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  • 101
    Electronic Resource
    Electronic Resource
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 14 (1998), S. 1-8 
    ISSN: 1069-8299
    Keywords: frequency condensation method ; large matrix systems ; Guyan method ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A frequency condensation method is presented for solving the eigenvalue problem of a large matrix system. The eigenproblem is reduced to a smaller problem by condensing the stiffness and mass matrices. As distinct from the Guyan method, the frequency condensation method is based on approximation preserving eigenvalues in a preset frequency range and effective procedure of master and slave selection. A numerical example and comparison with subspace iteration and Lanczos methods demonstrate good accuracy and the high performance of this method. © 1998 John Wiley & Sons, Ltd.
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  • 102
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    Communications in Numerical Methods in Engineering 14 (1998), S. 23-34 
    ISSN: 1069-8299
    Keywords: Gauss-Poisson equation ; exact discrete conservation ; discrete divergence ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We consider numerical methods which exactly preserve the Gauss-Poisson equation when solving the charge conservation and Maxwell-Ampère's equations. Apart from the well-known leap-frog method, we present two situations where this property is verified, one with rectangular mesh and functions defined at the center of the cells, and one with a finite volume type of formulation on triangles. © 1998 John Wiley & Sons, Ltd.
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  • 103
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    Communications in Numerical Methods in Engineering 14 (1998), S. 51-58 
    ISSN: 1069-8299
    Keywords: arc-length method ; user control ; accumulated arc-length control ; non-linear analysis ; predefined deformation states ; convergence ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The arc-length method has been widely used in non-linear analysis of structures. While it can handle complex load-deflection paths effectively, the analyst has no control over the load incrementation scheme to achieve convergence to specific locations along the load-deflection path. There are a number of situations in which such control is necessary, for example when convergence to predefined load levels, displacement levels or stress levels is required. This paper describes an improved arc-length method in which this deficiency is overcome. © 1998 John Wiley & Sons, Ltd.
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  • 104
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    Communications in Numerical Methods in Engineering 14 (1998), S. 43-49 
    ISSN: 1069-8299
    Keywords: lumped mass matrix ; Hermitian finite difference operators ; flexural wave propagation ; numeric velocity dispersion ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Hermitian finite difference operators are employed to formulate a new diagonal mass matrix for solving flexural wave propagation problems. The dynamic equilibrium for moments is considered. The consistency of the formulation is evident because these operators explicitly present second-order convergence. Numerical results indicate that the velocity dispersion is very close to the consistent mass matrix. © 1998 John Wiley & Sons, Ltd.
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  • 105
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    Communications in Numerical Methods in Engineering 14 (1998), S. 751-761 
    ISSN: 1069-8299
    Keywords: acoustic ; electromagnetic ; integral equations ; scattering ; time domain ; radar cross-section ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Analysis of high frequency scattering using pulsed illumination generates surface fields which are small over most of the scatterer most of the time. A reformulation of the usual integral equation time domain approach which exploits this is presented. It is shown that cost scaling can be reduced, with costs reduced by an order of magnitude for the examples presented, with negligible accuracy loss. © 1998 John Wiley & Sons, Ltd.
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  • 106
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    Communications in Numerical Methods in Engineering 14 (1998), S. 161-178 
    ISSN: 1069-8299
    Keywords: finite elements ; unsteady incompressible flows ; fractional-step ; equal-order-interpolation ; pQ2Q1 element ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Implementation of an equal-order-interpolation velocity-pressure element pair is presented for the finite element solution of incompressible viscous flows. A fractional-step method is employed for temporal discretization. The element pair, also called a pseudo-biquadratic velocity/bilinear pressure element (pQ2Q1), consists of a bilinear pressure element and bilinear velocity elements defined on subdivisions of the pressure element. This pair satisfies the so-called ‘Ladyzhenskaya-Babuska-Brezzi’ condition. Considerable savings in computational cost are achieved due to the reduced number of elements for pressure. A modification of the element is realized for a better representation of curved surfaces. Two test cases, namely the lid-driven cavity flow and impulsively started circular cylinder in cross-flow, are used to assess the accuracy and efficiency of the element compared to a regular bilinear velocity-pressure (Q1Q1) element pair. Computational results presented show that the pQ2Q1 element solutions require less memory and CPU time compared to Q1Q1 element solutions, for at least the same accuracy. © 1998 John Wiley & Sons, Ltd.
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  • 107
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    Communications in Numerical Methods in Engineering 14 (1998), S. 195-208 
    ISSN: 1069-8299
    Keywords: differential quadrature method ; elastic torsion ; numerical solution ; Poisson equation ; Laplace equation ; geometric mapping ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The governing equation of an elastic prismatic shaft is the two-dimensional Poisson equation defined on the cross-sectional area of the shaft. In this paper, the differential quadrature method (DQM) is employed to solve the Poisson equation on some non-rectangular domains. Singularities, which may appear in the expression of stress components or boundary conditions at a degenerated point of the grid, are removed by means of the Taylor expansion. The results of three examples are compared with the exact solutions. It is shown that accurate results can be achieved by the DQM. In addition, three geometric transformations are conducted in the third example so that the effect of mapping on the convergence and accuracy of results is investigated. It is found that rapid convergence can be fulfilled if the degenerated point of the mesh falls on a Dirichlet boundary. The approach addressed in the paper can be extended to other potential problems governed by either the Poisson equation or the Laplace equation. © 1998 John Wiley & Sons, Ltd.
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  • 108
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    Communications in Numerical Methods in Engineering 14 (1998), S. 209-218 
    ISSN: 1069-8299
    Keywords: three-dimensional convection-diffusion equation ; fourth-order compact scheme ; iterative methods ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We present an explicit fourth-order compact finite difference scheme for approximating the three-dimensional convection-diffusion equation with variable coefficients. This 19-point formula is defined on a uniform cubic grid. We compare the advantages and implementation costs of the new scheme with the standard 7-point scheme in the context of basic iterative methods. Numerical examples are used to verify the fourth-order convergence rate of the scheme and to show that the Gauss-Seidel iterative method converges for large values of the convection coefficients. Some algebraic properties of the coefficient matrices arising from different discretization schemes are compared. We also comment on the potential use of the fourth-order compact scheme with multilevel iterative methods. © 1998 John Wiley & Sons, Ltd.
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  • 109
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    Communications in Numerical Methods in Engineering 14 (1998), S. 241-251 
    ISSN: 1069-8299
    Keywords: finite elements ; pollutant ; saturated porous medium ; semi-implicit method ; velocity correction ; mass transfer ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A practical problem of pollutant migration has been studied for different concentration differences and mass diffusivities using the finite element method. The results indicate that the pollutant takes years to travel 10 m into the water-saturated soil when the mass diffusivity and concentration differences are less. © 1998 John Wiley & Sons, Ltd.
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  • 110
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    Communications in Numerical Methods in Engineering 14 (1998), S. 907-920 
    ISSN: 1069-8299
    Keywords: Stokes problem ; meshless ; MLS interpolant ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper a numerical solution for incompressible Stokes equations using moving least-squares interpolators is developed. This approach does not require an element discretization; just a cloud of points is necessary. This is very attractive for 3D problems and deformable domains. First, taking into consideration that Dirichlet boundary conditions are not applicable a posteriori as in finite elements, a variational weak formulation that includes all kinematic restrictions (Dirichlet and incompressibility) is derived. Then the discretized resultant equations using a moving least-squares (MLS) interpolant for velocity and pressure fields are presented. Finally, the performance of the MLS interpolation is analysed by comparing numerical and analytical solutions, paying attention to the selection of the polynomial degree for the basis function and its orthogonalization. Different aspects of numerical implementation are discussed. © 1998 John Wiley & Sons, Ltd.
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  • 111
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    Communications in Numerical Methods in Engineering 14 (1998), S. 921-930 
    ISSN: 1069-8299
    Keywords: non-linear Richards equation ; mixed-hybrid finite elements ; quasi-Newton methods ; fast-secant methods ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Groundwater flow in partially saturated porous media is modelled by using the non-linear Richards equation, which is discretized in the present work by using linear mixed-hybrid finite elements.The discretization produces an algebraic non-linear system, which can be solved by an iterative fixed-point algorithm, the Picard method. The convergence rate is linear, and may be too poor for practical applications. A superlinear convergence rate is obtained by considering a Broyden-type approach, based on the Shermann-Morrison formula.The local character of the Broyden method can be overcome by an accurate estimate of the initial solution, that is by appropriately initializing the computation via some (relaxed) Picard iterations. This strategy needs a convergence criterion to decide when switching from the Picard to the quasi-Newton method, which is crucial for the effectiveness of the scheme, as illustrated by some numerical experiments.We also consider the non-linear algebraic problem from a different viewpoint. Instead of applying the quasi-Newton method directly to such a non-linear system, we applied it to the non-linear function tied to the Picard scheme. Each function evaluation requested by such an algorithm corresponds to a local step of the Picard method, which is then used to compute a Broyden displacement. The present technique can be seen as an accelerated Picard algorithm.We compare the performances of these algorithms when applied to a stationary and a time-dependent benchmark problem. © 1998 John Wiley & Sons, Ltd.
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  • 112
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    Communications in Numerical Methods in Engineering 14 (1998), S. 897-906 
    ISSN: 1069-8299
    Keywords: preconditioning ; conjugate gradient ; finite elements ; shells ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Preconditioned Krylov subspace methods have proved to be efficient in solving large, sparse linear systems in many areas of scientific computing. The success of these methods in many cases is due to the existence of good preconditioning techniques. In problems of structural mechanics, like the analysis of heat transfer and deformation of solid bodies, iterative solution of the linear equation system can result in a significant reduction of computing time. Also many preconditioning techniques can be applied to these problems, thus facilitating the choice of an optimal preconditioning on the particular computer architecture available.However, in the analysis of thin shells the situation is not so transparent. It is well known that the stiffness matrices generated by the FE discretization of thin shells are very ill-conditioned. Thus, many preconditioning techniques fail to converge or they converge too slowly to be competitive with direct solvers. In this study, the performance of some general preconditioning techniques on shell problems is examined. © 1998 John Wiley & Sons, Ltd.
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  • 113
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    Communications in Numerical Methods in Engineering 14 (1998), S. 959-961 
    ISSN: 1069-8299
    Keywords: Stokes problem ; finite element ; eigensystems ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The use of mixed finite element methods in discretizing the Stokes equations leads to systems involving the so-called pressure matrix. Some new spectral properties of this important matrix are here presented for the Q1-P0 element. © 1998 John Wiley & Sons, Ltd.
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  • 114
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    Communications in Numerical Methods in Engineering 14 (1998), S. 941-957 
    ISSN: 1069-8299
    Keywords: geometrical non-linear ; UL formulation ; drilling degrees of freedom ; generalized conforming ; arc-length method ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An updated Lagrangian formulation of the generalized conforming flat shell element with drilling degrees of freedom is derived based on the incremental equation of virtual work of a three-dimensional (3D) continuum for a purely geometric non-linear analysis of the space structure. While solving the non-linear equations, the Euler-Newton method and modified Euler-Newton method are used in static analyses, and the modified arc-length method and Newton arc-length method for post-buckling problems. A number of numerical examples are given to demonstrate the effectiveness of the proposed approach and programs. © 1998 John Wiley & Sons, Ltd.
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  • 115
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    Communications in Numerical Methods in Engineering 14 (1998), S. 963-975 
    ISSN: 1069-8299
    Keywords: cylindrical gears ; load distribution ; stress ; deformation ; stiffness ; FEM ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A method, namely the normal stiffness matrix along contact line (NSMACL) method, for analysing gears is proposed. Three-dimensional (3D) finite element (FE) solid models have been established. Cylindrical gears - spur and helical, external and internal, hobbling and slotting, different parameters and materials - can be analysed using these models. Results such as load distribution along the contact lines, deformations and stiffness at any contact position, and contact stresses are presented. Calculation results show that the trend of gear tooth deformation coincides with the tested ones using the dynamic speckle photography method. © 1998 John Wiley & Sons, Ltd.
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  • 116
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    Communications in Numerical Methods in Engineering 14 (1998), S. 1013-1025 
    ISSN: 1069-8299
    Keywords: assumed natural strain ; plate bending ; triangular ; finite element ; six-node ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper, a six-node triangular C0 plate bending element is developed by the assumed natural strain method. In the element, all the sampled natural transverse shear strains are chosen such that the latter has a favourable constraint index and the strains are optimized with respect to a linear pure moment field. The element passes the patch tests, yields satisfactory accuracy and shows no sign of shear locking in all the problems considered. Copyright © 1998 John Wiley & Sons, Ltd.
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  • 117
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    Communications in Numerical Methods in Engineering 14 (1998), S. 463-472 
    ISSN: 1069-8299
    Keywords: multibody systems ; non-linear analysis ; static equilibrium ; constrained optimization ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper presents four different approaches to the solution of the non-linear static-equilibrium problem in complex linkages, including rigid and elastic elements. The error function is based on the potential of the system, and includes rigid elements by means of non-linear constraints. To this end use is made of Lagrange multipliers, along with the primal-dual method, penalty functions and weighted stiffness, comparisons being made between them. A Newton-Raphson method is used in seeking function minima for equilibrium positions. This procedure is also directly applicable to the other linkage and multibody position problems. © 1998 John Wiley & Sons, Ltd.
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  • 118
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    Communications in Numerical Methods in Engineering 14 (1998), S. 473-491 
    ISSN: 1069-8299
    Keywords: temporal acceleration ; viscoelastic ; recovery ; Taylor-Galerkin ; finite elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper presents a finite element study based on a technique associated with time extrapolation to accelerate the convergence rate to the steady state for viscoelastic flows. The approach adopted is a local extrapolation method attributed to Neville. Temporal extrapolation is embedded within a time-marching Taylor-Galerkin/pressure-correction scheme as applied to the solution of model channel flow, 4:1 plane contraction flow and flow past a circular cylinder. In particular, consideration is given to obtaining steady-state solutions for an Oldroyd-B model. When extrapolation is performed for stress and velocity or pressure, then stress and velocity overshoot, which consequently leads to divergence. In contrast, a stable numerical scheme emerges when only the stress is extrapolated. © 1998 John Wiley & Sons, Ltd.
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  • 119
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    Communications in Numerical Methods in Engineering 14 (1998), S. 505-517 
    ISSN: 1069-8299
    Keywords: hypersingular integral equation ; continuous elements ; stress analysis ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper, a method for the evaluation of boundary stresses directly from the BEM solution of elastostatic problems using the static boundary integral equation is presented. The technique is valid for corners and edges as well as smooth points on the boundary, and involves defining a new interpolation function for continuous elements which incorporate certain continuity conditions arising from the hypersingular nature of the integrals involved. An integration technique based on the singularity subtraction method using series expansions is adopted for the hypersingular intergrals. Results are shown to be more accurate than those obtained with conventional techniques. © 1998 John Wiley & Sons, Ltd.
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  • 120
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    Communications in Numerical Methods in Engineering 14 (1998), S. 539-547 
    ISSN: 1069-8299
    Keywords: boundary-only element analysis ; thermal cracking ; crack trajectory ; singular elements ; traction formula ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A boundary element procedure is developed concerning the prediction of the quasistatic crack growth in uniformly heated bimaterials. This procedure assumes the existence of an initial small crack in one of the two phases, and further cracking progress from this point due to thermal loading. The resulting mixed boundary value problem is solved by applying an incremental boundary-only method in conjunction with the multidomain technique. Fracture characterizing parameters are evaluated utilizing special crack tip singular elements and appropriate formulas. The crack path is predicted using the strain energy release rate criterion, and the mesh is updated at the end of each increment. The presented results are in good agreement with previously reported experimental results and those obtained by the finite element method. Various numerical studies were conducted and interpreted concerning crack-path dependence on individual material property mismatch. © 1998 John Wiley & Sons, Ltd.
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  • 121
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    Communications in Numerical Methods in Engineering 14 (1998), S. 1119-1131 
    ISSN: 1069-8299
    Keywords: Reissner-Mindlin plate ; finite element method ; p version ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We consider the finite element (FE) approximation of the Reissner-Mindlin (RM) plate model, and indicate how to design meshes that yield accurate results when the p/hp version of the standard FE method is used. These guidelines allow quantities of engineering interest to be predicted numerically with great confidence near the boundary. We illustrate this through numerical computations in the case when both boundary layers and corner singularities are present. Copyright © 1998 John Wiley & Sons, Ltd.
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  • 122
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    Communications in Numerical Methods in Engineering 14 (1998), S. 1133-1141 
    ISSN: 1069-8299
    Keywords: differential quadrature element method ; rectangular plate ; static analysis ; free vibration analysis ; numerical method ; stepped plate ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The differential quadrature (DQ) element method proposed by Wang and Gu in 1997 has been extended to analyse rectangular plate problems. The methodology is worked out in detail and some numerical examples are given. Copyright © 1998 John Wiley & Sons, Ltd.
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  • 123
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    Communications in Numerical Methods in Engineering 14 (1998), S. 1143-1154 
    ISSN: 1069-8299
    Keywords: super-element ; cracked plate ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: On the formulation of a super-element for the dynamic problem of a cracked plate, a geometric series of similar elements is designed. This group of elements is generated layer by layer, approaching infinitely small size around the point of singularity. The relations between similar elements for the stiffness matrix and the mass matrix are established. The concept of matrix condensation is applied to formulate the super-element by using these relationships. This method presents a feasible approach to the solution of the cracked problem with an arbitrary order of stress singularity. The method is computationally economical. Copyright © 1998 John Wiley & Sons, Ltd.
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  • 124
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    Communications in Numerical Methods in Engineering 14 (1998), S. 1171-1179 
    ISSN: 1069-8299
    Keywords: linear θ method ; time-domain BEM ; stability ; scalar wave ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A linear θ method is used in this paper to improve the stability of the standard time-domain BEM formulation. The time-stepping procedure is similar to that of the Wilson θ method; however, unlike in the FEM, where linear time variation of acceleration (for elastodynamic problems) is assumed, here linear time variation for both potential and flux (for scalar waves) is assumed in the time interval θΔt. A comparison between numerical results obtained from the standard formulation and from the linear θ method studied here shows the latter to be more stable than the former. The effect of varying θ for different values of time steps is also studied in this paper. Copyright © 1998 John Wiley & Sons, Ltd.
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  • 125
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    Communications in Numerical Methods in Engineering 14 (1998), S. 1155-1170 
    ISSN: 1069-8299
    Keywords: shell problem ; Naghdi model ; finite elements ; mixed elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The subject of this work is the construction of some special finite elements for the numerical solution of Naghdi cylindrical shell problems. The standard numerical approximation of the shell problem is subjected to the shear and membrane locking phenomenon, i.e. the numerical solution degenerates for low thickness. The most common way to avoid locking is the use of modified bilinear forms to describe the shear and membrane energy of the shell. In this paper we build a family of special finite elements that still follow the above strategy by introducing a linear operator that reduces the influence both of the shear and membrane energy terms. The main idea comes from the non-standard mixed interpolated tensorial components (MITC) formulation for Reissner-Mindlin plates. The performance of the new elements is then tested for solving benchmark problems involving very thin shells. The results show both the properties of convergence and robustness. Copyright © 1998 John Wiley & Sons, Ltd.
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  • 126
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    Numerical Linear Algebra with Applications 5 (1998), S. 33-55 
    ISSN: 1070-5325
    Keywords: eigenvalues ; Arnoldi ; Krylov subspaces ; preconditioning ; Davidson's method ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The problem of finding interior eigenvalues of a large nonsymmetric matrix is examined. A procedure for extracting approximate eigenpairs from a subspace is discussed. It is related to the Rayleigh-Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other approximate eigenvalues are generated. This procedure can be applied to the Arnoldi method, to preconditioning methods, and to other methods for nonsymmetric eigenvalue problems that use the Rayleigh-Ritz procedure. The subject of estimating the boundary of the entire spectrum is briefly discussed, and the importance of preconditioning for interior eigenvalue problems is mentioned. © 1998 John Wiley & Sons, Ltd.
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  • 127
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    Numerical Linear Algebra with Applications 5 (1998), S. 79-99 
    ISSN: 1070-5325
    Keywords: bidiagonalization ; least squares ; minimum norm solution ; rank-deficient ; regularization ; Riley-Golub iteration ; singular value decomposition ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we consider the solution of linear least squares problems          minx∥Ax - b∥22 where the matrix A ∊ Rm × n is rank deficient. Put p = min{m, n}, let σi, i = 1, 2,…, p, denote the singular values of A, and let ui and vi denote the corresponding left and right singular vectors. Then the minimum norm solution of the least squares problem has the form x* = ∫ri = 1(uTib/σi)vi, where r ≤ p is the rank of A.The Riley-Golub iteration,          xk + 1 = arg minx{∥Ax - b∥22 + λ∥x - xk∥22} converges to the minimum norm solution if x0 is chosen equal to zero. The iteration is implemented so that it takes advantage of a bidiagonal decomposition of A. Thus modified, the iteration requires only O(p) flops (floating point operations). A further gain of using the bidiagonalization of A is that both the singular values σi and the scalar products uTib can be computed at marginal extra cost. Moreover, we determine the regularization parameter, λ, and the number of iterations, k, in a way that minimizes the difference x* - xk with respect to a certain norm. Explicit rules are derived for calculating these parameters.One advantage of our approach is that the numerical rank can be easily determined by using the singular values. Furthermore, by the iterative procedure, x* is approximated without computing the singular vectors of A. This gives a fast and reliable method for approximating minimum norm solutions of well-conditioned rank-deficient least squares problems. Numerical experiments illustrate the viability of our ideas, and demonstrate that the new method gives more accurate approximations than an approach based on a QR decomposition with column pivoting. © 1998 John Wiley & Sons, Ltd.
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    Numerical Linear Algebra with Applications 5 (1998), S. 253-274 
    ISSN: 1070-5325
    Keywords: stochastic matrix ; aggregation/disaggregation iterative methods ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: An aggregation/disaggregation iterative algorithm for computing stationary probability vectors of stochastic matrices is analysed. Two convergence results are presented. First, it is shown that fast, global convergence can be achieved provided that a sufficiently high number of relaxations is performed on the fine level. Second, local convergence is shown to take place with just one relaxation performed on the fine level. The convergence proofs are general and require no assumptions on the magnitude of off-diagonal elements (blocks). Furthermore, a relationship between the errors on the fine and on the coarse level is described. To illustrate the theory, the results of some numerical experiments are presented. © 1998 John Wiley & Sons, Ltd.
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  • 129
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    Numerical Linear Algebra with Applications 5 (1998), S. 321-345 
    ISSN: 1070-5325
    Keywords: second order elliptic problems ; Dirichlet boundary conditions ; mixed finite elements ; preconditioning ; domain embedding ; auxiliary space method ; non-conforming finite elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we study block diagonal preconditioners for mixed systems derived from the Dirichlet problems for second order elliptic equations. The main purpose is to discuss how an embedding of the original computational domain into a simpler extended can be utilized in this case. We show that a family of uniform preconditioners for the corresponding problem on the extended, or fictitious, domain leads directly to uniform preconditioners for the original problem. This is in contrast to the situation for the standard finite element method, where the domain embedding approach for the Dirichlet problem is less obvious. Copyright © 1999 John Wiley & Sons, Ltd.
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  • 130
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    Numerical Linear Algebra with Applications 5 (1998), S. 377-399 
    ISSN: 1070-5325
    Keywords: iterative methods ; operator trigonometry ; anti-eigenvector ; Dirichlet problem ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The new operator-trigonometric theory for iterative linear solvers is illustrated by working out its details for the classical model problem for numerical partial differential equations: the Dirichlet problem on the unit square. Copyright © 1999 John Wiley & Sons, Ltd.
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  • 131
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    Numerical Linear Algebra with Applications 5 (1998), S. 363-376 
    ISSN: 1070-5325
    Keywords: convergence ; multigrid ; sparse grids ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The convergence behaviour of solution algorithms is investigated for the anisotropic Poisson problem on partially ordered, sparse families of regular grids in 3D. In order to study multilevel techniques on sparse families of grids, first we consider the convergence of a two-level algorithm that applies semi-coarsening successively in each of the coordinate directions. This algorithm shows good convergence, but recursive application of the successive semi-coarsening is not sufficiently efficient. Therefore we introduce another algorithm, which uses collective 3D semi-coarsened coarse grid corrections. The convergence behaviour of this collective version is worse, due to the lack of correspondence between the solutions on the different grids. By solving for the trivial solution we demonstrate that a good convergence behaviour of the collective version of the algorithm can be retained when the different solutions are sufficiently coherent. In order to solve also non-trivial problems, we develop a defect correction process. This algorithm makes use of hierarchical smoothing in order to deal with the problems related to the lack of coherence between the solutions on the different grids. Now good convergence rates are obtained also for non-trivial solutions. All convergence results are obtained for two-level processes. The results show convergence rates which are bounded, independent of the discretisation level and of the anisotropy in the problem. Copyright © 1999 John Wiley & Sons, Ltd.
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    Numerical Linear Algebra with Applications 5 (1998), S. 319-319 
    ISSN: 1070-5325
    Keywords: Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    Numerical Linear Algebra with Applications 5 (1998), S. 401-440 
    ISSN: 1070-5325
    Keywords: thin shell problems ; linear partial differential equations ; parallel computing ; multilevel methods ; additive splittings ; finite element methods ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In recent years multilevel preconditioners like BPX have become more and more popular for solving second-order elliptic finite element discretizations by iterative methods. P. Oswald has adapted these methods for discretizations of the fourth order biharmonic problem by rectangular conforming Bogner-Fox-Schmit elements and non-conforming Adini elements and has derived optimal estimates for the condition numbers of the preconditioned linear systems. In this paper we generalize the results from Oswald to the construction of parallel BPX and multilevel diagonal scaling (MDS-BPX) preconditioners for the elasticity problem of thin smooth shells in connection with Koiter's shell theory. We use the two discretizations mentioned above and the preconditioned conjugate gradient method as iterative method. The parallelization concept is based on a non-overlapping domain decomposition data structure. We describe the implementations of the parallel multilevel preconditioners. Finally, we show numerical results for some shells representing elliptic, parabolic, hyperbolic and more complicated classes. In addition, the influence of the thickness parameter and the loading on the preconditioner are investigated experimentally. Copyright © 1999 John Wiley & Sons, Ltd.
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  • 134
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    Numerical Linear Algebra with Applications 5 (1998), S. 441-459 
    ISSN: 1070-5325
    Keywords: iterative methods for linear systems ; acceleration of convergence ; preconditioning ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider algebraic multilevel preconditioning methods based on the recursive use of a 2 × 2 block incomplete factorization procedure in which the Schur complement is approximated by a coarse grid matrix. As is well known, for discrete second-order elliptic PDEs, optimal convergence properties are proved for such basic two-level schemes under mild assumptions on the PDE coefficients, but their recursive use in a simple V-cycle algorithm does not generally lead to optimal order convergence.In the present paper, we analyse the combination of these techniques with a smoothing procedure much the same as the one used in standard multigrid algorithms, except that smoothing is not required on the finest grid. Theoretical results prove optimal convergence properties for the V-cycle under an assumption similar to the ‘approximation property’ of the classical multigrid convergence theory. On the other hand, numerical experiments made on both 2D and 3D problems show that the condition number is close to that of the two-level method. Further, the method appears robust in the presence of discontinuity and anisotropy, even when the material interfaces are not aligned with the coarse grid. Copyright © 1999 John Wiley & Sons, Ltd.
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  • 135
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    Numerical Linear Algebra with Applications 5 (1998), S. 461-474 
    ISSN: 1070-5325
    Keywords: diagonally dominant matrices ; matrix cones ; Dykstra's algorithm ; Kuhn-Tucker conditions ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We solve the problem of minimizing the distance from a given matrix to the set of symmetric and diagonally dominant matrices. First, we characterize the projection onto the cone of diagonally dominant matrices with positive diagonal, and then we apply Dykstra's alternating projection algorithm on this cone and on the subspace of symmetric matrices to obtain the solution. We discuss implementation details and present encouraging preliminary numerical results. Copyright © 1999 John Wiley & Sons, Ltd.
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  • 136
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    Numerical Linear Algebra with Applications 5 (1998), S. 57-71 
    ISSN: 1070-5325
    Keywords: preconditioning ; sparse approximate inverse ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We investigate different methods for computing a sparse approximate inverse M for a given sparse matrix A by minimizing ∥AM - E∥ in the Frobenius norm. Such methods are very useful for deriving preconditioners in iterative solvers, especially in a parallel environment. We compare different strategies for choosing the sparsity structure of M and different ways for solving the small least squares problem that are related to the computation of each column of M. Especially we show how we can take full advantage of the sparsity of A. © 1998 John Wiley & Sons, Ltd.
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  • 137
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    Numerical Linear Algebra with Applications 5 (1998), S. 299-311 
    ISSN: 1070-5325
    Keywords: iterative methods ; singular linear systems ; index of a matrix ; M-splittings ; multi-splittings ; Q-matrices ; Markov chains ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Given a singular M-matrix of a linear system, convergent conditions under which iterative schemes based on M-multisplittings are studied. Two of those conditions, the index of the iteration matrix and its spectral radius are investigated and related to those of the M-matrix. Furthermore, a parallel multisplitting iteration scheme for solving singular linear systems is suggested which can be applied to practical problems such as Poisson and elasticity problems under certain boundary conditions, the Neumann problem, and in Markov chains. A discussion of that multisplitting scheme, based on Gauss-Seidel type splittings is given for computing the stationary distribution vector of Markov chains. In this case a computational viable algorithm can be constructed, since only the nonsingularity of one weighting matrix of the multisplitting is needed. © 1998 John Wiley & Sons, Ltd.
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    Numerical Linear Algebra with Applications 5 (1998), S. 275-297 
    ISSN: 1070-5325
    Keywords: GMRES method ; Krylov subspace methods ; non-symmetric linear systems ; homotopy curve tracking ; sparse matrix ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: GMRES(k) is widely used for solving non-symmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram-Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(k) procedure outperforms standard GMRES(k), several other GMRES-like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified Gram-Schmidt process in GMRES, can fail causing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(k) with the orthogonalization performed by Householder transformations succeeds whenever GMRES(k) with the orthogonalization performed by the modified Gram-Schmidt process fails, and the extra cost of computing Householder transformations is justified for these applications. © 1998 John Wiley & Sons, Ltd.
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    Numerical Linear Algebra with Applications 5 (1998), S. 347-362 
    ISSN: 1070-5325
    Keywords: almost incompressible elasticity ; finite elements ; semi-coarsening refinement ; algebraic multilevel ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The constant γ in the strengthened Cauchy-Buniakowski-Schwarc (CBS) inequality plays a key role in the convergence analysis of the multilevel iterative methods. We consider in this paper the approximation of the two-dimensional elasticity problem by bilinear rectangle finite elements. Two semi-coarsening refinement procedures are studied. We prove for both cases new estimates of the constant γ, uniformly on the Poisson ratio.As a result of the presented analysis we obtain an optimal order algebraic multiLevel iteration (AMLI) method for the case of balanced semi-coarsening mesh refinement. The total computational complexity of the algorithm is proportional to the size of the discrete problem with a proportionality constant independent of the Poisson ratio, that is, the algorithm is of optimal order for almost incompressible elasticity problems. Copyright © 1999 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 59-63 
    ISSN: 1069-8299
    Keywords: Delaunay triangulation ; mesh colouring ; mesh coarsening ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new algorithm is described for automatic coarsening of Delaunay triangulations. The method automatically identifies the nodes of the mesh which are candidates for deletion in each region to be coarsened. The mesh coarsening procedure is to regenerate a Delaunay triangulation for all the remaining nodes in the solution domain. The adapted mesh retains a high quality for two reasons: (i) because a set of nodes have been used to obtain the coarse mesh, which are equidistant apart in some sense defined by the original mesh; (ii) because the triangulation is Delaunay. A two-dimensional example demonstrates the quality of the mesh after coarsening. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 35-42 
    ISSN: 1069-8299
    Keywords: potential problems ; rectangular finite element ; consistency criteria ; rigid body modes ; constant straining modes ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This short paper demonstrates how the matrix for a 4-node potential finite element can be derived from the conditions of ‘rigid body movement’, symmetry and consistency, and the requirement that the element solve exactly the case of φ=xy. This derivation is intended to be a simple illustration of the ideas of Bergan and others. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 135-149 
    ISSN: 1069-8299
    Keywords: multiphase immiscible flow ; deforming porous media ; fully coupled ; soil displacements ; fluid pressures ; finite element method ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A fully coupled numerical model has been developed which describes multiphase fluid flow through soil: namely gas, water and a non-aqueous phase liquid (NAPL) in a deforming porous media for subsurface systems. A multiphase flow model, based on the two-phase flow model of Brooks and Corey, is presented to express the dependence of saturation and relative permeability on the capillary pressure. Non-linear saturation and relative permeability functions are incorporated into a Galerkin finite element model which is subsequently used to simulate multiphase immiscible fluid flow under saturated and unsaturated conditions in porous media. The governing partial differential equations, in terms of soil displacements and fluid pressures, which are coupled and non-linear, are solved by the finite element method. Numerical implementation of the formulation is discussed, and example problems demonstrate the model and solution procedure. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 151-160 
    ISSN: 1069-8299
    Keywords: genetic algorithms ; evolutionary-genetic analogy ; heat transfer ; finite elements ; numerical analysis ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new approach was developed for the analysis of problems of physics governed by laws of conservation. The methodology establishes an analogy between various aspects of evolution and genetics of life processes and inanimate physical systems. This analogy is then used to generate solutions to physical problems in a similar way to which organisms in nature solve problems imposed by environmental constraints. Finite difference and finite elements approximations are used in the numerical implementation of the method and a simple two-dimensional problem is solved to illustrate the method. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 181-194 
    ISSN: 1069-8299
    Keywords: oscillating flow ; Keulegan-Carpenter number ; finite element method ; in-line force ; transverse force ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The results of a numerical study of the viscous oscillating flow around a circular cylinder at Reynolds number=200 and Keulegan-Carpenter number=20 are presented. The finite element method was used for the solution of the Navier-Stokes equations, in the formulation where the streamfunction and the vorticity are the field variables. The computation revealed a complicated flow pattern at which the flow is not periodic but switches between different modes at various oscillation cycles. The vortex shedding patterns occurring at different cycles were identified by means of numerical flow visualization. The traces of the hydrodynamic forces were derived from the solution, and the hydrodynamic coefficients of the in-line force were evaluated. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 219-228 
    ISSN: 1069-8299
    Keywords: boundary integral ; spectral strip ; curvature geometries ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The non-element boundary spectral strip method is developed for solution of problems in curvature domains. The method, which uses a spectral Fourier approximation for expansion of the variables along the boundaries, shows an exponential convergence behaviour. A remarkable reduction in problem size and improved accuracy of results is achieved by the method as compared to the boundary element method. The method is illustrated by some physical problems from elastostatics. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 229-240 
    ISSN: 1069-8299
    Keywords: filling of thin section ; finite element method ; surface tension ; interface element ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An interface element to model the pressure discontinuity due to surface tension when applied to the filling of a thin section cavity is presented. The equations used to form the element matrix for the interface element are the line integral form of the continuity and momentum equations. During the development of the finite element model, the pressure difference across the free surface due to surface tension is treated as an additional traction and is applied to all element sides which form the free surface. Simple numerical examples are then presented to illustrate the technique on the filling of a rectangular thin section cavity. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 103-134 
    ISSN: 1069-8299
    Keywords: computational quantifier elimination ; edge crack ; feasibility (consistency) conditions ; fracture mechanics ; numerical methods in engineering ; parametric inequality constraints ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper, an attempt is made to show the usefulness of computational quantifier elimination (CQE) techniques in computer algebra inside classical numerical methods in engineering for the derivation of feasibility (consistency) conditions in problems with weakly parametric linear inequality constraints (with the parameters appearing only in their right-hand sides). A simple, but non-trivial, straight edge-crack problem in fracture mechanics under linear inequality constraints both on the applied loading along the crack faces and on the value of the stress intensity factor at the crack tip (associated with the Green/weight function method, numerical approximations and classical numerical integration) is used for an elementary illustration of the proposed approach. In this application, the method used tries to imitate the theoretical principle of the linear programming methods. The manually obtained related result is expressed as a disjunction of conjunctions of inequalities (as is frequently the case in similar CQE problems), and concrete numerical results are also displayed. The related influence of various approximations and the application of the trapezoidal quadrature rule are also considered in some detail. Further possibilities could concern the application of the approach to other numerical methods in engineering (such as to the finite and the boundary element methods, to singular and hypersingular integral equation methods, etc.) combined with efficient algorithms for linear inequality constraints such as the old Fourier and the recent Weispfenning elimination methods. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 797-808 
    ISSN: 1069-8299
    Keywords: mesh equidistribution ; area preserving map ; singular BVP ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper, an adaptive mesh method is employed to solve a class of singular boundary value problems. The approach is based on an area-preserving map and some mesh shape control in two-dimensional space. Two benchmark problems, which both involve singularities in physical domains, are tested. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 839-847 
    ISSN: 1069-8299
    Keywords: coupled models of dynamic electroelasticity ; generalized solutions ; metrics weaker than L2 ; spectrum of a priori estimates ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper we deal with a model of coupled non-stationary electroelasticity with displacements and potential given on the boundary of a body. We construct a numerical scheme for modelling electromechanical interactions in the body, and present a spectrum of a priori estimates for the solution of this scheme. Such estimates allow us to prove the convergence of the scheme to a generalized solution of the differential problem from W22(QT) with the second order of accuracy in a weaker than L2 metric. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 821-837 
    ISSN: 1069-8299
    Keywords: diagonally implicit Runge-Kutta methods ; parallelism ; stiffness ; stability ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper we propose a parallel diagonal iteration process for solving a low-order implicit Runge-Kutta method of Lagrange type. The resulting scheme can be regarded as a parallel singly diagonally implicit Runge-Kutta (PSDIRK) method and it is strongly A-stable when the classical linear test model is used. On a two-processor computer, this method requires the solution of two implicit relations (sequential time units) per step and per processor. We compare our method with some sequential and parallel methods from the literature for solving linear as well as non-linear stiff problems and semidiscretized convection-diffusion equations. The numerical experiments show the behaviour of our method with regard to the other methods. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 871-877 
    ISSN: 1069-8299
    Keywords: nearest neighbours ; Voronoi diagram ; Delaunay triangulation ; computational geometry ; algorithms ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper deals with the largest interior angle property of the Voronoi nearest neighbours and the Delaunay triangulation in two dimensions. It is indicated that each interior angle opposite to an edge of a Delaunay triangle is the largest of the interior angles opposite the same edge of all possible triangles on the same side. Based on this property, the algorithms for finding the Voronoi neighbours of a point (site) and for constructing the Delaunay triangulation are presented. The algorithms are especially useful for the meshless methods developed in recent years. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 277-285 
    ISSN: 1069-8299
    Keywords: numerical simulation ; steel ; quenching ; finite volume method ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: On the basis of the control volume method the algorithm and computer program for prediction of the hardness distribution in quenched steel specimens with complex geometries have been developed. The algorithm and computer program are designed to solve 2D situation problems such as the quenching of complex cylinders, cones, spheres, etc. The computer program consists of three parts: automatic computation of domain and grid generation, computation of cooling curve in grid-points, and computation of hardness in grid-points. The mathematical model has been tested experimentally. The test showed that the model describes the hardness distribution in a quenched steel specimen of a complex form, with quite satisfactory accuracy. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 271-275 
    ISSN: 1069-8299
    Keywords: basis transformation ; interpolations ; finite elements ; thin plates ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The 9 df thin plate element of Mohr and Mohr uses cubic interpolation to obtain values of w at the third points of the element sides, in turn interpolating from these and the vertex values within the element. Recently this element has been modified and successfully applied to ‘potential’ problems. Subsequently it was found that the interpolations of the element of Bazeley et al. (1965, 1968) gave identical results for potential problems. In the present paper it is shown that this is because the interpolations of the two elements are exactly equivalent. © 1998 John Wiley & Sons, Ltd.
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  • 154
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    Communications in Numerical Methods in Engineering 14 (1998), S. 313-319 
    ISSN: 1069-8299
    Keywords: fluid mechanics ; vortex dynamics ; viscous flow ; Navier-Stokes equation ; vortex methods ; splitting procedure ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A numerical scheme has been obtained rigorously from the initial-boundary value problem for the Navier-Stokes (NS) equations in the context of the vortex method. The technique is based on a transformation of the NS equation into a parabolic equation which has an exact solution. The numerical scheme is derived by expanding the exact solution in Taylor series in powers of a small time interval. Numerical implementation is developed with use of vortex particles to represent the vortex flow domain. The method is used to solve practical engineering problems. The technique can also incorporate turbulence modelling. © 1998 John Wiley & Sons, Ltd.
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  • 155
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    Communications in Numerical Methods in Engineering 14 (1998), S. 347-354 
    ISSN: 1069-8299
    Keywords: thin wire loop antenna ; integro-differential equation ; frequency domain ; current distribution ; the weak formulation ; finite element technique ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The circular loop antenna is analysed by using the electric field integro-differential equation in the frequency domain. The weak form of the integro-differential is derived and then the current distribution along the circular loop antenna is calculated by solving the resulting equation via the finite element technique. Accurate results are obtained using the linear shape and test functions. © 1998 John Wiley & Sons, Ltd.
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  • 156
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    Communications in Numerical Methods in Engineering 14 (1998), S. 355-365 
    ISSN: 1069-8299
    Keywords: transient dynamics ; interaction of the crack sides ; crack propagation ; time domain formulation ; boundary element method ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: For two-dimensional solids the multiregion concept and the time-domain approach in the boundary element method are employed to model cracks in transient dynamics. The propagation of cracks and the closing and opening of crack sides are simulated by disconnecting and connecting degrees of freedom of a stationary mesh, as was also demonstrated recently in statics by Beer. An iterative technique is developed to determine the changes of the extent of the area where there is connection, contact or no contact at the interface between dynamically loaded regions. © 1998 John Wiley & Sons, Ltd.
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  • 157
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    Communications in Numerical Methods in Engineering 14 (1998), S. 381-392 
    ISSN: 1069-8299
    Keywords: porous medium ; fluid flow ; electromechanical coupling ; FEM ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We consider a coupled problem of the deformation of a porous solid, flow of a compressible fluid and the electrical field in the mixture. The governing equations consist of balance of the linear momentum of solid and of fluid, continuity equations of the fluid and current density, and a generalized form of Darcy's law which includes electrokinetic coupling. The compressibility of the solid and the fluid are taken into account. We transform these equations to the corresponding finite element relations by employing the principle of virtual work and the Galerkin procedure. The nodal point variables in our general formulation are displacements of solid, fluid pore pressure, relative velocity of the fluid and electrical potential. Derivation of the FE equations is presented for small displacements and elastic solid, which can further be generalized to large displacements and inelastic behaviour of the solid skeleton.According to this formulation we can include general boundary conditions for the solid, relative velocity of the fluid, fluid pressure, current density and electrical potential. The dynamic-type non-symmetric system of equations is solved through the Newmark procedure, while in the case of neglect of inertial terms we use the Euler method.Numerical examples, solved by our general-purpose FE package PAK, are taken from biomechanics. The results are compared with those available in the literature, demonstrating the correctness and generality of the procedure presented. © 1998 John Wiley & Sons, Ltd.
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  • 158
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    Communications in Numerical Methods in Engineering 14 (1998), S. 397-407 
    ISSN: 1069-8299
    Keywords: 3D mesh generation ; adaptive refinement ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An adaptive finite element (FE) method for the solution of three-dimensional elasto-static problems is described. The computational domain is represented by an assembly of tetrahedral elements and the mesh adaptation is achieved by a 3D bisection method using an error estimator procedure coupled with an automatic 3D mesh generator. The performance of the method is demonstrated using a number of examples. © 1998 John Wiley & Sons, Ltd.
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  • 159
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    Communications in Numerical Methods in Engineering 14 (1998), S. 429-436 
    ISSN: 1069-8299
    Keywords: stress intensity factor (SIF) ; singular finite element ; double cracks ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Using the quasi-compatible finite element method advanced by Zhichao and Lisu (1990), the influence of the size of a plate with double cracks on the stress intensity factor is studied. Many numerical computations are carried out and various data, curves and formulae are given. It is shown clearly how the stress intensity factors of double cracks are affected by the size of the cracked body. © 1998 John Wiley & Sons, Ltd.
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  • 160
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    Communications in Numerical Methods in Engineering 14 (1998), S. 559-567 
    ISSN: 1069-8299
    Keywords: gust ; buffet ; random ; vibration ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A convenient and effective finite-element-based method for buffeting analysis of complex structures is presented. This method is very efficient for structures with the natural frequencies closely spaced because their corresponding modes and the cross-correlation terms between them should be included in the mode-superposition analysis. The imperfect correlation between gust excitations can also be dealt with conveniently. A numerical example is given which shows that by using this method the buffeting analysis of complex structures with thousands of degrees of freedom, dozens of imperfectly related excitations and a number of participating modes can be executed quite easily on an ordinary IBM/486 (or 586) personal computer. © 1998 John Wiley & Sons, Ltd.
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  • 161
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    Communications in Numerical Methods in Engineering 14 (1998), S. 519-527 
    ISSN: 1069-8299
    Keywords: shell element ; large strain ; sheet metal stamping ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper presents a general method of large strain determination over the deformed surface of a sheet metal stamping. It is demonstrated that the conventional degenerated shell element with two normal rotation degrees of freedom is not suitable for large deformation, especially when large element rotation is present. This inaccuracy is primarily caused by the fact that the displacement field description used in the degenerated shell element is only a first-order approximation with respect to the two rotation degrees of freedom, and is therefore suitable only for small rotation angles. The new method presented in this paper replaces the two rotation DOFs with three new degrees of freedom to describe the rotation of the surface normal so that the element deformation can be accurately described with no limitation on the amount of deformation and rotation involved. The advantages of this new method are: (i) a linear and accurate expression of the displacement field in terms of nodal DOFs is obtained; (ii) the formulation is easily incorporated into any existing degenerated shell elements; (iii) the strain calculation is accurate for any amount of element rigid body rotation; (iv) if the method is used in surface grid analysis, the algorithm will not only provide correct surface strains, but also their variation through the thickness direction, i.e. the bending deformation. © 1998 John Wiley & Sons, Ltd.
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  • 162
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    Communications in Numerical Methods in Engineering 14 (1998), S. 597-608 
    ISSN: 1069-8299
    Keywords: inverse problems ; dual systems ; vibrating rod ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The discretization of the differential equation governing the axial vibration of a rod with varying cross-section leads to a specially structured matrix pencil. This paper deals with the reconstruction of this pencil from its spectrum. An iterative algorithm for this problem and an analytic characterization of complementary solutions are given. The method is demonstrated on some examples. © 1998 John Wiley & Sons, Ltd.
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  • 163
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    Communications in Numerical Methods in Engineering 14 (1998), S. 621-632 
    ISSN: 1069-8299
    Keywords: segmentation ; finite elements ; adaptive ; image ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The image segmentation problem in computer vision is considered. Given a two-dimensional domain D and a function defined on it (the original image), the problem is to obtain a ‘cartoon’ associated with this function, namely to find a set of inner boundaries which divide D into subdomains (objects) in an optimal way. The optimality criterion used here is given by the Mumford-Shah (MS) and Blake-Zisserman model, which leads to a strongly non-linear problem. Related problems appear in multiphase continuum mechanics. An iterative procedure based on an h-adaptive finite element method is proposed for the solution of this problem. The mesh adaptivity enables an efficient solution technique, with the use of basic coarse discretization and a few local regions of high resolution where needed. © 1998 John Wiley & Sons, Ltd.
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  • 164
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    Communications in Numerical Methods in Engineering 14 (1998), S. 633-645 
    ISSN: 1069-8299
    Keywords: error estimation ; boundary elements ; adaptive mesh ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A posteriori error estimation for the boundary element method is developed. The nodal error is estimated from the difference of two solutions - one from the original BEM solution, and the other from interpolation of the original solution. Based on nodal errors, a matrix analysis is carried out to calculate the corresponding errors at source points. Two formulae for estimating global percentage errors are proposed in the paper. The first formula uses nodal errors directly to estimate element error, and the second uses an integral form to calculate element error to eliminate extremely high mesh concentration near areas with singular solutions. An h-version adaptive mesh refinement is implemented to study the accuracy of the proposed error estimation. Numerical examples show that the error estimator can correctly guide mesh refinement, and a final mesh can be obtained in a few iterations. © 1998 John Wiley & Sons, Ltd.
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  • 165
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    Communications in Numerical Methods in Engineering 14 (1998), S. 647-655 
    ISSN: 1069-8299
    Keywords: finite difference methods ; stability and convergence of numerical methods ; advection schemes ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper discusses implementation strategies for second-order finite difference discretizations of advection. Purely explicit and implicit methods both have disadvantages, and we consider semi-implicit schemes in which the flux is split into a primary implicit part and a secondary explicit correction. One-dimensional scalar advection is used as a model problem to analyse the leading order error terms and the stability of the schemes. Some of the splittings turn out to be unconditionally stable, but accuracy or monotonicity may deteriorate for large time steps. © 1998 John Wiley & Sons, Ltd.
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  • 166
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    Communications in Numerical Methods in Engineering 14 (1998), S. 697-708 
    ISSN: 1069-8299
    Keywords: incompressible flow ; generalized streamline operator ; upwinding tensor ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the present work the backward-facing step problem is analysed in the framework of the finite element method. It is a very well-known benchmark problem for the numerical resolution of the incompressible Navier-Stokes equations. In particular, a generalized streamline operator technique (GSO) is used in the numerical approach to these equations. The results show very good agreement with those reported by other authors using different methodologies. © 1998 John Wiley & Sons, Ltd.
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  • 167
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    Communications in Numerical Methods in Engineering 14 (1998), S. 671-679 
    ISSN: 1069-8299
    Keywords: advection-diffusion ; semi-Lagrangian ; stability ; accuracy ; QUICKEST ; DISCUS ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper describes the incorporation of diffusive transport into the original semi-Lagrangian DISCUS algorithm for pure advection. An explicit treatment of diffusion is adopted following the approach used in the QUICKEST algorithm for advection-diffusion. The semi-Lagrangian treatment of the advection term relaxes the small time step restriction normally associated with Eulerian treatments of advection, but the Eulerian treatment of the diffusion term imposes conventional limitations on the scheme. Numerical experiments of advection-diffusion, however, indicate that DISCUS has advantages over the QUICKEST scheme for advection-diffusion in three key areas: stability, accuracy and computational efficiency. © 1998 John Wiley & Sons, Ltd.
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  • 168
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    Communications in Numerical Methods in Engineering 14 (1998), S. 709-717 
    ISSN: 1069-8299
    Keywords: fluid dynamics ; multiphase-flow ; lattice gases ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Nearly all CFD methods can be considered as discretization methods for partial differential equations, such as finite difference, finite volume, finite element, spectral or boundary integral element methods. Virtually unrecognized by the scientific mainstream in computational fluid dynamics (CFD) during the last decade, a completely different approach to flow simulation has been developed in computational physics.The basic idea of lattice-gas solvers (LGS) goes back to the cellular automation concept of John von Neumann. LGS use objects (‘cells’), being extremely simple compared to finite boxes or finite elements. The state of a cell is usually described by only a few bits therefore often two orders of magnitude more cells are used for a simulation with LGS than ‘elements’ in a finite element computation. LGS are explicit time-stepping procedures; no equation systems have to be solved. Thus every time-step is extremely cheap in terms of CPU power compared to standard procedures, yet again much shorter time-steps have to be used. LGS are inherently parallel and are suitable to coarse-grain as well as to fine-grain parallelization.The paper will discuss some advantages and disadvantages of lattice-gas solvers and present LG simulation results of two-phase flow with moving boundaries on a microscope scale for a two-dimensional test geometry of randomly distributed equally sized disks where the effect of surface tension on the steady-state saturation will be demonstrated. © 1998 John Wiley & Sons, Ltd.
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  • 169
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    Communications in Numerical Methods in Engineering 14 (1998), S. 731-749 
    ISSN: 1069-8299
    Keywords: superconvergence ; corrected conjoint polynomial ; weight functions ; element patch ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A superconvergent element patch based stress extraction strategy is proposed for general FE postprocessing and/or error estimation procedures in adaptive finite element analysis. Generalized weight functions are proposed for discrete least-square functionals, and a corrected conjoint polynomial fitting procedure is presented to ensure accurate stress extraction from the element domain once the primary level stress parameters have been evaluated. A numerical example is presented to fix the parameters of the weight functions. Several plane stress examples are solved using QUAD4 elements, and results are compared to those of the node patch based extraction method using the conventional conjoint polynomial method. It is shown that the proposed technique yields more accurate results with enhanced local convergence. Although the method may be applied to higher-order elements too, only QUAD4 elements are chosen for the present examples since the present secondary refitting strategy assumes superconvergent points only to be at element centroids. © 1998 John Wiley & Sons, Ltd.
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  • 170
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    Numerical Linear Algebra with Applications 5 (1998), S. 165-202 
    ISSN: 1070-5325
    Keywords: diffusion problem ; discontinuous coefficients ; indefinite Helmholtz equation ; multigrid method ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Convergence theory for a multigrid method with matrix-dependent restriction, prolongation and coarse-grid operators is developed for a class of SPD problems. It motivates the construction of improved multigrid versions for diffusion problems with discontinuous coefficients. A computational two-level analysis method for a class of separable problems is also available. It motivates the design of matrix-dependent multigrid algorithms and, in particular, multiple coarse-grid correction algorithms for highly indefinite equations. Numerical experiments show the advantage of the present methods for several examples. © 1998 John Wiley & Sons, Ltd.
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  • 171
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    Numerical Linear Algebra with Applications 5 (1998), S. 219-247 
    ISSN: 1070-5325
    Keywords: non-linear programming ; sparse problems ; equality constraints ; inexact Newton method ; augmented Lagrangian function ; indefinite systems ; indefinite preconditioners ; conjugate gradient method ; residual smoothing ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: An inexact Newton algorithm for large sparse equality constrained non-linear programming problems is proposed. This algorithm is based on an indefinitely preconditioned smoothed conjugate gradient method applied to the linear KKT system and uses a simple augmented Lagrangian merit function for Armijo type stepsize selection. Most attention is devoted to the termination of the CG method, guaranteeing sufficient descent in every iteration and decreasing the number of required CG iterations, and especially, to the choice of a suitable preconditioner. We investigate four preconditioners, which have 2 × 2 block structure, and prove theoretically their good properties. The efficiency of the inexact Newton algorithm, together with a comparison of various preconditioners and strategies, is demonstrated by using a large collection of test problems. © 1998 John Wiley & Sons, Ltd.
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    Numerical Linear Algebra with Applications 5 (1998), S. 475-482 
    ISSN: 1070-5325
    Keywords: nearest doubly stochastic matrix ; alternating projections ; first moment ; normal cone ; RC1 ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let T be an arbitrary n × n matrix with real entries. We consider the set of all matrices with a given complex number as an eigenvalue, as well as being given the corresponding left and right eigenvectors. We find the closest matrix A, in Frobenius norm, in this set to the matrix T. The normal cone to a matrix in this set is also obtained. We then investigate the problem of determining the closest ‘doubly stochastic’ (i.e., Ae = e and eT A = eT, but not necessarily non-negative) matrix A to T, subject to the constraints ${\bf e}_{1}^{\rm T} A^{k} {\bf e}_{1} = {\bf e}_{1}^{\rm T}T^{k}{\bf e}_{1}$, for k = 1, 2, … A complete solution is obtained via alternating projections on convex sets for the case k = 1, including when the matrix is non-negative. Copyright © 1999 John Wiley & Sons, Ltd.
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    Numerical Linear Algebra with Applications 5 (1998), S. 11-31 
    ISSN: 1070-5325
    Keywords: least squares ; QR factorization ; iterative solvers ; parallel algorithms ; multisplitting ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The linear least squares problem, minx∥Ax - b∥2, is solved by applying a multisplitting (MS) strategy in which the system matrix is decomposed by columns into p blocks. The b and x vectors are partitioned consistently with the matrix decomposition. The global least squares problem is then replaced by a sequence of local least squares problems which can be solved in parallel by MS. In MS the solutions to the local problems are recombined using weighting matrices to pick out the appropriate components of each subproblem solution. A new two-stage algorithm which optimizes the global update each iteration is also given. For this algorithm the updates are obtained by finding the optimal update with respect to the weights of the recombination. For the least squares problem presented, the global update optimization can also be formulated as a least squares problem of dimension p. Theoretical results are presented which prove the convergence of the iterations. Numerical results which detail the iteration behavior relative to subproblem size, convergence criteria and recombination techniques are given. The two-stage MS strategy is shown to be effective for near-separable problems. © 1998 John Wiley & Sons, Ltd.
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    Numerical Linear Algebra with Applications 5 (1998), S. 101-121 
    ISSN: 1070-5325
    Keywords: GMRES ; preconditioning ; invariant subspace ; deflation ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper compares the performance on linear systems of equations of three similar adaptive accelerating strategies for restarted GMRES. The underlying idea is to adaptively use spectral information gathered from the Arnoldi process. The first strategy retains approximations to some eigenvectors from the previous restart and adds them to the Krylov subspace. The second strategy also uses approximated eigenvectors to define a preconditioner at each restart. This paper designs a third new strategy which combines elements of both previous approaches. Numerical results show that this new method is both more efficient and more robust. © 1998 John Wiley & Sons, Ltd.
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    Numerical Linear Algebra with Applications 5 (1998), S. 123-139 
    ISSN: 1070-5325
    Keywords: elasticity problem ; separate displacement component ; preconditioned conjugate gradient method ; parallel algorithm ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The discretized linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorization pcg-method, i.e., we perform inner iterations. A finite element discretization and node numbering giving a high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorization requires an M-matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg-method is analysed for different preconditionings based on the sdc-part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects. © 1998 John Wiley & Sons, Ltd.
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    Numerical Linear Algebra with Applications 5 (1998), S. 147-164 
    ISSN: 1070-5325
    Keywords: eigenvalue problem ; Lanczos method ; power method ; Krylov information ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study Lanczos and polynomial algorithms with random start for estimating an eigenvector corresponding to the largest eigenvalue of an n × n large symmetric positive definite matrix. We analyze the two error criteria: the randomized error and the randomized residual error. For the randomized error, we prove that it is not possible to get distribution-free bounds, i.e., the bounds must depend on the distribution of eigenvalues of the matrix. We supply such bounds and show that they depend on the ratio of the two largest eigenvalues. For the randomized residual error, distribution-free bounds exist and are provided in the paper. We also provide asymptotic bounds, as well as bounds depending on the ratio of the two largest eigenvalues. The bounds for the Lanczos algorithm may be helpful in a practical implementation and termination of this algorithm. © 1998 John Wiley & Sons, Ltd.
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  • 177
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    Numerical Linear Algebra with Applications 5 (1998), S. 203-217 
    ISSN: 1070-5325
    Keywords: boundary value problem ; boundary element method ; preconditioning ; fast Fourier transform ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The algebraic properties of the matrix arising for the three-dimensional Dirichlet problem for Lamé equations in a rotational domain by the boundary element method are considered. The use of the special basis leads to a matrix having a block structure with sparse blocks. The possible strategies for the efficient solution of the above problem are discussed. © 1998 John Wiley & Sons, Ltd.
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 1-26 
    ISSN: 1082-5010
    Keywords: anisotropy ; failure criterion ; schist ; sedimentary rock ; rock joints ; laboratory testing ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: This paper is devoted to the assessment of some representative failure criteria in the framework of modelling the failure behaviour of strongly anisotropic geomaterials. Experimental data concerning the failure behaviour of a typical strongly anisotropic rock; the schist of Angers are first presented. Nine widely used failure criteria are then selected and classified into three groups, the mathematical continuous models, the empirical continuous models and the discontinuous weakness planes based models. This classification is made up according to the main assumptions and techniques used in each criterion to describe the strength anisotropy. The calibration of each one is carried out with respect to the laboratory data of Angers schist. Qualitative and quantitative comparisons between the selected criteria and with the experimental data are provided. © 1998 John Wiley & Sons, Ltd.
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 89-103 
    ISSN: 1082-5010
    Keywords: transversely isotropic material ; elliptic paraboloid failure surface ; strength differential effect ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: A coordinate-free formulation of a failure criterion for transversely isotropic solids is proposed. In the three-dimensional stress space the criterion is represented by an elliptic paraboloid. The anisotropic form of the proposed criterion is based on generalization of the second invariant of the deviatoric stress and of the mean stress obtained through the introduction of a unique fourth-order tensor. For isotropic conditions, the criterion reduces to the Mises-Schleicher failure condition. It is shown that the criterion satisfactorily predicts the strength anisotropy of transversely isotropic rocks subjected to an axisymmetric stress state. The procedure for the identification of the parameters of the criterion from a few simple laboratory tests is outlined. © 1998 John Wiley & Sons, Ltd.
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  • 180
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 27-39 
    ISSN: 1082-5010
    Keywords: creep ; effective spring concept ; Kelvin chain model ; load bearing volume ; micromechanical model ; relaxation ; softening spring ; solidification ; tension softening ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: An analytical constitutive model similar to the Kelvin chain rheological model associated with solidification theory, is developed for time-dependent tension softening of ageing materials like concrete. The stiffness of spring elements is allowed to vary with time via a function of load bearing volume fraction as in the solidification theory. The development of cracks reduces the load bearing volume fraction with time, so that the overall behaviour of springs is a softening type. A reduction in the load bearing volume with time ensures a gradual reduction in the spring stiffness without violating prescribed retardation times. In some circumstances, depending on the boundary conditions, the stress in a structure can remain unchanged over a period of time. During this period, any cracks in the structure will continue to experience an increased opening due to creep. In other parts of the structure, again depending on the boundary conditions, the crack opening displacements may remain unchanged over a period of time, so that the stress will relax over these parts of the structure. In a large concrete structure, creep and relaxation may be taking place simultaneously in different parts or in the same part but at different times. There is thus a need for a visco-elastic tension softening model for ageing concrete that will cater for both creep and relaxation. © 1998 John Wiley & Sons, Ltd.
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  • 181
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 41-63 
    ISSN: 1082-5010
    Keywords: elastoplasticity ; damage ; bonded geomaterials ; triaxial testing ; intact and remoulded specimen ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: In order to model the various phenomena which govern the mechanical response of bonded geomaterials under monotonic loadings, an elastoplastic model coupled with an elastic model with damage was developed, taking into account both the frictional and cohesive aspects of these materials. First, the principles at the base of the model are presented, as well as the physical meaning of the parameters which were used in the elastic model with damage. In order to illustrate the capabilities of the model to reproduce the mechanical behaviour of bonded geomaterials, we simulated triaxial tests on various materials: a deep cemented clay, whose heterogeneity from one specimen to another appeared mainly due to the calcium carbonate content; an assembly of sintered glass balls; and an artificially cemented sand. In this last example, various initial mean stresses allowed us to enlighten the brittle-ductile transition which was modelled by introducing the mean confining pressure in the damage evolution law. We demonstrated that the parameters of each part of the model could be adapted to reproduce the observed general trends. For weak bonds, the elastoplastic part played the predominant role, whereas for strong bonds, the elastic part with damage governed the mechanical behaviour. © 1998 John Wiley & Sons, Ltd.
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  • 182
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 65-87 
    ISSN: 1082-5010
    Keywords: elastoplasticity ; sand ; constitutive modelling ; controllability ; instability ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The paper investigates the possibility of the occurrence of a multiplicity of homogeneous responses of soil specimens under the same incremental loading, and the associated non-controllability of the imposed loading path.It is shown that for non-associated elastoplastic strain-hardening constitutive laws, such a possibility exists even in the hardening regime. Only when the stiffness matrix is positive-definite is the incremental solution unique for any loading programme.A particular constitutive model is chosen and the shape of the loci of non-uniqueness is calculated for various loading programmes. It is shown that the conditions for the occurrence of a shear band and of a peak of the deviator stress in a triaxial undrained test can be seen as special cases of the theory presented. Accordingly, the so-called ‘instability line’ concept in undrained conditions is generalized. © 1998 by John Wiley & Sons, Ltd.
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  • 183
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 105-125 
    ISSN: 1082-5010
    Keywords: localization ; non-local damage ; Perzyna viscoplasticity ; mesh adaptivity ; Arbitrary Lagrangian-Eulerian technique ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The computational modelling of localization of deformation in cohesive-frictional materials must be carried out in a proper, accurate and efficient manner. A proper solution can be obtained by using an enriched material description such as a non-local damage model or Perzyna's viscoplasticity model. However, still a large number of finite elements is needed for an accurate description of the localization zone. To improve efficiency, mesh adaption is applied here by means of the Arbitrary Lagrangian-Eulerian (ALE) technique. The ALE technique must typically be applied in combination with an enriched material model. Otherwise, the use of a standard model gives a zero-width solution of the localization zone and therefore remeshing results in a continuing decrease of finite element size. Implementation of this method requires the addition of convective forces in the equations of motion, transport of the state variables and the formulation of a remeshing strategy. The remeshing strategy heavily determines the success of the ALE method. Three examples of wave propagation have been treated and it is shown that a suitable remeshing strategy in combination with an enriched material model leads to proper, accurate and efficient analyses of the localization process. © 1998 John Wiley & Sons. Ltd.
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  • 184
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 127-153 
    ISSN: 1082-5010
    Keywords: constitutive equations ; post-localization behaviour ; large strain ; interface model ; bifurcation ; shear band ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: This paper addresses developments on a constitutive model able to describe the post-localized behaviour of structures composed by geomaterials. The behaviour of the shear band is defined by means of a specific non-linear constitutive equation in the framework of large strain. Concept of ‘consistency’ of this model with the CLoE model from which it is derived at the onset of localization is discussed. During the post-localization regime this model evolves in a specific way by introducing the concept of critical void ratio. After some recalls about the CLoE framework and about a Rice bifurcation analysis involving CLoE models, the basic concepts of the Daphnis model are introduced. The last part of this paper is devoted to numerical results on an initially homogeneous frictionless biaxial test. The behaviour of the sample is modelled by means of a plane Mohr Coulomb model defined in the CLoE framework and its associated Daphnis model is used to characterize the specific behaviour of the shear band in the post-localization regime. © 1998 John Wiley & Sons, Ltd.
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  • 185
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 155-180 
    ISSN: 1082-5010
    Keywords: reinforced concrete ; bond model ; plasticity ; interface ; yield surface ; modelling ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The development of an interface, non-associative, plasticity model for bond between ribbed, steel bars and concrete is discussed. The model relates average local slip and radial dilation to average bond shear stress and radial confinement stress. The model partially accounts for the response of the damaged, finite-thickness region around the bar - the bond zone. The model is developed for standard steel bars that are initially unstrained. With simplifying assumptions, data for the components of a plasticity law are extracted from a key set of experimental results. In this paper, we emphasize the development of an expression for the yield surface for monotonic loading. While the forms of the model's components are empirically derived, they qualitatively reflect the mechanics of the mechanical interaction of ribbed bars with the adjacent concrete. A characteristic length, related to the rib pattern, helps quantify this interaction. The mechanics of the bond are difficult to characterize in a simple form, but the calibrated model only requires four physical properties and reproduces with acceptable accuracy experimental results with various levels of radial confinement stress. Model refinements are suggested for future work. © 1998 John Wiley & Sons, Ltd.
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  • 186
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 181-205 
    ISSN: 1082-5010
    Keywords: cemented ; carbide ; finite ; element ; compaction ; sintering ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: We have developed a tool for simulating the press cycle and the sintering cycle of cemented carbides. A constitutive model for compaction and tensile fracture has been implemented in the explicit FE programme LS-DYNA2D, and it has been tuned to experiments of hydrostatic compression, uniaxial strain and conventional triaxial compression. The programme simulates the complete compaction cycle in a robust manner. Friction against the tooling is properly taken into account. We have further developed a constitutive model for the stress-free solid-state sintering to also account for a mechanical stress field. The sintering model is based on physical concepts, diffusion parameters, free surface energy, dimensions of carbide grains, etc. of the constituents, and was tuned to successfully simulate the unloaded and loaded sintering of simple specimens. The models of compaction and sintering have a common set of state variables, such that after the compaction cycle, and at raising the temperature the sintering starts out with proper account of the residual stress field. Finally, the potential of the model is demonstrated by simulating the production cycle of a simple WC-Co tool piece until sintering to 75% of full density. The final geometrical shape of the specimen is compared to experimental data. © 1998 John Wiley & Sons, Ltd.
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  • 187
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 229-256 
    ISSN: 1082-5010
    Keywords: finite element method ; viscous constitutive equations ; time-discrete scheme ; stability ; superstability ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The general framework of the paper deals with the finite element modelling of mechanical problems involving viscous materials such as bitumen or bituminous concrete. Its aim is to present a second-order-accurate discrete scheme which remains unconditionally superstable when used for the time discretization of the linear and non-linear viscoelastic constitutive equations considered. After stating the space- and time-continuous mechanical problem we focus on the time discretization of these equations, considering three different schemes. For both of them sufficiently small values of the time step are required in order to ensure the superstability, whereas the third remains unconditionally superstable. Eventually, some numerical results are presented. © 1998 John Wiley & Sons, Ltd.
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  • 188
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 257-276 
    ISSN: 1082-5010
    Keywords: impacts ; structural damage ; discrete element method ; dynamic loading ; scabbing ; perforation ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Impacts of rigid spherical nose shaped missiles on concrete beams are simulated with a discrete element method. The history of the structural damage of the beam is followed from the very first crushing and fracturing processes to the ultimate fragmentation step. Spalling, scabbing, penetration and perforation events are characterized and compared with real test cases of the civil engineering industry. It is found that these phenomena depend not only on the kinetic energy of the missile as predicted by some recent empirical laws but also on its momentum. This means that the dynamic impact loading time is determinant in the scabbing and perforation thicknesses. When the numerical data is plotted against the empirical rules determined from experimental data, the fit is good although the model is two dimensional. © 1998 John Wiley & Sons, Ltd.
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  • 189
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 207-228 
    ISSN: 1082-5010
    Keywords: jointed rock masses ; stability ; failure ; yield design ; homogenization ; strength anisotropy ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Conceived as a potential alternative to the methods usually employed for evaluating the stability of jointed rock masses, the homogenization approach developed in this paper stems from the intuitive idea that, from a macroscopic point of view, a rock mass intersected by a regular network of joint surfaces may be perceived as a homogeneous continuum. The failure criterion of such an equivalent medium is theoretically determined from the knowledge of the failure conditions of the individual constituents, namely the intact rock matrix and joint interfaces. Owing to the existence of privileged material directions associated with the joint orientation distribution, this criterion turns out to be of the anisotropic frictional type, as shown by the closed-form expression obtained in the particular case of two mutually orthogonal joint sets. This criterion is then applied to the investigation of two illustrative engineering problems: the calculation of the load-bearing capacity of a shallow foundation on a jointed rock half-space and the stability analysis of an excavation. Both problems are handled by means of the kinematic method of yield design, making use of the previously determined macroscopic failure condition. The upper bound estimates thus derived appear to be significantly better than those obtained from a direct analysis in which the intact rock matrix and joints are considered separately. In spite of some limitations regarding its range of applicability, which are underlined throughout the paper, the homogenization approach may constitute an appropriate general framework for the design of densely jointed rock structures. © 1998 John Wiley & Sons, Ltd.
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  • 190
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 277-290 
    ISSN: 1082-5010
    Keywords: experimental analysis ; slabs on soil ; reinforced concrete ; steel fibres ; fracture energy ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The cracking control of plain concrete slabs on soil foundation requires the execution of joints with mechanisms of load transfer between adjacent panels. These joints increase the construction costs and, often, are the source of local damage and loss of service performance. Slabs reinforced with steel wire mesh have been used in order to increase the load-carrying capacity and to enhance the cracking control. However, the use of this conventional reinforcement increases the costs, mainly due to labour time spent on the arrangement and positioning of the reinforcement.Fibre-reinforced concrete is a recent material well fitted for applications in industrial floors on soil foundation. The cost of fibres is compensated by a faster construction process and a reduction in the number of expansion joints. The fatigue, impact and flexural strength are significantly improved when steel fibres are added to the concrete mix.The work developed aims to contribute to the on going research effort to clarify the behaviour of fibre-reinforced concrete slabs on soil foundation. For this purpose, an experimental and numerical investigation were carried out. The present article deals basically with the experimental work developed, describing the tests performed and discussing the main results obtained. © 1998 John Wiley & Sons, Ltd.
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  • 191
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 291-303 
    ISSN: 1082-5010
    Keywords: poroelasticity ; isotropic damage ; brittle geomaterials ; enhanced consolidation ; saturated geomaterials ; computational modelling ; indentation of geomaterials ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: This paper examines consolidation behaviour of saturated geomaterials with a matrix component which is susceptible to damage. Finite-element-based computational model accounts for the alteration in both the deformability and permeability characteristics of the porous material due to damage evolution. The isotropic damage criteria governing the evolution of elastic stiffness and hydraulic conductivity parameters are characterized by the dependency of the damage variable on the distortional strain invariant. The computational procedure is utilized to evaluate the extent to which the time-dependent axisymmetric indentation behaviour of a rigid circular punch on a poroelastic half-space can be influenced by the damage evolution in the porous skeleton. © 1998 John Wiley & Sons, Ltd.
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  • 192
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 305-322 
    ISSN: 1082-5010
    Keywords: strain localization ; finite element ; mesh sensitivity ; regularization ; gradient plasticity ; multilayered concrete beams ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: A gradient-dependent plasticity theory is applied to finite element solutions of static strain localization problems. Assuming weak satisfaction of constitutive equations, a multilayered beam finite element with a mixed character is developed. The plastic strain field is discretized in addition of the displacement field. A consistent Newton-Raphson algorithm for this approach is presented. Some examples are carried out in order to illustrate the approach and verify the performances of the element. © 1998 John Wiley & Sons, Ltd.
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  • 193
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 365-365 
    ISSN: 1082-5010
    Keywords: Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: No Abstract
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  • 194
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 323-342 
    ISSN: 1082-5010
    Keywords: concrete fracture ; computational modelling ; continuum damage mechanics ; localization ; regularization ; gradient enhancement ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Classical continuum damage theory for quasi-brittle fracture exhibits an extreme sensitivity to the fineness and orientation of the spatial discretization in finite element simulations. This sensitivity is caused by the fact that the mathematical description becomes ill-posed at a certain level of accumulated damage. The ill-posedness can be removed by the use of a gradient-enhanced damage model. In this model, higher-order deformation gradients give rise to a non-local effect, which regularizes the localization of deformation and thus renders numerical analyses mesh-objective.The mesh objectivity of the gradient-enhanced damage approach is demonstrated by the application to two concrete fracture experiments: a double-edge notched bar subjected to a uniaxial, tensile load and a single-edge notched beam under anti-symmetric four-point loading. Both the initiation and the propagation of damage can be simulated. Particularly the latter aspect calls for an appropriate definition of the strain measure which governs the evolution of damage. © 1998 John Wiley & Sons, Ltd.
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  • 195
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    Mechanics of Cohesive-frictional Materials 3 (1998), S. 343-364 
    ISSN: 1082-5010
    Keywords: concrete modelling ; microplane model ; anisotropic damage ; consistent linearization ; localization analysis ; Engineering ; Civil and Mechanical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Architecture, Civil Engineering, Surveying , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The paper addresses the microplane model in the context of localization analysis. Capable of reproducing experimental results of concrete specimens, the microplane model includes anisotropic damage in a natural and conceptually simple and explicit way. However, the efficiency of former microplane implementations suffers from the expense of the solution procedure being based on the secant stiffness method. Within this paper, the macroscopic constitutive equation derived by kinematically constraining the microplane strains to the macroscopic strain tensor is consistently linearized resulting in quadratic convergence of the Newton-Raphson iteration for the equilibrium equations. A fully three-dimensional model will be presented and linearized incorporating the two-dimensional case in a natural fashion. Furthermore, the localization criterion is analysed, indicating locally the onset of localization in terms of the acoustic tensor. Several examples demonstrate the features of the microplane model in predicting the material behaviour of concrete in tension and compression as well as in shear. © 1998 John Wiley & Sons, Ltd.
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  • 196
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    Communications in Numerical Methods in Engineering 14 (1998), S. 581-593 
    ISSN: 1069-8299
    Keywords: finite elements ; electromagnetics ; numerical simulation ; electric furnaces ; electrodes ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper we give a numerical method based on finite element discretizations to simulate the thermoelectrical behaviour of electrodes for electric reduction furnaces. After introducing the mathematical model we take advantage of the cylindrical symmetry of the problem to compute boundary conditions for the Maxwell equations. Thermal and electrical problems are coupled and non-linear because of the Joule effect and the fact that thermal conductivity and electrical resistivity depend on temperature. A classical piecewise linear finite element method on a triangular mesh is used to discretize weak formulations in cylindrical co-ordinates for the two problems. Then an iterative algorithm is proposed to solve the coupled discrete system. Numerical results are shown both for an analytical test and for a real industrial electrode. © 1998 John Wiley & Sons, Ltd.
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  • 197
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    Communications in Numerical Methods in Engineering 14 (1998), S. 657-670 
    ISSN: 1069-8299
    Keywords: flux intensity factors ; singularities ; multiple singular points ; eliptic PDEs ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A simple method for computing the flux intensity factors associated with the asymptotic solution of elliptic equations having a large convergence radius in the vicinity of singular points is presented. The Poisson and Laplace equations over domains containing boundary singularities due to abrupt change of the boundary geometry or boundary conditions are considered. The method is based on approximating the solution by the leading terms of the local symptotic expansion, weakly enforcing boundary conditions by minimization of a norm on the domain boundary in a least-squares sense. The method is applied to the Motz problem, resulting in extremely accurate estimates for the flux intensity factors. It is shown that the method converges exponentially with the number of singular functions and requires a low computational cost. Numerical results to a number of problems concerned with the Poisson equation over an L-shaped domain, and over domains containing multiple singular points, demonstrate accurate estimates for the flux intensity factors. © 1998 John Wiley & Sons, Ltd.
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  • 198
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    Communications in Numerical Methods in Engineering 14 (1998), S. 681-692 
    ISSN: 1069-8299
    Keywords: finite elements ; contact/friction problem ; Lobatto integration ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Because the 20-node solid element of the serendipity family does not fulfil the physical contact requirements properly (the equivalent nodal forces include negative values) a 21-node three-dimensional transient element for the non-linear contact/friction problem is investigated. Construction of the shape functions is described. The proposed transition element is established by adding one node to the top or bottom face of the basic 20-node solid serendipity element for an effective connection between the contact region (21-node elements) and the rest of the structure (20-node elements) with minimum degrees of freedom possible. Comparisons with results calculated for the 3D-contact problem using the combinations of the 8-node element connected to 20-node finite elements prove the high accuracy and overall superiority of the present method. The numerical examples are shown to illustrate the validity and efficiency of the developed technique. The approach may be employed easily in existing computer codes. © 1998 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 14 (1998), S. 719-730 
    ISSN: 1069-8299
    Keywords: phase-change problems ; conduction-advection equation ; upwind weight function ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A finite element formulation for solving transient multidimensional phase-change problems considering advective effects is presented. This temperature-based formulation includes the definition of a phase-change function able to deal with classical isothermal and non-isothermal phase-change cases. Moreover, a new upwind weight function is defined in order to avoid numerical oscillations in problems with dominant advective effects. Further, some important aspects related to its numerical implementation are also addressed. The ability of this methodology is illustrated, firstly, in the solution of a one-dimensional test example. Finally, the numerical simulation of a direct-chill continuous casting process is performed. © 1998 John Wiley & Sons, Ltd.
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  • 200
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    Communications in Numerical Methods in Engineering 14 (1998), S. 763-772 
    ISSN: 1069-8299
    Keywords: coupled vibrations ; Timoshenko beam ; boundary integral equation method ; symbolic computation ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Symbolic computer algebra systems relieve one from the tedious task of different mathematical operations which are essential to obtain solutions. Due to their highly advanced features they have come to be used widely in computational mechanics. This paper describes an application of the modern computer algebra system Mathematica to the derivation of fundamental solutions necessary for the application of the boundary integral equation method. The problem treated is an asymmetric cross-section Timoshenko beam in free vibration. For this problem, the derivation of fundamental solutions involves lengthy mathematical operations which are very tedious if performed explicitly by hand. Therefore, using Mathematica we derive the fundamental solutions and generate the influence matrices from which the natural frequencies can be found. © 1998 John Wiley & Sons, Ltd.
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