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1、2011 International Conference on Modeling, Simulation and Control I PC SIT vol.10 (2011) (2011) IACSIT Press, Singapore Finite Elements for Engineering Analysis: A Brief Review Logah Perumal1 and Daw Thet Thet Mon 2 l, 2 Faculty of Engineering and Technology, Multimedia University, Malacca Campus, J
2、alan Ayer Keroh Lama, Bukit Beruang, 75450 Melaka, Malaysia Abstract, Finite element analysis (FEA) employs piecewise approximation in which the continuum (domain) of interest is divided into several sub regions called finite elements. Each finite element is solved independently and later, overall s
3、olution for the continuum is obtained by combining these individual finite element results. Various finite elements have been proposed to facilitate analysis of new phenomena and to improve existing methods. This paper is written to provide brief introduction to FEA and evolution of finite elements
4、used in engineering analysis. From the review reported in this paper, it can be seen that finite element has undergone vast development since its formulation. There are still rooms for further development to suite for analysis of new phenomena and more specific elements are being developed to facili
5、tate new engineering application. Keywords： Element, Finite element, finite element analysis, element formulation. 1. Introduction Finite elements play an important role in FEA since it enables a continuum to be analysed with ease, by discretizing the continuum into several sub domains. Apart from t
6、hat, finite elements enable analysis of a continuum with complex geometries, since finite elements can be formulated with arbitrary shape. This enables a continuum to be represented without need for any actual geometry simplification. Finite elements are formulated to carry out specific analysis and
7、 they are used in various engineering applications, especially in engineering design. Use of FEA enables verification of a proposed design to be safe and meets required specifications even before the design is manufactured. Finite elements play an important role in providing accurate and reliable re
8、sults, since it would impact important decisions which are taken based on the FEA results. Early studies showed that accuracy of a FEA depends on the elements geometry, geometry distortion such as isoparametric elements, shape function used in development of finite element, principles and laws used
9、in developing the governing equation and material to be analysed. Since there are many choices of elements to be used for an analysis, studies have been conducted to ease selection of suitable element for an analysis. For 1-dimensional analysis, specific elements can be directly used to solve certai
10、n analysis. Subsequent research results indicated that for 2-dimensional analysis, bilinear quadrilateral elements are superior compared to simple linear triangular elements in terms of meshing and accuracy. Accuracy of simple linear triangular element can be improved by using higher order elements,
11、 but it leads to a problem known as mesh locking, which is main drawback of triangular element when the analysis is done onto incompressible materials 1. Later, h-p adaptive techniques were introduced to increase accuracy of finite element method. It is found out that quadrilateral elements give bet
12、ter accuracy compared to triangular elements when tested using the newly formulated h-p adaptive technique 2. As for 3-dimensional analysis, hexahedron elements are preferred compared to tetrahedron elements. For same degree of order, hexahedron elements give more accurate results compared to tetrah
13、edron elements. Nevertheless, both quadratic hexahedron and tetrahedron elements give similar performance and accuracy. 60 This paper is hoped to give a brief idea to the reader on FEA and finite elements that have been developed so far. Organization of the paper is as follows. Following section 2 i
14、s used to provide brief introduction to FEA. Formulation of a finite element is explained in section 3. Section 4 is used to highlight evolution of finite elements since its formation and the paper is finally concluded in section 5. 2. Finite element analysis Behaviour of a phenomenon can be represe
15、nted using mathematical models (approximate models), which are derived based on principles and laws. There are several principles which are used to formulate finite elements. Principle of static equilibrium (also known as direct method) is used for phenomena which can be represented by simple govern
16、ing equation, and theorem of Castigliano and principle of minimum potential energy are applied for complicated elastic structural systems. Higher mathematical principles, known as variational methods are used to formulate finite element analysis for phenomena governed by complex mathematical model,
17、involving derivative terms. There are several variational methods such as Ritz, Galerkin, collocation, and least-squares methods 3. Once approximate model has been developed using principles and laws described above, shape functions are then applied according to element geometry to complete the fini
18、te element formulation. General equation for a single finite element is represented in form shown below: where k is the matrix representing characteristics of the continuum, is the column matrix representing nodal values (output variable of interest), and Q represents input to the continuum. In case
19、 of stress analysis, k represents stiffness matrix, q represents vector of nodal displacements, and Q represents vector of nodal forces. Once individual finite elements are formulated, these finite elements would then be assembled to form global/assemblage equations which are represented generally i
20、n form below: W W = W ( 2 ) where K represents assemblage property matrix, r represents assemblage vector of nodal unknowns, and /? represents assemblage vector of nodal forcing parameters. Figure 1 summarises steps involved in FEA. Assemble element equations to obtain global assemblage equations in
21、 the foim of equation (2) Fig. 1: Steps involved in finite element analysis. 61 3. Formulation of a finite element Finite element equation, which is represented by equation (1), is obtained by incorporating governing equation into element equation. Element equation is derived from shape function and
22、 geometry equation. Number of equations for an element depicts number of nodes for the element. For example, considering two degree of freedom for each node, linear shape function for triangular and rectangular elements will consists of 2 unknowns (represents 2 degree of freedom). Number of element
23、equations for a triangular element would be 3 whereas rectangular element would have 4 equations. In other words, number of equations for an element represents total number of nodes for the particular element and number of unknowns in each element equation represents degree of freedom of the particu
24、lar node. Figure 2 below shows example of triangular and rectangular elements with 2 degree of freedom for each node. Assuming one field variable to be analysed, linear shape function for elements in figure 2 are given by equations 3 and 4. 3(X3=J3) 4(X4， .V4) 1(0:0) 2(X2, ) 1 (1=1) 2 Fig. 2: Triang
25、ular and rectangular elements. Shape function for linear triangular element (3 nodes): 3(x， j) = “0 + (3) Shape function for linear rectangular element (4 nodes): 3(又， j)二 “0 + w (4) where coefficients a , a j , a 2, and a 3 are the unknowns. Element equations for triangular element are later obtain
26、ed by incorporating nodal conditions into the shape function and rewriting: j) = x2 3 - 3- + (-3 X l ) y 2x,y)-yx-xzy y ) = 少 3 (5) Where x and y represents the coordinate systems and N j , N 2 and N 3 represents element equations. Similarly for rectangular element: (6) where r and 5 represent the n
27、ormalized coordinate systems. Governing equation is then incorporated into the element equation to form the specific finite element. For example, governing equations for 2-dimensional plane stress are given by: + L - d.x d.x (7) dy dy Substituting governing equations above into the element equation
28、for a triangular element produces the finite element formula: 62 小 dN dN2 dN3 0 0 0 ux dx dx dx 2 0 0 0 dN dN2 dN3 u3 y 17 dN dN2 dN3 dN、 dN2 dN3 y dx dx dx -=剛 similarly for cr = 5 (8) Where matrix s represents element strain matrix, B represents strain displacement matrix, 8 represents element dis
29、placement matrix and D represents material property matrix. Application of minimum energy principle later yields the following equation: m s = / (9) k = VeBrDB (10) Where V represents volume of the element and / represents nodal forces applied onto the element. Figure 3 below summarises steps involv
30、ed in development of a finite element equation. Fig. 3: Summary of development of an element equation. 4. Evolution of finite elements Basically, there are three groups of elements and those are; line elements (for one dimensional analysis), planar elements (also known as membrane elements; for two
31、dimensional analyses) and solid elements (for three dimensional analyses). There are various types of elements formulated under each group. For instance, spring element, bar element, and flexure elements are specific elements categorised under line elements. Specific elements are used in particular
32、cases in which the element has been formulated for. On the other hand, general elements can be used for any cases, by simply changing governing equation according to problem type. Triangular elements and rectangular elements are examples of general planar elements. Pentahedron (also known as triangu
33、lar prism or wedge), hexahedron (also known as brick) and tetrahedron (also known as tet) elements are examples of general solid elements. Since then, more types of elements have been formulated as described below. 4.1. One dimensional elements Simplest line element is the spring element (a specific
34、 element developed using principle of static equilibrium) and can be used to analyse linear member within its elastic range. Since its formulation, spring elements have been successfully applied in various applications. One of the applications which benefited plenty from line-spring element is simul
35、ation and analysis of surface crack growth. Line-spring model was developed and used to carry out stress analysis for cracked surface, but limited to linear elastic analysis 4. Nonlinear elastic-line-spring model was then developed 5. In order to incorporate plastic deformation, the 63 authors in 6
36、introduced elastic-plastic model of the line-spring finite element using the incremental theory of plasticity. Eventually, fully plastic crack growth has been formulated using line spring element in 7. Analysis of surface crack requires large computational time, especially for 3-dimensional analysis
37、. It is found out that this problem can be solved by coupling shell and line-spring finite elements together 8 (Shell elements are used to model the surface while line-spring elements are used to model the crack growth), which reduces 3-dimensional problem to equivalent simpler 2-dimensional problem
38、 and thus reduces computational effort. These observations are reported in 9. Another application of coupled shell and linespring finite elements is found in fracture mechanic analysis on welded wide plates 10. Spring element has also been used in seismic response analysis, by representing the conta
39、ct bodies by finite number of small rigid bodies which are connected with springs distributed over the contact area of two neighbouring bodies 11 . This method is called rigid body spring element method (RBSM) and proved to be more economical and reduces computational time compared to conventional f
40、inite elements 12. RBSM is also used in structural analysis, fracture analysis and damage analyses. Spring element has also been used in analysis of fiber-reinforced composites by representing the fiber with longitudinal spring element and the matrix with transverse shear spring element. The mathema
41、tical model representing fiber-reinforced composites using spring elements as described above is known as spring element model (SEM) 13. New types of spring elements have also been formulated. Such new development is shear spring elements which were formulated to carry out stress and energy analysis
42、 of a centre-crack panel reinforced by a rectangular patch 14. In 15, the authors have formulated generalized beam/spring track element by coupling spring elements at the periodic rail/tie intersections to analyze natural vibration of rail track within its elastic range. Concept of spring element ha
43、s then been extended to bar element, (also known as spar, link, beam or truss element, which are also specific elements) by introduction of “interpolation function， which is also known as “shape function or blending function” to evaluate values in between boundaries of element. From this point onwar
44、ds， the term “shape function” will be used. Bar elements are used to analyse skeletal-type systems such as planar trusses, space trusses, beams, continuous beams, planar frames, grid systems, and space frames. One of applications which benefitted a lot from this element is analysis of truss and beam
45、 structures. The authors in 16 have developed a beam finite element to analyse dynamic problems especially vibration, which is arising in beam-like truss structures. Later, elastic beam finite element was formulated based on elasto-plastic fracture mechanics to simulate formation and effects of crac
46、ks in beams 17. Analysis of hyperbolic cooling tower which was formerly focussing solely on the tower was made complete by using beam elements to include the supporting columns in the analysis 18. A new element known as rigid-ended beam element was formulated to represent brackets in beam modelling
47、of ship structures (Formerly, the brackets were represented by rigid elements). The newly formulated rigid-ended beam element has been found to increase computational efficiency as compared to previously used rigid elements 19. Beam element has been also found to be applied for linear static analysi
48、s of laminated composites and in wavelets analysis. Another type of beam element is cubic beam elements which are used in practical analysis and design of steel frames 20. Bar elements discussed above takes only axial loads and can be used to analyse simple trusses, since bending effects are not inc
49、luded in the analysis. Later, elementary beam theory is used to include transverse bending effect and thus new element known as flexure element was developed. Flexure element developed earlier is then enhanced using shear-deformable theory in order to avoid “locking” problem which arises in analysis of thin beam/plate 21. Flexure elements are also used as a design tool to d