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  • GEOPHYSICS  (1,320)
  • Engineering General
  • 1995-1999
  • 1980-1984  (700)
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  • 1925-1929
  • 1980  (700)
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  • 1
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    New York, NY [u.a.] : Wiley-Blackwell
    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 121-129 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: This paper examines the problem of the eccentric loading of a rigid circular disc-shaped foundation embedded in bonded contact with an istropic elastic medium of infinite extent. The solution of this problem is achieved by using a Hankel integral transform technique which reduces the problem to two sets of dual integral equations. These systems of dual integral equations represent the equations which govern the axisymmetric and asymmetric states of deformation induced by the loaded foundation. Explicit results are derived both for the displacement and rotation of the circular foundation and for the contact stress at the interface.
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  • 2
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 277-284 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 3
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    New York, NY [u.a.] : Wiley-Blackwell
    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980) 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 4
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 285-289 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 5
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 291-292 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 6
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. i 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 7
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 293-311 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: Hysteretic and viscous material damping are compared in the context of soils. Popular assumptions about damping are shown to lead to different results for the rocking mode of surface foundations at low frequency.
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  • 8
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 313-332 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A technique is developed for the analysis of multiple underream anchor systems resting in an elastic soil. This technique may be used to consider anchor systems involving arbitrary anchor inclination and depth beneath the soil surface, as well as arbitrary number, shape, size and spacing of underreams. The approach is largely analytical in nature and involves only a fraction of the computation required for a finite element analysis.Consideration is given to the effects of anchor depth and inclination to the soil surface, and the spacing and number of underreams upon the elastic response of anchor systems. On the basis of the result from this study, a simple, approximate method for estimating the response of multiple underream anchors is proposed. This approach involves the use of several interaction charts, which are presented in the paper, and can be used as a hand method for estimating the load-displacement behaviour of quite general anchor systems to sufficient accuracy for most practical purposes. The use of the approximate approach is illustrated by an example.
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  • 9
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 361-375 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A new concept based on the use of a function expressed as a (complete) polynomial expansion in terms of the three invariants of the stress tensor is proposed for deriving yield, failure and plastic potential functions for use in plasticity based constitutive laws. A mathematical interpretation and physical meaning of the proposed concept are provided by using the idea of the singular nature of constiutive matrices in incremental hypoelastic laws. It is suggested that the proposed function and (polynomial) forms of material moduli can be synonymous. A number of specialized forms of the general function are adopted and their values at failure from advanced three-dimensional tests for a number of (geological) media are evaluated. The results indicate the possibility that there exist invariant numbers associated with the functions(s) that may apply to a wide range of materials. Some ideas on implementation of the proposed concept are also presented.
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  • 10
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 333-359 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: The analysis of the stability of slopes using limiting equilibrium considerations necessitates the determination of the critical slip surface which yields the minimal factor of safety. The numerous methods currently available for slope stability analysis provide a procedure for assigning a factor of safety to a given slip surface, but do not consider the problem of identifying the critical conditions.This paper presents an effective minimization procedure based on dynamic programming by which the minimal factor of safety, and the corresponding surface, are determined simultaneously. This procedure SSDP (Slope Stability by Dynamic Programming), couples the minimization scheme with Spencer's method of slope stability analysis. It may be applied to slopes of any geometry, layering, pore pressure and external load distributions. No arbitrary restrictions are placed on the shape of the slip surfaces, and the analysis satisfies all equilibrium equations.Application of the procedure to slope stability problems reported in the literature shows that for a given slip surface the procedure yields factors of safety which are almost identical to those reported, but in every case a more critical slip surface, with a lower factor of safety, may be found.
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  • 11
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 377-387 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A new and simple concept based on the idea of correcting for non-associative characteristics of (geologic) media is presented. A special form of the concept is adopted and introduced in a critical state plasticity model. An example problem of behaviour of a soil tested under triaxial conditions is included. The concept can permit a simplified treatment of non-associativeness and under certain assumptions can allow use of existing formulations of plasticity by maintaining symmetry of the associated matrices.
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  • 12
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 389-389 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 13
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 389-391 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 14
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 15
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 131-142 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A non-linear distribution of vertical displacement versus aquifer depth is calculated in the case of a partially penertrating well. For a fully penetrating well, however, a linear distribution is observed. The solution exhibits a vertically uniform horizontal displacement in the case of a fully penetrating well and, for a partially penetrating well, the maximum horizontal displacement occurs at the elevation of the well bottom.
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  • 16
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 143-158 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: This paper analysis the stability of several methods for obtaining numerical solutions of second-order ordinary differential equations. The methods are popular in structural and geotechnical engineering applications and are direct, that is they do not require the transformation of the second-order equation into a first-order system. They include Newmark's method in both implicit and explicit forms, Wilson's θ-method, Houbolt's method and some variants on this latter method. We shall examine the stability of the methods when applied to the second-order scalar test equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \ddot x + 2a\dot x + (a^2 + c^2)x = 0 $$\end{document} where a and c are real.
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  • 17
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 175-184 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 18
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 185-189 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A method for determining the load-deformation response of a pile in a softening soil is presented. The results demonstrate the degree to which the flexibility of the pile can affect its ultimate capacity.
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  • 19
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 159-174 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: An exact formulation is presented for the problem of a rigid circular body performing harmonic vibrations on an elastic half-space whose shear modulus increases linearly with depth and is interrupted at some finite depth by a frictionless horizontal plane. The static case is derived in the limit of zero frequency vibrations while the known result for the uninterrupted half-space is recovered in either extreme limit of the horizontal frictionless plane coinciding with the surface or when it is pushed down to an infinite depth.It is shown that the maximum effect of the interruption occurs when the frictionless plane is at a depth where the shear modulus is about 1·6 times the surface shear modulus. Furthermore, this maximum effect is equivalent to a reduction of about 5 per cent of the surface shear modulus or a reduction of about 2½ per cent in the natural frequency of the rigid body on an uninterrupted half-space. The important conclusion, therefore, is that irrespective of the depth at which a half-space isso interrupted, the surface shear modulus is still the dominant parameter and that both the increase in shear modulus with depth and the interruption are not only secondary but also opposing effects.
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  • 20
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 191-196 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: Finite element solutions for steady-state flow of fluids through porous media do not include an explicit develpoment of the flow net, and many engineers find this lack frustrating. To solve for the stream function, the analyst can use essentially the same finite element routines as he used for the potential, but the matrix corresponding to the hydraulic conductivity is not the inverse of the conductivity. Contouring the potential and the stream function is done element by element; certain anomalous situations are described in an appendix.
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  • 21
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. i 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 22
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980) 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 23
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 1-23 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: One major problem encountered when using the finite element method to simulate the load-deformation behaviour of an elasto-plastic soil mass is that the theoretical collapse load is generally exceeded, and in some cases the numerical solution fails to exhibit a collapse load. A mixed variational principle is used as the basis for developing the governing equations of deformation and the results from this are compared with those obtained from the use of the virtual power equation. The former is found to give improved results and is generalized to include description of finite deformation. An Eulerian frame of reference is used. The method of approximating configuration changes in the numerical solution procedure is found to determine the shape of the load-deformation curve. Finally, a simple method of accounting for anisotropy of yield is presented.
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  • 24
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 25-43 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A finite element model for the prediction of discrete fracture propagation in rock structures loaded in compression is presented. The model integrates any one of three theories for mixed-mode fracture initiation; it contains an energy balance algorithm for predicting crack increment length, and incorporates recent developments in finite element stress-intensity factor computation. The predictions of the model are compared with the observed fracture response of a real rock structure. Results show that the model accurately predicts both stable and unstable fracture progagations observed experimentally.
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  • 25
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 45-55 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A comparative discussion of constitutive rate equations generated by using the hypoelasticity and plasticity theories is presented. If is shown that these two approaches are equivalent for the loading process but differ essentially in defining the unloading (or reverse loading) paths. The multisurface plasticity formulation proves to be more convenient in incorporating memory rules of particular loading events, whereas loading-unloading criteria used in hypoelasticity may result in an unrealistic prediction of material response for some loading paths, and in particular, for cyclic loading. The violation of the continuity condition between loading-unloading domains in proposed constitutive relations3,4 limits their applicability to some particular classes of loading paths.
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  • 26
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 57-72 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: The settlement and excess pore pressure are calculated for a column of water-saturated clay in which the permeability and/or shear modulus vary with depth. Several fairly general laws of variation are investigated. In each case, the soil is assumed to have a permeable top surface and to rest on an impervious substrate.For all the variations examined, in which the shear modulus increases with depth below the surface but the permeability of the soil remains constant, the deflection at a given instant after imposition of the load decreases and the degree of settlement increases with increasing rate at which the change with depth takes place. It is found that, when the variation over the height of the column is one order of magnitude or less, the curve of degree-of-consolidation versus time is reasonably wellapproximated by the curve for a homogeneous soil whose modulus equals the depth-averaged value. When, in addition, the permeability decreases with depth, the approximation becomes less accurate.The same approximation, for moderate increase in shear modulus with depth, predicts satisfactorily the process of dissipation of excess pore pressure at a given depth, but breaks down when the variation in modulus is large.Finally, it is shown that the effect of variable modulus on the settlement is greater than of a comparable variation in permeability.
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  • 27
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980) 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
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  • 28
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 89-101 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: When using numerical methods in soil mechanics, one often needs to define a yield surface in three-dimensional principal-stress space. A special class of yield surfaces, given by J = (p+a)α(1-β sin 3ν)n, where ν is the Lode angle, is considered from the point of view of convexity and agreement with experimental data. Some recently proposed yield functions which belong to this class are compared. It is shown that the model with n = -0.229 is optimal as regards convexity, and can give reasonable agreement with the data.
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  • 29
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 73-88 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: We continue our study of the consolidation of inhomogeneous clay soils with an analysis of the response of a soil mass, whose shear modulus increases linearly with depth, to axially symmetric pressure on its surface. This extends the work reported in Part I on the corresponding one-dimensional problem of a clay column, and generalizes researches of Gibson and coworkers on the elastic model of the soil (i.e., the initial response).Because of great mathematical complexity when Poisson's ratio is allowed to assume arbitrary values, we have confined ourselves (as did Gibson in several of his papers) to variations of the modulus from a zero surface value, this being in reasonable accord with published test results on London clay. The deflection of points within this region, the deflection relative to the central one (that is, the shape of the deformed loading area), as well as the absolute deflection of surface points exterior to the load.
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  • 30
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 103-119 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: The spontaneous shear band formation in the biaxial test on dry sand samples with constant cell pressure is treated as a bifurcation problem. The constitutive response of sand is described in terms of mobilized friction and dilatancy. Dilatancy is looked upon as an internal constraint and the hardening rule is expressed in terms of an adequate dimensionless stress measure. Owing to fail of normality in sand, localization always occurs in the hardening regime. The theoretical solution of the shear band inclination is a geometrical mean of the classical Coulomb and Roscoe solutions and is in good agreement with the experimental data. The incipient shear modulus is proportional to the stress level and can be estimated to be also proportional to these cant modulus.
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  • 31
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 215-232 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: The elastostatic problem of an infinite elastic medium containing an axially-loaded rigid cylindrical inclusion is investigated. This problem is of interest in connection with the geotechnical study of the time-independent, load-deflection characteristics of deep rigid anchors embedded in cohesive soil or rock media. The problem is formulated by means of Hankel integral transforms and reduced to a system of four coupled sing ular integral equations, where the unknown quantities are the normal and the shear stresses acting on the entire surface of the anchor. Numerical solutions are investigated for various Poisson's ratios and several values of the aspect ratio of radius to length of the cylindrical anchor.
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  • 32
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 255-276 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A finite element theory, suitable for describing the long-term transient thermal creep of geomechanical structures where the material obeys an arbitrary type of creep law, is presented. The method takes into account large deformation effects, is stable for the large time steps required to model geophysical phenomena and accurately simulates changing, incompressible, plastic flow fields. Applications of the theory to the prediction of long-term creep and creep rupture of simple engineering structures, are given. The theory is also applied to predict the thermal creep of layered media to study the mechanics of folding and rift formation.
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  • 33
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 199-213 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A numerical technique called Summary Representation was used to solve the problem of seepage through a leaky sheetpile; this particular method was selected owing to its accuracy and computational speed. Seepage parameters were evaluated quantitatively and the influence of the leakiness of sheetpiles in a porous medium underlain by an impervious horizontal stratum was determined. The case of a single sheetpile was first considered, and the effect of the depth of penetration and coefficient of leakiness on the total rate of flow and on the values of the exit gradient was analysed. Next, the flow under an impervious dam was considered to assess the accuracy of the method. Finally, the case of seepage under a dam with leaky sheetpiles at both ends was studied, including the effect of the leaky sheetpiles on the uplift pressure acting along the foundation of the dam. Patterns of seepage through the flow line deformation that is caused by the leakiness of the sheetpiles.
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    International Journal for Numerical and Analytical Methods in Geomechanics 4 (1980), S. 233-254 
    ISSN: 0363-9061
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences
    Notes: A new one-dimensional model for penetration analysis of a rigid projectile into a soil target is presented. The soil medium is represented by a set of discs of constant thickness, responding in the radial direction under plain strain regime. When penetration through a typical disc occurs, the projectile displaces soil material and a radial plastic shock wave propagates in the disc. The interaction pressure between the projectile and soil material is compatible with the motion of the contact boundary, between them. The instantaneous resistive force is obtained by summation of the contributions of all the discs which are in contact with the nose surface of the projectile. Soil is considered as a ‘floating’ ideally locking material, in which the locking volumetric strain is adjusted to the physical stress-strain properties of the soil material. Friction between projectile and soil is assumed to be negligible. Comparison is made with both twodimensional computer program results and experimental data, and very good agreement is observed. Comparison with other analytical methods reveals that the present model most closely resembles existing experimental results and permits further analysis of the problem as predictions of target response and of stress distribution on the projectile nose. The calculations require very small amounts of computer time. Analysis with the present model yields an efficient and comprehensive means to analyse penetration and perform parameter analysis.
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    International Journal for Numerical Methods in Engineering 9 (1975) 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 1-15 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: A simple mixed finite element method is developed. The finite element is a rectangular triangle and rectangle. In the element the deflections are assumed to be simple four-element polynomials, bending moments, Mx and My with a partially linear distribution, and a constant, Mxy, expressed in terms of the node deflections. The element matrix is of the order of 8 × 8. It is derived in a common engineering way. The unknowns are the deflections at the nodes and mid-diagonal, the two moments at the end of the diagonal and the two moments on the cathetus. The results obtained by this method show good convergence and an improvement in the accuracy of the moments as well as in the deflections, compared with results obtained by similar methods, such as those of Herrmann.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 39-50 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Finite element buckling analyses of plates are performed by means of quardrilateral elements having 12 degrees-of-freedom. It is found that if the plate is capable of appreciable transverse shear deformation, initial-stress stiffness matrices that represent the lateral dispalcement by cubic polynomials do not yield convergence to correct results. An initial-stress stiffness that works satisfactorily for both homogeneous and sandwich plates is formulated. A Fortran listing of the associated subroutine is provided.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 17-38 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: The results of a finite difference analysis are presented for the problem of incompressible laminar flow heat transfer in concentric annuli with simultaneously developing hydrodynamic and thermal boundary layers, the boundary conditions of one wall being isothermal and the other wall adiabatic. This corresponds to the fundamental solution of the third kind according to the four fundamental solutions classified by Reynolds, Lundberg and McCuen1.Firstly, the hydrodynamic entry length problem, based on the boundary layer simplifications of the Navier-Stokes equations, was solved by means of an extension of the linearized finite difference scheme used previously by Bodia and Osterle2 to solve a similar problem between parallel plates. The energy equation is then solved, using the velocity profiles previously obtained, by means of an implicit finite difference technique.The accuracy of the numerical solution was checked by comparing results for the annulus of radius ratio 0.25 with the avaiable solution of Shumway and McEligot3.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 51-62 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Accounting for shear deformations and rotary inertia effects, necessary condition for optimum fundamental frequency of a vibrating beam of constant volume and with a given distribution of non-structural mass, is obtained through the calculus of variations. Minimum cross-section of the beam is controlled by the introduction of an inequality constraint. A finite element displacement formulation is then used in an iterativve manner to arrive at the optimum fundamental frequency and the corresponding material distribution for the discretized beam models with various boundary conditions. A comparison is then made with the corresponding results of an Euler beam.
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    Notes: This paper describes a practical implementation of the Galerkin finite element procedure for solving systems of parabolic partial differential equations with non-linear boundary conditions. This technique consists of finding an approximation in the form of a finite sum of cubic B-splines, which yield high-order accurate results, and consequently, solutions can be developed with remarkable precision and speed up to the steady state region where conventional finite difference methods often fail. In addition, the choice of mesh width and nodal spacing can be automatically determined for a predictor-corrector routine, thereby relieving the engineer of a great deal of ‘guess work’ that is normally characteristic of solving such problems.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 250-252 
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    Keywords: Engineering ; Engineering General
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 245-250 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 337-351 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: The method of moments is a semidiscrete numerical method for solving partial differential equations. The method approximates the solution of a partial differential equation by a finite sum of products of two functions. One function in the product is an unknown function of a single variable and the other function (moment function) is a prescribed function in the remaining variables. Using variational technique we obtain a finite system of boundary value problems of ordinary differential equations for the unknown functions.The main goal of this paper is the study of the theoretical background and numerical effectiveness of the method of moments for solving linear partial differential equations on rectangular-like domains.The mathematical formulation of the method together with error estimates and the theory of optimal moment functions are given. If for the one-dimensional moment functions piecewise polynomials of degree K are used then finite element type error bounds are obtained for the approximate solution in two dimensions.We also consider the numerical implementation of the method through the factorization method and efficient initial value methods. Several numerical examples showing the efficiency of the method are presented.
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    Notes: A method is described whereby the formulation for non-conforming bending elements includes the work dissipated across the inter-element boundaries. For the examples considered, this correction is shown to lead to a significant improvement in accuracy at little extra cost. Since the solution procedure is based on an iterative technique, the method would be particularly suitable if applied to non-linear problems.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 679-689 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: This paper investigates the phenomenon of ‘noise’ which is common in most time-dependent problems. The emphasis is on the achievable accuracy that is obtained with various time-stepping algorithms and how this can be improved if noise is artificially damped to an acceptable level. A series of experiments are made where the space domain is discretized using the finite element method and the variation with time is approximated by several finite difference methods. The conclusion is reached that the Crank-Nicolson scheme with a simple averaging process is superior to the other methods investigated.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 691-699 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: The utility of the method of orthogonal collocation for solving stiff boundary value problems is investigated. To this end, the method is applied to a number of very stiff problems from the engineering and applied mathematics literature. The examples include both linear and non-linear problems defined on both finite and infinite intervals, a singular perturbation problem, and an inherently unstable problem. The examples demonstrate the usefulness of the method for a wide variety of stiff problems, as well as some of its limitations.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 711-720 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 701-709 
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    Notes: A symmetric central finite difference operator is derived for the small deflection analysis of transversely loaded, ‘thin’, elastic plates, whose stiffnesses vary as functions of the in-plane co-ordinates. The operator is written here in Cartesian co-ordinates for a square mesh, but the same technique can also be used to construct an operator for a rectangular mesh.
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    Notes: It is shown that a singularity occurs in isoparametric finite elements if the mid-side nodes are moved sufficiently from their normal position. By choosing the mid-side node positions on standard isoparametric elements so that the singularity occurs exactly at the corner of an element it is possible to obtain quite accurate solutions to the problem of determining the stress intensity at the tip of a crack. The solutions compare favourably with those obtained using some types of special crack tip elements, but are not as accurate as those given by a crack tip element based on the hybrid principle. However, the hybrid elements are more difficult to use.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 509-533 
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    Keywords: Engineering ; Engineering General
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    Notes: This paper describes the computational aspects of an earlier analytical study1,2 of the problem of maximizing a linear combination of the natural frequencies of vibration of a turbine disc idealization subject to weight and fabrication constraints. The problem was formulated as an optimal control problem and solutions to the adjoint equations of the Pontryagin formalism obtained using perturbation techniques. These techniques transformed the problem to a non-linear program. This paper gives detailed computational solutions to the non-linear program which is characterized by pockets of relative optima. In addition, the dual problem of minimizing the weight subject to a frequency constraint is considered within the optimal control framework. The duality results are confirmed computationally for the two non-linear programs.A description of some practical procedures for improving the convergence characteristics of these highly non-linear programs is included.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 535-550 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: Incremental equations of motion are derived from a Lagrangian variational formulation for the large displacement elastic-plastic and elastic-viscoplastic dynamic analysis of deformable bodies. The material constitutive behaviour is described in terms of the symmetric Piola-Kirchhoff stress and Lagrangian strain tensors. Degenerate isoparametric elements, permitting relaxation of the Kirchhoff-Love hypothesis, are used in a finite element formulation specialized for the analysis of shells of revolution subjected to axisymmetric loading. The linearized incremental equations of motion are solved using direct integration procedures, with added accuracy obtained from application of equilibrium correction at each step. The effectiveness of the numerical techniques is illustrated by the dynamic response analyses carried out on a shallow spherical cap subjected to uniform external step pressure loadings.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 551-561 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: A finite element method is developed by which it is possible to obtain the general solution of an ordinary differential equation directly. The procedure consists of approximating the differential equation with a rectangular matrix equation and of solving the latter equation by using generalized matrix inversion. It is shown in the paper that the homogeneous and inhomogeneous solutions of the two systems correspond and that the approximate solutions produced form the complete general solution of the original differential equation.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 581-599 
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    Notes: A discretization process is described by which it is possible to generate finite difference formulas for arbitrary linear two-dimensional partial differential equations. The process is based on a novel approach to finite difference analysis in which differential operators are approximated by rectangular matrices. In this approach, the discrete form of a complicated operator is obtained by performing simple numerical operations on elementary matrix factors. The analysis is augmented by a listing of a computer program based on the method for the automatic generation of finite difference formulas.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 601-615 
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    Notes: Alternate hybrid stress finite element models in which the internal equilibrium equations are satisfied on the average only, while the equilibrium equations along the interelement boundaries and the static boundary conditions are adhered to exactly a priori, are developed. The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed. Triangular elements for a moderately thick plate and a doubly-curved shallow thin shell are developed. Kinematic displacement modes, convergence criteria and bounds for the direct flexibility-influence coefficient are examined.
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    International Journal for Numerical Methods in Engineering 9 (1975), S. 563-580 
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    Notes: The typical numerical problem associated with finite element approximations is a quadratic programming problem with linear equality constraints. When nodal variables are employed, the coefficient matrix of the constraint equations, [A], acquires a block-diagonal structure. The transformation from polynomial coefficients to nodal variables involves finding a basis for [A] and computing its inverse.Simultaneous satisfaction of completeness and C1 (or higher) continuity requirements establishes linear relationships among the nodal variables and precludes inversion of the basis by exclusively element-level operations.Linear dependencies among the constraint equations and among the nodal variables can be evaluated by the simplex method. The computational procedure is outlined.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1337-1342 
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    Keywords: Engineering ; Engineering General
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    Notes: A finite difference perturbation scheme is developed which allows a simple solution to a wide class of non-linear difurcation problems. The analysis shows that in order to determine the inital post bucking behaviour accurately, it is not necessary to solve more than the linear eigenvalue difference equation with similar accuracy.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1346-1346 
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1347-1358 
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    Notes: Three algorithms are used to compute polynomial approximate solutions to initial value problems. One algorithm is based on the Roberts' minimal degree algorithm, whereas the other two algorithms utilize variants of the second algorithms of Remes. Our analysis indicates that the algorithm based on the Roberts' minimal degree algorithm is superior to the others considered.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1375-1382 
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    Notes: An optimal steady-state control problem governed by an elliptic state equation is solved by several finite element methods. Finite element discretizations are applied to different variational formulations of the problem yielding accurate numerical results as compared with the given analytical solution. It is sated that, for minimum computational effort and high accuracy, ‘mixed’ finite elements requiring only C° continuity, and approximating the control and state functions simultaneously are better suited to similar ‘fourth order’ problems.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1383-1397 
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    Notes: The boundary-integral equation medthod is particularly well suited for solution of stress concentration and elastic fracture mechanics problems. The method was not previously applicable to anisotropic three dimensional problems because no efficient technique existed for calculation of the required point load solution for an infinite body. A technique has been developed to evaluate numerically the anisotropic point load soultions, and used to generate data bases for various materials. An intrpolation technique is used to evaluate the point load solutions efficiently within a higher order boundary-intgral equation code. The effectiveness of the technique is verified by solution of problems involving both uniaxial stress states and stress concentrations.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1407-1413 
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    Notes: An algoritm is presented for point relaxation of an elliptic partial diferential equation at a grid station where its coefficient are discontinuous. Such equations can arise from the use of skewed co-ordinate system based on discotinuous reference functions (lengths) to map a complicated physical geometry onto a simple computational domain. Efficary of the scheme is illustrated by examples involving potential flow in a sharply bent channel and in a conical diffuser.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1493-1506 
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    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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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1-10 
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    Notes: A numerical technique for solving unsteady gas dynamic equations is presented. The technique is based on least squares finite element concepts with elements that are constructed in both space and time. Both linear and quadratic interpolation is used on individual elements. The technique is tested against a problem whose solution is known so that numerical accuracy can be ascertained.
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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. 1630-1630 
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1635-1650 
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    Notes: A diagonal lumped mas formulation with non-zero inertia terms is presented for Ahmad's general shell element. The effect of co-ordinate transformation of the mas matrix is demonstrated. Due to arbitray co-ordinate transformation on nodal variables, the diagonal lumped mas matrix becomes a banded matrix of half bandwidth three. It is shown that with some approximations this matrix can be made diagnoal without effecting the results appreciably. Numerical examples are presented to illustrate the accuracy of the formulation. Similar lumped mass matrix formulation is also given for axisymmetric shell elements.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1651-1657 
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    Notes: A Procedure is presented for generating a class of conformal mappings useful in the formulation of finite-difference problems involving curved boundaries. The method provides a systematic approach that is capable, in principle, of reducing the geometry to a nearly-rectangular domain for a wide variety of partical problems. The introduction of sheared co-ordinates in this computational domain then provides for solution of the problem in a nearly-orthogonal (in fact, nearly-conformal) co-ordinate system, with its boundaries corresponding to co-ordinate lines. The method is based upon the Schwarz-Christoffel transformation and is quite simple to apply. Several examples illustrating the types of geometries which can be treated in this manner are presented.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1677-1696 
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    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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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1705-1715 
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    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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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1743-1759 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A minimum-time, thermal-to-thermal sailplane trajectory optimization problem is formulated as a non-linear optimal control problem. Numerical solutions are obtained using a gradiaent projection algorithm which incorporates conjugate directions of search. Further insight into the nature of the solutions and the computational process is ontained through an analysis of the linearized sailplane dynamics and the necessary conditions for optimality. Numerical results are presented for two sailplane types and various values of thermal strength and distance between thermals. An additional problem is formulated and solved for the case of bounded control rate.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1727-1742 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: For the 12-node quadrilateral isoparametric elements, it is shown that the inverse square root singularity of the strain field at the crack tip can be obtained by the simple technique of collapsing the quadrilateral elements into triangular elements around the crack tip and placing the two side nodes of each side of the triangles at 1/9 and 4/9 of the length of the side from the tip. This is analogous to placing the mid-side nodes at quarter points in the vicinity of the crack tip for the quadratic isoparametric elements. The advantages of this method are that the displacement compatibility is satisfied throughout the region and that there is no need of special crack tip elements. The stress intensity factors can be accurately obtained by using general purpose programs having isoparametric elements such as NASTRAN.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1764-1771 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Upwinded parabolic and cubic elements are derived on a uniform grid of size h for the finite element Galerkin method applied to the solution of the model conduction-convection problem εu″  -  Ku′ = 0, ε, K 〉 0, subject to the boundary conditions u(0) = 1, u (1) = 0. Extension of the results to more complicated problems is indicated. Finally numerical results are given comparing some of the methods derived for a range of L(=Kh/2ε), the grid Peclet number.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1772-1773 
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    Keywords: Engineering ; Engineering General
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1799-1807 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: In this paper, the application of a particular class of computational methods, the alternating direction implicit methods applied to the solution of time dependent heat conduction problems in regions involving circular symmetry is presented.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1819-1839 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: The present paper describes a hybrid stress finite element formulation for geometrically non-linear analysis of thin shell structures. The element properties are derived from an incremental form of Hellinger-Reissner's variational principle in which all quantities are referred to the current configuration of the shell. From this multi-field variational principle, a hybrid stress finite element model is derived using standard matrix notation. Very simple flat triangular and quadrilateral elements are employed in the present study. The resulting non-linear equations are solved by applying the load in finite increments and restoring equilibrium by Newton-Raphson iteratioin. Numerical examples presented in the paper include complete snap-through buckling of cylindrical and spherical shells. It turns out that the present procedure is computationally efficient and accurate for non-linear shell problems of high complexity.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 203-227 
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    Keywords: Engineering ; Engineering General
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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. 229-235 
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    Keywords: Engineering ; Engineering General
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    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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    International Journal for Numerical Methods in Engineering 12 (1978), S. 237-244 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: The various ‘universal’ matrices from which finite element matrices for triangular elements are assembled in many electromagnetics and acoustics problems, can all be derived from a basic set of three fundamental matrices. These represent, respectively, the metric of the linear manifold spanned by the triangle interpolation polynominals, the finite differentiation operator on that same manifold, and a product-embedding operator for the corresponding manifold for interpolation polynomials one order higher. Two of these have already been tabulated and published; the required method for computing the third is given in this paper, along with tables of low-order matrices.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 245-266 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    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 13 (1978) 
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    International Journal for Numerical Methods in Engineering 13 (1978), S. 35-48 
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    Keywords: Engineering ; Engineering General
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    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 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 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. 181-196 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    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), S. 203-228 
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    Notes: A brief review of the structural optimization literature and a description of the present status of optimization methods are presented. The scope of the review is limited and does not profess to cover the entire literature.Results of the optimization of two structures (by the optimality criteria approach) are presented. There are some conclusions and recommendations for future developments in structural optimization.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 977-999 
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    Notes: Two-dimensional C° finite elements of arbitrary polynomial order are developed for a general functional which includes a least squares or potential energy functional as special cases. The elements are ‘hierarchal’ in the sense that the nodal variables for polynomial order p constitute a subset of the nodal variables for order p+1. It is shown that the elemental arrays for high polynomial order may be efficiently computed by using hierarchal elements together with precomputed arrays - i.e., arrays which are computed once, stored on permanent file, and then re-used in all subsequent applications of the program. A number of example problems are solved. Comparisons are made of the relative efficiency of finite element convergergence with mesh refinement (polynomial order held fixed) and with increasing polynomial order (mesh held fixed). The latter approach is found superior.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1001-1024 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: The paper describes a simplified finite element method for high temperature creep problems involving cyclic load and temperature. The principal assumptions, that the cycle time is small compared with characteristic material time scales, is appropriate for the moderate load levels which occur when structures are designed for life times measured in years and decades. Because of the simplicity of the method, thermal loading problems may be studied in some detail. A parametric study of a plane stress problem is presented. Characteristic deformation rates are related to reference stress and timperature quantities, thereby reducing the main features of the solution to a few important structural and material parameters.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1025-1030 
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    Notes: This paper proposes a procedure for the extension of a widely used numerical method1-4 for the solution of systems of singular integral equations, from mixed boundary value problems with two or three parts to those having many parts. The procedure is applied to three previously solved mixed boundary value problems and the result are presented in graphical form. The effectiveness of the given procedure is observed through the results for some limiting and special cases found in the literature.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 387-404 
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    Notes: In the last decade or so finite element techniques have been applied with increased frequency to contaminant transport problems. Whereas most of the attention has focused on finite element approximations of spatial derivatives, standard finite difference techniques are generally used for approximation of the time derivative. Such an approach yields a scheme which is at best second order correct in time. In this study several higher order approximations of the time derivative are developed and analyzed using a finite difference approximation, and Galerkin-type finite element approximations in conjunction with several sets of basis functions. Results obtained with the different schemes exhibit significant improvements in the numerical solution of the convective-dispersive equation.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 405-414 
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    Keywords: Engineering ; Engineering General
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 415-428 
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    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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    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. 437-451 
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    Keywords: Engineering ; Engineering General
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 453-469 
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    Notes: A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The mehod retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency i.e., the resulting matrix system is always symmetric and positive definite,. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1151-1160 
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    Keywords: Engineering ; Engineering General
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    Notes: This paper deals with the numerical solution of linear differential equations of fourth order by finite differences. It points out significant (but usually overlooked) errors which result from the conventional method of imposing the boundary conditions in such problems. Revised finite difference formulas are derived which apply near the boundaries and which eliminate the above errors. Three commonly encountered boundary conditions ar econsidered. These correspond, in the terminology of beam analysis, to a clamped end, to a simply supported end and to a free end. The improvement in accuracy that can be achieved with the revised formulas is illustrated by two representative examples. The revised formulas are shown to reduce the overall error of the numerical solution by a factor of about five in a typical case.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1161-1169 
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    Keywords: Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: An explicit formulation for the dynamic analysis of rivers is presented. The model is based on the finite element method but works like an explicit finite difference scheme. It includes the advantages of both methods: high flexibility and low computer costs.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1171-1184 
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    Notes: The conventional dynamic variational approach and finite element base functions lead to non-diagonal consistent mass matrics which are inappropriate for use with an explicit time integration scheme. In this work, it is shown that if orthogonal base function are used with a mixed variational formulation, then consistant diagonal mass matrices and corresponding sets of spatially discretized field equations are obtained. Although the approach is quite general, the theory is purposely illustrated by a detailed development for one set of base functions. Central difference time integration is incorporated for applications to one-dimensional wave propagation and to Euler-Bernoulli beams. Numerical examples are provided for elastic and elastic-plastic materials.
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    International Journal for Numerical Methods in Engineering 12 (1978), S. 1185-1190 
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    Keywords: Engineering ; Engineering General
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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 13 (1978), S. 17-33 
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    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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