김경종
615 lines
30 KiB
Markdown
615 lines
30 KiB
Markdown
<!-- source-page: 191 -->
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<details>
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<summary>text_image</summary>
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φ
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φ = φ(ξ)
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1
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φ₁
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-1 0 +1 ξ
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</details>
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(a)
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<details>
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<summary>text_image</summary>
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φ
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←ξ₁→←ξ₂→
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1
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2
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φ₁
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φ₂
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-1 0 +1 ξ
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</details>
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(b)
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<details>
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<summary>text_image</summary>
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φ
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←ξ₁ → ←ξ₃ →
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1 2 3
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φ₁ φ₂ φ₃
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-1 0 +1 ξ
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</details>
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(c)
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Figure 6.4-1. Gauss quadrature to compute the shaded area under the curve $\phi = \phi(\xi)$ , using (a) one, (b) two, and (c) three sampling points (also called Gauss points).
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2, so that $I \approx 2\phi_1$ . This result is exact if $\phi = \phi(\xi)$ happens to describe a straight line of any finite slope.
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Generalization of the foregoing procedure leads to the quadrature formula
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$$
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I = \int_ {- 1} ^ {1} \phi d \xi \approx W _ {1} \phi_ {1} + W _ {2} \phi_ {2} + \dots + W _ {n} \phi_ {n} \tag {6.4-2}
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$$
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Thus, to approximate I, we evaluate $\phi = \phi(\xi)$ at each of several locations $\xi_{i}$ to obtain ordinates $\phi_{i}$ , multiply each $\phi_{i}$ by an appropriate weight $W_{i}$ , and add. In the preceding one-point example, where $I \approx 2\phi_{1}$ , we have n = 1 and $W_{1} = 2$ . Gauss was able to prescribe the locations $\xi_{i}$ and weights $W_{i}$ such that greatest accuracy is achieved for a given n.
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Sampling points are located symmetrically with respect to the center of the integration interval. Symmetrically paired points have the same weight $W_{i}$ . Data appear in Table 6.4-1. These data are sometimes called Gauss–Legendre coefficients because sampling point locations happen to be roots of Legendre polynomials. Much more extensive tabulations are available [6.2]. In programming,
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TABLE 6.4-1. SAMPLING POINTS AND WEIGHTS FOR GAUSS QUADRATURE OVER THE INTERVAL $\xi = -1$ TO $\xi = +1$ .
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<table><tr><td>Order n</td><td>Location $\xi_{i}$ of Sampling Point</td><td>Weight Factor $W_{i}$ </td></tr><tr><td>1</td><td>0.</td><td>2.</td></tr><tr><td>2</td><td> $\pm 0.57735 \ 02691 \ 89626 = \pm 1/\sqrt{3}$ </td><td>1.</td></tr><tr><td>3</td><td> $\pm 0.77459 \ 66692 \ 41483 = \pm \sqrt{0.6}$ </td><td> $0.55555 \ 55555 \ 55555 = \frac{5}{9}$ </td></tr><tr><td></td><td>0.</td><td> $0.88888 \ 88888 \ 88888 = \frac{8}{9}$ </td></tr><tr><td>4</td><td> $\pm 0.86113 \ 63115 \ 94053 = \pm \left[ \frac{3 + 2r}{7} \right]^{1/2}$ </td><td> $0.34785 \ 48451 \ 37454 = \frac{1}{2} - \frac{1}{6r}$ </td></tr><tr><td></td><td> $\pm 0.33998 \ 10435 \ 84856 = \pm \left[ \frac{3 - 2r}{7} \right]^{1/2}$ </td><td> $0.65214 \ 51548 \ 62546 = \frac{1}{2} + \frac{1}{6r}$ </td></tr><tr><td></td><td>where $r = \sqrt{1.2}$ </td><td></td></tr></table>
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<!-- source-page: 192 -->
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we first check the correctness of tabulated data, then code the $\xi_{i}$ and $W_{i}$ with as many digits as the computer allows, in order to avoid unnecessary rounding error.
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Example. Consider the polynomial $\phi = a_{1} + a_{2}\xi + a_{3}\xi^{2} + a_{4}\xi^{3}$ , where the $\cdot a_{i}$ are constants. The exact integral is
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$$
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I = \int_ {- 1} ^ {1} \phi d \xi = 2 a _ {1} + \frac {2}{3} a _ {3} \tag {6.4-3}
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$$
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The approximate integral given by a one-point rule is
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$$
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I _ {1} \approx 2 a _ {1} \tag {6.4-4}
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$$
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The integral given by a two-point rule is, with $\xi_1 = -p$ , $\xi_2 = p$ , and $p = 1 / \sqrt{3}$ ,
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$$
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I _ {2} = 1. 0 \left(a _ {1} - a _ {2} p + a _ {3} p ^ {2} - a _ {4} p ^ {3}\right) + 1. 0 \left(a _ {1} + a _ {2} p + a _ {3} p ^ {2} + a _ {4} p ^ {3}\right) \tag {6.4-5}
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$$
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$$
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I _ {2} = 2 a _ {1} + \frac {2}{3} a _ {3}
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$$
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In the foregoing example we see an instance of a general rule: a polynomial of degree $2n - 1$ is integrated exactly by $n$ -point Gauss quadrature. Use of more than $n$ points will still produce the exact result. The degree of precision of a quadrature rule is the degree of the highest-order polynomial that is exactly integrated. Thus a second-order Gauss rule has degree of precision 3.
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If the function $\phi = \phi(\xi)$ is not a polynomial, Gauss quadrature is inexact, but becomes more accurate as more points are used. Here we refer to the accuracy of integration, not to the accuracy of the results of finite element analysis. We will see, for example, that some elements used for stress analysis are improved by using a Gauss rule of lower order than would be chosen if the goal were accurate integration. The question of what quadrature order is best is addressed in Sections 6.11 and 6.12.
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It is important to realize that the ratio of two polynomials is in general not a polynomial, and therefore will not be integrated exactly by Gauss quadrature.
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Two and Three Dimensions. Multidimensional Gauss rules, called Gaussian product rules, are formed by successive application of one-dimensional Gauss rules. In two dimensions, consider the function $\phi = \phi(\xi, \eta)$ . We elect to integrate first with respect to $\xi$ and then with respect to $\eta$ :
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$$
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I = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \phi (\xi , \eta) d \xi d \eta \approx \int_ {- 1} ^ {1} \left[ \sum_ {i} W _ {i} \phi (\xi_ {i}, \eta) \right] d \eta \tag {6.4-6}
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$$
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$$
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\approx \sum_ {j} W _ {j} \left[ \sum_ {i} W _ {i} \phi (\xi_ {i}, \eta_ {j}) \right] = \sum_ {i} \sum_ {j} W _ {i} W _ {j} \phi (\xi_ {i}, \eta_ {j})
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$$
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For the four-point rule depicted in Fig. 6.4-2a, $W_{i}W_{j} = 1$ , and Eq. 6.4-6 becomes
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$$
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I \approx \phi_ {1} + \phi_ {2} + \phi_ {3} + \phi_ {4} \tag {6.4-7}
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$$
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where $\phi_{i}$ is the numerical value of $\phi$ at the ith Gauss point. For the nine-point rule depicted in Fig. 6.4-2b, Eq. 6.4-6 yields
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<!-- source-page: 193 -->
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<details>
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<summary>text_image</summary>
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ξ = -1/√3
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η
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ξ = +1/√3
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2
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4
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η = +1/√3
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ξ
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1
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3
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η = -1/√3
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</details>
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(a)
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<details>
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<summary>text_image</summary>
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ξ = -√0.6
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η
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ξ = +√0.6
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η = +√0.6
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3
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6
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9
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2
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5
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8
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ξ
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1
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4
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7
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η = -√0.6
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</details>
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(b)
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Figure 6.4-2. Gauss point locations in a quadrilateral element using (a) four points (order 2 rule), and (b) nine points (order 3 rule).
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$$
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I \approx \frac {2 5}{8 1} \left(\phi_ {1} + \phi_ {3} + \phi_ {7} + \phi_ {9}\right) + \frac {4 0}{8 1} \left(\phi_ {2} + \phi_ {4} + \phi_ {6} + \phi_ {8}\right) + \frac {6 4}{8 1} \phi_ {5} \tag {6.4-8}
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$$
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In three dimensions, the Gauss quadrature rule has the form
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$$
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I = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \phi (\xi , \eta , \zeta) d \xi d \eta d \zeta \approx \sum_ {i} \sum_ {j} \sum_ {k} W _ {i} W _ {j} W _ {k} \phi (\xi_ {i}, \eta_ {j}, \zeta_ {k}) \tag {6.4-9}
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$$
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In an equation such as Eq. 6.3-20, each coefficient of the integrand $[B]^{T}[E][B]tJ$ is in general a function of $\xi$ and $\eta$ . There are 64 coefficients (or, 36 different coefficients, owing to symmetry), and each must be integrated as $\phi$ is integrated in Eq. 6.4-6: by evaluation at specific points, multiplication by weight factors, and addition. It is not necessary to use the same Gauss rule in both directions, but doing so is most common.
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# 6.5 COMPUTER SUBROUTINES FOR THE BILINEAR ISOPARAMETRIC ELEMENT
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The stiffness matrix of Eq. 6.3-20 and the load vector of Eq. 6.3-21 are evaluated by the subroutines presented here. Gauss quadrature is used. Of course, the coding is not unique: other procedures may be more compact, more general, or more efficient [6.3,6.4]. Nevertheless, the subroutines show precisely what must be done. A thorough understanding of these subroutines makes it easier to understand the isoparametric formulation in general.
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The notation, assumptions, and procedures are as follows. Shape functions of Eqs. 6.3-2 and their derivatives can be written in the form
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$$
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N _ {i} = (1 + \xi \xi_ {i}) (1 + \eta \eta_ {i}) / 4 \tag {6.5-1a}
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$$
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$$
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N _ {i, \xi} = \xi_ {i} (1 + \eta \eta_ {i}) / 4 \tag {6.5-1b}
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$$
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$$
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N _ {i, \eta} = \eta_ {i} (1 + \xi \xi_ {i}) / 4 \tag {6.5-1c}
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$$
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<!-- source-page: 194 -->
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```fortran
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SUBROUTINE SHAPE (PXI,PET,XL,YL,EXI,EYI,ESI,THC,THK,DETJAC)
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IMPLICIT DOUBLE PRECISION (A-H,O-Z)
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DOUBLE PRECISION NXI(4),NET(4),JAC
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DIMENSION XII(4),ETI(5)
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DIMENSION DUP(1),XL(4),YL(4),EXI(4),EYI(4),ESI(4),THC(4)
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COMMON /Q4/ EN(4),JAC(2,2),B(3,9),E(3,3),SE(8,8),RE(8)
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EQUIVALENCE (XII(1),ETI(2)),(DUP(1),JAC(1))
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DATA ETI /-1.D0,-1.D0,+1.D0,+1.D0,-1.D0/
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C --- Find shape functions (EN) and their derivatives (NXI,NET).
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DO 10 L=1,4
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DUM1 = (1. + XII(L)*PXI)/4.
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DUM2 = (1. + ETI(L)*PET)/4.
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EN(L) = 4.*DUM1*DUM2
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NXI(L) = XII(L)*DUM2
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10 NET(L) = ETI(L)*DUM1
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C --- Clear arrays JAC and B (words in common are stored in sequence).
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DO 20 L=1,31
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20 DUP(L) = 0.
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C --- Find Jacobian JAC and its determinant. Replace JAC by its inverse.
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DO 30 L=1,4
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JAC(1,1) = JAC(1,1) + NXI(L)*XL(L)
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JAC(1,2) = JAC(1,2) + NXI(L)*YL(L)
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JAC(2,1) = JAC(2,1) + NET(L)*XL(L)
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30 JAC(2,2) = JAC(2,2) + NET(L)*YL(L)
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DETJAC = JAC(1,1)*JAC(2,2) - JAC(2,1)*JAC(1,2)
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DUM1 = JAC(1,1)/DETJAC
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JAC(1,1) = JAC(2,2)/DETJAC
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JAC(1,2) = -JAC(1,2)/DETJAC
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JAC(2,1) = -JAC(2,1)/DETJAC
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JAC(2,2) = DUM1
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C --- Form strain-displacement matrix [B] (zero entries are already set).
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DO 40 J=1,4
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L = 2*J
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K = L-1
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B(1,K) = JAC(1,1)*NXI(J) + JAC(1,2)*NET(J)
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B(2,L) = JAC(2,1)*NXI(J) + JAC(2,2)*NET(J)
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B(3,K) = B(2,L)
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40 B(3,L) = B(1,K)
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C --- Interpolate initial strains from corner values, and store as column
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C --- 9 of [B] (already cleared to zero). Similarly, get thickness THK.
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THK = 0.
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DO 50 L=1,4
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B(1,9) = B(1,9) + EN(L)*EXI(L)
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B(2,9) = B(2,9) + EN(L)*EYI(L)
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B(3,9) = B(3,9) + EN(L)*ESI(L)
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50 THK = THK + EN(L)*THC(L)
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RETURN
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END
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```
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Figure 6.5-1. Fortran subroutine SHAPE. For the plane stress element of Section 6.3, it calculates the shape functions and their derivatives, the Jacobian matrix, its inverse and determinant, matrix [B], and initial strains and element thickness, all at the point whose $\xi$ and $\eta$ coordinates are PXI and PET.
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where i is the number of the shape function, and
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$$
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\xi_ {i} = - 1., 1., 1., - 1. \quad \text { for } \quad i = 1, 2, 3, 4 \tag {6.5-2}
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$$
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$$
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\eta_ {i} = - 1., - 1., 1., 1. \quad \text { for } \quad i = 1, 2, 3, 4
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$$
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In Fig. 6.5-1, $\xi_{i}$ and $\eta_{i}$ are placed in arrays XII and ETI by DATA and EQUIVALENCE statements. $N_{i}, N_{i,\xi}$ and $N_{i,\eta}$ are computed and stored in arrays EN, NXI, and NET. Because of the EQUIVALENCE statement containing DUP(1), the loop on statement 20 neatly initializes arrays JAC and B to zero. Coordinates $\xi$ and $\eta$ in Eqs. 6.5-1 are called PXI and PET in Fig. 6.5-1 and are transmitted as formal parameters. PXI and PET are Gauss point coordinates if SHAPE is called by QUAD4 (Fig. 6.5-2), but other coordinates could be prescribed by a subsequent calling routine (as, for example, when evaluating [B] for use in stress calculation by means of Eq. 4.7-1).
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<!-- source-page: 195 -->
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```fortran
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SUBROUTINE QUAD4 (NGAUSS,XL,YL,EXI,EYI,ESI,THC,PLACE,WGT,
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1 BODYFX,BODYFY)
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IMPLICIT DOUBLE PRECISION (A-H,O-Z) .
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DOUBLE PRECISION JAC
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C E = material property matrix. Data at element nodes is as follows:
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C XY,YL = Cartesian coordinates. (Used in subroutine SHAPE)
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C EXI,EYI,ESI = initial strains (x,y,shear). (Used in subroutine SHAPE)
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C THC = thicknesses in z direction. (Used in subroutine SHAPE)
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C The calling program must supply the following data in 3 by 3 arrays:
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C [PLACE] = 0. -.57735--- -.77459--- | [WGT] = 2. 1. .555--- |
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C 0. +.57735--- 0. | 0. 1. .888--- |
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C 0. 0. +.77459--- | 0. 0. .555--- |
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COMMON /Q4/ EN(4),JAC(2,2),B(3,9),E(3,3),SE(8,8),RE(8)
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DIMENSION XL(4),YL(4),EXI(4),EYI(4),ESI(4),THC(4),BTE(8,3),
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1 PLACE(3,3),WGT(3,3)
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C --- Clear load vector {r} and upper triangle of stiffness matrix [k].
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DO 10 K=1,8
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RE(K) = 0.
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DO 10 L=K,8
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10 SE(K,L) = 0.
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C --- Start Gauss quadrature loop. Use NGAUSS by NGAUSS rule.
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DO 90 NA = 1,NGAUSS
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PXI = PLACE(NA,NGAUSS)
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DO 80 NB = 1,NGAUSS
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PET = PLACE(NB,NGAUSS)
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CALL SHAPE (PXI,PET,XL,YL,EXI,EYI,ESI,THC,THK,DETJAC)
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DV = WGT(NA,NGAUSS)*WGT(NB,NGAUSS)*THK*DETJAC
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C ---- Store [B]-transpose times [E] in 8 by 3 work array [BTE].
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DO 30 J=1,4
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L = 2*J
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K = L-1
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C. ---- Do only multiplications that give a nonzero product.
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DO 20 N=1,3
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BTE(K,N) = B(1,K)*E(1,N) + B(3,K)*E(3,N)
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20 BTE(L,N) = B(2,L)*E(2,N) + B(3,L)*E(3,N)
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C ---- Add contribution of body forces to nodal load array {r}.
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RE(K) = RE(K) + EN(J)*BODYFX*DV
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30 RE(L) = RE(L) + EN(J)*BODYFY*DV
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C ---- Loop on rows of [k] (array SE) and {r} (array RE).
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DO 70 NROW=1,8
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C ---- Add contribution of initial strains to load array {r}.
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DO 40 J=1,3
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40 RE(NROW) = RE(NROW) + BTE(NROW,J)*B(J,9)*DV
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C ---- Loop to add contribution to element stiffness matrix [k].
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DO 60 NCOL=NROW,8
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DUM = 0.
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C ---- Loop for product [B]T*[E]*[B]. Zeros in [B] not skipped.
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DO 50 J=1,3
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50 DUM = DUM + BTE(NROW,J)*B(J,NCOL)
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60 SE(NROW,NCOL) = SE(NROW,NCOL) + DUM*DV
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70 CONTINUE
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80 CONTINUE
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90 CONTINUE
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C --- Fill in lower triangle of element stiffness matrix by symmetry.
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DO 100 K=1,7
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DO 100 L=K,8
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100 SE(L,K) = SE(K,L)
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RETURN
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END
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```
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Figure 6.5-2. Fortran subroutine QUAD4. It generates $[k]$ and $\{r_{e}\}$ for the plane stress element of Section 6.3 by Gauss quadrature. We store $[k]$ in array SE and $\{r_{e}\}$ in array RE.
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Through statement 40, subroutine SHAPE follows exactly the development in Section 6.3. In the DO 50 loop, initial strains $(\epsilon_{x0} = \mathsf{EXI}, \epsilon_{y0} = \mathsf{EYI}, \gamma_{xy0} = \mathsf{ESI})$ and element thickness $(t = \mathsf{THC})$ are prescribed at the four nodes. Values of these quantities at coordinates PXI and PET are found by interpolation (write $t = \Sigma \cdot N_i t_i$ , analogous to Eqs. 6.3-1). Initial stresses $\{\sigma_0\}$ are not included but can be added as an exercise.
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The foregoing calculations in SHAPE must be carried out at every Gauss point used by QUAD4 (Fig. 6.5-2).
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<!-- source-page: 196 -->
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QUAD4 requires as input data the global nodal coordinates XL and YL, nodal thicknesses and initial strains, the material property matrix [E] (presumed full, as for a general material, and constant over the element), body forces $ F\_{x} = \text{BODYFX} $ and $ F\_{y} = \text{BODYFY} $, and the quadrature order NGAUSS. As indicated by comments in the listing, Gauss quadrature data must be coded before the subroutine is used. Although perhaps less obvious than subroutine SHAPE, subroutine QUAD4 is a straightforward application of Gauss quadrature to Eqs. 6.3-20 and 6.3-21. DV represents the product $ W\_{i}W\_{j}tJ $, which is a common multiplier of each coefficient to be integrated—that is, of each coefficient in [B]$ ^{T} $[E][B], in [B]$ ^{T} $[E]{\epsilon\_{0}} \), and in [N]$ ^{T} ${F}.
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Cost is reduced by using quadrature to generate only the upper triangle of [k], leaving the lower triangle to be completed by symmetry as the last step. Coding of the DO 20 loop is an attempt to exploit the sparsity of [B]. Additional economies have been proposed [6.3,6.4].
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# 6.6 QUADRATIC PLANE ELEMENTS
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One, two, or more nodes can be placed on each side of a four-node quadrilateral. The resulting elements are called quadratic, cubic, and so on. Here we discuss the quadratic quadrilateral element (Fig. 6.6-1). Quadratic triangular elements are discussed in Section 6.8.
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As with the bilinear element, sides of the quadratic element are at $\xi = \pm 1$ and at $\eta = \pm 1$ . Two of the side nodes are at $\xi = 0$ and two are at $\eta = 0$ . Axes $\xi$ and $\eta$ may be curved in a quadratic element. As shown in Fig. 6.6-1, sides of an undeformed element may be straight lines or quadratic curves. Similarly, displacements may be linear or quadratic. (A side capable of deforming quadratically need not actually do so; it would deform only linearly in passing a constant-strain patch test.)
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Shape functions of a quadratic element can be generated systematically [3.4], or by inspection and trial, as follows. In Fig. 6.6-2a, one obtains $N_5$ by interpolating quadratically in $\xi$ and linearly in $\eta$ , taking care that $N_5 = 1$ at node 5 and $N_5 = 0$ at all other nodes. Similarly, $N_8$ is obtained in Fig. 6.6-2b. Next, one observes that $N_{(c)}$ (which is $N_1$ of a bilinear element) has ordinate 0.5 at nodes 5 and 8. The function $N_{(c)} - \frac{1}{2} N_5 - \frac{1}{2} N_8$ is therefore zero at all nodes but node 1, where
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<details>
|
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<summary>text_image</summary>
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η
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7
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3
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||
4
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8
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6
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ξ
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1
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||
5
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2
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(a)
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</details>
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<details>
|
||
<summary>text_image</summary>
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||
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1
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2
|
||
3
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||
4
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||
5
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6
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7
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8
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η
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ξ
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(b)
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</details>
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Figure 6.6-1. Quadratic plane elements. Those shown have (a) straight sides and midside nodes, and (b) some curved sides and off-center side nodes.
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||
<!-- source-page: 197 -->
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||
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
1
|
||
8
|
||
4
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||
7
|
||
3
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φ₁ = 1
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5
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||
2
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||
6
|
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ξ
|
||
η
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||
N₁
|
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(d)
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||
</details>
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(a) $N_{5} = \frac{1}{2} (1 - \xi^{2})(1 - \eta)$
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(b) $N_{8} = \frac{1}{2} (1 - \xi)(1 - \eta^{2})$
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(c) $N_{(c)} = \frac{1}{4} (1 - \xi)(1 - \eta)$
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(d) $N_{1} = N_{(c)} - \frac{1}{2} N_{5} - \frac{1}{2} N_{8}$
|
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|
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(a) $N_{5} = \frac{1}{2} (1 - \xi^{2})(1 - \eta)$
|
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(b) $N_{8} = \frac{1}{2} (1 - \xi)(1 - \eta^{2})$
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(c) $N_{(c)} = \frac{1}{4} (1 - \xi)(1 - \eta)$
|
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|
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(d) $N_{1} = N_{(c)} - \frac{1}{2} N_{5} - \frac{1}{2} N_{8}$
|
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|
||
Figure 6.6-2. Selected shape functions for the quadratic element in Fig. 6.6-1, shown normal to square elements in $\xi \eta$ coordinates.
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|
||
it is unity; therefore, it is shape function $N_{1}$ . The complete set of shape functions is
|
||
|
||
$$
|
||
N _ {1} = \frac {1}{4} (1 - \xi) (1 -, \eta) - \frac {1}{2} \left(N _ {8} + N _ {5}\right) \quad N _ {5} = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta)
|
||
$$
|
||
|
||
$$
|
||
N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta) - \frac {1}{2} \left(N _ {5} + N _ {6}\right) \quad N _ {6} = \frac {1}{2} (1 + \xi) \left(1 - \eta^ {2}\right) \tag {6.6-1}
|
||
$$
|
||
|
||
$$
|
||
N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta) - \frac {1}{2} \left(N _ {6} + N _ {7}\right). \quad N _ {7} = \frac {1}{2} \left(1 - \xi^ {2}\right) (1 + \eta)
|
||
$$
|
||
|
||
$$
|
||
N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) - \frac {1}{2} (N _ {7} + N _ {8}) \quad N _ {8} = \frac {1}{2} (1 - \xi) (1 - \eta^ {2})
|
||
$$
|
||
|
||
The foregoing element, and other isoparametric elements having boundary nodes only, are sometimes called “serendipity” elements. Addition of an internal node (node 9) at $\xi = \eta = 0$ in Fig. 6.6-1 makes the element a “Lagrange” quadratic element, Fig. 6.6-3a. Element sides may be straight or curved. The name “Lagrange” is used because the element shape functions can be obtained by taking
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y,v
|
||
η
|
||
4
|
||
7
|
||
3
|
||
8
|
||
9
|
||
6
|
||
ξ
|
||
1
|
||
5
|
||
2
|
||
x,u
|
||
</details>
|
||
|
||
(a)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
η
|
||
ξ
|
||
φ₉ = 1
|
||
N₉
|
||
</details>
|
||
|
||
(b)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
η
|
||
ξ
|
||
φ₅ = 1
|
||
N₅
|
||
</details>
|
||
|
||
(c)
|
||
Figure 6.6-3. (a) Nine-node Lagrange element in Cartesian coordinates. (b,c) Shape functions $N_{9}$ and $N_{5}$ , shown normal to square elements in $\xi\eta$ coordinates.
|
||
|
||
<!-- source-page: 198 -->
|
||
|
||
products of one-dimensional Lagrange interpolants (Eqs. 3.12-6). Computed results are usually most accurate if node 9 is assigned the location given by $\xi = \eta = 0$ in Eqs. 6.6-1—that is,
|
||
|
||
$$
|
||
x _ {9} = \sum N _ {i} x _ {i} = - \frac {1}{4} \left(x _ {1} + x _ {2} + x _ {3} + x _ {4}\right) + \frac {1}{2} \left(x _ {5} + x _ {6} + x _ {7} + x _ {8}\right) \tag {6.6-2}
|
||
$$
|
||
|
||
and similarly for $y_9$ . Thus the geometry of the Lagrange element is completely defined by coordinates of the eight boundary nodes.
|
||
|
||
The shape function associated with node 9 of the quadratic Lagrange element is
|
||
|
||
$$
|
||
N _ {9} = (1 - \xi^ {2}) (1 - \eta^ {2}) \tag {6.6-3}
|
||
$$
|
||
|
||
which may be called a “bubble function” because it resembles a bubble blown over a quadrilateral opening in a plate, as shown in Fig. 6.6-3b. The first eight shape functions of the Lagrange quadratic element can be obtained by modifying the $N_{i}$ of Eqs. 6.6-1 so that each is zero at $\xi = \eta = 0$ (compare $N_{5}$ in Fig. 6.6-2a with $N_{5}$ in Fig. 6.6-3c). The results are shown in Table 6.6-1, which is explained as follows.
|
||
|
||
With all nine nodes included, all nine $N_{i}$ of Table 6.6-1 are used. If only node 9 is omitted, the $N_{i}$ reduce to those of Eqs. 6.6-1. If nodes 5 through 9 are omitted, the element becomes bilinear and the $N_{i}$ reduce to those of Eqs. 6.3-2. If nodes 6 through 9 are omitted, the element has three linear sides and one quadratic side, and $N_{3}$ and $N_{4}$ become bilinear shape functions. (A five-node element is a “transition” element that can be connected to both bilinear and biquadratic elements without incompatibility.) Alternative shape functions for the nine-node element are explained in connection with Eqs. 8.1-4 and 8.1-5.
|
||
|
||
Let it be required to determine the characteristic matrix [k] of a scalar field element (see Eq. 6.3-5). To do so, we express $J$ and [B] in terms of $\xi$ and $\eta$ , then perform Gauss quadrature as described in Section 6.4. [J] is given by Eq. 6.3-11, in which index $i$ runs from 1 to $n$ , where $n$ is the number of nodes in the element; that is,
|
||
|
||
TABLE 6.6-1. SHAPE FUNCTIONS OF A PLANE QUADRILATERAL THAT HAS FROM FOUR TO NINE NODES. EXAMPLE (NOTE THAT $N_{5}$ THROUGH $N_{8}$ CONTAIN $N_{9}$ ): $N_{1} = \frac{1}{4}(1 - \xi)(1 - \eta) - \frac{1}{4}(1 - \xi^{2})(1 - \eta) - \frac{1}{4}(1 - \xi)(1 - \eta^{2}) + \frac{1}{4}N_{9}$ . NODE 9 IS AT $\xi = \eta = 0$ .
|
||
<table><tr><td rowspan="2"></td><td colspan="5">Include Only If Node i Is Present in the Element</td></tr><tr><td>i = 5</td><td>i = 6</td><td>i = 7</td><td>i = 8</td><td>i = 9</td></tr><tr><td> $N_1 = \frac{1}{4}(1 - \xi)(1 - \eta)$ </td><td> $-\frac{1}{2}N_5$ </td><td></td><td></td><td> $-\frac{1}{2}N_8$ </td><td> $-\frac{1}{4}N_9$ </td></tr><tr><td> $N_2 = \frac{1}{4}(1 + \xi)(1 - \eta)$ </td><td> $-\frac{1}{2}N_5$ </td><td> $-\frac{1}{2}N_6$ </td><td></td><td></td><td> $-\frac{1}{4}N_9$ </td></tr><tr><td> $N_3 = \frac{1}{4}(1 + \xi)(1 + \eta)$ </td><td></td><td> $-\frac{1}{2}N_6$ </td><td> $-\frac{1}{2}N_7$ </td><td></td><td> $-\frac{1}{4}N_9$ </td></tr><tr><td> $N_4 = \frac{1}{4}(1 - \xi)(1 + \eta)$ </td><td></td><td></td><td> $-\frac{1}{2}N_7$ </td><td> $-\frac{1}{2}N_8$ </td><td> $-\frac{1}{4}N_9$ </td></tr><tr><td> $N_5 = \frac{1}{2}(1 - \xi^2)(1 - \eta)$ </td><td></td><td></td><td></td><td></td><td> $-\frac{1}{2}N_9$ </td></tr><tr><td> $N_6 = \frac{1}{2}(1 + \xi)(1 - \eta^2)$ </td><td></td><td></td><td></td><td></td><td> $-\frac{1}{2}N_9$ </td></tr><tr><td> $N_7 = \frac{1}{2}(1 - \xi^2)(1 + \eta)$ </td><td></td><td></td><td></td><td></td><td> $-\frac{1}{2}N_9$ </td></tr><tr><td> $N_8 = \frac{1}{2}(1 - \xi)(1 - \eta^2)$ </td><td></td><td></td><td></td><td></td><td> $-\frac{1}{2}N_9$ </td></tr><tr><td> $N_9 = (1 - \xi^2)(1 - \eta^2)$ </td><td></td><td></td><td></td><td></td><td></td></tr></table>
|
||
|
||
<!-- source-page: 199 -->
|
||
|
||
$$
|
||
[ \mathbf {J} ] = [ \mathbf {D} _ {N} ] \left[ \begin{array}{l l} x _ {1} & y _ {1} \\ x _ {2} & y _ {2} \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ x _ {n} & y _ {n} \end{array} \right] \tag {6.6-4}
|
||
$$
|
||
|
||
where
|
||
|
||
$$
|
||
\left[ \mathbf {D} _ {N} \right] = \left[ \begin{array}{l l l l l} N _ {1, \xi} & N _ {2, \xi} & N _ {3, \xi} & \dots & N _ {n, \xi} \\ N _ {1, \eta} & N _ {2, \eta} & N _ {3, \eta} & \dots & N _ {n, \eta} \end{array} \right] \tag {6.6-5}
|
||
$$
|
||
|
||
Hence, Eq. 6.3-14 gives $[\Gamma] = [\mathbf{J}]^{-1}$ and Eq. 6.3-15 gives $J = \operatorname{det}[\mathbf{J}]$ . Derivatives of $\phi$ are
|
||
|
||
$$
|
||
\left\{ \begin{array}{l} \phi_ {, x} \\ \phi_ {, y} \end{array} \right\} = [ \Gamma ] \left\{ \begin{array}{l} \phi_ {, \xi} \\ \phi_ {, \eta} \end{array} \right\} \quad \text { and } \quad \left\{ \begin{array}{l} \phi_ {, \xi} \\ \phi_ {, \eta} \end{array} \right\} = [ \mathbf {D} _ {N} ] \{\phi_ {e} \} \tag {6.6-6}
|
||
$$
|
||
|
||
where $\{\phi_e\}$ are nodal values of $\phi$ . From Eqs. 6.6-6, we obtain
|
||
|
||
$$
|
||
\left\{ \begin{array}{l} \phi_ {, x} \\ \phi_ {, y} \end{array} \right\} = [ \mathbf {B} ] \{\phi_ {e} \}, \quad \text { where } \quad [ \mathbf {B} ] = [ \Gamma ] [ \mathbf {D} _ {N} ] \tag {6.6-7}
|
||
$$
|
||
|
||
The characteristic matrix, to be integrated numerically, is
|
||
|
||
$$
|
||
\underset {n \times n} {[ \mathbf {k} ]} = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \underset {n \times 2} {[ \mathbf {B} ] ^ {T}} k [ \mathbf {B} ] t J d \xi d \eta \tag {6.6-8}
|
||
$$
|
||
|
||
Except that $n$ may now be greater than 4, the foregoing argument is identical to that in Section 6.3.
|
||
|
||
Remarks. By adding yet more nodes, quadrilateral elements become successively cubic, quartic, and so on. Examination of the polynomial expansions for serendipity elements (which have no internal nodes) shows that serendipity elements “leave out the middle” of a Pascal triangle, Fig. 6.6-4. In contrast, Lagrange elements (which have one or more internal nodes) use a square block of terms, which allows Lagrange elements to have better accuracy.
|
||
|
||
All isoparametric elements lose. accuracy when distorted from a rectangular
|
||
|
||
|
||
|
||
<details>
|
||
<summary>flowchart</summary>
|
||
```mermaid
|
||
graph TD
|
||
A["Constant term"] --> B["1"]
|
||
C["Linear terms"] --> B
|
||
D["Quadratic terms"] --> B
|
||
E["Cubic terms"] --> B
|
||
F["Quartic terms"] --> B
|
||
G["Serendipity"] --> B
|
||
H["Lagrange"] -.-> I["ξ³η²"]
|
||
I --> J["ξ²η²"]
|
||
J --> K["ξη²"]
|
||
K --> L["η²"]
|
||
L --> M["Linear element"]
|
||
L --> N["Quadratic element"]
|
||
L --> O["Cubic element"]
|
||
I --> P["ξ³η³"]
|
||
P --> Q["ξ²η³"]
|
||
Q --> R["ξη³"]
|
||
R --> S["η³"]
|
||
S --> T["Linear element"]
|
||
S --> U["Quadratic element"]
|
||
S --> V["Cubic element"]
|
||
```
|
||
</details>
|
||
|
||
Figure 6.6-4. Polynomial coefficients in plane serendipity elements (boundary nodes only) and plane Lagrange elements (boundary and internal nodes).
|
||
|
||
<!-- source-page: 200 -->
|
||
|
||
shape. However, the nine-node element is much less sensitive than the eight-node element to nonrectangularity, to curvature of sides, and to placing side nodes away from midsides. Indeed, it has been found that a nonrectangular nine-node element can still represent a state of pure bending, provided that sides are straight and side nodes are at midsides as in Fig. 6.6-3a [4.6]. The eight-node element does not have this capability.
|
||
|
||
In dynamic problems, it is often desirable to use lumped (diagonal) mass matrices. All popular lumping schemes result in positive nodal masses for the nine-node element. Such is not the case for the eight-node element, which may have some negative nodal masses (see Chapter 13).
|
||
|
||
It is easy to alter the subroutines in Section 6.5 to accommodate a quadratic element. The major change is revision of the shape functions in subroutine SHAPE. Otherwise, one increases the size of some arrays and increases the range of most DO loops. The Jacobian matrix remains 2 by 2. In forming [B] from Eqs. 6.3-17 through 6.3-19, the rectangular matrix in Eq. 6.3-19 contains 16 or 18 columns, depending on whether node 9 is omitted or included.
|
||
|
||
# 6.7 HEXAHEDRAL (SOLID)
|
||
|
||
# ISOPARAMETRIC ELEMENTS
|
||
|
||
In three dimensions, the isoparametric procedure closely resembles the two-dimensional development in Section 6.3. The general procedures used for isoparametric elements have little to do with the specific shape functions of a particular element.
|
||
|
||
For element geometry and the field quantity $\phi$ of a solid isoparametric element, we write
|
||
|
||
$$
|
||
x = \sum N _ {i} x _ {i} \quad y = \sum N _ {i} y _ {i} \quad z = \sum N _ {i} z _ {i} \quad \phi = \sum N _ {i} \phi_ {i} \tag {6.7-1}
|
||
$$
|
||
|
||
where i ranges over the number of nodes in the element. Shape functions $N_{i}$ are functions of isoparametric coordinates $\xi$ , $\eta$ , and $\zeta$ . Faces of the element lie at $\xi = \pm 1$ , $\eta = \pm 1$ , and $\zeta = \pm 1$ . The Jacobian matrix is defined analogously to Eq. 6.3-11, but is now 3 by 3:
|
||
|
||
$$
|
||
[ \mathrm{J} ] = \left[ \begin{array}{l l l} x, _ {\xi} & y, _ {\xi} & z, _ {\xi} \\ x, _ {\eta} & y, _ {\eta} & z, _ {\eta} \\ x, _ {\zeta} & y, _ {\zeta} & z, _ {\zeta} \end{array} \right] = \sum \left[ \begin{array}{l l l} N _ {i, \xi} x _ {i} & N _ {i, \xi} y _ {i} & N _ {i, \xi} z _ {i} \\ N _ {i, \eta} x _ {i} & N _ {i, \eta} y _ {i} & N _ {i, \eta} z _ {i} \\ N _ {i, \zeta} x _ {i} & N _ {i, \zeta} y _ {i} & N _ {i, \zeta} z _ {i} \end{array} \right] \tag {6.7-2}
|
||
$$
|
||
|
||
With $[\Gamma] = [\mathrm{J}]^{-1}$ ,
|
||
|
||
$$
|
||
\left\lfloor \phi_ {, x} \quad \phi_ {, y} \quad \phi_ {, z} \right] ^ {T} = [ \Gamma ] \left\lfloor \phi_ {, \xi} \quad \phi_ {, \eta} \quad \phi_ {, \zeta} \right] ^ {T} \tag {6.7-3}
|
||
$$
|
||
|
||
For a scalar field problem in three dimensions, with $n$ the number of nodes per element, Eq. 6.3-5 becomes
|
||
|
||
$$
|
||
[ \mathbf {k} ] _ {n \times n} = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} [ \mathbf {B} ] ^ {T} k [ \mathbf {B} ] J d \xi d \eta d \zeta \tag {6.7-4}
|
||
$$
|
||
|
||
The Jacobian determinant $J = \det[\mathbf{J}]$ expresses the ratio of volume $dx \, dy \, dz$ to $d\xi \, d\eta \, d\zeta$ . In general, [J] and [B] are functions of $\xi, \eta,$ and $\zeta$ .
|