Method of Area Coordinate From Triangular to Quadrilateral Elements *

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1 Method of Area Coordinate From Triangular to Quadrilateral Elements * Yuqiu Long, Zhifei Long 2 and Song Cen 3 Department of Civil Engineering, Tsinghua University, Beijing, 00084, P. R. China 2 China University of Mining and Technology, Beijing, 00083, P. R. China 3 Department of Engineering Mechanics, Tsinghua University, Beijing, 00084, P. R. China ABSTRACT: In this paper, the area coordinate method is generalized to formulate quadrilateral elements. The general theory of area coordinate for quadrilateral elements is presented and new quadrilateral elements for membrane and plate bending are formulated by using the area coordinate method.. INTRODUCTION The triangular area coordinate method, 2 has been successfully applied in the construction of triangular elements. The coordinate transformation between triangular area coordinates (L,, ) and Cartesian coordinates (x, y) is L i 2A } (a i + b i x + c i y) () A is the area of the triangle, a i, b i and c i are constants determined by nodal coordinates, e.g. a i = x j y m - x m y j, b i = y j - y m, (2) c i = x m - x j (i, j, m =,2,3,) Advantages of the triangular area coordinate method are: (a) The coordinate transformation between (L i ) and (x, y) is linear. (b) The area coordinate L i is natural and invariant. (c) The stiffness matrix of the triangular element constructed by the coordinate method can be easily formulated with exact integration. The integral formula for arbitrary power function of area coordinate over the triangular element is rather simple: e e A L m! n! p! m L n 2 L p 3 da } 2A (3) (m + n + p + 2)! On the other hand, quadrilateral elements are conventionally formulated by using the isoparametric coordinate method 3, 4. The coordinate transformation between isoparametric coordinates (x, h) and Cartesian coordinate (x, y) is x 4 } ^ 4 x i (+x i x) ( + h i h) i= y 4 } ^ 4 y i (+x i x) ( + h i h) (4) i= (x i, y i ) and (x i, h i ) are the respective coordinates of node i. Although the isoparametric coordinate method has been broadly applied, there are still some disadvantages: (a) The coordinate transformation (4) is nonlinear and its inverse transformation is complicated to utilize 5. (b) The stiffness matrix of the quadrilateral element constructed by the isoparametric coordinate method has to be evaluated by numerical integration instead of exact integration. (c) The serendipity type of isoparametric elements perform very badly when distorted 6. In this paper, the area coordinate method is generalized to formulate quadrilateral elements. The general theory of area coordinate for quadrilateral * This project is supported by Natural Science Foundation of China ( ) Advances in Structural Engineering Vol. 4 No. 200

2 Method of Area Coordinate from Triangular to Quadrilateral Elements Figure. Definition of g, g2, g3 and g4 elements is presented and new quadrilateral elements for membrane and plate bending are formulated by using the area coordinate method. 2. THE QUADRILATERAL AREA 3, 4 COORDINATES 2. The Characteristic Parameters of A Quadrilateral The dimensionless parameters g, g 2, g 3 and g 4 (Figure a,b) are defined as the characteristic parameters of a quadrilateral. is the quadrilateral area. Obviously, we have g + g 3 =, g 2 + g 4 = (5) therefore only two of the parameters g, g 2, g 3, and g 4 are independent. 2.2 Definition of Quadrilateral Area Coordinates In a quadrilateral, the area coordinates (L,,, ) of any point P are defined as Ai L i } (i =,2,3,4) (6) A A, A 2, A 3 and A 4 are the areas of the four triangles formed by point P and two adjacent vertices in the quadrilateral element, respectively (Figure 2). 2.3 Two Identical Equations Satisfied by Area Coordinates The four area coordinates L,, and should satisfy the following two identical equations: L = (7) g 4 g L - g g 2 + g 2 g 3 - g 3 g 4 = 0 (8) In the four area coordinates, only two are independent. Figure 2. The definition of quadrilateral area coordinates 2 Advances in Structural Engineering Vol. 4 No. 200

3 Yuqiu Long, Zhifei Long and Song Cen Figure 3. Eight-node quadrilateral membrane element 2.4 Area Coordinates Expressed by Cartesian Coordinates The transformation formula from Cartesian coordinates to area coordinates is L i 2A } (a i + b i x + c i y) (i =,2,3,4) (9) A is the area of the quadrangular element. a i = x j y k - x k y j b i = y j - y k c i = x k - x j (i,j,k =,2,3,4) (0) The transformation formula (9) is linear and similar to formula (). 2.5 Area Coordinates Expressed by Isoparametric Coordinates The transformation formula is as follows: L 4 } (-x)[g 2 (-h) + g 3 (-h)] 4 } (-h)[g 4 (-x) + g 3 (+x)] 4 } (+x)[g (-h)+g 4 (+h)] 4 } (+h)[g (-x)+g 2 (+x)] () 2.6 Differential Formula The transformation of of the first order derivatives is given by { } x y 2A 3 b b 2 b 3 b 4 { } (2) c c 2 c 3 c 4 4 Figure 4. Pure bending test of a cantilever Advances in Structural Engineering Vol. 4 No

4 ^ ^ Method of Area Coordinate from Triangular to Quadrilateral Elements Figure 5. Percentage error (%) of the deflection at A in constant bending problem due to mesh distortion Figure 6. Linear-bending problem for cantilever { } = 3 } } } } } } } } 4 L T (3) e e A L m n p q da = m!n!p!q! } } } (m + n + p + q + 2)! 2A 2.7 Integral Formula The integral formula for arbitrary power function L m n p q of area coordinates is given as follows? [g 3 m+n+^ + g p+q+ ^ q p k=0 i=0 m n k=0 j=0 C m m+q-k Cn n+p-j Ck k+j g 2 k g 4 j g p+q-k-j C q m+q-k Cp n+p-j Ck k+j g 2 k g 4 j g 3 m+n-k-j ] 4 Advances in Structural Engineering Vol. 4 No. 200

5 Yuqiu Long, Zhifei Long and Song Cen Figure 8. Plate bending element C i k = k! } } (5) (k-i)!i! If a quadrilateral element degenerates into a triangular element, then Eq.(4) degenerates into Eq.(3). 3. QUADRILATERAL MEMBRANE ELEMENT ACQ8 FORMULATED IN AREA COORDINATES The eight-node quadrilateral membrane element ACQ8 is formulated by using the area coordinate method (Figure 3). 3. Shape Functions Eight shape functions are given as follows: For the mid-side nodes: 4 g i + g j N 4+i } L i L k (L j - L m + } } ) g 2 i g j For the corner nodes: ( i,j,k,k =,2,3,4) N i = N 0 i - (} 2 } gi - g j g k ( gj - g - } } )N 4+i - } } N 4+j 8g 8g g m m i k ) (6) g k (gk g + } } N 4+k - (} 8g ī g 2 } g m - g i + } } )N 4+m (7) 8g j m ) j ( i,j,k,k =,2,3,4) N 0 i } L i L k ( + } } L j - } } L m g g k + g i + i g j g m g j (8) ( i,j,k,k =,2,3,4) 3.2 Mesh Distortion Sensitive Analysis In order to test the effects of element distortions on the performance of the ACQ8 element, two demonstrative example problems are solved with distorted ACQ8 element, and compared with the isoparametric serendipity elements Q8 and Q4. Example 3.. Pure bending of a cantilever beam (Figure 4). a varies from 0 to 0.99l. The curves of the percentage error are plotted in Figure 5. Example 3.2. Linear-bending problem of a cantilever beam (Figure 6). The curves of the percentage error are plotted in Figure QUADRILATERAL THIN PLATE BENDING ELEMENT ACQP 4. Element deflection field Element displacement field is composed of two parts: w = w 0 + w * (9) the low-order displacement field can be defined as: w 0 = ^ 4 i= N i 0 w i = [N 0 ]{q} e (20) Advances in Structural Engineering Vol. 4 No

6 } Method of Area Coordinate from Triangular to Quadrilateral Elements Figure 9. Meshes of symetric quadrant of a circular plate g +} g k 0 N 0 i =} 4 } Li Lj L 3 k -L +} } +} } -} g g g j m } (g i L i L j -g j L j L k +g k L k L m -g m L m L i )4 (2) i m ( i,j,k,k =,2,3,4) [L]=[L L L L ( -L ) ( - ) 2 4 ] (26) {l}=[l l 2 l 3 l 4 l 5 l 6 l 7 l 8 ] T (27) g 0 =g g 2 g 3 g 4 (22) [N 0 ]=[N N N N ] (23) {q} e =[w y x y y w 2 y x2 y y2 w 3 y x3 y y3 w 4 y x4 y y4 ] T (24) The high-order displacement field w * can be defined as: w * =[L]{l} (25) The low-order displacement field w 0 in equation (20) and the total displacement field w in equation (9) have already satisfied 4-point compatibility conditions at the nodes: (w- ~ w) j =0 (j=,2,3,4 indicate the element nodes)(28) The eight unknown coefficients l i in total displacement field w are determined by eight generalized conforming conditions, w 4-line compatibility conditions for the deflection and 4-line compatibility conditions for the normal slope y n : E (w- w)ds=0 ~ (29a) d i Table. Central displacement and moment for circular plate under uniformly distrbuted load q Simply-supported plate Mesh Central displacement (3 qr 4 /D) * ) * Central moment (3 qr 2 ) a (0.6%) (2.0%) b (0.6%) (0.56%) Analytical solutions Clamped plate a (-5.34%) (.5%) b (-.30%) (0.43%) analytical solutions * r is the radius of the plate. D=Eh 3 /2(-m 2 ), E is Young s moduli, h is the thickness of the plate. 6 Advances in Structural Engineering Vol. 4 No. 200

7 Yuqiu Long, Zhifei Long and Song Cen w E (} } -y ~ n )ds=0 (i=,2,3,4) (29b) d i n d is the length of the ith side. Thus, {l} can be expressed as follows: {l }=[M ]{q} e, (30a,b) {l }=[l l 2 l 3 l 4 ] T, {l }=[l 5 l 6 l 7 l 8 ] T (3a,b) [M ]=[P ]+[R][M ], [M ]=[F] - [H] (32) [P ]=[[P ] [P 2 ] [P 3 ] [P 4 ]] (33) Figure. Sensitivity to mesh distortion Figure 2. Mesh of quarter square plate (O is the central of the plate) Advances in Structural Engineering Vol. 4 No

8 3 Method of Area Coordinate from Triangular to Quadrilateral Elements Figure 0. Distorted mesh for sensitive test ql4 Table 2. Central displacement (} } )of SS2 square plate under uniformly distributed load q 00D Thickness-span Mesh Analytical ratio solution A B C A DKQ Q4BL B C A PQI Q4BL B C A PQI Q4BL B C A B C A B C A B C Advances in Structural Engineering Vol. 4 No. 200

9 Yuqiu Long, Zhifei Long and Song Cen (36) [F]=[F ]+[F ][R] [H]=[P]-[F ][P ] (37) (38) (34) b i =y i+ -y i+22, c i =x i+2 -x i+ (35) (39) Advances in Structural Engineering Vol. 4 No

10 Method of Area Coordinate from Triangular to Quadrilateral Elements parameter varies D from 0 to 0.5L. The results are plotted in Figure, and are compared with element CRB, CRB2, S 7 and DKQ QUADRILATERAL THIN-THICK PLATE BENDING ELEMENT TACQP 5. Element deflection field The deflection field of the element TACQP is the same as the element ACQP, which has been given in section Element shear strain field Firstly, the nodal shear strains (g xi, g yi ) (i=,2,3,4) are determined based on Timoshenko s beam theory. Then the shear fields g x and g y within the element can be interpolated from the nodal shear strains g xi and g yi (i=,2,3,4) in the form: g x =g x N 0 +g x2 N 2 0 +g x3 N 3 0 +g x4 N 4 0 {g y =g y N 0 +g y2 N 2 0 +g y3 N 3 0 +g y4 N 4 0 (30) Finally, we obtain (40) F ij =b i b j +c i c j i,j=,2,3,4 (4) w=w 0 +w * =[N]{q} e (42) [N]=[N 0 ]+[L][M] (43) [ M ] [M]=3 } } [ M ] 4 (44) Then, the strain matrix and the stiffness matrix of the element can be obtained according to the standard procedure. 4.2 Numerical Examples Example 4.. Simply supported and clamped circular plate (refer to Figure 9) under uniformly distributed load q. The Possion s ratio is m=0.3. The results are given in table. Example 4.2. Sensitivity to mesh distortion. Analyze a clamped square plate under uniformly distributed load q by using a distorted mesh (Figure 0). The distortion (i=,2,3,4) are given in Eq.(2). The nodal shear strain, (g xi, g yi ) (i=,2,3,4), are determined by the procedure proposed in the reference 9. When the thickness of the plate tends to be zero, the shear fields g x and g y also tend to be zero. So no shear locking phenomenon will take place. 5.3 Element rotation field The rotation field is given as follows: (3) We can see that the element TACQP will degenerate to the thin plate element ACQP when the shear fields g x and g y tend to be zero. 5.4 Numerical Examples Example 5.. Simply supported (SS2, the displacement boundary conditions are w=0, y s =0) square plate under uniformly distributed load q, the Possion s ratio m=0.3. The results of central displacement are given in table CONCLUSION In this paper, a new approach based on the concept of area coordinates for quadrilateral elements is presented. Three new quadrilateral elements ACQ8 for membrane, ACQP for thin plate bending and TACQP for thin-thick 0 Advances in Structural Engineering Vol. 4 No. 200

11 Yuqiu Long, Zhifei Long and Song Cen plate bending are formulated by using the area coordinate method. Numerical examples show that these presented elements have the advantage of high accuracy, good convergence. They are insensitive to geometric distortion. This indicates that the area coordinate method is an effective tool to construct quadrilateral elements. REFERENCES. J B Mertie. Transformation of trilinear and quadriplanar coordinates to and from Cartesian coordinates. The American Mineralogist, 964, 49(7/8): 926~ M A Eisenberg and L E Malvern. On finite element integration in natural coordinates. Int.J.Numer.Methods Eng., 973, 7(4): 574~ I C Taig. Structural analysis by the matrix displacement method, Engl. Electric Aviation Report, 96; S07 4. B M Irons. Engineering application of numerical integration in stiffness method. J. AIAA, 966, 4: 2035~ C Hua. An inverse transformation for quadrilateral isoparametric elements: analysis and application. Finite Elements in Analysis and Design, 990, 7: 59~66 6. N S Lee and K J Bathe. Effects of element distortions on the performance of isoparrametric elements. Int.J.Numer.Methods Eng., 977, 36: 3553~ S L Weissman and R L Taylor. Resultant fields for mixed plate bending elements. Computer Methods in Applied Mechanics and Engineering, 990, 79: 32~ J L Batoz and M B Tahar. Evaluation of a new quadrilateral thin plate bending element. Int.J.Numer.Methods Eng., 982, 8: 655~ A K Soh, Long Zhifei and Cen Song. A Mindlin plate triangular element with improved interpolation based on Timoshenko s beam theory. Communication in Numerical Methods in Engineering, 999, 5(7): J L Batoz, M B Tahar. Evaluation of a new quadrilateral thin plate bending element. Int.J.Num.Meth.Eng., 982;8: O C Zienkiewicz, Z Xu, L F Zeng, A Samuelson and N E Wiberg. Linked interpolation for Ressner-Mindlin plate elements: Part I A simple quadrilateral. Int. J. Num. Meth. Eng., 993;36: A Ibrahimbegovic. Quadrilateral finite elements for analysis of thick and thin plates. Computer Methods in Applied Mechanics and Engineering, 993;0: Yuqiu Long, Juxuan Li, Zhifei Long and Song Cen. Area coordinates used in quadrilateral elements. Communications in Numerical Methods in Engineering. 999, 5(8): 533~545 4.Zhifei Long, Juxuan Li, Song Cen and Yuqiu Long. Some basic formulae for area coordinates used in quadrilateral elements. Communications in Numerical Methods in Engineering. 999, 5(2): 84~852 YUQIULONG Born on January 5, 926. Native of Hunan Province, China. Member of the Chinese Academy of Engineering since 995. Professor of Tsinghua University. Expert in Civil Engineering and Structural Mechanics. Graduated from the Department of Civil Engineering of Tsinghua University in 948. Member of the Council of China Civil Engineering Society since 985. Editor-in-Chief of the Journal of Engineering Mechanics, the Chinese Society of Theoretical and Applied Mechanicals, since 992. ZHIFEI LONG Born on December 3, 957. Native of Beijing, China. Expert in Computational Structural Mechanics. Graduated from Beijing Institute of Civil Engineering in 982. Received Master of Solid Mechanics at Tsinghua University in 986. Serving now as Professor of China University of Mining and Technology. SONG CEN Song Cen is fulfilling a post-doctoral Research Program at the Department of Engineering Mechanics, Tsinghua University. He obtained his Bachelor degree in Engineering Mechanics in 994 from Hohai University; his Master of Science Degree in 997 and PhD degree in 2000 from the Department of Civil Engineering, Tsinghua University. Dr Cen s research interests mainly cover Finite Element Method and Computational Solid Mechanics. Advances in Structural Engineering Vol. 4 No. 200