김경종
313 lines
25 KiB
Markdown
313 lines
25 KiB
Markdown
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The Gauss-Legendre integration procedure is commonly used in isoparametric finite element analysis. However, it should be noted that other integration schemes, in which both the weights and sampling positions are varied to obtain maximum accuracy, have also been derived (see C. E. Fröberg [A] and A. H. Stroud and D. Secrest [A]).
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# 5.5.4 Integrations in Two and Three Dimensions
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So far we have considered the integration of a one-dimensional function $F(r)$ . However, two- and three-dimensional integrals need to be evaluated in two- and three-dimensional finite element analyses. In the evaluation of rectangular elements, we can apply the above one-dimensional integration formulas successively in each direction. $^{8}$ As in the analytical evaluation of multidimensional integrals, in this procedure, successively, the innermost integral is evaluated by keeping the variables corresponding to the other integrals constant. Therefore, we have for a two-dimensional integral,
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$$
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\int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} F (r, s) d r d s = \sum_ {i} \alpha_ {i} \int_ {- 1} ^ {+ 1} F (r _ {i}, s) d s \tag {5.151}
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$$
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or $\int_{-1}^{+1}\int_{-1}^{+1}F(r,s)drds = \sum_{i,j}\alpha_i\alpha_jF(r_i,s_j)$ (5.152)
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and corresponding to (5.133), $\alpha_{ij} = \alpha_i\alpha_j$ , where $\alpha_{i}$ and $\alpha_{j}$ are the integration weights for one-dimensional integration. Similarly, for a three-dimensional integral,
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$$
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\int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} F (r, s, t) d r d s d t = \sum_ {i, j, k} \alpha_ {i} \alpha_ {j} \alpha_ {k} F (r _ {i}, s _ {j}, t _ {k}) \tag {5.153}
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$$
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and $\alpha_{ijk} = \alpha_{i}\alpha_{j}\alpha_{k}$ . We should note that it is not necessary in the numerical integration to use the same quadrature rule in the two or three dimensions; i.e., we can employ different numerical integration schemes in the r, s, and t directions.
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EXAMPLE 5.40: Given that the $(i,j)$ th element of a stiffness matrix K is $\int_{-1}^{+1}\int_{-1}^{+1}r^{2}s^{2}dr ds$ . Evaluate the integral $\int_{-1}^{+1}\int_{-1}^{+1}r^{2}s^{2}dr ds$ using (1) Simpson's rule in both r and s, (2) Gauss quadrature in both r and s, and (3) Gauss quadrature in r and Simpson's rule in s.
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1. Using Simpson's rule, we have
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$$
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\begin{array}{l} \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} r ^ {2} s ^ {2} d r d s = \int_ {- 1} ^ {+ 1} \frac {1}{3} [ (1) (1) + (4) (0) + (1) (1) ] s ^ {2} d s \\ = \int_ {- 1} ^ {+ 1} \frac {2}{3} s ^ {2} d s = \frac {1}{3} \left[ (1) \left(\frac {2}{3}\right) + (4) (0) + (1) \left(\frac {2}{3}\right) \right] = \frac {4}{9} \\ \end{array}
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$$
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2. Using two-point Gauss quadrature, we have
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$$
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\begin{array}{l} \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} r ^ {2} s ^ {2} d r d s = \int_ {- 1} ^ {+ 1} \left[ (1) \left(\frac {1}{\sqrt {3}}\right) ^ {2} + (1) \left(\frac {1}{\sqrt {3}}\right) ^ {2} \right] s ^ {2} d s \\ = \int_ {- 1} ^ {+ 1} \frac {2}{3} s ^ {2} d s = \frac {2}{3} \left[ (1) \left(\frac {1}{\sqrt {3}}\right) ^ {2} + (1) \left(\frac {1}{\sqrt {3}}\right) ^ {2} \right] = \frac {4}{9} \\ \end{array}
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$$
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$^{8}$ This results in much generality of the integration, but for special cases somewhat less costly procedures can be designed (see B. M. Irons [C]).
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3. Finally, using Gauss quadrature in $r$ and Simpson's rule in $s$ , we have
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$$
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\begin{array}{l} \int_ {- 1} ^ {+ 1} \left[ (1) \left(\frac {1}{\sqrt {3}}\right) ^ {2} + (1) \left(\frac {1}{\sqrt {3}}\right) ^ {2} \right] s ^ {2} d s \\ = \int_ {- 1} ^ {+ 1} \frac {2}{3} s ^ {2} d s = \frac {1}{3} \left[ (1) \left(\frac {2}{3}\right) + (4) (0) + (1) \left(\frac {2}{3}\right) \right] = \frac {4}{9} \\ \end{array}
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$$
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We should note that these numerical integrations are exact because both integration schemes, i.e., Simpson's rule and two-point Gauss quadrature, integrate a parabola exactly.
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The above procedure is directly applicable to the evaluation of matrices of quadrilateral elements in which all integration limits are -1 to +1. Hence, in the evaluation of a two-dimensional finite element, the integrations can be carried out for each entry of the stiffness and mass matrices and load vectors as illustrated in Example 5.40. Based on the information given in Table 5.6, some common Gauss quadrature rules for two-dimensional analysis are summarized in Table 5.7.
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Considering next the evaluation of triangular and tetrahedral element matrices, however, the procedure given in Example 5.40 is not applicable directly because now the integration limits involve the variables themselves. A great deal of research has been spent on the development of suitable integration formulas for triangular domains, and here, too, formulas of the Newton-Cotes type (see P. Silvester [A]) and of the Gauss quadrature type are available (see P. C. Hammer, O. J. Marlowe, and A. H. Stroud [A] and G. R. Cowper [A]). As in the integration over quadrilateral domains, the Gauss quadrature rules are in general more efficient because they yield a higher integration accuracy for the same number of evaluations. Table 5.8 lists the integration stations and integration weights of the Gauss integration formulas published by G. R. Cowper [A].
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# 5.5.5 Appropriate Order of Numerical Integration
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In the practical use of the numerical integration procedures presented in the previous section, basically two questions arise, namely, what kind of integration scheme to use, and what order to select. We pointed out that in using the Newton-Cotes formulas, $(n + 1)$ function evaluations are required to integrate without error a polynomial of order n. On the other hand, if Gauss quadrature is used, a polynomial of order $(2n - 1)$ is integrated exactly with only n function evaluations. In each case of course any polynomial of lower order than n and $(2n - 1)$ , respectively, is also integrated exactly.
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In finite element analysis a large number of function evaluations directly increases the cost of analysis, and the use of Gauss quadrature is attractive. However, the Newton-Cotes formulas may be efficient in nonlinear analysis for the reasons discussed in Section 6.8.4.
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Having selected a numerical integration scheme, the order of numerical integration to be used in the evaluation of the various finite element integrals needs to be determined. The choice of the order of numerical integration is important in practice because, first, the cost of analysis increases when a higher-order integration is employed, and second, using a different integration order, the results can be affected by a very large amount. These considerations are particularly important in three-dimensional analysis.
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The matrices to be evaluated by numerical integration are the stiffness matrix K, the mass matrix M, the body force vector $R_{B}$ , the initial stress vector $R_{I}$ , and the surface load
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TABLE 5.7 Gauss numerical integrations over quadrilateral domains
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<table><tr><td>Integration order</td><td>Degree of precision</td><td>Location of integration points</td></tr><tr><td>2×2</td><td>3</td><td><img src="images/102e817203885f9455c333c488c472129e89ee2cf11074b550a8dba8c58b3c82.jpg"/></td></tr><tr><td>3×3</td><td>5</td><td><img src="images/a473a6aff239f029ff7db81583a974c7814e281119c6205b328e0c4315b7615e.jpg"/></td></tr><tr><td>4×4</td><td>7</td><td><img src="images/48a129bf66a3c829430ca3ca4117cbfeed1c46c006c50679b0ef52d1058bae2f.jpg"/></td></tr></table>
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$^{(1)}$ The location of any integration point in the x, y coordinate system is given by: $x_{p} = \Sigma_{i}h_{i}(r_{p}, s_{p})x_{i}$ and $y_{p} = \Sigma_{i}h_{i}(r_{p}, s_{p})y_{i}$ . The integration weights are given in Table 5.6 using (5.152).
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vector $R_{s}$ . In general, the appropriate integration order depends on the matrix that is evaluated and the specific finite element being considered. To demonstrate the important aspects, consider the Gauss numerical integration order required to evaluate the matrices of the continuum and structural elements discussed in Sections 5.3 and 5.4.
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A first observation in the selection of the order of numerical integration is that, in theory, if a high enough order is used, all matrices will be evaluated very accurately. On the other hand, using too low an order of integration, the matrices may be evaluated very inaccurately and, in fact, the problem solution may not be possible. For example, consider an element stiffness matrix. If the order of numerical integration is too low, the matrix can have a larger number of zero eigenvalues than the number of physical rigid body modes. Hence, for a successful solution of the equilibrium equations alone, it would be necessary
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TABLE 5.8 Gauss numerical integrations over triangular domains $\left[\iint F \, dr \, ds = \frac{1}{2} \Sigma w_{i} F(r_{i}, s_{i})\right]$
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<table><tr><td>Integration order</td><td>Degree of precision</td><td>Integration points</td><td>r-coordinates</td><td>s-coordinates</td><td>Weights</td></tr><tr><td>3-point</td><td>2</td><td><img src="images/be55e0ddeafd0947b6a4f56f307bee44256b49f1d1abae6c699621a7c75834a0.jpg"/></td><td> $r_1 = 0.16666666666667$ $r_2 = 0.66666666666667$ $r_3 = r_1$ </td><td> $s_1 = r_1$ $s_2 = r_1$ $s_3 = r_2$ </td><td> $w_1 = 0.3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333</td></tr><tr><td>7-point</td><td>5</td><td><img src="images/2b6b4a520e97c1f4f096055b61966d4f3a7faf8fa1e0a1285aa8bcdeb5186859.jpg"/></td><td>\( r_1 = 0.1012865073235$ $r_2 = 0.7974269853531$ $r_3 = r_1$ $r_4 = 0.4701420641051$ $r_5 = r_4$ $r_6 = 0.0597158717898$ $r_7 = 0.33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333</td><td>\( s_1 = r_1$ $s_2 = r_1$ $s_3 = r_2$ $s_4 = r_6$ $s_5 = r_4$ $s_6 = r_4$ $s_7 = r_7$ </td><td> $w_1 = 0.1259391805448$ $w_2 = w_1$ $w_3 = w_1$ $w_4 = 0.1323941527885$ $w_5 = w_4$ $w_6 = w_4$ $w_7 = 0.225$ </td></tr><tr><td>13-point</td><td>7</td><td><img src="images/ff66ed28b48a9da745baa5e5a2851aef71b889aef9400680a2443c3ae7566d38.jpg"/></td><td> $r_1 = 0.0651301029022$ $r_2 = 0.8697397941956$ $r_3 = r_1$ $r_4 = 0.3128654960049$ $r_5 = 0.6384441885698$ $r_6 = 0.0486903154253$ $r_7 = r_5$ $r_8 = r_4$ $r_9 = r_6$ $r_{10} = 0.2603459660790$ $r_{11} = 0.4793080678419$ $r_{12} = r_{10}$ \( r_{13} = 0.33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333</td><td></td><td></td></tr></table>
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that the deformation modes corresponding to all zero eigenvalues of the element be properly restrained in the assemblage of finite elements because otherwise the structure stiffness matrix would be singular. A simple example is the evaluation of the stiffness matrix of a three-node truss element. If one-point Gauss numerical integration is used, the row and column corresponding to the degree of freedom at the midnode of the element are null vectors, which may result in a structure stiffness matrix that is singular. Therefore, the integration order should in general be higher than a certain limit.
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The integration order required to evaluate a specific element matrix accurately can be determined by studying the order of the function to be integrated. In the case of the stiffness matrix, we need to evaluate
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$$
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\mathbf {K} = \int_ {V} \mathbf {B} ^ {T} \mathbf {C} \mathbf {B} \det \mathbf {J} d V \tag {5.154}
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$$
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where C is a constant material property matrix, B is the strain-displacement matrix in the natural coordinate system r, s, t, det J is the determinant of the Jacobian transforming local (or global) to natural coordinates (see Section 5.3), and the integration is performed over the element volume in the natural coordinate system. The matrix function F to be integrated is, therefore,
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$$
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\mathbf {F} = \mathbf {B} ^ {T} \mathbf {C B} \det \mathbf {J} \tag {5.155}
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$$
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The matrices $\mathbf{J}$ and $\mathbf{B}$ have been defined in Sections 5.3 and 5.4.
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A case for which the order of the variables in F can be evaluated with relative ease arises when the four-node two-dimensional element studied in Example 5.5 is used as a rectangular or parallelogram element. It is instructive to consider this case in detail because the procedure of evaluating the required integration order is displayed clearly.
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EXAMPLE 5.41: Evaluate the required Gauss numerical integration order for the calculation of the stiffness matrix of a four-node displacement-based rectangular element.
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The integration order to be used depends on the order of the variables $r$ and $s$ in $\mathbf{F}$ defined in (5.155). For a rectangular element with sides $2a$ and $2b$ , we can write
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$$
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x = a r; y = b s
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$$
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and consequently the Jacobian matrix $\mathbf{J}$ is
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$$
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\mathbf {J} = \left[ \begin{array}{c c} a & 0 \\ 0 & b \end{array} \right]
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$$
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Since the elements of J are constant, referring to the information given in Example 5.5, the elements of the strain-displacement matrix B are therefore functions of r or s only. But the determinant of J is also constant; hence,
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$$
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\mathbf {F} = f (r ^ {2}, r s, s ^ {2})
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$$
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where $f$ denotes "function of."
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Using two-point Gauss numerical integration in the r and s directions, all functions in r and s involving at most cubic terms are integrated without error; e.g., for integration order n, the order of r and s integrated exactly is $(2n - 1)$ . Hence, two-point Gauss integration is adequate.
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Note that the Jacobian matrix J is also constant for a four-node parallelogram element; hence, the same derivation and result are applicable.
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In an analogous manner, the required integration order to evaluate exactly (or very accurately) the stiffness matrices, mass matrices, and element load vectors of other elements can be assessed. In this context it should be noted that the Jacobian matrix is not constant for nonrectangular and nonparallelogram element shapes, which may mean that a very high integration order might be required to evaluate the element matrices to high accuracy.
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In the above example, a displacement-based element was considered, but we should emphasize that, of course, the same numerical integration schemes are also used in the evaluation of the element matrices of mixed formulations. Hence, in mixed formulations the required integration order must also be identified using the procedure just discussed (see Exercise 5.57).
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In studying which integration order to use for geometrically distorted elements, we recognize that it is frequently not necessary to calculate the matrices to very high precision using a very high order of numerical integration. Namely, the change in the matrix entries (and their effects) due to using an order of l instead of $(l - 1)$ may be negligible. Hence, we need to ask what order of integration is generally sufficient, and we present the following guideline.
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We recommend that full numerical integration $^{9}$ always be used for a displacement-based or mixed finite element formulation, where we define “full” numerical integration as the order that gives the exact matrices (i.e., the analytically integrated values) when the elements are geometrically undistorted. Table 5.9 lists this order for elements used in two-dimensional analyses.
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Using this integration order for a geometrically distorted element will not yield the exactly integrated element matrices. The analysis is, however, reliable because the numerical integration errors are acceptably small assuming of course reasonable geometric distortions. Indeed, as shown by P. G. Ciarlet [A], if the geometric distortions are not excessive and are such that in exact integration the full order of convergence is still obtained (with the provisions discussed in Section 5.3.3), then that same order of convergence is also obtained using the full numerical integration recommended here. Hence, in that case, the order of numerical integration recommended in Table 5.9 does not result in a reduction of the order of convergence. On the other hand, if the element geometric distortions are very large, and in nonlinear analysis of course, a higher integration order may be appropriate (see Section 6.8.4).
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Figure 5.39 shows some results obtained in the solution of the ad hoc test problem described in Fig. 4.12. These results were obtained using sequences of distorted, quasi-uniform meshes. Figure 5.39(a) describes the geometric distortions used, and Fig. 5.39(b) and (c) show the convergence results obtained with the eight-node and nine-node elements using the Gauss integration order in Table 5.9. These results show that the order of convergence (the slopes of the graphed curves when h is small) is approximately 4 in all cases (as is theoretically predicted). However, the actual value of the error for a given value of h is larger when the elements are distorted. That is, the constant c in (4.102) increases as the elements are distorted.
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The reason for recommending the numerical integration orders in Table 5.9 is that the reliability of the finite element procedures is of utmost concern (see Section 1.3), and if an
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TABLE 5.9 Recommended full Gauss numerical integration orders for the evaluation of isoparametric displacement-based element matrices (use of Table 5.7)
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<table><tr><td colspan="2">Two-dimensional elements (plane stress, plane strain and axisymmetric conditions)</td><td>Integration order</td></tr><tr><td>4-node</td><td><img src="images/3d94a87776063c72bf40bb27c17df7b51ace1c11ca28a63cc7c5962b4dd80580.jpg"/></td><td>2 × 2</td></tr><tr><td>4-node distorted</td><td><img src="images/abd39431b9b1c4ed3d7bc870a020c11fcb32d7821a036cabdfdc03f384c9f3db.jpg"/></td><td>2 × 2</td></tr><tr><td>8-node</td><td><img src="images/cd5d5b13fcb6245db9e201197cf747a92f2d9f2307c9f192262582bf67e61fd9.jpg"/></td><td>3 × 3</td></tr><tr><td>8-node distorted</td><td><img src="images/1b6cc50dca99ab1beca3b376eacce2228670813b64e4e0b82b0f40901c999604.jpg"/></td><td>3 × 3</td></tr><tr><td>9-node</td><td><img src="images/f94e9dce1323d7eb4f9bbebf875195e3f74d8140609e7d0de3d7f4062a537cc1.jpg"/></td><td>3 × 3</td></tr><tr><td>9-node distorted</td><td><img src="images/ad216dba3035a70423c84bd9872b45875ddeb5f9c6e408d541eae4d38113b1d5.jpg"/></td><td>3 × 3</td></tr><tr><td>16-node</td><td><img src="images/def90c2fda9eb50138b29fbc46a72c01c40160c632cde229d7640c24390200ae.jpg"/></td><td>4 × 4</td></tr><tr><td>16-node distorted</td><td><img src="images/56192a1956886a186d409d2a6ed36f848fa7648c2dfd128a1de7abafa121fea9.jpg"/></td><td>4 × 4</td></tr></table>
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(Note: In axisymmetric analysis, the hoop strain effect is in all cases not integrated exactly, but with sufficient accuracy.)
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<details>
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<summary>text_image</summary>
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0.25
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1.0
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A
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C
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2.0
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B
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O
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B
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0.25
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A
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2.0
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</details>
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Case A
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<details>
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<summary>text_image</summary>
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1.0
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0.5
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A
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C
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B
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0.5
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2.0
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B
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O
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A
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2.0
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</details>
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Case B
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<details>
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<summary>text_image</summary>
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1.0
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0.75
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A
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C
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B
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0.75
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2.0
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B
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O
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A
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2.0
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</details>
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Case C
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<details>
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<summary>text_image</summary>
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Element sides of
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equal length = 0.50/4
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C
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B
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O
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B
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A
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</details>
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$8 \times 8$ mesh, case $B, h = 2/8$
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The lines AA and BB are drawn, and then the sides AC, CB, BO, OA are subdivided into equal lengths to form the elements in the domain ACBO. Similarly for the other three domains.
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(a) Distortions used
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<details>
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<summary>line</summary>
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| log10 h | log10(E - E_h) for Case A | log10(E - E_h) for Case B | log10(E - E_h) for Case C |
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| ------- | -------------------------- | -------------------------- | -------------------------- |
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| -2 | 0 | 0 | 0 |
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| -1 | 3 | 2.5 | 2.5 |
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| 0 | 5 | 5 | 5 |
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</details>
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(b) Results using 8-node elements
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<details>
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<summary>line</summary>
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| log10 h | log10(E - E_h) - Case A | log10(E - E_h) - Case B | log10(E - E_h) - Case C | log10(E - E_h) - Nondistorted elements |
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| ------- | ------------------------ | ------------------------ | ------------------------ | -------------------------------------- |
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| -2 | ~0.5 | ~0.5 | ~0.5 | ~0.5 |
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| -1 | ~2.5 | ~2.5 | ~2.5 | ~2.5 |
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| 0 | ~5.0 | ~5.0 | ~5.0 | ~5.0 |
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</details>
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(c) Results using 9-node elements
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Figure 5.39 Solution of test problem of Fig. 4.12 with geometrically distorted elements, and the Gauss integration order of Table 5.9. $E = a(\mathbf{u}, \mathbf{u})$ , $E_{h} = a(\mathbf{u}_{h}, \mathbf{u}_{h})$
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integration order lower than the “full” order is used (for a displacement-based or a mixed formulation), the analysis is in general unreliable.
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An interesting case is the rectangular two-dimensional plane stress eight-node displacement-based isoparametric element evaluated with $2 \times 2$ Gauss integration. This integration order yields an element stiffness matrix with one spurious zero energy mode (see Exercise 5.56); that is, the element matrix not only has three zero eigenvalues (corresponding to the physical rigid body motions) but also has one additional zero eigenvalue that is purely a result of using too low an order of integration. Figure 5.40 shows a very simple analysis case using a single eight-node element with $2 \times 2$ Gauss integration in which the model is unstable; that is, if the solution is obtained, the calculated nodal point displacements are very large and have no resemblance to the correct solution. $^{10}$ In this simple analysis it is readily seen that the eight-node element using $2 \times 2$ Gauss integration is inadequate, and it can be argued that in more complex analysis the (single) spurious zero energy mode is usually adequately restrained in an assemblage of elements. However, in a large, complex model, in general, elements with spurious zero energy modes in an uncontrolled manner improve the overall solution results, introduce large errors, or result in an unstable solution.
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P
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A
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B
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Figure 5.40 Eight-node plane stress element supported at B by a spring. Analysis unstable with $2 \times 2$ Gauss integration.
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As an example, let us consider the dynamic analysis of the cantilever bracket shown in Fig. 5.41 and use the nine-node displacement-based element with $2 \times 2$ Gauss integration, in which case each element stiffness matrix has three spurious zero energy modes. We have considered this bracket already in Fig. 4.20, but with two pin supports instead of the fixed condition used now. (As noted there, the 16-element model of the pin-supported bracket using $2 \times 2$ Gauss integration for the element stiffness matrices was unstable). The frequency solution of the 16-element mesh of nine-node displacement-based elements representing the clamped cantilever bracket gives the results listed in Table 5.10. This table shows that the use of $2 \times 2$ Gauss integration (referred to as reduced integration; see Section 5.5.6) does not result in a spurious zero energy mode of the complete model (because the bracket is clamped at its left end) but in one spurious nonzero energy mode that is part of the predicted smallest six frequencies. Such modes of no physical reality—which we refer to also as “phantom” modes—can introduce uncontrolled errors into a
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10 mm
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5 mm
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E = 55 N/mm²
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v = 0.3
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ρ = 1.3 × 10⁻⁹ N·sec² / mm⁴
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(a) Geometry and material data
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Pure geometric diagram of a T-shaped structure with dots and lines, no text or symbols present
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(b) Sixteen element mesh of 9-node elements
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Figure 5.41 Frequency solution of clamped cantilever bracket
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dynamic step-by-step solution $^{11}$ that may not be easily detectable, and even if these errors are detected, the analysis would require additional solution attempts all of which may result in extensive and undesirable numerical experimentation.
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For these reasons any element with a spurious zero energy mode should not be used in engineering practice, in linear or in nonlinear analysis, and we therefore do not discuss such elements in this book. However, we should mention that to prevent the deleterious effects of spurious modes, significant research efforts have been conducted to control their
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