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  • Life and Medical Sciences  (963)
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  • 1995-1999
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  • 1995-1999
  • 1980-1984  (1,395)
  • 1975-1979  (1,006)
  • 1910-1914  (18)
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  • 101
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 17 (1981), S. 1877-1881 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 102
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 17 (1981), S. 1890-1890 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 103
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 12 (1978), S. 11-33 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The discretization of the media at all spatial co-ordinates but one is presented here. This partial discretization leads to continuous finite elements as opposed to fully discrete ones and the problem resolves, for the cases presented here, into a set of linear differential equations rather than algebraic equations. The general problem of first derivative functionals in elastostatics is considered and it is shown, in general, how the continuous finite elements required for the solution may be obtained. Plane states, axisymmetric and three-dimensional continuous elements are obtained to illustrate application to particular cases.Different methods of solution for the set of differential equations are discussed and it is shown that several existing and widely used finite element related techniques are particular cases of this local partial discretization. Three numerical examples are solved to demonstrate the good comparison obtained between the numerical and the exact solutions. The semi-infinite examples included also illustrate the treatment of these types of problems without the use of fictitious boundaries.
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  • 104
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 12 (1978), S. 67-76 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: For a wide calss of finite element matrices integrated numerically rather than exactly, a definable number of sampling points is found to be sufficient for keeping their theoretical properties unchanged. A systematic criterion limiting the number of possible point configurations for numerical quadrature formulas on triangles is established. Some new high order formulas are presented. Tables containing optional formulas with respect to minimum number of sampling points and required degrees of accuracy are given. They are arranged so as to assist with selection of suitable quadrature formulas for finite element computer programming.
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  • 105
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 12 (1978), S. 115-127 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper presents the finite element method of the analysis of tidal flow. Assuming that tidal flow is periodic, the Galerkin approach is employed as the numerical integration procedure in time using a trigonometric function as the interpolation function. The present method has shown to be suitable for computation especially from the point of computing time and numerical stability.
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  • 106
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 169-181 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The visco-plastic model is used in the context of large displacement (geometrically non-linear) analysis. As the process now involves in numerical computation an updating of relevant stiffness matrices, formulation which is stable and computtionally more efficient. The applications shown demonstrate the applicability of the process to large displacement elasto-plastic shell analysis as well as to problems of creep buckling.
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  • 107
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 12 (1978), S. 185-186 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 108
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 12 (1978), S. 194-194 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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  • 109
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 229-235 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The approximation of normal derivatives along a curved boundary becomes a major difficulty in obtaining finite difference solutions for irregular regions. A number of authors (for-example, Allen,1 Fox,2 Greenspan3 and Parker and Ma4) have suggested various solutions to this problem. However, most of these suggested methods are fairly cumbersome to use since they require the use of different formulas for various combinations of mesh and boundary geometries. This note shows a different way to approximate a quite general boundary condition along curved boundaries. The method uses a uniformly spaced mesh, is simple, general, relatively easy to program and appears to give accurate results.
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  • 110
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 267-278 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In many systems it is known a priori, that some states are ony weakly coupled with others. If such systems are solved on parallel processor it is possible to partition the states in such a way that one set of states is assigned to one processor and the other set of weakly coupled states to another. Computations can then be done in parallel. Because of the weak coupling it may not be necessary for information from the two sets to be communicated to each other for many integration steps. This can result in significant cost savings. In this paper, a partitined Runge - Kutta scheme is formulated for use on weakly coupleld systems of ordinary differential equations. An error expression is derived which provides a means for predicting the number of step sizes over which the partitioned formulas can be used in terms of a prespecified error tolerance. Numerical examples are presented both to verify the error expressions and to compare solutions using partitioned and unpartitioned schemes.
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  • 111
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 319-329 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The diferential equations governing two-phase flow through a porous medium are derived and the space discretization is carried out by the Galerkin finite element method. An examination of the eigenvalues of the resulting system follows in order to decide on a suitable stability criterion to be satisfied by the time-stepping algorithm. A class of time-stepping schemes is introduced and their behaviour when applied to a specific problem is analyzed. The aim is to achieve an adequate modelling of the water-oil interface and to preserve reasonable values of the capillary pressure. A technique to clustering the mesh around the interface is used to improved the modelling, and a boundary condition based on the relative mobilities of the two fiuids is imposed to enable the program to calculate the water-oil ratio after breakthrough. The effect of lumping the mass matrix is then examined with a view to reducting the surge in the values obtained for the capillary pressure. Finally an optimum method is suggested for use as a starting algorithm for more sophisticated time-stepping schemes.
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  • 112
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 384-384 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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  • 113
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 415-428 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A Finite Element Method using rectangular elements is applied to solve Burgers' non-linear equation in one space dimension. The non-linear algebraic equations thus obtained are solved by an iteration which is equivalent to the generalized Newton method. Numerical experiments indicate that the order of accuracy is equal to 2. In two space dimensions, the method is more efficient than other methods we have tested.
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  • 114
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 12 (1978), S. 471-477 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A global variational method (GVM) not requiring the boundary conditions to be incorporated in the trial functions is described. The method is outlined for a linear differential equation and the extensions necessary to handle the non-linear problem of the indirect design of a compressible fluid channel described. The method requires a suitable estimate of a parameter, but the results obtained for the non-linear problem indicate that the actual value of this parameter is not critical (provided that a sufficeintly large number of trial functions is used). Comparisons are made with the results from an alternative GVM approach described in Davies and Hendry.1
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  • 115
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 479-486 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The problem of confined seepage of water under a hydraulic structure, through a porous medium, is considered in an appropriate abstract space. Existence and uniqueness of the solution are proved. The numerical solution is derived by means of the finite element method. An error estimate is given. A sample problems in included.
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  • 116
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1485-1486 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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  • 117
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1487-1488 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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  • 118
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    International Journal for Numerical Methods in Engineering 12 (1978) 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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  • 119
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1491-1491 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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  • 120
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1492-1492 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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  • 121
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1493-1506 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An iterative algorithm is presented: it builds a continuous stress and displacement solution starting from the solution of a classical displacement finite element analysis. The modified solution satisfies the virtual work principle, and is much better than the starting solution. The algorithm is very easily included in existing programs. A number of examples shows the efficiency of the method.
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  • 122
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 12 (1978), S. 1507-1541 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the stressed-skin analysis of metal structures it is vital to have a simple yet accurate treatment of the shear deformation of profiled metal sheeting. A suitable theory is developed in this paper for the two important cases of sheeting fastened to the supporting structure through every and alternate troughs of the corrugations. Various practical factors are taken into account. The results of an extensive series of numerical investigations are presented as a simple equation and a set of tables.
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  • 123
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1677-1696 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper presents a classification of mathematical commonly encountered in connection with solution of non-linear finite element problems. The principal methods for numerical solution of the non-linear equations are surveyed and discussed. Special emphasis is placed upon the description of an automatic load incrementation procedure with equilibrium iterations. It is shown how this algorithm can be adapted for solving problems involving instabilities, snap-through and snap-back. A simple scalar quantity denoted the current stiffness parameter is suggested; this parameter is used to characterize the overall behaviour of non-linear problems. It can also be used as a steering parameter in the solution process. The use of the present technique is illustrated by several examples.
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  • 124
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1705-1715 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new and efficient method of dynamic condensation is presented. It reduces the order of dynamic matrices without intoroducing further approximation by representing the passive co-ordinates in terms of the active ones exactly. The resulting frequency dependent eigenvalue problem is solved by a combined technique of Sturm sequence and subspace iteration. Examples are given for a space frame and a substructure problem. Results are compared to the complete solution.
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  • 125
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1841-1851 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A common method for numerically approximating two-point parabolic boundary value problems of the form ut = L[u]+f(u) defined of the semi-infinite strip S = [0, 1]×[0, ∞] is to first discretize the spatial operator in the differential equation and then solve for the time evolution. Such an approach typically involves solving a system of algebriaic equations at a sequence of time steps. In this paper we take a different approach and subdivide S into a collection of semi-infinite substrips Si = [xi, xi+1]×[0, ∞], and use blending function techniques to derive finite parameter functions ei(x, t) defined on Si. Spectral matching methods are used in deriving ei to ensure that (u - ei) can be made small on Si. Galerkin's method, with associated integration sover the entire space-time domain S, is then used to generate approximations to u(x, t) based upon the so defined infinite element (ei, Si). Approximations are hence found for all (x, t) in S by solving one well structed system of algebraic equations. We apply the method to several linear and non-linear problms.
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  • 126
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1873-1878 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new and a simple algorithm for approximate calculation of optimal output-constrained feedback regulators is presented. computational simplicity is achieved by using properties of weakly coupled linear systems. The algorithm has been shown to be very effective in solving high dimensional control problems.
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  • 127
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1879-1882 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 128
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1853-1871 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper a technique previously developed for the analysis of rigid-plastic structural frames is adapted to provide a formal procedure for finding the limit load of any structure formed from rigid blocks. In this procedure the load factor is maximized subject to the euilibrium equations of the structure and linear constraints imposed by criteria of failure at the block interfaces.In part I of the paper it is assumed that the limiting shear force associated with sliding at a block interface is independent of the normal component of force across the interface. This assumption means that the normality rule is satisfied, so that the upper- and lower-bound theorems of classical limit analysis apply. This part includes a description of computer program for the collapse analysis of masonry arches with joints incapable of carrying tension.In part II the limit on the Shear force at a block interface is assumed to be that associated with Coulomb friction. It is not difficult to extend the computational algorithm described in part I to deal with this situation. However, the failure mechanism computed by the algorithm will not necessarily satisfy the normality rule. The corresponding limit load may therefore be an over-estimate of the true failure load, even though it is computed by a lower-bound (equilibrium) approach. A criterion is established for testing the validity of a failure load computed in these circumstances.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1897-1897 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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  • 130
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    International Journal for Numerical Methods in Engineering 13 (1978) 
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 1-16 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper presents an introduction to two general approaches used in the soluation of coupled structures and fluid systems in which effects of large scale flow are excluded. In the first approach, the Lagrangian, the fluid is simply treated as a ‘solid’ with a negligible shear modulus. In the second method, Eulerian, a single pressure variable is used in the fluid.The numerical problems posed, discretization methods used and possible simplifications are discussed.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 17-33 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The fluid forces acting on a uniform cylinder with an infinitely long axis, heaving in the free surface or an infinite ideal fluid, are described in terms of its ‘added mass’ and ‘damping coefficient’. The techniques of multipole expansion and multiparameter conformal transformation are adopted for such calculations and applied to shapes which cannot be adequately represented by the conventional, and more rudimentary, ‘Lewis form fit’. The shapes referred to are both relevant to ship bows, one being a ‘fine section’ and the other a ‘bulbous section’. The parameters which influence the accuracy of the solution are examined. Results for these two sections are computed and compared with results based on (a) the ‘Lewis form approximation’ and (b) the ‘Frank's close fit method’ which employs a singularity representation.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 35-48 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The hydrodynamic mass matrix for a vibrating ship is evaluated using a three-dimensional finite element discretization of the surrounding water. For the important case of vertical vibrations of the ship, values of the added masses obtained in this way are compared with those given by the conventional semi-empirical formulas currently in use in the shipbuilding industry. A numerical example is presented. It is concluded that conventional two-dimensional methods, the usefulness of which is confirmed when the first few modes of the hull girder only are sought, do not apply to higher modes, for which significant deformation of the cross-section may occur. On the other hand, a finite element approach has the advantage of its generality as well as flexibility. Suggestions for future research are given.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 49-72 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A computer program, originally developed to predict wave loads on large-diameter fixed offshore structures, has been modified to compute added masses, damping coefficients and response motions of free-floating and moored rigid systems. It is now being extended to predict multi-degree-of-freedom responses of articulated structures, particularly wave energy absorbing devices. The structure is modelled by surface singularities (point sources) in potential flow. The boundary conditions, fluid and structure equations of motion are linearized. Engineering applications here include the predictions, fluid and structure equations of motion are linearized. Engineering applications here include the prediction of wave loads on a gravity platform, response motions of a moored barge, and comparisons with laboratory data on the mooring loads and response of a tethered buoyant platform, and on the wave loading, response and efficiency of two wave energy absorbing devices.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 73-92 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper is concerned with the development of transfer functions for the dynamic response of multidegree-of-freedom offshore structures, excited by sinusoidal waves. The problem is formulated in terms of the principal modes of the structure vibrating freely in vacuo. The hydrodynamic analysis is performed through use of a distribution of wave sources on the submerged surface of the structure, which leads to an approximate solution satisfying the boundary condition on the fluid-structure interface at a discrete set of points. Generalized forces are obtained corresponding both to excitation by waves and to the hydrodynamic reactions induced by motions of the structure: the latter are expressed in terms of added mass and damping matrices. Results are given for simple geometries to illustrate the approach.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 93-107 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A method of fluid-structure connection is presented for solutions which mainly involve the calculation of the dynamic response of structures by using the method of modal analysis. This connection method is based upon finite element structural and fluid representation and is efficiently employed when used with the inverse iteration procedure for eigenvalue, eigenvector evaluation. The inverse iteration procedure is described in conjunction with the connection method. Also a method of non-linear structural analysis and a method of frequency domain analysis are discussed with reference to the presented fluid-structure connection procedure.Applications are considered for both model and prototype situations. Efficient use of the mini computer for solving dynamic problems by the proposed method is also considered.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 123-138 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper presents a numerical technique for the linear dynamic analysis of a finite elastic structure immersed in an infinite homogeneous acoustic medium. It is required to determine the vibrational motion of the structure and also the associated acoustic field in the fluid, when the structure is either subjected to internal applied forces or is acting as a scatterer of an incident acoustic wave. A finite element analysis of the structure is matched at the structure-fluid interface with an integral equation representation of the exterior acoustic field, leading to a coupled system of equations which may be cast in either acoustic or structural from. The former approach is preferred here for which numerical results are presented when the method is applied to plane wave scattering by thick and thin elastic spherical shells.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 109-122 
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    Keywords: Engineering ; Engineering General
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    Notes: A finite element method is presented for predicting the coupled structural-acoustic response of a flexible cylinder, which contains an acoustic medium and is excited by mechanical forces. The cylinder is represented by an existing axisymmetric, cylindrical shell element. The acoustic space inside the cylinder is modelled using a new axisymmetric, acoustic ring element. The cross-secton of the ring takes the form of an eight node, isoparametric element. The coupled equations of motion for the cylinder and acoustic field are solved using modal analysis techniques. The numerical results obtained are compared with results from an experimental investigation which is also described.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 139-150 
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    Keywords: Engineering ; Engineering General
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    Notes: A variational principle in terms of displacements in the fluid and the structure with a penalty for irrotationality of displacement in the fluid is developed for the analysis of harmonic vibrations of ideal compressible fluid and elastic structure systems. Its Discretization by the finite element method leads to an algebraic eigenvalue problem with a positive definite symmetric banded matrix. Numerical examples obtained for pure acoustic cases and coupled cases show the efficiency of the method.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 151-164 
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    Notes: A theory is developed whereby acoustic and structural modes are coupled by means of geometrical distortions in a circular cylindrical shell filled with liqued.The prediction is that a circularly symmetric discrete frequency acoustic wave in the liquid will excite in the shell not only a circular symmetric ‘breathing mode’ (which would be the only one present in an undistorted shell) but also short wavelength flexural waves at the excitation frequency. It is possible, even for quite small distortions, for the displacement amplitude of some of these flexural waves to exceed that of the ‘breathing mode’.The results of calculations for a particular shell are shown to be consistent with experimental results.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 165-180 
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    Keywords: Engineering ; Engineering General
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    Notes: The motion of a terthered buoyant platform (TBP) in a regular train is investigated numerically using the Zienkiewicz four-time-level scheme.5 Wave forces are calculated using Morison's equation7 and linear wave theory. Results are presented for the theoretical response to the periodic forcing function for an optimized method,6 Wood, A.,6 Wilson θ =1.412 and Houbolt13 and compared with the actual computed behaviour of the TBP.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 197-201 
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 181-196 
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    Keywords: Engineering ; Engineering General
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    Notes: The seismic behaviour of an offshore pipeline, which may be totally or partially in contact with the sea-bed, is characterized by the confluence of structural dynamics, hydrodynamics and soil mechanics and also the possible dynamic interactions between these aspects. This paper contains a study dealing with the seismic response analysis of such a pipeline including pressure drag effects. A numerical analysis basesd on the assumption that the pipe behaves linearly elastically and that pipe deffection is elastoplastically related, without loss, to sea-bed resistance shows that pipe response in significantly attenuated by contact, as aspected, the amount of attenuation depending upon contact length and the parameters defining idealized sea-bed behaviour.
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    International Journal for Numerical Methods in Engineering 13 (1978) 
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 71-80 
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    Keywords: Engineering ; Engineering General
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    Notes: Techniques for computing the sensitivity of temperatures to changes in design variables needed for designing thermal protection systems are considered. It is shown that the choice of the most efficient technique depends on the ratio of number of temperature constraints to the number of design variabies, as well as on the thermal analysis method employed. The analysis is specialized to the case of a structure modelled by finite elements, and an example of an insulation panel is used demonstrate the techniques.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 229-246 
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    Keywords: Engineering ; Engineering General
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    Notes: Prestressed concrete members, because of the use of high strength materials and the efficient disposition of tensile forces, need considerably less material. This and other advantages, however, are offset by the high unit costs of materials and the associated specialized labour. Thus, it is important to make the most of the savings in materials through the application of optimum design procedure. Among the available optimization techniques, generalized geometric progamming (GGP) is ideally suited because the design equations of prestressed concrete members are expressible in the form of polynomials. Using the GGP, in this study, a set of minimum prestress force designs are obtained for various depths of one-way slab. From the data thus obtained a minimum cost design procedure is explained. Moreover, the need for supplementary non-prestressed reinforcement to obtain economical designs for depths of slab beyond a certain depth is also shown.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 471-476 
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    Notes: This paper describes how general-purpose finite element structural analysis computer programs can be used, without modification, to solve various scalar field equations such as the wave equation, the Helmholtz equation, Laplace's equation, Poisson's equation, the heat equation and the telegraph equation, as well as mixed field problems (such as coupled structural-acoustic problems) which involve these equations.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 480-482 
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 427-444 
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    Notes: A finite difference method for the large deformation elastic-plastic analysis of spherical caps is applied to predict the collapse strength of initially imperfect deep spherical shells. Twelve uniformly loaded hemispherical shell models with flat spots at their apex are analysed. For each model, a number of shallow spherical regions containing the flat spot are selected from its domain. One of these selected shallow regions yields a minimum buckling pressure; this minimum value is taken as the theoretical buckling load for the shell model under consideration. Present solutions are in good agreement with existing experimental and empirical results. The good comparison suggests that initially imperfect deep spherical shells may be analysed by using a much simpler mathematical model - the spherical cap - and thus the analytical cost may be greatly reduced. This also demonstrates that the collapse of imperfect spherical shells is primarily a local phenomenon and therefore dependent on local geometry. Consequently, the presence of initial imperfections must be fully taken into consideration in any large deformation inelastic buckling analysis before such analysis can be expected to quantitatively predict the collapse strength of practical shell structures.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 478-478 
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 615-631 
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    Notes: Simple mixed models are developed for the geometrically nonlinear analysis of deep arches. A total Lagrangian description of the arch deformation is used and the analytical formulation is based on a form of the nonlinear deep arch theory with the effects of transverse shear deformation included. The fundamental unknowns consist of the six internal forces and generalized displacements of the arch, and the element characteristic arrays are obtained by using Hellinger-Reissner mixed variational principle. The polynomial interpolation functions used in approximating the forces are one degree lower than those used for approximating the displacements, and the forces are discontinuous at the interelement boundaries.The equivalence between the mixed models developed herein and displacement models based on reduced integration of both the transverse shear and extensional energy terms is discussed. The advantages of mixed models over equivalent displacement models are outlined. Numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed, and compare their performance with other mixed models reported in the literature.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 679-706 
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    Notes: The treatment of zero energy modes which arise due to one-point integration of first-order isoparametric finite elements is addressed. A method for precisely isolating these modes for arbitrary geometry is developed. Two hourglass control schemes, viscous and elastic, are presented and examined. In addition, a convenient one-point integration scheme which analytically integrates the element volume and uniform strain modes is presented.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 633-644 
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    Notes: Often, finite element solutions of thin plate/shell elements become very stiff and the displacement field solutions diverge from those predicted by Kirchhoff's theory. This phenomenon is known as the locking phenomenon. A theoretical fomulation demonstrating its existence is developed, and results of finite element analysis of a single element and mesh are discussed. This leads to a sufficient and necessary criterion which must be satisfied to avoid the locking phenomenon.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 663-678 
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    Notes: This paper deals with constructing finite difference formulaes for approximating ordinary derivatives in a uniform or non-uniform mesh. The corresponding order of truncation error is also established. The accuracy and versatility of the proposed formulae are illustrated through numerical solution to two-point boundary value problems.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 311-336 
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    Keywords: Engineering ; Engineering General
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    Notes: This paper treats shape optimal design of two-dimensional structures. Sensitivity of cost and constraint functions to changes in the shape of the structure are obtained by applying theorems from the calculus of variations and by using consistent first-order approximations of functions arising in an optimal design problem. A generalized steepest descent algorithm is presented for determination of the optimum shape. The method is applied to a load carrying shear plate that models a component of the high pressure seal in a gun launched high velocity projectile.
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 373-384 
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    Notes: Derivation of an optimality criterion for finite element structural representations using constant-moment plate-bending triangular elements ispresented. Numerical examples of minimum-mass design of both simply-supported and clamped elastic plates for concentrated and distributed loading conditions under a single displacement constraint serve to illustrate the suggested procedure. Interpretation of the solution obtained and difficulties involved are discussed in the light of recent application of optimal control theory to structural optimization.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 923-930 
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    Notes: This paper presented an efficient computational algorithm for the eigensolution for real symmetric positive definite matrices. The algorithm is based on the Cholesky decomposition and will be referred to as the Cholesky algorithm. The efficiency of the algorithm is due to the combination of several iteration cycles into one, as well as to a judicious choice of shifts. The latter is accomplished by combining an approach based on Sturm's theorem for the initial stages of the process with another approach based on Gersch-gorin's theorem for the subsequent and final stages of the process. The efficiency of the proposed algorithm was tested against that of the QL method and was found to be superior. Moreover, its superiority increases with an increase in the order of the eigenvalue problem. For an eigenvalue problem of order 100, the proposed algorithm produced a solution in about one-half the time required by the QL method.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. i 
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 991-1014 
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    Notes: The application of the mixed finite element method to two-dimensional elastic contact problems is investigated. Since in the mixed method, both displacements and stresses are retained as variables, it is found that all the contact conditions - displacement as well as stress - can be approximated directly. A significant novelty is that some of the displacement variables are treated as natural boundary conditions in the contact region. In cases where the contact region is independent of the applied loading, an iterative procedure is used to establish the sliding and adhering portions of the contact region. In cases where the contact region is a function of the applied loading, for example progressive contact, an incremental formulation is employed to describe the discretized contact stages before invoking the former iterations. Several numerical examples are presented and the results are compared with those from the conventional potential energy or displacement finite element method.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1015-1044 
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    Notes: Finite element preprocessor programs have been developed in recent years in order to expedite the task of data preparation. A key decision in the design of these programs is the choice of a method for automated mesh generation. Three popular methods (Laplacian, isoparametric and transfinite mappings) are compared. Transfinite mappings with boundary information represented in discrete form are found to possess distinct advantages for the task of mesh generation.A two-dimensional preprocessor program utilizing the discrete transfinite mappings is described. Interactive computer graphics techniques are used extensively to facilitate data preparation and display. The geometry-generating routines are general, and may be used in any finite element application. A specialized graphical ‘attribute editor’ for structural mechanics problems is also described. This editor provides an efficient method of specifying boundary conditions, material properties, loads, etc.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1120-1121 
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1187-1200 
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    Notes: The paper demonstrates that Fredholm integral equations of the first kind with the kernel K(x, s)=(x-s)k/[1+(x-s)2]n occur while solving problems of synthesis of electrostatic and magnetic fields. Numerical results for the solution of one of the equations obtained by the regularization method are provided, with two ways of regularization being employed - that of Tikhonov and that of Lavrentev.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1177-1185 
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    Notes: A parametric infinite element is presented for solving axisymmetric problems under non-axisymmetric forcing functions where the considered domain is assumed infinite. The element accuracy is compared against the exact solution for two classical problems: Boussinesq's and Cerruti's. Boussinesq's problem for a rigid circular plate is also presented. For these problems, using relatively simple displacement functions, the element provides a reasonable infinite domain representation.
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    International Journal for Numerical Methods in Engineering 17 (1981) 
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1285-1312 
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    Notes: This paper is a user's guide for the stand-alone explicit direct time integration package STINT/CD for structural dynamics analysis. STINT/CD uses an automatic variable time increment central difference method. The purpose, function, limitations and usage of the package are described. A FORTRAN listing of STINT/CD is given along with a sample problem which illustrates its usage and performance.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1355-1384 
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    Notes: Two of the most promising localized finite element methods are compared: the boundary series element method, in which a series of eigenfunctions is used to represent the far field solution; and the boundary integral element method, in which an integral equation is satisfied at the boundary between localized finite element and outer regions. The methods are applied to water of arbitrary depth. The theory of the two methods is summarized, and typical numerical results are discussed. Consideration is given to the well-known hydrodynamical reciprocal relations, and to the phenomenon of ‘irregular’ frequencies. The relative merits of the two methods are established.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1525-1538 
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    Notes: A numerical scheme is developed to study the stability of the linearized vorticity equations. The linear growth rates of Kelvin-Helmholtz instabilities resulting from a velocity shear are calculated numerically. The theory of neutrally stable waves, in two dimensions, is extended to include the cases where the stream function is zero or periodic at the boundary, and is verified numerically for the case of a velocity profile having the shape of a cosine function. The instability growth rates of the oscillations in the neighbourhood of the neutrally stable waves are calculated numerically and are shown to be in very good agreement with the theory. The results are also of interest in relation to studies on the instabilities of a two-dimensional guiding-centre plasma, and also in the study of the diocotron instability in electronics.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1497-1514 
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    Notes: A finite element formulation to deal with the flow of metals coupled with thermal effects in presented. The deformation process of the metal is treated using the visco-plastic flow approach and the solution technique for the coupled problem implies a simultaneous solution for velocities and temperatures. Some aspects of the numerical solution of the problem are given and in the last part of the paper some examples of steady-state extrusion and rolling problems showing the applicability of the method are shown.
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1589-1590 
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    International Journal for Numerical Methods in Engineering 17 (1981), S. 1605-1611 
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    Notes: A method is presented for the approximate solution of the axisymmetric problem of an annular crack Embedded in an infinite elastic solid and subjected to a normal internal pressure. The numerical results Obtained for the stress intensity factors are in good agreement with those obtained by Smetanin2.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 195-201 
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    Notes: A Finite Element formulation for the axisymmetrical analysis of rotational shells with geometric non-linearity is developed, using a mixed finite element model of a curved rotational shell type, the two global components of displacement (ur, uz) and the two stress couples (Mr, MtH) being the free and independent unknowns of the problem. The prescribed geometry at every nodal circle comprises the co-ordinates and meridional slope angle and curvature. Newton-Raphson Iteration is used in solving the non-linear system of equations. Circular plates and a spherical cap are used as examples to test the formulation; good results were achieved which are presented graphically in comparison to the analytical solution.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 279-288 
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    Notes: Shape functions are given for a six node triangular element in which the √r displacement filed varies quadratically in the circumferential direction. Using a convenient co-ordinate system the stiffness development is sketched and the integrations re done exactly in the radial co-ordinate direction. Brief numerical results are presented. Shape functions for a 15 node three-dimensional element are given explicitly. The shape functions provide both constant strain capability and conformability with quadratic isoparametric elements.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 331-351 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: Numerical analysis of tsunamis applying the finite element method is presented based on the shallow water wave equation. To discretize time, a two step explicit method is used. The scheme is the extension of the Lax-Wendroff finite difference method. The present finite element method is used for the analysis of the Tokachi-oki Earthquake tsunami problem and compared with the tide gauge records. The conclusion of this paper is that the present method is suitable for the prediction of the tsunami wave propagation problem.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 141-158 
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    Notes: The convergence properties of the conjugate gradient method are discussed in relation to relaxation methods and Chebyshev accelerated Jacobi iteration when applied to the solution of large sets of linear equations which have a sparse, symmetric and positive definite coefficient matrix. The conclusion is reached that its convergence rate is unlikely to be much worse than these methods, and may be considerably better. The conjugate gradient method may either be applied to the basic unscaled or scaled equations or alternatively to various transformed equations. Preconditioning, block elimination and partial elimination methods of transforming equations are considered, and some comparative tests given for six problems.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 187-190 
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    Keywords: Engineering ; Engineering General
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 35-45 
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    Notes: Formulation of a transition element is presented. Such an element makes it possible to match one element with two elements side-by-side. A modified quadrature formula is used to numerically integrate discontinuous functions in the expression of the stiffness matrix. Numerical examples show the applicability of the element.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 77-83 
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    Keywords: Engineering ; Engineering General
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    Notes: Blending function interpolants on rectangular elements are used to construct an overall interpolant which exactly matches function and normal derivative on the perimeter of a rectangular region. This overall interpolant is incorporated into the Ritz-Galerkin version of the finite element method and a test problem involving the biharmonic equation is solved. The numerical results obtained demonstrate the considerable increase in accuracy of the exact boundary methods as compared with the usual method using interpolated boundary conditions. A similar investigation of second order elliptic problems with Dirichlet boundary conditions where the rectangular region is divided up into triangular elements yields the perhaps surprising result that blended interpolants on triangular elements do not necessarily improve the accuracy of the Ritz-Galekin version of the Finite Element Method.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 129-139 
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    Keywords: Engineering ; Engineering General
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    Notes: The paper presents a method of identification of material behaviour parameters. The method is applied to a simple example: the identification of orthotropic parameters using a plate bending test. A number of results are presented concerning the choice of numerical algorithm, the experimental protocole (number and locations of test points, loading) and the sensitivity of the method on experimental errors.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 159-167 
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    Keywords: Engineering ; Engineering General
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    Notes: The flexural stiffness for a horizontally curved beam element is developed for nodal parameters wi, (∂w/∂s)i, βi, (∂β/∂s)i, (∂2β/∂s2)i, (∂3β/∂s3)i by assuming a cubic displacement function for vertical deflection w, and a seventh order polynomical function for the angle of twist β, along the length of the curved beam. The element gives good accuracy with even coarse mesh for the analysis of horizontally-curved steel beam with open thin-walled bisymmetrical cross-section.Warping restraint is considered in the derivation of the stiffness matrix, which has considerable influence on the behavior and design of steel bridges curved in-plan. The results obtained from such a new element are compared with those given by a lower order finite element, developed previously by the first author, and with theoretical solutions for the particular problem.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 191-191 
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    Keywords: Engineering ; Engineering General
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 203-227 
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    Notes: Shell-like assumptions introduced in a three-dimensional isoparametric finite element provide a kinematic description of smooth thick shell structures. Elastoplastic solutions are obtained via the stationary response of an elastic/viscoplastic model. Special attention is devoted to the probem of numerical integration across the depth of the shell, as well as the applicability of the method to thin shell analysis.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 245-266 
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    Notes: A hybrid element method developed recently for two-dimensional problems of water waves in an infinite fluid is extended to three dimensions. In this method only a limited fluid domain close to irregular bodies is discretized into conventional finite elements, while the remaining infinite domain is treated as one element with analytical representations of high accuracy. Continuity at the junction surface is treated as natural boundary conditions in a variational principle. Computation experience and numerical results for several ocean structures are presented.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 299-317 
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    Keywords: Engineering ; Engineering General
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    Notes: A computational algorithm is developed and applied for optimization of beam and plate structures, subject to constraints on transient dynamic response. A continuous design formulation is retained, with dynamic response governed by partial differential operator equations. Adjoint equations are employed for sensitivity analysis and a function space gradient projection optimization approach is presented. Finite element analysis methods are applied for solution of the system dynamic and adjoint differential equations. Displacement constrained beam and plate minimum weight examples are solved, with a variety of boundary conditions.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 429-435 
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    Notes: The problem of cold rolling of a composite sheet which consists of alternate layers of viscoelastic and elastic materials is studied by the finite element method. Results computed and presented graphically include the pressure distribution at the contact surface, the normal stress distribution at the middle surface and the asymmetric deformation of the surface of the sheet in contact with the roller.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 487-504 
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    Notes: This paper present a numerical method to obtain optimal quadrature formulas of Gauss type and Radau type in the sense of Sard. Using the relation between optimal quadrature formulas and nonospline functions, the optimal quadrature formula can be obtained by solving a set of no-linear simultaneous algebraic equations induced from the interpolatory conditions of the monospline. In attempting to solve this set of non-linear algebraic equations for numbers of knots and degrees of interpolution required in estimation problem applications insurmountable numerical errors were encountered. This paper solves the numerical problem by first reducing the number of unknowns and equations to approximately one half the original number. This is accomplished by showing and then using a symmetry property of the monospline. Second an iteration scheme which partitions the reduced order set of non-linear algebraic equations into a linear subsystem and a non-linear subsystem is developed to numerically solve the equations. This iteration algorithm provides the advantages of reducing the computational complexity, dynamically checking the convergence and explicitly evluating the resulting accuracy.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 551-560 
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    Keywords: Engineering ; Engineering General
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    Notes: The strain energy release rate (G) converges rapidly in finite element approximations in which the finite element mesh is fixed and the order of polynomial displacement interpolations (p) is increased. Numerical experiments indicate that the error inG is very closely estimated, even for small pand very coarse finite element meshes, by an expression of the form k (NDF)-1 in which k is a mesh dependent constant and NDF is the number of degrees-of-freedom. The method provides for very efficient and accurate computation of G without the use of special techniques.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 531-550 
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    Notes: Computational techniques for the treatment of fluid-structure interaction effects by discrete boundary integral methods are examinde. Attention is focused on the computation of the added mass matrix by finite element methods for a structure submerged in an infinite, inviscid, incompressible fluid. A general computational procedure is presented that is based upon a variational approach involving the assumption of constant source strength over each surface element. This is followed by an analysis of the discretization error for a spherical body that is then used to develop a hierarchy of computational schemes. These schemes are than evaluated numerically in terms of ‘fluid boundary modes’ for a submerged spherical surface. One scheme has been found to be surprisingly accurate in relation to its computational demands.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 505-529 
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    Keywords: Engineering ; Engineering General
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    Notes: Considerable effort has been invested lately in the application of isoparametric finite elements for numerical solution of a wide range of applied mechanics problems. In fact, several general purpose computer programs are now available which are based upon such finite elements. In the present paper, a new application of the isoparametric finite element concept is introduced which significantly extends its usefulness for many practical structural configurations. In this application, final working or architectural drawings of the structure are made from the same (or similar) finite element model as was utilized in a structural integrity analysis. The hardware necessary to produce such drawing, a computer driven plotter or automated drafting machine, is available commercially or through most data centres, and the software concepts required are described herein.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 561-566 
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    Notes: A non-compatible rectangular element with 12 degrees-of-freedom (DF) has been developed by Wilson et at.5 for analyzing 2-dimensional problems. The superior convergence properties of this element as compared with the 8 DF element for linear elasticity problems with important bending have been demonstrated in References 2, 5. The purpose of this study is to assess the compartive efficiency of these two rectangular elements for an estimation of collapse loads in plasticity problems. For convenience, the 12 DF and 8 DF element will be represented by Q6 and Q4 respectively.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 573-576 
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 567-572 
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    Notes: Variational formulations are given for the stationary heat conduction and thermodelasticity problems of an orthotropic, non-homogeneous, finite length, cylinder. It is demonstrated that a single-element approach based upon Ritz approximations and the method of Lagrange multipliers offers and efficient means for computing the temperature and thermal stress distributions.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 579-580 
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    Keywords: Engineering ; Engineering General
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 589-595 
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 581-588 
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    Notes: We consider finite elements that have internal degrees-of-freedom. These d.o.f. are condensed before elements are assembled into the structure, and must be recovered later for use in element stress computation. References 1 and 2 describe two different ways to do the calculations. This paper compares the computational costs of the two methods. Test runs were made to assess costs. Relative costs depend on several factors. The factors are discussed, and suggestions are made as to the better method in various circumstances.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 613-624 
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    Notes: The general inverse problem is one for which measurements are made on some of the state variables and it is desired to find the forcing functions. This problem is a natural one for the method of dynamic programming. The formulation and solution of the problem are presented in matrix form, together with two illustrative examples of an inverse heat conduction problem and an inverse structural dynamics problem.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 597-612 
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    Notes: A general-purpose technique has been developed for solving non-linear partial differential equations. A set of approximating functions with undetermined parameters is used to evaluate the differential equation and boundary conditions at discrete points, forming a set of residuals to be minimized. The parameters which minimize the sum of squared residuals are determined by a non-linear least-squares minimization technique. Initial value problems are solved by integrating the equations with respect to time at the fitting points by a predictor-corrector algorithm. The resulting formulation is independent of the form of the problem and the approximating functions, so that a broad class of problems may be solved with a single computer program. The technique is applied to several boundary and initial value problems in one and two spatial dimensions. The tecnique is applied to several boundary and initial value problems in one and two spatial dimensions.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 625-634 
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    Notes: In a previous paper1 a method was described for computing the current distribution and inductance of systems of infinitely-long, perfectly-conducting, straight conductors. The method relied on the solution of a Fredholm Integal equation of the first kind, and in the present paper the method is extended to equations of the second kind in order to study the inductance and a.c. resistance factor of resistive for which approximate formulas and experimental results exist, with which the method is in good agreement over a range of grequencies of particular practical practical interest. Results have also been obtained for rectangular conductors connected in parallel.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 635-642 
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    Notes: A correlation is geven between two notations currently used in the formulation of geometrically non-linear analysis in a Lagrangian co-ordinate system. The first referred to as the B-notation results in the linear elastic, initial displacement and initial stress matrices, whilst the second, here called the N-notation features matrices that repeat in the total potential energy, equilibrium and incremental equilibrium equations. An explicit example of the correlation is presented for a pin-jointed bar element.
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