Universal Format of Shape Function for Numerical Analysis using Multiple Element Forms

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1 , pp Universal Format of Shape Function for Numerical Analysis using Multiple Element Forms Yuting Zhang, Yujie Li, Xiuli Ding Yangtze River Scientific Research Institute, Wuhan, China Abstract. For mesh discretization of complex engineering structures, it is difficult to solely use hexahedrons. So multiple forms of elements, such as tetrahedron, rectangular pyramid, triangle prism are introduced. A universal format of shape function is put forward according to degeneration relationships between non-hexahedrons and hexahedron. Based on the linear shape function of hexahedral element, shape functions for non-hexahedral elements are presented. Corresponding programming techniques are also put forward, thus greatly facilitating the coding process. An example is given and proves that the proposed method is rational with sufficient accuracy, indicating that it is reliable and efficient. Keywords: universal format; shape degeneration; shape function; finite element analysis Introduction For stress and strain analysis of engineering structures, the finite element method is so far the most sophisticated method and wide adopted in practical projects []. It is recommended use linear hexahedral element during mesh discretization [2] as it has higher accuracy comparing to other liner elements. However, when conducting numerical simulations on stability analysis of underground caverns, it is difficult to include geological faults into finite element meshes due to the complexity of cavern profile and the arbitrariness of fault distribution. So multiple forms of elements should be introduced for convenience. The author proposed element reconstruction technique for mesh discretization of faults [3]. The essence of this technique is to further divide hexahedral elements into tetrahedron, rectangular pyramid and triangle prism elements to simulate faults. As the hexahedral elements still accounts for a large proportion in the reconstructed mesh, its calculation accuracy is higher compared to those meshes that are solely tetrahedrons. A fundamental procedure for finite element analysis is shape function definition for each included element type. Although the shape functions of non-hexahedral elements can be constructed based on certain rules, it inevitably leads to significant amount of coding work if each element type is specifically considered. Based on the degeneration relationships among different linear element types, this paper proposes universal format of shape function for calculation of meshes containing multiple element forms. ISSN: ASTL Copyright 206 SERSC

2 2 Universal Format of Shape Function using Linear Hexahedron 2. Degeneration Pattern for Non-hexahedral Elements Linear hexahedral element has simple form and comparatively higher calculation accuracy. It is the most commonly adopted element form in finite element analysis. Based on this element as standard form, by coinciding 4, 3, and 2 nodes, tetrahedron, rectangular pyramid, triangle prism can be obtained, respectively (Figure.). Based on above degeneration relationships, by repeating numbering the coinciding nodes, these non-hexahedral elements can be stored in hexahedral element format. (a) (b) (c) (d) Fig.. Standard form element and degeneration elements: (a) hexahedron, (b) tetrahedron, (c) rectangular pyramid, (d) triangle prism. 2.2 Universal Format of Shape Function for Multiple Forms of Elements The shape function of linear hexahedral element is: Ni ( 0 )( 0 )( 0 ) (i=,2,,8) () 8 where ξ 0 =ξ i ξ, η 0 =η i η, ζ 0 =ζ i ζ. ξ, η, and ζ are local coordinates. ξ i, η i, and ζ i are local coordinates for each node i. By coinciding node 2 and 4, coinciding node 5, 6, 7, and 8, hexahedron is degenerated into tetrahedron. As node 2 and 4 represent a same node, the shape function on node 2 and 4, namely N 2 and N 4, can be merged and collectively named as N 2. N 4 is then valued zero. The same treatment can be performed on N 5 N 6 N 7 and N 8. Therefore the shape function for tetrahedron can be written as: N ( )( )( ), N2 ( )( ), N3 ( )( )( ), N5 ( ), N4 N6 N7 N8 0 tetrahedron 2 (2) 64 Copyright 206 SERSC

3 In the same way, shape functions of rectangular pyramid and triangular prism can be obtained: 2.3 Characteristics of Universal Shape Function The shape function is a continuous weighting function defined within an element and has following characteristics [4]: (). Shape function related variables are continuous between neighboring elements; (2). Arbitrary linear item is included; (3). N i = at node i and N i = 0 at other node. The sum of N i is. It can be seen that Equation () ~ (4) have all characteristics mentioned above. So the universal shape function can be theoretically grounded. 3 Program Implementation As the shape functions of non-hexahedral elements adopt shape function form originally intended for hexahedral element, there is no need to specifically construct new shape functions. Therefore the coding work is greatly facilitated. Only the procedures that are associated with shape function and node coinciding should be reviewed. From program implementation point of view, shape functions are used in their original form and partial derivative form. The node coinciding operation affects the assembling of element stiffness matrices. Therefore these two aspects should be altered accordingly. The partial derivative form for shape function defined for tetrahedrons is altered as shown in Equation (3). N N N N 2 ( )( ), ( )( ), ( )( ), ( ), N2 N2 N3 N3 ( ), 0, ( )( ), ( )( ), N3 N5 N5 N5 N j N j N j ( )( ), 0, 0,, = 0 ( j 4,6,7,8) 8 2 (3) The partial derivative forms of shape functions for other elements can be obtained in the same way. Further, for a linear hexahedral element, its element stiffness matrix is a matrix, in which each value represents the stiffness of one degree of freedom to another degree of freedom. As for non-hexahedral elements, the coinciding nodes should be identified to prevent repeat stiffness accumulation. As other procedures of finite element analysis, such as calculation of degree of freedom, stiffness matrix decomposition, and equation solving, are conducted based on nodes rather than elements [5], no additional treatments are needed. Copyright 206 SERSC 65

4 4 Verification A cantilever beam example is used to verify the accuracy of proposed method. A cantilever beam with dimension of 0m 0.5m.0m (long thick height). One end is fixed and the other end is free. A downward load P of 3 000N is imposed on the free end. The elastic modulus of beam is 0GPa and its Poisson s ratio is The downward displacement, based on analytical method, is 2.40mm. For numerical approaches, the beam is discretized using hexahedron, triangular prism, rectangular pyramid and tetrahedron, respectively and four meshes are obtained. Each mesh of the beam contains only one form of element. The meshes are calculated using ANSYS and the proposed method. For ANSYS analysis, only linear elements are chosen. It can be seen from Table. that the proposed method achieves favorable accuracy compared to ANSYS and the calculated values are all close to analytical solution. It indicates that the proposed element method is reliable. Table. Comparison of vertical displacement of free edge (mm). Element type ANSYS results Proposed method s results Hexahedron Triangular prism Rectangular pyramid Tetrahedron Conclusion Based on a mesh gridding technique previously proposed by the author, this paper put forward a universal format of shape function for finite element meshes containing multiple element forms. The presented universal format of shape function is theoretically complete based on definition and characteristics of shape function concept. During program implementation, only shape function related modules and element stiffness assembling module should be altered to adapt proposed shape function, thus greatly reducing the coding work amount. It is verified that the proposed method achieves favorable accuracy and provides a new way for numerical analysis containing multiple element forms. Acknowledgments. Financial supports from National Natural Science Foundation of China (Nos , ) are greatly acknowledged References. Xu, G.C., Cai, B. H.: Supporting Structures for Underground Engineering. China Water & Power Press, Beijing (2002) (in Chinese) 66 Copyright 206 SERSC

5 2. Zheng, Y.L., Xiao, M.: Realization of 3D FEM mesh subdivision for complicated underground cavity group in CAD. Chinese Journal of Rock Mechanics and Engineering. 23, (2004) (in Chinese) 3. Zhang, Y.T., Xiao, M., Zuo, S.Y.: Methodology for modeling of complex geological faults in geotechnical engineering based on element reconstruction, Chinese Journal of Rock Mechanics and Engineering. 28(9), (2009) (in Chinese) 4. Zhu, B.F.: The finite element method theory and applications, Third Edition. China Water & Power Press, Beijing (2009) (in Chinese) 5. Smith, I.M., Griffiths, D.V.: Programming the Finite Element Method, Third Edition, Publishing House of Electronics Industry, Beijing, (2003) (in Chinese) Copyright 206 SERSC 67