basis is employed, the differentiated value of the interpolation function of EFGM becomes continuous. Contrary to EFGM, FEM results become a constant derivatives. This is one of the important characteristics of the MLSM. sin(efgm) cos(efgm) sin(regular) sin(irregular) Fig.7. EFGM (C continuity) Fig.8. EFGM(irregular node) Domain of influence is defined at every evaluation point. Fig.7 shows the result that the size of domain of influence is changed in large amount of distance at the one evaluation point. As for EFGM, it is important how to select the domain of influence. As for MLSM, C continuity is not satisfied automatically. In case of the size of domain of influence is changed in relatively long distance next to the evaluation point, the weighting value is changed at that node. Then C continuity is not satisfied. From Fig.7, it is clear that the C continuity is not satisfied. The condition of satisfying C continuity is that the size of domain of influence should not change in long distance cos(efgm) cos(fem) It is also important in EFGM, the data of nodal space coordinates is changed into the data of the distance. Therefore, it is very important to select the well-balanced node for MLSM. Fig.8 shows the result that the nodal distribution is irregular. Comparing with the regular nodal distribution, there are some errors in computation. Fig.9 shows the comparison of the differentiated value(cosine curve) between EFGM and FEM. Fig.9. Comparison of cosine curve 235

5 4 Effective Scheme for Searching Nodes As for EFGM, because the interpolation function is approximated by MLSM, it takes a lot of calculation time to search neighborhood of nodes at every integration point. Generally, for making the shape function, the distance between integration point and all nodes should be computed. The distance are sorted in the ascent. Then, optimum nodes can be found to be closest distance. The domain of influence dm can be computed. In case of the present scheme, the node number which is contained in the cell is investigated. Therefore, only to create the distance between integration point and some nodes which exist in the neighboring of cells (Fig.) can be sorted. If there are no optimal nodes in the neighboring cells, continue to adding the next neighboring cells(fig.). By reason to apply this scheme, it is able to reduce the calculation time as the number of integration points is increasing. Domain of influence :integration point :candidate node :not reffered node :integration point :candidate node :not reffered node Fig. < Image of Present Scheme > Fig. 5 Penalty Function Method As for EFGM, the state value of the approximated function at every node does not correspond to the practical state value of every node, which means the approximated function does not pass the node as shown in Fig., because of the least square approximation. Then, in this paper, the Penalty Function Method (PFM) is employed to satisfy the Dirichlet boundary condition. Using PFM, derivatives of an approximated solution at any point can directly coincide with nodal state values. PFM is solved by the one dimensional FEM. 6 Sparse Method Using PFM, the large scale matrix is resulted, so that the iterative method such as Conjugate Gradient Method is unsuitable for solving the PFM formulation. Therefore, it is important to select a direct method which is able to perform the rapid computation. In this paper, the sparse method and improved Cholesky decomposition scheme are employed to solve the EFGM equation. As for the sparse method, only non-zero factor can be memorized. Therefore, the calculation time can be drastically reduced. 236

6 7 Numerical Study 7. Diffusion equation As the numerical study, two-dimensional diffusion equation using EFGM is analyzed. The following diffusion equation is introduced, φ κ( 2 φ x 2 2 φ )=, y2... (in V ) (2) φ = φ,... (on Γ ) (2) where φ is contaminant density, κ is diffusion coefficient, respectively over-bar in Eq.(2) denotes the given function on Γ. For the discretization in time, the Crank Nicolson scheme, which is capable of taking the large time increment and superior in stability, is employed and expressed as follows. φ n φ n κ( 2 φ n 2 t x 2 2 φ n 2 y 2 )=,... (in V ) (22) where φ n 2 = 2 (φn φ n ). (23) Applying the PFM, weighting residual equation written as follows, (φ φ n )dv V 2 tκ φ ( 2 φ n V x 2 2 φ n y 2 )dv 2 α (φ φ n )dγ = γ (φ φ n )dv V 2 tκ φ ( 2 φ n V x 2 2 φ n y 2 )dv 2 α (φ φ n )dγ α (φ φ)dγ, (24) γ γ α is a penalty number to ease the basic boundary condition. α is very large number. 7.2 Numerical example Fig.2 shows the initial state of function φ. Fig.3 represents the nodal distribution. The total number of node is 68. The distribution is referred to as NODE.. As for the domain of influence, fixed value is obtained for all integration points. Time increment t is.. Diffusion coefficient is.. In the following analysis, point- expresses the node of which is the center of the corn. initial state Fig.2. Initial State Fig.3. NODE. 237

7 7.3 Numerical result Fig.4 shows the result of NODE.. The cross axis shows the elapsed time. The vertical axis is the concentration φ at point-. In this case, the nodal distribution is regular. The nodal distribution has a significant influence on the result. From Fig.4, the result by EFGM brings well in agreement with by FEM. The regular nodal distribution such as NODE. is suitable for EFGM analysis. state value FEM EFGM time Fig.4. point-(node.) 7.4 Numerical example 2 Because EFGM can perform the analysis without element data, it is easy to make the input data of EFGM. Figs.6 and 7 show the computational domain and the nodal distribution. The special technique needs not to make the mesh, simple procedure can be used for mesh generation. Fig.6. Computational Domain Fig.7. NODE.2 Fig.8. NODE.3 By the characteristics of MLSM interpolation, EFGM analysis needs the node adjusting. The nodal distribution must keep well-balance at every domain of influence. As for NODE.2, the nodal distribution is irregular at the center of the computational domain. The distance of nodes becomes short, MLSM can not make the fine shape function, because MLSM changes the data of nodal space coordinates into the data of distance. Fig.8(NODE.3) represents the nodal distribution. This figure illustrates the distribution after adjusting of the NODE Numerical result 2 After the node adjusting, the NODE.3 keeps the well-balance at every domain of influence. Figs.9 and 2 represent the result of the final contour of NODE.2 and NODE.3. Fig.9. contour(node.2) Fig.2. contour(node.3) 238

8 From Figs.9 and 2, it is clear that adjusting the nodal distribution, the EFGM can analyze the every computational domain, and the adjustment of the nodal distribution is not hand-working task. Figs.2 and 22 show the result of the point- based on NODE.3. Even if diffusion coefficient is changed into large number, it keeps good stability. FEM EFGM.9 K=. K=.3 K= Fig.2. point-(node.3) Fig.22. point-(node.3) 8 Conclusions In this paper, the analysis of diffusion equation using EFGM is presented. The balance of the nodal distribution is investigated. From the numerical results, nodal balance inside the domain of influence has a significant influence on the computational results. To obtain the fine results, the well-balanced nodal distribution is necessary for EFGM analysis. If the well-balanced nodal distribution is prepared, then EFGM can analyze various fields of numerical analysis. Considering the time-consuming mesh generation in FEM analysis, the more research of EFGM is expected. In the future, although the technology of computer will progress, the problem of mesh generation is not expected to reduce by large amount. As for FEM, the incompatible mesh for the input data is not applicable. To adjust the incompatible mesh in FEM, it is hand-working task. This work takes a lot of time in computation. Even if the computer will progress, the time of hand-working task is not expected to large reduction[3],[4]. In the point of total computational time, EFGM is expected as the effective method. Especially, EFGM is hopeful for those problems depending on the mesh, requiring re-mesh and making use of the characteristics of continuity of derivatives such as C continuity. References [] T.Belytschko, Y.Y.Lu and L.Gu : Element-free galerkin method, International Journal for Numerical Method in Engineering, Vol.37,pp ,(994). [2] H. OKUDA, T. NAGASHIMA and G. YAGAWA : Basic study on Element-Free Galerkin Method(st report, application to ordinary differential equation) and (2st report, application to two-dimensional potential problems), JSME International Journal, Vol.A, No95-395,(995). [3] H. Sakurai and M. KAWAHARA (22a): The element-free Galerkin method for three-dimensional groundwater flow analysis. WCCM V, 5th World Cong. Comp. Mech., Vienna(publ. on congress website) [4] H. Sakurai and M. KAWAHARA (22b): Three-dimensional groundwater flow analysis system using the element-free Galerkin method. Proc. 6th. Japan-US Intern. Symp. Flow Sim. Modeling, Fukuoka, pp