Key Engineering Materials Online: 2011-01-20 ISSN: 1662-9795, Vols. 462-463, pp 1008-1012 doi:10.4028/www.scientific.net/kem.462-463.1008 2011 Trans Tech Publications, Switzerland Effect of Mesh Size of Finite Element Analysis in Modal Analysis for Periodic Symmetric Struts Support Weibing Liu 1,a, Mamtimin Geni 1,b and Lie Yu 2 1 School of Mechanical Engineering, Xinjiang University, Urumqi, China,830008 2 School of Mechanical Engineering, Xi an Jiaotong University, Xi an, China,710049 a lwbhaiyun@163.com, b mgheni58@gmail.com Keywords: Gas Turbine; Finite Element Analysis; Modal Analysis; Hexahedral Mesh; Tetrahedral Mesh Abstract: Meshing for finite element analysis accuracy plays a very important part in numerical simulation of Periodic Symmetric Struts Support (PSSS). Different accuracy can be obtained by different element sizes or types. Three element types and eight element sizes are used for comparing the accuracy of modal analysis in this paper. Comparing with the mutual relations of different accuracy, the scientific basis is provided for selecting the correct mesh size and improves the efficiency of numerical calculation in modal analysis. Introduction The Periodic Symmetric Struts Support is an important component in gas turbine, Advantages and disadvantages of structural design play a key role in the stability of the gas turbine shaft and working efficiency of gas turbine. In order to avoid resonance that happened between PSSS and gas turbine shaft and service for Dynamic Response Analysis (DRA). The necessary method of the modal analysis is used for PSSS of gas turbine. But the difficulty of technology will be needed and the huge cost will be spent for dynamic response of PSSS measured directly. The quite necessary and effective method of the finite element analysis (FEA) in modal analysis is used to instead the test in field. The mathematical exposition of FEA is the differential equation discrete into much amount of algebraic equations and solved by using the computer technology [1-4]. Different accuracy of FEA can be obtained by different element types and number of elements. Meanwhile, accuracy can be raised by decreased the mesh size. However, after mesh size decreased, number of elements and computing time will be increased and accuracy will be decreased by calculation of the accumulated error that caused by elements increased[5]. The three types of elements (4 points Tetrahedral, 10 points Tetrahedral and 20 points Hexahedral) are selected in this paper for modal analysis of PSSS. The effect of each element on the accuracy is found by comparing this numerical solution. Meanwhile, the effect of different element sizes are compared too. By this numerical solution which contains the effect of different element types and element sizes, the scientific basis which is for selecting the correct mesh size and improving the efficiency of numerical calculation in modal analysis is provided. Evaluation Method The 4 points tetrahedral, 10 points tetrahedral and 20 points hexahedral element types are chose for a conducting large bending deformation of PSSS by considering the thermal deformation and dynamic response. Fig.1 shows the three types of elements. The picture of Fig.1(a) shows the 4 points tetrahedral mesh types, Fig.1(b) is10 point tetrahedral mesh type and 20 point hexahedral mesh type is shown as in Fig.1(c). All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (#69822597, Pennsylvania State University, University Park, USA-19/09/16,00:24:03)
Key Engineering Materials Vols. 462-463 1009 (a) 4 points tetrahedral (b) 10 points tetrahedral (c) 20 points hexahedral Fig. 1 Element type Element shape functions of 4 points tetrahedral, 10 points tetrahedral and 20 points hexahedral are expressed as following equation respectively. 1 N = ( a + b x+ c y+ d z) (1) i 6V i i i i where i= 1, 2,3 4,V is volume element. ( 2 1) ( 1, 2,3,4) Ni = Li Li i= N5 = 4 L1 L2, N6 = 4 L2L3, N7 = 4L1L 3 N = 4 L L, N = 4 L L, N = 4L L 8 4 1 9 4 2 10 4 3 (2) 1 Ni = ( 1 + ξ0)( 1 + η0)( 1 + ζ 0) (3) 20 Where, ξ0 = ξξ, η0 = ηη, ζ 0 = ζζ, = 1,,20 i i i i Under the principle of virtual work which is principle of internal forces do work equal to outside do work. Dynamic response equation is wrote as [ M]{ U} + [ C]{ U} + [ K]{ U} = { F( t) } (4) When system does simple harmonic motion in modal analysis, { U} { U } ( ωt ϕ) = 0 sin + (5) Equation (4) can be changed as ([ K] ω 2 i [ M] ){ Ui} { 0} = (6) In this formula,[ M] is mass matrix; [ K ] is stiffness matrix; U is deformation ;ω is frequency. The Effect of Different Element Types Results of modal analysis in ensuring the freedom of three mesh types of similar circumstances can be calculated. Modal frequency an important data for modal analysis is got last. By comparing modal frequency of PSSS, the effect of different element types can be found. Fig.2(a) is one kind of PSSS with 8 tangential struts. Fig.2(b) is a hexahedral with 20 points. Fig.2(c) is the modal shape at the first - order mode of PSSS.
1010 Fracture and Strength of Solids VII (a) PSSS model (b) Mesh of hexahedral (c) Modal shape at first-order Fig. 2 Modal Analysis of PSSS Change of the modal frequency that under the different s obtained by modal analysis of PSSS. 150 52.2 125 100 75 52 50 1 2 3 4 5 6 7 8 9 10 11 51.8 0 1 2 (a) Modal frequency compared (b) Partial enlargement (first modal) 73.5 72.5 2 3 Modal order 112 111 110 109 3 4 5 6 7 8 9 10 (c) Partial enlargement (d) Partial enlargement Fig. 3 Calculation results on different element types Modal frequency (0-150Hz) curves under three different element types are very close for each other in Fig.3(a). But after Fig.3(a) is partial enlarged, Through the study found that the modal frequency value under 4 points tetrahedral is biggest in all, then followed by 10 points tetrahedral, and finally is 20 points hexahedral in Fig.3(d). The difference of maximum and minimum frequency is about one Hz. As shown in Fig.3(b) and Fig.3(c) the formal third modal frequency is same exchanges. The frequency of 4 points tetrahedral and 10 points tetrahedral are higher than 20 points hexahedral. But the difference between 4 points tetrahedral and 10 points tetrahedral may be seems very small. However, negative impact which is for determining the modal frequency correctly, the stability of shaft and the best match between PSSS and machine will be produced. Evaluation of
Key Engineering Materials Vols. 462-463 1011 tetrahedral is too high compared with hexahedral. So 20 points hexahedral element type which adapt to the larger deformation and bending deformation is chose to discrete PSSS of gas turbine. The Effect of Different Element Sizes In order to ensure sub-modes under different levels of the stress concentration region with the full breakdown of mesh size, eight different mesh sizes of 20 point hexahedral are divided for modal analysis of PSSS. The number of nodes and degree of freedom (DOF) of different element sizes are listed in Table 1. After meshing which under the different mesh sizes of 20 point hexahedral and modal analysis for PSSS. Different modal frequency distributions under different mesh sizes can be found. By comparing the modal frequency of PSSS, the law of the calculated results and the time can be found. Table 1 Number of nodes and DOF of different mish sizes M S NOD DOF M S NOD DOF 30mm 975668 55mm 230542 40mm 453321 60mm 186801 45mm 340273 70mm 127531 50mm 273701 100mm 74755 150 125 100 75 50 0 1 2 3 4 5 6 7 8 9 10 (a) Modal frequency compared 52.7 52.5 52.3 52.1 51.9 0 1 2 (b) Enlargement (first mode) frequency(hz) 74 73.5 73 72.5 72 1 2 3 4 modal 114 113 112 111 110 109 3 4 5 6 7 8 9 10 (c) Partial enlargement (1) (d) Partial enlargement (2) Fig. 4 Calculation results on different element sizes The overall modal frequency curves are similar with the gradual reduction in mesh size or with the gradual increase in the number of DOF shown in Fig.4(a). However, after partial enlargement,
1012 Fracture and Strength of Solids VII modal frequency gradually decreases when mesh size is deceased at the first-order mode. Fig.4(b) shows the difference between maximum modal frequency value and minimum is 0.7Hz at the first-order mode. With the increases, the most difference of modal frequency is about 3Hz, as shown in Fig.4(d). The inversely relationship between frequency curve and mesh size shows in Fig.4. Modal frequencies are close to steady-state and modal frequency range is getting smaller and smaller with the better meshing. The maximum of frequency gap between 30mm and 100mm mesh size (between DOF and DOF) reaches to 0.04%. But the frequency gap between 30mm and 40mm (between DOF and DOF) is less than 0.01%. Modal frequency gradually decreases when mesh size is deceased. However, the big change occurs when the modal frequency under 50mm mesh size ( DOF) and 60mm mesh size ( DOF). The primarily reason is the mismatch among meshing quality, binding effect and modal shape is unworthy. So, overall optimization of meshing or the best choosing of mesh size is one of an important issue in the future. Anyway, calculation accuracy of modal analysis can be satisfied under the 40mm mesh size ( DOF). Conclusions As curve and surface boundary of higher-order element can approach structure boundary accurately, calculation accuracy under hexahedral element is higher than tetrahedral element. Meanwhile, calculation accuracy of modal analysis can be improved by increasing the number of nodes because of the order of algebraic equations obtained by discrete differential equation is increased. After mesh type is selected, calculation accuracy of modal analysis can also be increased by changing element size. But when mesh size is changed to a point or a place, the changes of calculation accuracy become stabled. There is no significance for improving calculation of modal analysis by changing mesh size continuously. Acknowledgement The study is supported by the National Key Basic Research and Development Program (973 Program No. 2007CB707706). References [1] S.M. Beden, S. Abdullah, A.K. Ariffin, M.M. Rahman and N. A. Al-Asady: (Engineering Mathematics Group, 2008). [2] A. Turon,C.G. Davila, P.P. Camanho, J. Costa: Engineering Fracture Mechanics (2007) [3] C.A. Felippa: Advanced finite element analysis. Department of Aerospace. Engineering Sciences, (2001) [4] Y. Fukuizumi et. al, Journal of Engineering for Gas Turbines and Power, Vol.127 (2005), p.369 [5] Suhe Gao: Mesh density and Finite Element Analysis of Accuracy (Heavy Industry Science and Technology, 2006).