김경종
262 lines
31 KiB
Markdown
262 lines
31 KiB
Markdown
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which are orthonormal with respect to [B] in the sense of Eq. C.3-5b. Note that [L] generalizes the diagonal transformation matrix [T] of Eq. C.2-2 to the case in which [B] need not be diagonal. When [A] is positive definite, all $\lambda_{i} > 0$ . When [A] is positive semidefinite, $\lambda_{i} \geq 0$ . In structural problems, the $\lambda_{i} = 0$ are associated with rigid-body motions or mechanisms.
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Either ad hoc or optimal mass lumping may lead to a problem in which [A] is positive semidefinite and [B] is diagonal with some $B_{ii} = 0$ (i.e., some massless d.o.f.). Consideration of the generalized Rayleigh quotient (see Eqs. 13.5-4 and 13.5-5),
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$$
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\lambda_ {\min} \leq \frac {\{\mathbf {x} \} ^ {T} [ \mathbf {A} ] \{\mathbf {x} \}}{\{\mathbf {x} \} ^ {T} [ \mathbf {B} ] \{\mathbf {x} \}} \leq \lambda_ {\max} \tag {C.3-9}
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$$
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usually indicates that there are as many infinite eigenvalues, $\lambda_{\max} = \infty$ , as there are $B_{ii} = 0$ . It is possible to construct pathological forms of [A] and [B] that violate this rule. Pathological forms will not arise in finite element analysis if rigid-body motions (or mechanisms) are associated with positive kinetic energy [C.2]. It is up to the analyst to ensure that this condition prevails. If there are no pathological forms, [A] is positive semidefinite, and there are $m$ massless d.o.f. ( $B_{ii} = 0$ ), then there are $m$ infinite eigenvalues and $n - m$ finite eigenvalues associated with $n - m$ independent eigenvectors $\{\mathbf{x}\}$ . Thus $q(\lambda)$ has degree $n - m$ . The infinite eigenvalues are associated with $m$ independent eigenvectors $\{\mathbf{x}\}$ such that $[\mathbf{B}]\{\mathbf{x}\} = \{\mathbf{0}\}$ .
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Infinite eigenvalues can be observed in a more general context by considering the finite eigenvalues of the inverse eigenproblem, whose pencil is
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$$
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[ \mathbf {B} ] - \mu [ \mathbf {A} ] \tag {C.3-10}
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$$
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Many of the following statements are also true in a more general context, but if we limit our attention to the diagonalizable case, it is easy to see that the correspondence to the pencil of Eq. C.3-1 is that
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$$
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\frac {1}{\mu_ {i}} = \lambda_ {i} \tag {C.3-11}
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$$
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where $\mu_{i}=0$ corresponds to $\lambda_{i}=\infty$ . The complete set of eigenvectors is the same for both pencils. Equation C.3-11 defines an infinite eigenvalue even when [B] is nondiagonal. The pencils of Eqs. C.3-1 and C.3-10 may not be diagonalizable in general. When one is, the other clearly will be also, and this will be true whenever
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(a) Either [A] or [B] is positive definite.
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(b) [Å] and [B] are positive semidefinite and there are no ill-disposed vectors (i.e. vectors such that [A]{x} = {0} and [B]{x} = {0}).
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(c) [A] is semidefinite and [B] is an indefinite matrix arising from an optimal lumping formula, which leads to no indeterminate vectors [C.2] (i.e. vectors such that $\{\mathbf{x}\}^T [\mathbf{A}]\{\mathbf{x}\} = 0$ and $\{\mathbf{x}\}^T [\mathbf{B}]\{\mathbf{x}\} = 0$ ).
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Case (c) arises in optimal lumping in which some masses (quadrature weights) are negative, as explained in Section 13.3. No known optimal lumping scheme leads to indeterminate vectors. It should be noted that ill-disposed vectors are a special case of indeterminate vectors, and that in pathological cases indeterminate vectors are possible even without massless d.o.f., because an indefinite [B] always has vectors $\{\mathbf{x}\}$ such that $\{\mathbf{x}\}^7 [\mathbf{B}]\{\mathbf{x}\} = 0$ . Pencils that fall outside of restrictions (a) through (c) can be highly pathological but are almost never observed in finite element practice.
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In the most general diagonalizable cases—that is, (a) through (c) with [B] singular—we must revise our notion of orthogonality of [Q]. Simultaneous diagonalization of Eq. C.3-1 is possible, and
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$$
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[ \mathbf {Q} ] ^ {T} [ \mathbf {A} ] [ \mathbf {Q} ] = [ \Lambda^ {+} ] \tag {C.3-12a}
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$$
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$$
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[ \mathbf {Q} ] ^ {T} [ \mathbf {B} ] [ \mathbf {Q} ] = [ \Omega ] \tag {C.3-12b}
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$$
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where $\Lambda_{ii}^{+} = \lambda_{i}$ for $0 \leq \lambda_{i} < \infty$ , $\Lambda_{ii}^{+} = 1$ for $\lambda_{i} = \infty$ . In case (c), $\lambda_{i} \leq 0$ with $|\lambda_{i}|$ “very large” are introduced for each negative mass [C.1] and $\Lambda_{ii}^{+} = |\lambda_{i}|$ for $\lambda_{i} < 0$ . $[\Omega]$ is also diagonal, and $\Omega_{ii} = 1$ for $0 \leq \lambda_{i} < \infty$ , $\Omega_{ii} = 0$ for $\lambda_{i} = \infty$ , and $\Omega_{ii} = -1$ for $\lambda_{i} < 0$ . In case (c), the negative signs can only be transferred from $[\Omega]$ to $[\Lambda^{+}]$ by a transformation of the pencil of the form of Eqs. C.3-4 in which $[R] \neq [Q]^{T}$ . The zeros on the diagonal of $[\Omega]$ cannot be avoided. Thus the eigenvectors with $0 \leq \lambda_{i} < \infty$ are orthonormal in the usual sense. The other eigenvectors can have “negative lengths” or be “orthogonal to themselves.” Their physical meaning has been sacrificed in such cases.
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# C.4 REMARKS ON SPECIAL FORMS
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We consider the general eigenproblem,
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$$
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([ \mathbf {A} ] - \lambda [ \mathbf {B} ]) \{\mathbf {x} \} = \{\mathbf {0} \} \quad \text { or } \quad ([ \mathbf {K} ] - \omega^ {2} [ \mathbf {M} ]) \{\mathbf {D} \} = \{\mathbf {0} \} \tag {C.4-1}
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$$
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where the latter form uses the symbolism adopted for vibration problems in Section 13.5. In what follows we assume that [A] and [B] are symmetric, and consider how to proceed under circumstances that may arise in finite element modeling.
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If [A] and [B] are positive definite, there is no special difficulty. One may proceed directly to solve the eigenproblem (see Section C.5).
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[B] will be positive semidefinite and diagonal if “lumping” is used with $B_{ii} = 0$ for some d.o.f. $x_{i}$ ; for example, if mass particles have no rotary inertia or are not attached to all nodes (this is not a recommended practice). One can eliminate the $x_{i}$ for which $B_{ii} = 0$ in a diagonal [B]. The procedure is discussed in Section 8.1. In the condensed system, matrices are of lower order than initially and [B] is positive definite. In the case where $B_{ii} = 0$ because of optimal lumping, condensation is not recommended, since it will lose the accuracy the scheme is designed to retain. Several algorithms discussed in the next section and in Ref. C.1 are either not affected if [B] fails to be positive definite or can be easily modified to work in this case.
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In the structural context, a positive semidefinite [A] implies an unsupported structure such as a spacecraft. This circumstance presents no obstacle to some algorithms for solving the eigenproblem. Other algorithms may require that [A] be nonsingular. This restriction can be met by using an eigenvalue shift. Thus we select a scalar $c$ and substitute $\lambda = \lambda' - c$ into Eq. C.1-1, to obtain
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$$
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([ \mathbf {A} + c \mathbf {B} ] - \lambda^ {\prime} [ \mathbf {B} ]) \{\mathbf {x} \} = \{\mathbf {0} \} \tag {C.4-2}
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$$
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in which $[A + cB]$ is positive definite if $\{x\}_{i}^{T}[B]\{x\}_{i} > 0$ for every $\{x\}_{i}$ for which $\{x\}_{i}^{T}[A]\{x\}_{i} = 0$ . We solve for shifted eigenvalues $\lambda_{i}^{\prime}$ , then obtain actual eigenvalues $\lambda_{i} = \lambda_{i}^{\prime} - c$ . Eigenvectors are not changed by shifting. If $|c|$ is too small, $[A + cB]$ is almost singular. If $|c|$ is too large, convergence of the eigensolver may be slow. A possible choice is c = 0.01r, where r is the ratio of the trace of [A] to the trace of [B]. Surprisingly, perhaps, shifting usually yields a positive definite $[A + cB]$ even when [B] is indefinite, as when [B] arises from an optimal lumping scheme [C.1]. If the problem is structural and [B] is diagonal, one can interpret Eq. C.4-2 physically by saying that a spring of stiffness $cB_{ii}$ has been added between each d.o.f. $x_{i}$ and ground, thus preventing any rigid-body motion.
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# C.5 SOLUTION ALGORITHMS
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Algorithms for eigenvalue extraction are plentiful. The best choice for a particular eigenproblem depends on the order n of the matrices, their sparsity, the number of eigenvalues to be extracted, and where in the eigenspectrum the eigenvalues of interest are located. In what follows we indicate some choices commonly made in practice, but make little or no attempt to explain the workings of various algorithms. These details may be found in references such as $[C.4-C.12]$ . At the outset we note that the various methods are often used in combination rather than as stand-alone algorithms.
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Writing out the characteristic polynomial, $q(\lambda) = \det[A - \lambda B] = 0$ , and calculating its roots is a method suitable only for hand calculation, for example, when there are only one or two roots. If n is large, the method is costly and inaccurate.
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If all $n$ eigenvalues are required and $n$ is relatively small (roughly $n < 200$ ), the Jacobi method is a good choice.
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If [A] and [B] are narrowly banded and only a few eigenvalues are required (e.g., only $\lambda_{1}$ and $\lambda_{2}$ ), determinant search may be appropriate. Effectively, one seeks zeros of the characteristic polynomial by repeated factoring of [A - $\lambda$ B] for trial values of $\lambda$ . To ensure that no root is skipped, one can use the Sturm sequence property. $^{1}$
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The Sturm sequence property is also useful in calculating the eigenvalues that appear in a prescribed range. The associated eigenvectors must be calculated by a different method. A slightly modified Sturm sequence property holds for indefinite [B] resulting from optimal lumping [C.1].
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The inverse iteration or inverse power method computes the lowest eigenvalue. Or, when used with an eigenvalue shift, it computes the eigenvalue closest to the “shift point.” The associated eigenvector is automatically computed as part of the process. The inverse power method does not require a positive definite [B].
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The subspace iteration method uses k trial vectors that are iteratively improved as calculation proceeds. If m is the number of eigenvalues required, k may be taken as the smallest of the three numbers 2m, $m + 8$ , and n. Typically, k << n. There is a similarity between subspace iteration and mass condensation (Section 13.7).
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Householder reduction and Givens reduction transform a standard or general eigenproblem into one in which the symmetric [A] and [B] are tridiagonal, which then can be easily solved by one of the other methods cited here. The transformation involved can be chosen to respect band storage format.
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The Lanczos method is probably the most efficient algorithm available. Early difficulties with the method have been overcome. It is usually not available as a library package at computer installations, but appears destined to supplant other eigenproblem algorithms among the finite element community [C.5]. Like the Householder and Givens methods, the Lanczos method is a tridiagonal transformation method. It can be arranged so that it respects band or skyline/profile storage format. The resulting tridiagonal system is easily solved by other means.
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The QR and QZ methods can solve a general eigenproblem but are particularly suited to extracting all eigenvalues and eigenvectors of a tridiagonal system resulting from Givens, Householder, or Lanczos transformations. The QZ method is complicated, but is the only known method capable of handling cases where [A] and [B] are singular and do not satisfy restrictions (a) through (c) of Section C.3. Thus the QZ method is useful for testing small systems whose properties are unknown a priori.
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$^{1}$ We can use Subroutine SOLVER, Fig. B.2-3, to exploit the Sturm sequence property. Thus we select a numerical value of $\lambda$ , place [A] - $\lambda$ [B] in array A of Fig. B.2-3, and complete the loop on statement 790. At this time the number of eigenvalues exceeded by the selected $\lambda$ is equal to the number of negative diagonal coefficients (which appear in column 1 of array A in Fig. B.2-3).
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Standard library packages (e.g., EISPACK, IMSL, NAg; Refs. C.10–C.12) provide several eigensolvers, usually tailored to exploit matrix properties such as symmetry or bandedness. The storage formats used, particularly in the more specialized algorithms, often disagree with storage formats for [A] and [B] in the program that generates these arrays.
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An explanation of many of the methods described here, aimed at an engineering audience, can be found in Chapter 5 of [C.7]. A guide to obtaining software from EISPACK, IMSL, NAg, and other packages is given in the Appendix of [C.7].
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# REFERENCES
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# CHAPTER 1
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1.1 J. Ergatoudis, B. M. Irons, and O. C. Zienkiewicz, "Three-Dimensional Analysis of Arch Dams and Their Foundations," Research Report No. C/R/74/67, University of Wales, Swansea, 1968.
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1.2 K. Wieghardt, "Über einen Grenzübergang der Elastizitätslehre und seine Andwendung auf die Statik hochgradig statisch unbestimmter Fachwerke," Verhandlungen des Vereins z. Beförderung des Gewerbefleisses, Abhandlungen, Vol. 85, 1906, pp. 139–176.
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1.3 W. Riedel, “Beiträge zur Lösung des ebenen Problems eines elastichen Körpers mittels der Airyschen Spannungsfunktion,” Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 7, No. 3, 1927, pp. 169–188.
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1.4 A. Hrennikoff, "Solution of Problems in Elasticity by the Framework Method," J. Appl. Mech., Vol. 8, No. 4, 1941, pp. A169-A175.
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1.5 R. Courant, “Variational Methods for the Solution of Problems of Equilibrium and Vibrations,” Bulletin of the American Mathematical Society, Vol. 49, 1943, pp. 1–23.
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1.6 S. Levy, “Structural Analysis and Influence Coefficients for Delta Wings,” J. Aero. Sci., Vol. 20, No. 7, 1953, pp. 449–454.
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1.7 R. W. Clough, "The Finite Element Method After Twenty-Five Years: A Personal View," Computers & Structures, Vol. 12, No. 4, 1980, pp. 361–370.
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1.8 M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp, “Stiffness and Deflection Analysis of Complex Structures,” J. Aero. Sci., Vol. 23, No. 9, 1956, pp. 805–823.
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1.9 J. H. Argyris and S. Kelsey, Energy Theorems and Structural Analysis, Butterworths, London, 1960 (collection of papers published in Aircraft Engineering in 1954 and 1955).
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1.10 J. Robinson, Early FEM Pioneers, Robinson & Associates, Dorset, England, 1985.
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# CHAPTER 2
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2.1 K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982.
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2.2 G. C. Everstine, “A Comparison of Three Resequencing Algorithms for the Reduction of Matrix Profile and Wavefront,” Int. J. Num. Meth. Engng., Vol. 14, No. 6, 1979, pp. 837–853.
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2.3 S. W. Sloan, “An Algorithm for Profile and Wavefront Reduction of Sparse Matrices,” Int. J. Num. Meth. Engng., Vol. 23, No. 2, 1986, pp. 239–251.
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# CHAPTER 3
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3.1 H. L. Langhaar, Energy Methods in Applied Mechanics, John Wiley & Sons, New York, 1962.
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3.2 K. Washizu, Variational Methods in Elasticity and Plasticity, 3rd Ed., Pergamon Press, Oxford, England, 1982.
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3.3 P. Tong, "Exact Solution of Certain Problems by the Finite Element Method," AIAA Jnl., Vol. 7, No. 1, 1969, pp. 178-180.
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3.4 A. A. Ball, “The Interpolation Function of a General Serendipity Rectangular Element,” Int. J. Num. Meth. Engng., Vol. 15, No. 5, 1980, pp. 773–778.
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# CHAPTER 4
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4.1 R. J. Melosh, "Basis for Derivation of Matrices for the Direct Stiffness Method," AIAA Jnl., Vol. 1, No. 7, 1963, pp. 1631-1637.
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4.2 R. Narayanaswami and H. M. Adelman, "Inclusion of Transverse Shear Deformation in Finite Element Displacement Formulations," AIAA Jnl., Vol. 12, No. 11, 1974, pp. 1613–1614 (discussion: Vol. 13, No. 9, pp. 1253–1254; application to plates: Vol. 12, No. 12, pp. 1761–1763).
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4.3 R. J. Melosh and D. W. Lobitz, "On a Numerical Sufficiency Test for Monotonic Convergence of Finite Element Models," AIAA Jnl., Vol. 13, No. 5, 1975, pp. 675–678.
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4.4 W. E. Haisler and J. A. Stricklin, "Rigid-Body Displacements of Curved Elements in the Analysis of Shells by the Matrix Displacement Method," AIAA Jnl., Vol. 5, No. 8, 1967, pp. 1525-1527.
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4.5 G. P. Bazeley, Y. K. Cheung, B. M. Irons, and O. C. Zienkiewicz, “Triangular Elements in Plate Bending—Conforming and Nonconforming Solutions,” Proc. First Conf. on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, 1965 (AFFDL-TR-66-80, Nov. 1966; AD-646-300, N.T.I.S.), pp. 547–576.
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4.6 R. L. Taylor, J. C. Simo, O. C. Zienkiewicz, and A. C. H. Chan, "The Patch Test—A Condition for Assessing FEM Convergence," Int. J. Num. Meth. Engng., Vol. 22, No. 1, 1986, pp. 39–62.
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4.7 I. U. Ojalvo, “Improved Thermal Stress Determination by Finite Element Methods,” AIAA Jnl., Vol. 12, No. 8, 1974, pp. 1131–1132.
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4.8 J. Pittr and H. Hartl, "Improved Stress Evaluation Under Thermal Load for Simple Finite Elements," Int. J. Num. Meth. Engng., Vol. 15, No. 10, 1980, pp. 1507-1515.
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4.9 G. Loubignac, G. Cantin, and G. Touzot, "Continuous Stress Fields in Finite Element Analysis," AIAA Jnl., Vol. 15, No. 11, 1977, pp. 1645–1647.
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4.10 R. D. Cook and X. Huang, "Continuous Stress Fields by the Finite Element-Difference Method," Int. J. Num. Meth. Engng., Vol. 22, No. 1, 1986, pp. 229–240.
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4.11 R. T. Severn, "Inclusion of Shear Deflection in the Stiffness Matrix for a Beam Element," J. of Strain Analysis, Vol. 5, No. 4, 1970, pp. 239-241.
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4.12 R. E. Cornwell and R. D. Cook, "Improvement in Peak Stress Estimates Through Post-Processing," Finite Elements in Analysis and Design (to appear).
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4.13 I. C. Taig, "Finite Element Analysis in Industry—Expertise or Proficiency?," in Accuracy, Reliability, and Training in FEM Technology, J. Robinson, ed., Pitman Press, England, 1984.
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# CHAPTER 5
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5.1 K. Bell, "A Refined Triangular Plate Bending Finite Element," Int. J. Num. Meth. Engrg., Vol. 1, No. 1, 1969, pp. 101-122.
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5.2 N. M. Ferrers, An Elementary Treatise on Trilinear Coordinates, the Method of Reciprocal Polars and the Theory of Projections, Macmillan, London, 1861.
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5.3 J. B. Mertie, "Transformation of Trilinear and Quadriplanar Coordinates to and from Cartesian Coordinates," The American Mineralogist, Vol. 49, Nos. 7/8, 1964, pp. 926–936.
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5.4 R. H. Ghallagher, Finite Element Analysis: Fundamentals, Prentice-Hall, Englewood Cliffs, NJ, 1975.
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5.5 C. A. Felippa, "Refined Finite Element Analysis of Linear and Nonlinear Two-Dimensional Structures," Ph.D. dissertation, University of California, Berkeley, 1966 (also available as PB-178-418 and PB-178-419, N.T.I.S.; computer programs in the latter).
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5.6 G. Subramanian and C. Jayachandrabose, “Convenient Generation of Stiffness Matrices for the Family of Plane Triangular Elements,” Computers & Structures, Vol. 15, No. 1, 1982, pp. 85–89.
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# CHAPTER 6
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6.1 B. M. Irons, "Engineering Applications of Numerical Integration in Stiffness Methods," AIAA Jnl., Vol. 4, No. 11, 1966, pp. 2035-2037.
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6.2 A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, NJ, 1966.
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6.3 S. W. Sloan, “A Fast Stiffness Formulation for Finite Element Analysis of Two-Dimensional Solids,” Int. J. Num. Meth. Engng., Vol. 17, No. 9, 1981, pp. 1313–1323.
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6.4 A. K. Gupta, "Efficient Numerical Integration of Element Stiffness Matrices," Int. J. Num. Meth. Engng., Vol. 19, No. 9, 1983, pp. 1410-1413.
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6.5 D. A. Dunavant, "High Degree Efficient Symmetrical Gaussian Quadrature Rules for the Triangle," Int. J. Num. Meth. Engng., Vol. 21, No. 6, 1985, pp. 1129–1148.
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6.6 A. K. Noor and C. M. Anderson, "Computerized Symbolic Manipulation in Nonlinear Finite Element Analysis," Computers & Structures, Vol. 13, Nos. 1-3, 1981, pp. 379-403.
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6.7 I. Ergatoudis, B. M. Irons, and O. C. Zienkiewicz, "Curved Isoparametric, 'Quadrilateral' Elements for Finite Element Analysis," Int. J. Solids Structures, Vol. 4, No. 1, 1968, pp. 31–42.
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6.8 P. C. Hammer and A. H. Stroud, "Numerical Evaluation of Multiple Integrals II," Math. Tables and Other Aids to Comp., Vol. 12, No. 64, 1958, pp. 272-280.
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6.9 B. M. Irons, "Quadrature Rules for Brick Based Finite Elements," Int. J. Num. Meth. Engng., Vol. 3, No. 2, 1971, pp. 293-294.
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6.10 T. K. Hellen, "Effective Quadrature Rules for Quadratic Solid Isoparametric Finite Elements," Int. J. Num. Meth. Engng., Vol. 4, No. 4, 1972, pp. 597–599.
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6.11 B. Verhegghe and G. H. Powell, "Control of Zero-Energy Modes in 9-Node Plane Element," Int. J. Num. Meth. Engng., Vol. 23, No. 5, 1986, pp. 863–869.
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6.12 R. D. Cook and Z. H. Feng, “Control of Spurious Modes in the Nine-Node Quadrilateral Element,” Int. J. Num. Meth. Engng., Vol. 18, No. 10, 1982, pp. 1576–1580.
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6.13 W. K. Liu, J. S. J. Ong, and R. A. Uras, "Finite Element Stabilization Matrices: A Unification Approach," Comp. Meth. Appl. Mech. Engng., Vol. 53, No. 1, 1985, pp. 13–46.
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6.14 E. Hinton and J. S. Campbell, "Local and Global Smoothing of Discontinuous Finite Element Functions Using a Least Squares Method," Int. J. Num. Meth. Engng., Vol. 8, No. 3, 1974, pp. 461-480.
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6.15 J. Barlow, "Optimal Stress Locations in Finite Element Models," Int. J. Num. Meth. Engng., Vol. 10, No. 2, 1976, pp. 243-251 (discussion: Vol. 11, No. 3, p. 604).
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6.16 A. Peano, "Inadmissible Distortion of Solid Elements and Patch Test Results," Comm. in Appl. Num. Meth., Vol. 3, No. 2, 1987, pp. 97-101.
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# CHAPTER 7
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7.1 A. K. Gupta and P. S. Ma, "Error in Eccentric Beam Formulation," Int. J. Num. Meth. Engng., Vol. 11, No. 9, 1977, pp. 1473-1477.
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7.2 R. E. Miller, "Reduction of the Error in Eccentric Beam Modelling," Int. J. Num. Meth. Engng., Vol. 15, No. 4, 1980, pp. 575-582.
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# CHAPTER 8
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8.1 E. L. Wilson, "The Static Condensation Algorithm," Int. J. Num. Meth. Engng., Vol. 8, No. 1, 1974, pp. 198–203.
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8.2 R. D. Cook and V. N. Shah, “A Cost Comparison of Two Static Condensation-Stress Recovery Algorithms,” Int. J. Num. Meth. Engng., Vol. 12, No. 4, 1978, pp. 581–588.
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8.3 R. L. Taylor, P. J. Beresford, and E. L. Wilson, “A Non-Conforming Element for Stress Analysis,” Int. J. Num. Meth. Engng., Vol. 10, No. 6, 1976, pp. 1211–1219.
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8.4 M. Fröier, L. Nilsson, and A. Samuelsson, “The Rectangular Plane Stress Element by Turner, Pian and Wilson,” Int. J. Num. Meth. Engng., Vol. 8, No. 2, 1974, pp. 433–437.
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8.5 D. J. Allman, "A Compatible Triangular Element Including Vertex Rotations for Plane Elasticity Analysis," Computers & Structures, Vol. 19, No. 1-2, 1984, pp. 1-8.
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8.6 R. D. Cook, "On the Allman Triangle and a Related Quadrilateral Element," Computers & Structures, Vol. 22, No. 6, 1986, pp. 1065-1067.
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8.7 P. Tong and T. H. H. Pian, “A Variational Principle and the Convergence of a Finite-Element Method Based on Assumed Stress Distribution,” Int. J. Solids Structures, Vol. 5, No. 5, 1969, pp. 463–472.
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8.8 T. H. H. Pian, “Derivation of Element Stiffness Matrices by Assumed Stress Functions,” AIAA Jnl., Vol. 2, No. 7, 1964, pp. 1333–1336 (discussion: Vol. 3, No. 1, 1965, pp. 186–187).
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8.9 J. P. Wolf, "Alternate Hybrid Stress Finite Element Models," Int. J. Num. Meth. Engng., Vol. 9, No. 3, 1975, pp. 601-615.
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8.10 T. H. H. Pian and K. Sumihara, "Rational Approach for Assumed Stress Finite Elements," Int. J. Num. Meth. Engng., Vol. 20, No. 9, 1984, pp. 1685–1695.
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8.11 R. D. Cook, “A Plane Hybrid Element with Rotational D.O.F. and Adjustable Stiffness,” Int. J. Num. Meth. Engng., Vol. 24, No. 8, 1987, pp. 1499–1508.
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8.12 M. F. Kanninen and C. H. Popelar, Advanced Fracture Mechanics, Oxford University Press, New York, 1985.
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8.13 D. P. Rooke and D. J. Cartwright, Compendium of Stress Intensity Factors, Her Majesty's Stationary Office, London, 1976.
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8.14 R. S. Barsoum, "On the Use of Isoparametric Finite Elements in Linear Fracture Mechanics," Int. J. Num. Meth. Engng., Vol. 10, No. 1, 1976, pp. 25-37.
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8.15 R. S. Barsoum, "Letter to the Editor," Int. J. Num. Meth. Engng., Vol. 18, No. 9, 1982, pp. 1420–1422.
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8.16 L. P. Harrop, "The Optimum Size of Quarter-Point Crack Tip Elements," Int. J. Num. Meth. Engng., Vol. 18, No. 7, 1982, pp. 1101-1103.
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8.17 N. A. B. Yahia and M. S. Shephard, "On the Effect of Quarter-Point Element Size on Fracture Criteria," Int. J. Num. Meth. Engng., Vol. 21, No. 10, 1985, pp. 1911–1924.
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8.18 V. E. Saouma and D. Schwemmer, "Numerical Evaluation of the Quarter-Point Crack Tip Element," Int. J. Num. Meth. Engng., Vol. 20, No. 9, 1984, pp. 1629–1641.
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8.19 D. M. Parks and E. M. Kamenetzky, "Weight Functions From Virtual Crack Extension," Int. J. Num. Meth. Engng., Vol. 14, No. 11, 1979, pp. 1693–1706.
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8.20 A. D. Kerr, "Elastic and Viscoelastic Foundation Models," J. Appl. Mech., Vol. 31, No. 3, 1964, pp. 491-498.
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8.21 M. S. Cheung, "A Simplified Finite Element Solution for the Plates on Elastic Foundation," Computers & Structures, Vol. 8, No. 1, 1978, pp. 139–145.
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8.22 Z. Feng and R. D. Cook, "Beam Elements on Two-Parameter Elastic Foundations," J. of Engng. Mech., Vol. 109, No. 6, 1983, pp. 1390-1402.
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8.23 O. C. Zienkiewicz, P. Bettess, T. C. Chaim, and C. Emson, “Numerical Methods for Unbounded Field Problems and a New Infinite Element Formulation,” in Computational Methods for Infinite Domain Media Structure Interaction, A. J. Kalinowski, ed., Am. Soc. Mech. Engrs., New York, 1981.
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8.24 P. Bettess and J. A. Bettess, "Infinite Elements for Static Problems," Engineering Computations, Vol. 1, No. 1, 1984, pp. 4–16.
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8.25 J. M. M. C. Marques and D. R. J. Owen, "Infinite Elements in Quasi-Static Materially Nonlinear Problems," Computers & Structures, Vol. 18, No. 4, 1984, pp. 739–751.
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8.26 R. T. Fenner, “The Boundary Integral Equation (Boundary Element) Method in Engineering Stress Analysis,” J. of Strain Analysis, Vol. 18, No. 4, 1983, pp. 199–205.
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8.27 J. Mackerle and T. Andersson, "Boundary Element Software in Engineering," Advances in Engng. Software, Vol. 6, No. 2, 1984, pp. 66–102 (lists 656 references).
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8.28 S. J. Fenves et al., eds., Numerical and Computer Methods in Structural Mechanics, Academic Press, New York, 1973 (see papers by D. Bushnell, pp. 291–336, and by S. W. Key and R. D. Krieg, pp. 337–352).
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8.29 J. S. Arora, "Survey of Reanalysis Techniques," J. Struct. Div., Proc. ASCE, Vol. 102, No. ST4, 1976, pp. 783-802 (lists 89 references).
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8.30 U. Kirsch, Optimum Structural Design, McGraw-Hill, New York, 1981.
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8.31 J. S. Przemienicki, "Matrix Structural Analysis of Substructures," AIAA Jnl., Vol. 1, No. 1, 1963, pp. 138–147.
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8.32 A. K. Noor, H. A. Kamel and R. E. Fulton, “Substructuring Techniques—Status and Projections,” Computers & Structures, Vol. 8, No. 5, 1978, pp. 621–632.
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3.33 P. G. Glockner, "Symmetry in Structural Mechanics," J. Struct. Div., Proc. ASCE, Vol. 99, No. ST1, 1973, pp. 71–89.
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3.34 A. K. Noor and R. A. Camin, "Symmetry Considerations for Anisotropic Shells," Comp. Meth. Appl. Mech. Engng., Vol. 9, No. 3, 1976, pp. 317-335.
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3.35 P. D. Mangalgiri, B. Dattaguru, and T. S. Ramamurthy, "Specification of Skew Conditions in Finite Element Formulation," Int. J. Num. Meth. Engng., Vol. 12, No. 6, 1978, pp. 1037–1041.
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3.36 O. C. Zienkiewicz and F. C. Scott, “On the Principle of Repeatability and Its Application in Analysis of Turbine and Pump Impellers,” Int. J. Num. Meth. Engng., Vol. 4, No. 3, 1972, pp. 445–448.
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.37 R. H. MacNeal and R. L. Harder, "A Refined Four-Noded Membrane Element with Rotational Degrees of Freedom," Computers & Structures, Vol. 28, No. 1, 1988, pp. 75–84.
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# CHAPTER 9
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9.1 J. F. Abel and M. S. Shephard, "An Algorithm for Multipoint Constraints in Finite Element Analysis," Int. J. Num. Meth. Engng., Vol. 14, No. 3, 1979, pp. 464-467.
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9.2 O. C. Zienkiewicz, The Finite Element Method, 3rd Ed., McGraw-Hill, London, 1977.
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9.3 C. A. Felippa, "Iterative Procedures for Improving Penalty Function Solutions of Algebraic Systems," Int. J. Num. Meth. Engng., Vol. 12, No. 5, 1978, pp. 821-836.
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9.4 T. J. R. Hughes, R. L. Taylor, and W. Kanoknukulchai, "A Simple and Efficient
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Finite Element for Plate Bending," Int. J. Num. Meth. Engng., Vol. 11, No. 10, 1977, pp. 1529–1543.
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9.5 D. S. Malkus, Finite Element Analysis of Incompressible Solids, Ph.D. Dissertation, Boston University, Boston, 1976.
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9.6 D. S. Malkus and T. J. R. Hughes, "Mixed Finite Element Methods-Reduced and Selective Integration Techniques: A Unification of Concepts," Comp. Meth. Appl. Mech. Engng., Vol. 15, No. 1, 1978, pp. 68–81.
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9.7 T. J. R. Hughes, A Course in the Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987.
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9.8 R. L. Spilker and N. I. Munir, "A Hybrid Stress Quadratic Serendipity Displacement Mindlin Plate Element," Computers & Structures, Vol. 12, No. 1, 1980, pp. 11–21.
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9.9 Y. C. Fung, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1965.
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9.10 M. Engelman, R. L. Sani, P. M. Gresho, and M. Bercovier, "Consistent vs. Reduced Integration Penalty Methods for Incompressible Media Using Several Old and New Elements," Int. J. for Num. Meth. in Fluids, Vol. 2, No. 1, 1982, pp. 25–42.
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9.11 C. Johnson and J. Pitkäranta, "Analysis of Some Mixed Finite Element Methods Related to Reduced Integration," Mathematics of Computation, Vol. 38, No. 158, 1982, pp. 375-400.
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9.12 E. T. Olsen, Stable Finite Elements for Non-Newtonian Flows; First Order Elements Which Fail the LBB Condition, Ph.D. Dissertation, Illinois Institute of Technology, Chicago, 1983.
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9.13 J. Pitkäranta and R. Stenberg, "Error Bounds for the Approximation of the Stokes Problem Using Bilinear/Constant Elements on Irregular Quadrilateral Meshes," Report MAT-A222, Helsinki University of Technology, Espoo, Finland, 1984.
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9.14 T. J. R. Hughes, W. K. Liu, and A. Brooks, "Finite Element Analysis of Incompressible Viscous Flows by the Penalty Function Formulation," J. of Comp. Phys., Vol. 30, No. 1, 1979, pp. 1–60.
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|
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# CHAPTER 10
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|
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10.1 E. L. Wilson, "Structural Analysis of Axisymmetric Solids," AIAA Jnl., Vol. 3, No. 12, 1965, pp. 2269–2274.
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10.2 J. G. Close and R. M. Jones, SAAS III: Finite Element Analysis of Axisymmetric and Plane Solids with Different Orthotropic, Temperature-Dependent Material Properties in Tension and Compression, Aerospace Corp., San Bernardino, CA, 1971 (AD-729-188, N.T.I.S.).
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10.3 O. C. Zienkiewicz and Y. K. Cheung, "Stresses in Shafts," The Engineer (London), Vol. 224, No. 5835, 1967, pp. 696–697.
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10.4 T. Belytschko, "Finite Elements for Axisymmetric Solids Under Arbitrary Loadings with Nodes on Origin," AIAA Jnl., Vol. 10, No. 11, 1972, pp. 1532-1533 (discussion and closure: Vol. 11, No. 9, pp. 1357-1358).
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10.5 J. Padovan, "Quasi-Analytical Finite Element Procedures for Axisymmetric Anisotropic Shells and Solids," Computers & Structures, Vol. 4, No. 3, 1974, pp. 467–483.
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10.6 J. G. Crose, "Stress Analysis of Axisymmetric Solids with Asymmetric Properties," AIAA Jnl., Vol. 10, No. 7, 1972, pp. 866-871.
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10.7 M. Sedaghat and L. R. Herrmann, "A Nonlinear, Semi-Analytical Finite Element Analysis for Nearly Axisymmetric Solids," Computers & Structures, Vol. 17, No. 3, 1983, pp. 389–401.
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10.8 E. L. Wilson and P. C. Pretorius, "A Computer Program for the Analysis of
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