Appl. Math. Mech. -Engl. Ed. 31(1), 125–134 (2010) DOI 10.1007/s10483-010-0113-1 c Shanghai University and Springer-Verlag Berlin Heidelberg 2010
Applied Mathematics and Mechanics (English Edition)
A 17-node quadrilateral spline finite element using the triangular area coordinates ∗ Juan CHEN ()1 , Chong-jun LI ()1 , Wan-ji CHEN ()2,3 (1. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China; 2. State Key Laboratory for Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024, P. R. China; 3. Institute for Structural Analysis of Aerocraft, Shenyang Institute of Aeronautical Engineering, Shenyang 110136, P. R. China) (Communicated by Hong-qing ZHANG)
Abstract Isoparametric quadrilateral elements are widely used in the finite element method. However, they have a disadvantage of accuracy loss when elements are distorted. Spline functions have properties of simpleness and conformality. A 17-node quadrilateral element has been developed using the bivariate quartic spline interpolation basis and the triangular area coordinates, which can exactly model the quartic displacement fields. Some appropriate examples are employed to illustrate that the element possesses high precision and is insensitive to mesh distortions. Key words 17-node quadrilateral element, bivariate spline interpolation basis, triangular area coordinates, B-net method, fourth-order completeness Chinese Library Classification O241, O343 2000 Mathematics Subject Classification 65D07, 74S05
1
Introduction
The displacement-based finite element model means that displacements are supposed to be simple polynomials in the Cartesian coordinates, and the unknown coefficients are generalized coordinate parameters. One common practice to construct compatible quadrilateral elements is isoparametric transformation. In general, there are two types of quadrilateral isoparametric elements in the literature[1] . One is the Serendipity type (S-type), such as 8/12/17-node elements denoted by Q8/Q12/Q17. Another is the Lagrangian type (L-type), such as 9/16/25-node elements denoted by Q9/Q16/Q25. The S-type elements have been broadly used since they have no or less internal nodes compared with the L-type elements. Take the quartic elements for an example, Q25 has 8 more internal nodes than Q17. However, the Serendipity elements have a disadvantage of accuracy loss when the elements are distorted. Lee and Bathe[2] pointed out ∗ Received Jul. 27, 2009 / Revised Nov. 29, 2009 Project supported by the Natural Science Foundation of China China (Nos. 60533060, 10672032, and 10726067) and the Science Foundation of Dalian University of Technology (No. SFDUT07001) Corresponding author Juan CHEN, Ph. D., E-mail:
[email protected]
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that, when the element is distorted from rectangle to irregular quadrilateral, the transformation between the Cartesian coordinates (x, y) and the isoparametric coordinates (ξ, η) changes from linear to nonlinear, and then the complete order of (x, y) of displacement field will drop dramatically. As a result, the serendipity elements Q8 and Q12 only possess the first-order completeness in Cartesian coordinates (x, y), while Q17 only possesses the second-order, although they have some higher-order terms of the isoparametric coordinates (ξ, η). In order to overcome the shortcomings of the isoparametric coordinates, Long et al.[3] and Cen et al.[4] presented a family of quadrilateral membrane elements with 4–8 nodes by employing the quadrilateral area coordinate method. Li and Chen[5] proposed refined nonconforming 8node plane elements. Li et al.[6] and Chen et al.[7-8] constructed two spline elements with the same 8 and 12 nodes, which possess the second- and third-order completeness in Cartesian coordinates, respectively. These elements are more accurate and robust in various distorted meshes. Besides, the use and improvement of higher-order elements is another way to improve the accuracy of finite element analysis. Rathod and Kilari[9] derived the explicit expressions for interpolation functions of the Serendipity and Complete Lagrange family elements, which allow uniform spacing of nodes over the element domain for the elements of orders 4–10. A family of enriched elements with bubble functions is presented by Ho and Yeh[10] . Hence, the study on the finite element family with higher-order elements is a very useful and necessary part in finite element analysis. In mathematics, splines are piecewise polynomials satisfying certain continuous conditions[11-12] , which are applied widely in the finite element method. We have constructed several quadrilateral spline finite elements using the bivariate spline interpolate basis with completeness orders 1–3[6-8] . In this paper, we develop a 17-node spline element denoted by L17. The element possesses the fourth-order completeness in the Cartesian coordinates and is not affected by mesh distortions as well. Thus, we have a family of spline elements L4, L8, L12, and L17 of orders 1–4, which have the same completeness as the Lagrange family elements but with less nodes. The paper is organized as follows. First, the B-net method is introduced briefly in Section 2. Then, the 17-node spline element is given in Section 3. At last, some appropriate examples are employed to evaluate the performance of the proposed element compared with the isoparametric element Q17 in Section 4.
2
Triangular area coordinates and B-net method
The B-net method is one important tool for studying the bivariate splines defined on triangulations[13]. It is originated from the Bernstein polynomials, and is based on the triangular area coordinates. As shown in Fig. 1(a), P denotes an arbitrary point within the triangle P1 P2 P3 , which are specified by the Cartesian coordinates P1 = (x1 , y1 ), P2 = (x2 , y2 ), P3 = (x3 , y3 ), and P = (x, y). Let the area coordinates of P be (λ1 , λ2 , λ3 ). The Cartesian coordinates (x, y) can be expressed by the area coordinates λ1 , λ2 , and λ3 as follows: x = x1 λ1 + x2 λ2 + x3 λ3 , (1) y = y1 λ1 + y2 λ2 + y3 λ3 . There are (n+2)(n+1) domain points ξi,j,k equally locating in the triangle P1 P2 P3 with the 2 i j k area coordinates ( n , n , n ) (as shown in Fig. 1(b) when n = 4). The Bernstein polynomials of degree n on the triangle P1 P2 P3 are defined by n Bi,j,k (λ1 , λ2 , λ3 ) =
n! i j k λ λ λ , i!j!k! 1 2 3
i + j + k = n,
λ1 , λ2 , λ3 ≥ 0, λ1 + λ2 + λ3 = 1.
(2)
A 17-node quadrilateral spline finite element
Fig. 1
127
The area coordinates (a) and the quartic domain points in a triangle (b)
It is easy to know that all the Bernstein polynomials of degree n are linear independent. Besides, they satisfies the partition of unity n Bi,j,k (λ1 , λ2 , λ3 ) = (λ1 + λ2 + λ3 )n ≡ 1. i+j+k=n
A row vector composed of all Bernstein bases of degree n is denoted by B n . For example, 4 B 4 = [B4,0,0 4 B1,1,2
= [λ41
4 B3,1,0 4 B1,0,3
4λ31 λ2
12λ1 λ2 λ23
4 B3,0,1 4 B0,4,0
4λ31 λ3 4λ1 λ33
4 B2,2,0 4 B0,3,1
6λ21 λ22 λ42
4 B2,1,1 4 B0,2,2
4 B2,0,2 4 B0,1,3
12λ21 λ2 λ3
4λ32 λ3
6λ22 λ23
4 B1,3,0
4 B1,2,1
4 B0,0,4 ]
6λ21 λ23
4λ1 λ32
4λ2 λ33
λ43 ].
12λ1 λ22 λ3
For an given arbitrary polynomial of degree n in Cartesian coordinates, p(x, y) = ai,j xi y j .
(3)
i+j≤n
By substitution of Eq. (1) into Eq. (3), p(x, y) can be represented in the following B-net form by the area coordinates and the Bernstein polynomials as n p(x, y) = f (λ1 , λ2 , λ3 ) = bi,j,k Bi,j,k (λ1 , λ2 , λ3 ) = B n fb , (4) i+j+k=n
where bi,j,k are called the B´ezier coefficients corresponding to the domain points ξi,j,k , fb is a column vector composed of bi,j,k with the same order as B n . For example, a quartic polynomial f (λ1 , λ2 , λ3 ) can by expressed in B-net form as f = B 4 fb , where fb = [b4,0,0 b1,1,2
b3,1,0 b1,0,3
b3,0,1 b0,4,0
b2,2,0 b0,3,1
b2,1,1 b0,2,2
b2,0,2 b0,1,3
b1,3,0
b1,2,1 T
b0,0,4 ] .
There are some advantages of the computation on polynomials by B-net form[6,13] .
3
The 17-node quadrilateral spline element
For a convex quadrangle, as shown in Fig. 2(a), denote the corner nodes by P1 , · · · , P4 , and the intersection of two diagonals P1 P3 and P2 P4 by P0 . The quadrangle is divided into four subtriangles Δ1 , · · · , Δ4 .
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By the triangular area coordinates and the B-net method, each triangle has 15 domain points according to the quartic polynomial. Hence, there are 41 domain points located in the quadrangle. Their indexes are shown in Fig. 2(b).
Fig. 2
A convex triangulated quadrangle (a) and its domain points of degree 4 (b)
Consider a bivariate quartic spline space on Δ1 , · · · , Δ4 denoted by S43 (Δ). We define a spline s ∈ S43 (Δ) by a piecewise quartic polynomial with C 3 continuous on both diagonals P1 P3 and P2 P4 . Using the smoothing cofactor-conformality method[11-12] , we know that the dimension of S43 (Δ) is 17. We can obtain 17 linear independent quartic spline bases, which are denoted by L1 , L2 , · · · , L17 . These spline bases are represented in B-net form, and the vectors of B´ezier coefficients corresponding to the 41 domain points of each spline base are Lb1 = {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Lb2
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}T, = {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}T,
Lb3 = {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}T, Lb4 = {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}T, −b2 b −b2 −b3 −b3 b3 , , , , , , 2 6 2a 2d 2ad 6d2 3 3 3 2 2 3 3 3 ab ab ab −b ab ab ab ab ab ab ab2 T 3 , , , , a, , 0, 0, , −b } , , , , 0, , 0, 0, , 2cd 6cd2 2c2 d 2c 2c2 6c 3 a 3d2 c2 c 7ab 5a 5b −ab2 b2 −3ab2 , , , , , = {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, , 4 4 4 4d 4d 4d2 −5a2 b2 −3a2 b2 −3a2 b2 −a2 b −3a2 b a2 , , , , , , 4cd 4cd2 4c2 d 4c 4c2 4c 3a2 3a2 b 3b2 3ab2 −3a2 b2 −3a2 b2 −3a2 b2 −3a2 b2 T 0, , , 0, , , 0, , 0, } , , , 2 2 2 2 2 2d 2d 2c2 2c −a2 −a2 a a2 b ab a2 b , , , , , = {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, , 2 2b 6 2d 6d 2d2 a3 b a3 b a3 b −a3 a3 −a3 −a3 a3 b a3 b a3 b ab , , 2, , 0, , −a3 , b, , 0, 0, 2 , , 0, 2 , 0}T , , 2 , 2 2cd 2cd 6c d 2c 6c 2bc b 3 d d 3c
Lb5 = {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
Lb6
Lb7
A 17-node quadrilateral spline finite element
129
a2 d a2 d ad −a2 a −a2 , , , , , , 2b 2b2 6b 2 6 2d a3 d a3 d ad −a3 a3 d −a3 −a3 a3 a3 d a3 d a3 d 3 , , 2, , , 0, , d, , 0, 0, , −a , , , 0, , 0}T , 2c 2cd 6c 2bc 6bc2 2b2 c b2 b 3 d 3c2 −ad2 −3ad2 d2 7ad 5d 5a , , , , , , Lb9 = {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4b 4b2 4b 4 4 4 −a2 d a2 −3a2 d −5a2 d2 −3a2 d2 −3a2 d2 , , , , , , 4c 4c 4c2 4bc 4bc2 4b2 c −3a2 d2 −3a2 d2 3d2 3ad2 3a2 3a2 d −3a2 d2 −3a2 d2 T 0, , 0, , , 0, , , 0, } , , , 2b2 2b 2 2 2 2 2c2 2c −d3 d3 −d3 −d2 −d2 d Lb10 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, , , , , , , 2b 6b2 2ab 2 2a 6 ad2 ad ad2 ad3 ad3 ad3 ad3 ad3 ad3 T −d3 ad 3 , , 2, , , 0, , −d , 0, 0, } , , , 0, 0, , a, , 2c 6c 2c 2bc 2bc2 6b2 c 3b2 a 3 c2 c cd3 cd3 cd3 cd2 cd2 cd Lb11 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, , , , , , , 2ab 6ab2 2a2 b 2a 2a2 6a −d2 d −d2 −d3 −d3 d3 cd −d3 cd3 cd3 cd3 , , , , , 2 , 0, 2 , 0, 0, 2 , , c, , 0, 0, , −d3 }T , 2 6 2c 2b 2bc 6b 3b a a 3 c −5c2 d2 −3c2 d2 −3c2 d2 −c2 d Lb12 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, , , , , 4ab 4ab2 4a2 b 4a −3c2 d c2 7cd 5c 5d −cd2 d2 −3cd2 , , , , , , , , 4a2 4a 4 4 4 4b 4b 4b2 −3c2 d2 −3c2 d2 −3c2 d2 −3c2 d2 3c2 3c2 d 3d2 3cd2 T 0, , 0, , 0, , , 0, , } , , , 2b2 2b 2a2 2a 2 2 2 2 c3 d c3 d c3 d −c3 c3 −c3 , , , , , , Lb13 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2ab 2ab2 6a2 b 2a 6a2 2ad −c2 −c2 c c2 d cd c2 d c3 d c3 d c3 d −c3 cd , , , , , 2 , 0, 2 , , 0, 2 , 0, 0, , −c3 , d, , 0}T , 2 2d 6 2b 6b 2b b b 3a d 3 −c3 −c3 c3 bc3 bc3 bc3 Lb14 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, , , , , , , 2a 2ab 6a2 2ad 6a2 d 2ad2 bc2 bc2 bc −c2 c −c2 −c3 bc bc3 bc3 bc3 , 2, , , , , 0, , −c3 , 0, 2 , 0, 0, 2 , , b, , 0}T , 2d 2d 6d 2 6 2b b 3a d d 3 −bc2 c2 −3bc2 −5b2 c2 Lb15 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, , , , , 4a 4a 4a2 4ad −3b2 c2 −3b2 c2 −b2 c −3b2 c b2 7bc 5b 5c , , , , , , , , 4a2 d 4ad2 4d 4d2 4d 4 4 4 3c2 3bc2 −3b2 c2 −3b2 c2 −3b2 c2 −3b2 c2 3b2 3b2 c T 0, , , 0, , 0, , 0, , } , , , 2 2 2a2 2a 2d2 2d 2 2 b2 c bc b2 c b3 c b3 c b3 c Lb16 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, , , , , , , 2a 4a 2a2 2ad 2a2 d 6ad2 −b3 b3 −b3 −b2 −b2 b bc b3 c b3 c b3 c −b3 , 2, , , , , c, , 0, 0, 2 , , 0, 2 , 0, 0, , −b3 }T , 2d 6d 2cd 2 2c 6 3 a a 3d c 1 1 1 1 1 1 Lb17 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, , , , , , , 4ab 12ab2 12a2 b 4ad 12a2 d 12ad2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , , , , 0, 2 , , 0, 2 , , 0, 2 , , 0, 2 , }T , , , 4cd 12cd2 12c2 d 4bc 12bc2 12b2 c 6b 2b 6a 2a 6d 2d bc 2c
Lb8 = {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
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Juan CHEN, Chong-jun LI, and Wan-ji CHEN
where the coefficients a, b, c, and d are defined by the following ratios: a=
|P3 P0 | |P4 P0 | , b= , c = 1 − a, d = 1 − b. |P4 P2 | |P3 P1 |
(5)
It is easy to verify that the 17 spline bases satisfy the partition of unity. The piecewise quartic polynomials of each spline base restricted on four subtriangles can be obtained by the B-net form Eq. (4) (where n = 4) with corresponding B´ezier coefficients, respectively. Denote the area coordinates in each triangle Δk by (λk,1 , λk,2 , λk,3 ), and the corresponding quartic Bernstein polynomials by Bk4 (k = 1, 2, 3, 4). Denote the 41 B´ezier coefficients (i) (i) (i) (i) of each spline base Li by dj (i = 1, 2, · · · , 17; j = 1, 2, · · · , 41), i.e., Lbi = {d1 , d2 , · · · , d41 }T (i = 1, 2, · · · , 17). Then, the piecewise polynomials of Li restricted on Δ1 , Δ2 , Δ3 , and Δ4 are ⎧ (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) T 4 ⎪ ⎪ Li |Δ1 = B1 · {d17 , d32 , d35 , d31 , d18 , d34 , d30 , d19 , d20 , d33 , d1 , d5 , d6 , d7 , d2 } , ⎪ ⎪ ⎪ ⎨ L | = B 4 · {d(i) , d(i) , d(i) , d(i) , d(i) , d(i) , d(i) , d(i) , d(i) , d(i) , d(i) , d(i) , d(i) , d(i) , d(i) }T , i Δ2 2 17 35 38 34 21 37 33 22 23 36 2 8 9 10 3 (6) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) T 4 ⎪ ⎪ ⎪ Li |Δ3 = B3 · {d17 , d38 , d41 , d37 , d24 , d40 , d36 , d25 , d26 , d39 , d3 , d11 , d12 , d13 , d4 } , ⎪ ⎪ ⎩ (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) Li |Δ4 = B44 · {d17 , d41 , d32 , d40 , d27 , d31 , d39 , d28 , d29 , d30 , d4 , d14 , d15 , d16 , d1 }T . In fact, the piecewise representations are not needed in the computation of the finite element method, since the computations on products, integrals, and derivatives of the element shape functions can be simplified by using their B´ezier coefficients[6] . By a simple linear transformation, we can obtain another set of interpolation bases corresponding to the nodes Pi = (xi , yi ) (i = 1, 2, · · · , 17) as follows: ⎧ 13 1 1 13 13 Nb1 = Lb1 − 13 ⎪ 12 Lb5 + 18 Lb6 − 4 Lb7 − 4 Lb14 + 18 Lb15 − 12 Lb16 , ⎪ ⎪ ⎪ N = L − 1 L + 13 L − 13 L − 13 L + 13 L − 1 L , ⎪ b2 b2 ⎪ 4 b5 18 b6 12 b7 12 b8 18 b9 4 b10 ⎪ ⎪ ⎪ 1 13 13 13 13 1 ⎪ N = L − L + L − L − L + L − b3 ⎪ 4 b8 18 b9 12 b10 12 b11 18 b12 4 Lb13 , ⎪ b3 ⎪ ⎪ 1 13 13 13 13 1 ⎪ ⎪ Nb4 = Lb4 − 4 Lb11 + 18 Lb12 − 12 Lb13 − 12 Lb14 + 18 Lb15 − 4 Lb16 , ⎪ ⎪ 32 4 20 ⎨ Nb5 = 4Lb5 − Lb6 + Lb7 , Nb6 = −3Lb5 + Lb6 − 3Lb7 , 9 3 3 (7) 4 32 32 4 = L − L + 4L , N = 4L − L N ⎪ b7 b6 b7 b8 b8 b9 + 3 Lb10 , 3 b5 9 9 ⎪ ⎪ ⎪ 4 32 ⎪ N = −3Lb8 + 20 ⎪ ⎪ 3 Lb9 − 3Lb10 , Nb10 = 3 Lb8 − 9 Lb9 + 4Lb10 , ⎪ b9 ⎪ 32 4 20 ⎪ ⎪ ⎪ Nb11 = 4Lb11 − 9 Lb12 + 3 Lb13 , Nb12 = −3Lb11 + 3 Lb12 − 3Lb13 , ⎪ ⎪ ⎪ 32 4 ⎪ Nb13 = 43 Lb11 − 32 ⎪ 9 Lb12 + 4Lb13 , Nb14 = 4Lb14 − 9 Lb15 + 3 Lb16 , ⎪ ⎩ 4 32 Nb15 = −3Lb14 + 20 3 Lb15 − 3Lb16 , Nb16 = 3 Lb14 − 9 Lb15 + 4Lb16 , Nb17 = Lb17 . Then, Ni (Pj ) =
1, i = j; 0, i = j,
i, j = 1, 2, · · · , 17.
(8)
A 17-node quadrilateral element constructed by N1 , N2 , · · · , N17 is denoted by L17. The essential displacement fields related to element nodes can be written as 17 u = i=1 ui Ni , (9) 17 v = i=1 vi Ni . The quadrilateral element stiffness matrix K can be obtained by the sum of the stiffness matrix Kk on each triangle Δk (k = 1, 2, 3, 4). Due to Eq. (8), we know that the displacement is compatible. Furthermore, the L17 element (9) possesses the fouth-order completeness in
A 17-node quadrilateral spline finite element
131
the Cartesian coordinates. We have the following theorem by verifying quartic polynomials 1, x, y, · · · , x4 , x3 y, · · · , y 4 . Theorem 1 Let D be an arbitrary convex quadrilateral domain P1 P2 P3 P4 , (N f )(x, y) :=
17
f (xi , yi )Ni (x, y),
(10)
i=1
then, for all f (x, y) ∈ P4 , (N f )(x, y) ≡ f (x, y), (x, y) ∈ D.
(11)
Besides, we remark that the stress is C 2 continuous within the element since the displacement is C 3 continuous on both diagonals of the element. Note that the spline element functions are actually piecewise polynomials. Hence, no domain transformation or Jacobian is needed.
4
Numerical examples
In this section, some appropriate examples of plane problem of elasticity are employed to evaluate the performance of the proposed element L17, compared with the standard 17-node Serendipity isoparametric element Q17. Example 1 Patch test. A small patch is discretized into some arbitrary elements as shown in Fig. 3.
Fig. 3
Patch test
For a given arbitrary quartic displacement fields, ⎧ u = a0 + a1 x + a2 y + a3 x2 + a4 xy + a5 y 2 + a6 x3 + a7 x2 y + a8 xy 2 ⎪ ⎪ ⎨ + a y 3 + a x4 + a x3 y + a x2 y 2 + a xy 3 + a y 4 , 9
10
11
12
13
14
2 2 3 2 2 ⎪ ⎪ ⎩ v = b0 + b31 x + b2 y4 + b3 x 3+ b4 xy +2b52y + b6 x 3 + b7 x y4 + b8 xy + b9 y + b10 x + b11 x y + b12 x y + b13 xy + b14 y ,
(12)
where the coefficients a0 , b0 , . . . , a14 , b14 satisfy the conditions by the stress equilibrium equation. Especially, we choose a set of displacements of degree 1, 2, 3, and 4 as follows: ⎧ ⎪ ⎨ u = 1 + x + 3y, 4 (13) 1 ⎪ ⎩ v = 1 + x + 2y; 2
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⎧ ⎪ ⎨ u = 1 + x + 3y − 2x2 − 4xy + 5 y 2 , 4 2 2 3 1 17 ⎪ ⎩ v = 1 + x + 2y − x2 + xy + y 2 ; 2 3 5 2 ⎧ 1 5 1 ⎪ ⎨ u = + x + 3y − 2x2 − 4xy + y 2 − 2x3 + x2 y − 4xy 2 − y 3 , 4 2 3 ⎪ ⎩ v = 1 + 1 x + 2y − 2 x2 + 17 xy + 3 y 2 + 1 x3 + 12x2 y − xy 2 − 2 y 3 ; 2 3 5 2 3 3 ⎧ 1 5 1 ⎪ ⎪ u = + x + 3y − 2x2 − 4xy + y 2 − 2x3 + x2 y − 4xy 2 − y 3 ⎪ ⎪ 4 2 3 ⎪ ⎪ ⎪ 7 19 11 ⎪ ⎪ ⎨ − x4 − x3 y + x2 y 2 + xy 3 − y 4 , 32 24 96 2 3 1 17 1 2 ⎪ ⎪ ⎪ v = 1 + x + 2y − x2 + xy + y 2 + x3 + 12x2 y − xy 2 − y 3 ⎪ ⎪ 2 3 5 2 3 3 ⎪ ⎪ ⎪ 11 19 7 ⎪ ⎩ − x4 + x3 y + x2 y 2 − xy 3 − y 4 . 96 24 32
(14)
(15)
(16)
Table 1 shows the numerical results at a selected location (x1 , y1 ) = (0.04, 0.02) for patch test by the four given displacement fields Eq. (13), Eq. (14), Eq. (15), and Eq. (16). It demonstrates that Q17 only possesses the second-order completeness, while L17 possesses the fourth-order completeness. Table 1 (u1 , v1 ) ×
103
Eq. (13) of degree 1
Results of patch test
Eq. (14) of degree 2
Eq. (15) of degree 3
Eq. (16) of degree 4 (0.344 3, 1.062 5) N
Q17
(0.35, 1.06) Y
(0.344 6, 1.062 3) Y
(0.344 3, 1.062 6) N
L17
(0.35, 1.06) Y
(0.344 6, 1.062 3) Y
(0.344 4, 1.062 6) Y
(0.344 4, 1.062 6) Y
Exact
(0.35, 1.06)
(0.344 6, 1.062 3)
(0.344 4, 1.062 6)
(0.344 4, 1.062 6)
Note: the letter “Y” denotes that the element passes the patch test exactly, and “N” denotes not.
Example 2 Linear bending problem. For elements that have complete polynomial basis function of degree 3, we use the linear bending-moment problem described in Fig. 4 and Fig. 5 to demonstrate the effects of element distortions. Here, to allow the 2-D plane stress problem to behave like a beam, we set the edge nodes free and prescribe the correct shear and bending reactions[2].
Fig. 4
Fig. 5
Linear bending problem for cubic elements
Linear bending sensitivity test for mesh distortion
A 17-node quadrilateral spline finite element
133
Let L = 10, c = 2, and e varies from 0 to 4.99. As shown in Table 2, Q17 is affected by angular distortions, while the results by L17 always accurately satisfy the following cubic displacement fields given in [2]: ⎧ 138 2 120 2 92 3 60 2 240 46c ⎪ ⎨ u=( x y− y − x − xy + y + 120x − y)/E, cL cL L c L L (17) ⎪ ⎩ v = (− 40 x3 − 36 xy 2 + 120 x2 + 36 xy + 36 y 2 + 46c x − 36y)/E. cL cL c L c L Table 2
Q17
L17
Deflection at a selected location for linear bending problem with e varying e=0
e=1
e=2
e=3
e=4
e = 4.99
u(10, 0) × 104
0.6
0.6
0.601
0.602
0.598
0.612
0.6
v(10, 0) × 104
4.092
4.087
4.063
4.008
3.882
3.651
4.092
Exact
u(10, 0) × 104
0.6
0.6
0.6
0.6
0.6
0.6
0.6
v(10, 0) × 104
4.092
4.092
4.092
4.092
4.092
4.092
4.092
Example 3 Shear load sensitivity test. The mesh is shown in Fig. 6, where e varies from 0 to 4.5. The numerical results of the deflection at a selected location are shown in Table 3. The results show that L17 always has higher precision and better insensitivity than Q17 to the element angular distortion, even if the exact results are not quartic polynomials. Compared with the cubic spline element L12 presented in [7], L17 has higher precision indeed.
Fig. 6 Table 3 vC
e=0
Shear load sensitivity test for mesh distortion
Deflection at a selected location for shear problem with e varying e=1
e=2
e=3
e=4
e = 4.5
Exact[5]
Q17
102.606
102.608
102.291
101.195
97.962
95.460
102.62
L12
102.590
102.588
102.584
102.583
102.582
102.583
102.62
L17
102.619
102.617
102.613
102.612
102.615
102.606
102.62
5
Conclusions
A 17-node quadrilateral element is constructed by the spline method in B-net form. This element has the following properties: (i) it satisfies the partition of unity; (ii) it can interpolate nodal data; (iii) it has the fourth-order completeness in the Cartesian coordinates and is insensitive to mesh distortions; (iv) the displacement is compatible and the stress is C 2 continuous within the element; and (v) no domain transformation or Jacobian is needed in the computation. The numerical examples also show that the new spline element has higher accuracy and efficiency.
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Juan CHEN, Chong-jun LI, and Wan-ji CHEN
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