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  • 1
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    Springer
    Acta applicandae mathematicae 9 (1987), S. 175-198 
    ISSN: 1572-9036
    Keywords: 65D ; 65J ; 65L ; 65M ; 65N ; 65P ; 65R ; Superconvergence ; finite elements ; differential and integral equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A brief survey with a bibliography of superconvergence phenomena in finding a numerical solution of differential and integral equations is presented. A particular emphasis is laid on superconvergent schemes for elliptic problems in the plane employing the finite element method.
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  • 2
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    Applications of mathematics 44 (1999), S. 435-458 
    ISSN: 1572-9109
    Keywords: algebraic multigrid ; zero energy modes ; convergence theory ; finite elements ; computational mechanics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We generalize the overlapping Schwarz domain decomposition method to problems of linear elasticity. The convergence rate independent of the mesh size, coarse-space size, Korn's constant and essential boundary conditions is proved here. Abstract convergence bounds developed here can be used for an analysis of the method applied to singular perturbations of other elliptic problems.
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  • 3
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    Acta applicandae mathematicae 1 (1983), S. 121-150 
    ISSN: 1572-9036
    Keywords: 65N30 ; 65P05 ; 58E35 ; 73C35 ; Dual variational formulations ; finite elements ; Signorini's problem ; contact problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In some boundary-value problems the gradient or the cogradient of the solution is more important than the solution itself. Dual variational formulation of elliptic problems is utilized to define finiteelement approximations of the cogradient. A priori error estimates are presented for a class of second-order elliptic problems, including problems of elastostatics. If the boundary conditions are classical (i.e., of Dirichlet, Neumann. Newton, or mixed type), the primal and dual formulations are equivalent with variational equations, whereas the unilateral boundary conditions lead to variational inequalities. The paper has a surveyable character.
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  • 4
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    Advances in computational mathematics 9 (1998), S. 281-310 
    ISSN: 1572-9044
    Keywords: partial differential equations ; finite elements ; unstructured grids ; parallel solver ; domain decomposition ; additive Schwarz ; MPI ; general purpose software ; 65F10 ; 65N22 ; 65N55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper describes a parallel iterative solver for finite element discretisations of elliptic partial differential equations on 2D and 3D domains using unstructured grids. The discretisation of the PDE is assumed to be given in the form of element stiffness matrices and the solver is automatic in the sense that it requires minimal additional information about the PDE and the geometry of the domain. The solver parallelises matrix–vector operations required by iterative methods and provides parallel additive Schwarz preconditioners. Parallelisation is implemented through MPI. The paper contains numerical experiments showing almost optimal speedup on unstructured mesh problems on a range of four platforms and in addition gives illustrations of the use of the package to investigate several questions of current interest in the analysis of Schwarz methods. The package is available in public domain from the home page http://www.maths.bath.ac.uk/∼mjh/.
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  • 5
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    Applications of mathematics 45 (2000), S. 357-379 
    ISSN: 1572-9109
    Keywords: unilateral contact ; Coulomb friction ; finite elements ; existence proofs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A unilateral contact problem with a variable coefficient of friction is solved by a simplest variant of the finite element technique. The coefficient of friction may depend on the magnitude of the tangential displacement. The existence of an approximate solution and some a priori estimates are proved.
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  • 6
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    Applications of mathematics 42 (1997), S. 233-242 
    ISSN: 1572-9109
    Keywords: divergence-free functions ; finite elements ; internal approximation ; stream function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The space of divergence-free functions with vanishing normal flux on the boundary is approximated by subspaces of finite elements that have the same property. The easiest way of generating basis functions in these subspaces is considered.
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  • 7
    ISSN: 1573-4862
    Keywords: Modeling ; ultrasonic testing ; finite elements
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology , Mathematics
    Notes: Abstract To improve the ultrasonic inspection of the safe ends connecting the nuclear vessel to the piping, we are conducting modeling studies using two models: PROMANIS, a ray tracing code, and ULTSON, a 2-D finite element code. The study has shown a great interest in combining modeling and experiments to improve the codes and react to the experimental conditions.
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  • 8
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    Advances in computational mathematics 4 (1995), S. 171-206 
    ISSN: 1572-9044
    Keywords: Anisotropic elliptic equations ; finite elements ; wavelets ; preconditioners ; multi-level methods ; AMS(MOS) 65F10 ; 65F35 ; 65N20 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in ℝ d . Our analysis is based on the theory of additive subspace correction methods and applies to finite element and prewavelet schemes. We present multilevel- and prewavelet-based methods that are robust for anisotropic diffusion operators with additional Helmholtz term. Furthermore, the resulting convergence rates are independent of the discretization level. Beside their theoretical foundation, we also report on the results of various numerical experiments to compare the different methods.
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  • 9
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    Journal of nondestructive evaluation 14 (1995), S. 137-146 
    ISSN: 1573-4862
    Keywords: Ultrasonic transducers ; acoustic medium ; nearfield ; farfield ; finite elements
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology , Mathematics
    Notes: Abstract A finite element model is implemented for simulating the radiated fields of both planar and curved transducers in acoustic media. The approach is based on general finite element analysis developed for solving the governing equations of elastic wave propagation. The distributions of the wave fields are presented for both the nearfield and farfield regions of the transducer. Three excitation pulses with the same center frequency but different bandwidths are examined and the accuracy is indicated by a comparison of the simulation results for the axial and transverse fields with the analytical results for continuous wave excitation.
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  • 10
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    Journal of nondestructive evaluation 7 (1988), S. 55-69 
    ISSN: 1573-4862
    Keywords: Nondestructive evaluation (NDE) ; eddy currents ; finite elements ; computational modeling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology , Mathematics
    Notes: Abstract Quantitative eddy-current nondestructive evaluation is hampered by the lack of suitable models to describe the field/defect interaction. The governing partial differential equations admit of closed-form algebraic solution only in the most trivial of geometries. Finite-element methods hold out the hope of providing numerical solutions for more complex cases. In this paper certain effectively two-dimensional eddy-current problems are considered, previous analytical results being used to validate the computations wherever feasible. The extension to three-dimensional fields, which is outlined here, is treated in detail in a subsequent paper.
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  • 11
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    Nonlinear dynamics 13 (1997), S. 279-305 
    ISSN: 1573-269X
    Keywords: Nonlinear shell dynamics ; integration schemes ; chaos ; finite elements
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper is concerned with a dynamical formulation of a recently established shell theory capable to catch finite deformations and falls within the class of geometrically exact shell theories. A basic aspect is the design of time integration schemes which preserve specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the applied forces allow to. The integration method differs from the one recently proposed by Simo and Tarnow in being applicable without modifications to shell formulations with linear as well as nonlinear configuration spaces and in being independent of the nonlinearities involved in the strain-displacement relations. A finite element formulation is presented and various examples of nonlinear shell dynamics including large overall and chaotic motions are considered.
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  • 12
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    Journal of engineering mathematics 35 (1999), S. 405-426 
    ISSN: 1573-2703
    Keywords: finite elements ; mass lumping ; numerical quadrature ; wave equation ; acoustics ; seismics.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Technology
    Notes: Abstract The higher-order finite-element scheme with mass lumping for triangles and tetrahedra is an efficient method for solving the wave equation. A number of lower-order elements have already been found. Here the search for elements of higher order is continued. Elements are constructed in a systematic manner. The nodes are chosen in a symmetric way. Integration rules must be exact up to a certain degree to maintain an overall accuracy that is the same as without mass lumping. First, for given integration degrees, consistent rule structures are derived for which integration formulas are likely to exist. Then, as each rule structure corresponds to a potential element of certain order, the position of element nodes and the integration weights can be found by solving the related system of nonlinear equations. With this systematic approach, a number of new sixth-order triangular elements and a new fourth-order tetrahedral element have been found.
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  • 13
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    Journal of engineering mathematics 32 (1997), S. 27-40 
    ISSN: 1573-2703
    Keywords: slumping ; creeping flow ; molten glass ; Stokes flow ; finite elements
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Technology
    Notes: Abstract A layer of very viscous liquid (e.g. tar, molten glass) spans a chasm between two vertical walls. The slow fall or slump of this initially-rectangular liquid bridge is analysed. A semi-analytical solution is obtained for the initial motion, for arbitrary thickness/width ratios. The formal limits of large and small thickness/width ratios are also investigated. For example, the centre section of a thin bridge of liquid of density ρ and viscosity µ, with width 2w and thickness 2h≪2w falls under gravity g at an initial velocity ρgw4/(32μh2). A finite element technique is then employed to determine the slumping motion at later times, confirming in passing the semi-analytical prediction of the initial slumping velocity.
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  • 14
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    Nonlinear dynamics 11 (1996), S. 189-212 
    ISSN: 1573-269X
    Keywords: Geometric exact rods ; finite elements ; integration schemes ; chaotic motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper is concerned with a hybrid finite element formulation for the geometrically exact dynamics of rods with applications to chaotic motion. The rod theory is developed for in-plane motions using the direct approach where the rod is treated as a one-dimensional Cosserat line. Shear deformation is included in the formulation. Within the elements, a linear distribution of the kinematical fields is combined with a constant distribution of the normal and shear forces. For time integration, the mid-point rule is employed. Various numerical examples of chaotic motion of straight and initially curved rods are presented proving the powerfulness and applicability of the finite element formulation.
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  • 15
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    Nonlinear dynamics 3 (1992), S. 85-104 
    ISSN: 1573-269X
    Keywords: Multibody systems ; finite elements ; non-linear deformations ; large rotations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem of formulating and numerically solving the equations of motion for a multibody system undergoing large motion and clasto-plastic deformations is considered here. Based on the principles of continuum mechanics and the finite element method, the equations of motion for a flexible body are derived. It is shown that the use of a lumped mass formulation and the description of the nodal accelerations relative to a nonmoving reference frame lead to a simple form of these equations. In order to reduce the number of coordinates that describe a deformable body, a Guyan condensation technique is used. The equations of motion of the complete multibody system are then formulated in terms of joint coordinates between the rigid bodies. The kinematic constraints that involve flexible bodies are introduced in the equations of motion through the use of Lagrange multipliers. In this paper the following general rules will apply: (a) Matrices and higher order tensors are in boldface upper-case characters. (b) Column and algebraic vectors are in boldface lower-case characters. (c) Scalars are in lightface characters. (d) Summation convention is applied when tensors are written on component form. (e) Left superscript denotes the configuration in which an event occurs. (f) Left subscript denotes the configuration to which an event is refered to.
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  • 16
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    Nonlinear dynamics 9 (1996), S. 21-36 
    ISSN: 1573-269X
    Keywords: Beams ; finite elements ; multibody dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A spatial beam element for static and dynamic problems which involve large displacements and rotations is described. This beam element is applied to static linear buckling problems, the simulation of the motion of a slider-crank mechanism with a flexible connecting rod and a planar and spatial spin-up motion of a flexible beam. Results are compared with those from the open literature.
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  • 17
    ISSN: 1573-269X
    Keywords: rotating shaft ; moving load ; nonlinear boundary condition ; finite elements
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A C 0 continuity isoparametricfinite-element formulation is presented for the dynamic analysis of arotating or nonrotating beam with or without nonlinear boundaryconditions subject to a moving load. The nonlinear end conditions arisefrom nonlinear rolling bearings (both the nonlinear stiffness andclearance(s) are accounted for) supporting a rotating shaft. The shaftfinite-element model includes shear deformation, rotary inertia, elasticbending, and gyroscopic effect. Lagrange's equations are employed toderive system equations of motion which, in turn, are decoupled usingmodal analysis expressed in the normal coordinate representation. Theanalyses are implemented in the finite-element program ‘DAMRO 1’.Dynamic deflections under the moving load of rotating and nonrotatingsimply supported shafts are compared with those obtained using exactsolutions and other published methods and a typical coincidence isobtained. Samples of the results, in both the time and frequencydomains, of a rotating shaft incorporating ball bearings are presentedfor different values of the bearing clearance. And the results show thatsystems incorporating ball bearings with tight (zero) clearance have thesmallest amplitude-smoothest profile dynamic deflections. Moreover, fora system with bearing clearance, the vibration spectra of the shaftresponse under a moving load show modulation of the system naturalfrequencies by a combination of shaft rotational and bearing cagefrequencies. However, for a simply supported rotating shaft, the firstnatural frequency in bending dominates the response spectrum. The paperpresents the first finite-element formulation for the dynamic analysisof a rotating shaft with or without nonlinear boundary conditions underthe action of a moving load.
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  • 18
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    Nonlinear dynamics 4 (1993), S. 357-372 
    ISSN: 1573-269X
    Keywords: Backward Kolmogorov equation ; Duffing oscillator ; finite elements ; first passage problem ; Fokker-Planck equation ; linear oscillator ; nonlinear vibration ; probability distribution ; random vibration ; stochastic processes ; Van der Pol oscillator
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The finite element method is applied to the solution of the transient Fokker-Planck equation for several often cited nonlinear stochastic systems accurately giving, for the first time, the joint probability density function of the response for a given initial distribution. The method accommodates nonlinearity in both stiffness and damping as well as both additive and multiplicative excitation, although only the former is considered herein. In contrast to the usual approach of directly solving the backward Kolmogorov equation, when appropriate boundary conditions are prescribed, the probability density function associated with the first passage problem can be directly obtained. Standard numerical methods are employed, and results are shown to be highly accurate. Several systems are examined, including linear, Duffing, and Van der Pol oscillators.
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  • 19
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    Communications in Numerical Methods in Engineering 14 (1998), S. 253-269 
    ISSN: 1069-8299
    Keywords: potential flow ; optimization approach ; sensitivity analysis ; adjoint variable method ; finite elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Potential flow problems around immersed bodies have been treated by an optimization approach. When the stream function is used as the field variable, the boundary values may not be known a priori and may be taken as the decision parameters to minimize integral objective functionals. The circulation integrals around the immersed bodies or the Kutta condition at the trailing edges of the bodies may be used to construct the objective function of optimization. The sensitivity analysis needed for the minimization process is performed by the adjoint variable method, while the numerical solutions of the primary (flow) and adjoint equations have been obtained by the finite element method. Having checked the present method with exact solutions and the classical superposition method, several flow problems involving one or more immersed bodies with or without circulation are investigated numerically. © 1998 John Wiley & Sons, Ltd.
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  • 20
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    Communications in Numerical Methods in Engineering 14 (1998), S. 437-449 
    ISSN: 1069-8299
    Keywords: finite elements ; explicit dynamics ; near incompressible deformation ; large strains ; tetrahedron ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper presents a simple linear tetrahedron element that can be used in explicit dynamics applications involving nearly incompressible materials or incompressible materials modelled using a penalty formulation. The element prevents volumetric locking by defining nodal volumes and evaluating average nodal pressures in terms of these volumes. Two well-known examples relating to the impact of elasto-plastic bars are used to demonstrate the ability of the element to model large isochoric strains without locking. © 1998 John Wiley & Sons, Ltd.
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  • 21
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    Communications in Numerical Methods in Engineering 14 (1998), S. 549-558 
    ISSN: 1069-8299
    Keywords: finite elements ; adaptive analysis ; error estimate ; rh-method ; r-method ; h-method ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An rh-method, which combines r- and h-methods, is proposed for cost-effective adaptive FE analysis in two-dimensional linear elastic problems. Through various numerical test examples, the rh-method is compared with the h-method. From these examples it is concluded that the rh-method has the advantages of both the r- and h-methods, so that the rh-method can minimize the computational time required for the adaptive analysis. The algorithm of the rh-method can be implemented easily in existing h-adaptive FE codes, since the r-method is independently combined with the h-method. © 1998 John Wiley & Sons, Ltd.
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  • 22
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    Communications in Numerical Methods in Engineering 13 (1997), S. 949-961 
    ISSN: 1069-8299
    Keywords: consistent normal ; discrete normal ; slip boundary conditions ; finite elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Computational fluid dynamics simulations often require the imposition of slip boundary conditions: zero-normal velocity. When the geometry is complex, the imposition of this type of boundary condition is not trivial since the current implementations require the computation of consistent normals. The slip boundary condition can be viewed as a constraint to the solution. Hence in the framework of the Lagrange multiplier approach, we present an accurate general implementation of consistent normals. With this new approach, these consistent normals are not computed explicitly at the expense of adding new unknowns - the Lagrange multipliers of the constraint - which can be interpreted as normal forces on the flow. © 1997 John Wiley & Sons, Ltd.
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  • 23
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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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  • 24
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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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  • 25
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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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  • 26
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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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  • 27
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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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  • 28
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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. 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. 271-275 
    ISSN: 1069-8299
    Keywords: basis transformation ; interpolations ; finite elements ; thin plates ; Engineering ; Numerical Methods and Modeling
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    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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    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
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    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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    Communications in Numerical Methods in Engineering 12 (1996), S. 243-248 
    ISSN: 1069-8299
    Keywords: finite elements ; automatic generation ; Engineering ; Engineering General
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    Notes: Finite element generation methods, such as Delaunay, allow the meshing of shapes from a set of nodes. These nodes must previously exist. The authors present a simple, fast and reliable method to generate them in an unstructured way in N-dimensional space. Its main characteristic is to easily allow variation of the node density by introducing a density function, either analytical or discretized. This function can also be anisotropic. Some nodes can have a pre-fixed position. Inner lines and holes can also be specified in the domain to be meshed. Generated nodes are to be connected according to triangles in 2D and tetrahedrals in 3D.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 599-608 
    ISSN: 1069-8299
    Keywords: finite elements ; moving grids ; moving bodies ; mesh velocity ; ALE ; Engineering ; Engineering General
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    Notes: A Laplacian smoothing of the mesh velocities with variable diffusivity based on the distance from moving bodies is introduced. This variable diffusivity enforces a more uniform mesh velocity in the region close to the moving bodies. Given that in most applications these are regions where small elements are located, the new procedure decreases element distortion considerably, reducing the need for local or global remeshing, and in some cases avoiding it altogether.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 643-655 
    ISSN: 1069-8299
    Keywords: viscoelastic flows ; sphere ; annular flow ; adaptive meshing ; finite elements ; error analysis ; Engineering ; Engineering General
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    Notes: This work describes the implementation of an adaptive procedure for viscoelastic flows. Finite element simulations are conducted using a Taylor-Galerkin/pressure correction scheme. The constitutive equations considered are those for an Oldroyd-B and a Phan-Thien/Tanner model. The adaptive meshing technique is h-refined by grid remeshing, based on a Delaunay procedure. Results are presented for two benchmark problems, namely flow past a sphere in a cylindrical tube and flow in an annular converging tube. It is concluded from the results that the use of an adaptive procedure with mesh regeneration enables an optimized mesh to be devised and a predicted accuracy to be attained.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 683-702 
    ISSN: 1069-8299
    Keywords: unstructured grid generation ; finite elements ; advancing front ; Engineering ; Engineering General
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    Notes: We describe extensions and improvements to the advancing front grid generation technique that have proven useful over the years. The following areas are treated in detail: situations with thin or crossing surfaces, meshing of surfaces defined by triangulations, and ease of user input to define the desired element size in space. The first extension is important if one considers the generation of volumetric grids around shells, membranes, fabrics, or CAD-data that exhibit cusps. Traditional advancing front generators are likely to fail in these situations. We propose the introduction of a crossing environment variable attached to faces and points in order to filter out undesired or incorrect information during the grid generation process. The second extension is required for situations where the surfaces to be gridded are not defined analytically, but via a triangulation. Typical cases where such triangulations are used to define the domain are geophysical problems, climate modelling and medical problems. The third topic deals with the reduction of manual labour to specify element size in space. Sources, element size attached directly to CAD-data, and adaptive background grids are discussed. Adaptive background grids, in combination with surface deviation tolerances, are used to obtain surface triangulations that represent the geometry faithfully, and at the same time enable a smooth transition to volumetric meshes.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 731-743 
    ISSN: 1069-8299
    Keywords: explicit integration ; finite elements ; stiffness matrix ; symbolic manipulation ; Engineering ; Engineering General
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    Notes: The paper deals with the symbolic integration of a 4-noded isoparametric finite element for plane elasticity. An efficient approach to generate explicit formulas for computing the elementary stiffness matrix is discussed. The procedure is based on the use of the Derive symbolic manipulation code as well as in a posteriori manipulation of the expressions obtained. The accuracy of the results is tested in extremely distorted and geometrically ill-conditioned elements. Three practical engineering models are presented and the accuracy of the results is discussed. A computer time comparison between both numerical and symbolic integration approaches is also included, showing that relevant CPU savings are obtained when applying symbolic integration.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 755-766 
    ISSN: 1069-8299
    Keywords: finite elements ; isoparametric ; Jacobian ; transformation ; valid ; Engineering ; Engineering General
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    Notes: Isoparametric elements are only valid if the Jacobian determinant of the transformation between a given element and a master element does not change sign within or on the element boundary. Some algorithms are known which analyse Jacobians for various element types. Some necessary conditions are presented for determining the validity of an element.
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    Communications in Numerical Methods in Engineering 13 (1997), S. 285-299 
    ISSN: 1069-8299
    Keywords: error estimates ; remeshing strategies ; finite elements ; adaptive analysis ; elasto-plastic analysis ; Engineering ; Numerical Methods and Modeling
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    Notes: The major difficulty in applying the finite element method to practical problems is the design of a suitable mesh and the assessment of discretization errors. To overcome this difficulty considerable effort has been made in developing adaptive finite element methods, but most of the work has been limited to linear problems. In this paper, fundamental concepts related to error estimates and mesh refinement strategies for non-linear problems are addressed. A simple, but reliable, path-dependent error estimator is proposed. Based on the errors estimated by the present method, a mesh refinement strategy is also suggested. Numerical examples of two ideal plasticity plane stress problems are shown. The non-linear plasticity model is based on an incremental theory using the von Mises yield criterion. © 1997 by John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 13 (1997), S. 337-342 
    ISSN: 1069-8299
    Keywords: heat flow ; finite elements ; probabilistic analysis ; Engineering ; Numerical Methods and Modeling
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    Notes: A solution of the thermal problem in random conditions is presented. The heat flow is formulated in terms of finite elements. The theoretical formulation is described which presents probabilistic distributions for temperature, taking into account random initial and boundary conditions as well as thermal properties of material. An example of the thermal analysis is demonstrated.© 1997 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 135-140 
    ISSN: 1069-8299
    Keywords: finite elements ; QUAD4 elements ; RBF corrections ; shear force predictions ; Engineering ; Engineering General
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    Notes: The residual bending flexibility correction is an extra-variational trick used to enhance the convergence rate of shear flexible beam and plate elements. In the paper we show how this can affect the shear deflection and shear force estimates in the QUAD4 element.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 351-357 
    ISSN: 1069-8299
    Keywords: anisotropy ; finite elements ; layers ; Engineering ; Engineering General
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    Notes: A modification of the usual quadrangle (2D) and brick (3D) finite element for field problems is proposed. It allows the principal directions of the conductivity tensor (anisotropy directions) to be derived from the geometrical shape of the finite element and to change within it. An internal layering is assumed which is parallel to the boundary at two opposite faces (top and bottom) of the element. Without an explicit computation of angles the anisotropy directions are parallel and perpendicular to this layering. An application is the modelling of pore fluid flow in sedimentary basins.
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    Communications in Numerical Methods in Engineering 13 (1997), S. 207-217 
    ISSN: 1069-8299
    Keywords: plates ; finite elements ; mixed method ; error estimation ; superconvergent patch recovery ; adaptivity ; Engineering ; Numerical Methods and Modeling
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    Notes: The superconvergent patch recovery (SPR technique) has been shown to be an effective postprocessing procedure in which an improved solution is obtained based on the original finite element solution. The technique was applied to second order problems with success, but the method has not yet been studied on problems posed in mixed form. This paper demonstrates that the technique can be applied to beam and plate bending problems, characterized by fourth order differential equations. The differential equation is here written as two coupled differential equations of second order leading to a mixed finite element procedure based on approximations of the moment and displacement fields. Two elements of mixed type are handled, namely a triangular plate element with constant moment field and a rectangular element with linearly varying bending moments. Numerical examples are given to show that the postprocessed solution is more accurate and has a higher rate of convergence. © 1997 by John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 13 (1997), S. 511-513 
    ISSN: 1069-8299
    Keywords: finite elements ; arbitrary Lagrange-Euler ; free interfaces ; multiphase flow ; Engineering ; Numerical Methods and Modeling
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    Notes: A correction to the free interface condition given in P. Szabo and O. Hassenger (Int. J. Numer. Meth. Engng, 38, 717-734 (1995)) is presented. The corrections to the computations in the paper are found to be within numerical accuracy. © 1997 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 161-173 
    ISSN: 1069-8299
    Keywords: inverse identification ; non-linear behaviour ; material parameters ; sensitivity analysis ; finite elements ; Engineering ; Engineering General
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    Notes: The identification of materials rheological behaviour in the non-linear range is based on experimental tests. When using direct identification methods, one faces the problem of the interpretation of the experimental tests, which requires the assumption of deformation homogeneity and therefore the use of approximation methods. Since this assumption is often not satisfied in the case of non-linear behaviour, material parameters are not assessed precisely. In the paper, an inverse identification method is proposed to avoid the problems raised by interpretation of the experimental tests and to determine material parameters more accurately. The algorithm developed consists of both an optimization method and a finite element method. This method is applied to the inverse identification of viscoplastic parameters of an aluminium alloy, with an investigation on the effect of the initial guess and errors in experimental data on the identified values.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 181-184 
    ISSN: 1069-8299
    Keywords: finite elements ; infinite elements ; mapping functions ; unbounded domains ; Engineering ; Engineering General
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    Notes: A few of the published three-dimensional Serendipity infinite element mapping functions have been discovered to be in error. The paper gives corrected versions of the defective mapping functions. The problems only relate to three-dimensional elements of the Serendipity type, when they extend to infinity in two or more directions. The new version of the mapping functions have been tested and appear to be robust.
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    Communications in Numerical Methods in Engineering 11 (1995), S. 59-68 
    ISSN: 1069-8299
    Keywords: RLW equation ; finite elements ; splines ; Engineering ; Engineering General
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    Notes: A B-spline finite element method is used to solve the regularized long wave equation numerically. This approach involves a Galerkin method with quadratic B-spline finite elements so that there is continuity of the dependent variable and its first derivative throughout the solution range. Time integration of the resulting system of ordinary differential equations is effected using a Crank-Nicolson approximation. Standard problems are used to validate the algorithm, which is then used to model the smooth development of an undular bore.
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    Communications in Numerical Methods in Engineering 11 (1995), S. 179-185 
    ISSN: 1069-8299
    Keywords: finite elements ; fluid flow ; open boundaries ; Sommerfield boundary conditions ; Engineering ; Engineering General
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    Notes: We examine a method to impose boundary conditions on arbitrary boundaries, introduced to make domains of infinite extent finite for the purpose of numerical calculations, when a finite element discretization based on linear, bilinear or trilinear elements is used, in one, two or three dimensions, respectively. In particular, we look at the so-called ‘free’ boundary condition, which consists in retaining the boundary integrals generated by the weighted-residuals formulation along the open boundaries and adding them to the stiffness matrix. We show that this procedure is exactly equivalent to imposing on the boundary nodes a Sommerfeld radiation condition in one dimension, and a slightly modified form of the Sommerfeld boundary condition in two and three dimensions. We also show that the procedure is not applicable to the purely elliptic case.
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    Communications in Numerical Methods in Engineering 11 (1995), S. 199-211 
    ISSN: 1069-8299
    Keywords: SUPG ; Petrov-Galerkin methods ; finite elements ; explicit scheme ; Engineering ; Engineering General
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    Notes: This work is devoted to the simulation by finite elements of nearly incompressible inviscid flows in real 3D geometries, by means of an Euler code based on the SUPG (streamline upwind Petrov-Galerkin) method, explicit forward Euler pseudo-temporal time integration and periodic and absorbing boundary conditions, among other features. The main goal is the application to flow around turbomachinery, with special emphasis on the performance analysis of a given machine, that involves several numerical computations at different operation points. Finally, these results are summarized in the form of characteristic curves.
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    Communications in Numerical Methods in Engineering 11 (1995), S. 213-226 
    ISSN: 1069-8299
    Keywords: a posteriori error estimates ; asymptotic exactness ; finite elements ; Engineering ; Engineering General
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    Notes: A posteriori error estimation for typical Euler-Bernoulli beam bending problems is developed using a recovery operator based on a smoothing technique that is applied in the context of one-dimensional fourth-order problems. The development exploits a discrete superconvergence property of the ‘bending moments’ of the trial space interpolant of a polynomial solution of sufficiently low degree at certain Gauss points, and the superconvegence property of the energy norm difference of the trial space interpolant (of the weak solution) and the finite element approximation. An asymptotically exact error estimator is developed.
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    Communications in Numerical Methods in Engineering 11 (1995), S. 549-562 
    ISSN: 1069-8299
    Keywords: plate problem ; Reissner-Mindlin ; finite elements ; hierarchic elements ; Engineering ; Engineering General
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    Notes: The focus of the present work is directed towards the construction of a family of special finite elements for the numerical solution of the Reissner-Mindlin plate problem. The model describes the deformation of a plate when transverse shear deformation is taken into account. The model is widely used for thin to moderately thick plates. Despite its simple formulation, the numerical approximation is not straightforward. When standard low-order finite elements are used for the approximation the solution degenerates very rapidly for small thickness (locking phenomenon). To overcome such behaviour, non-standard formulations of the problem are usually combined with low-order finite elements to weaken or possibly eliminate the locking of the numerical solution.In recent years high-order finite elements have been introduced and successfully applied in several fields. Previously we have constructed a family of hierarchic high-order finite elements to solve the Reissner-Mindlin problem in its plain formulation. The locking was strongly reduced but was still active for very thin plates. Meanwhile some mixed-interpolated finite elements have been developed and shown to be locking free. In this paper we combine the two approaches, namely, the hierarchic high-order elements and the mixed-interpolated elements. The result is a family of special finite elements that exhibits both properties of convergence and robustness.
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    Communications in Numerical Methods in Engineering 11 (1995), S. 735-741 
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    Keywords: unbounded domain ; artificial boundary ; finite elements ; Engineering ; Engineering General
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    Notes: Exact non-local boundary conditions are derived and used on an artificial boundary to solve the two-dimensional Laplace and Helmholtz equations in an unbounded domain. The artificial boundary is chosen to be an ellipse, as opposed to previous works which employed circular boundaries. The use of elliptic artificial boundaries enables one to enclose slender obstacles very efficiently, especially in the case of Laplace's equation.
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    Communications in Numerical Methods in Engineering 11 (1995), S. 857-868 
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    Keywords: adaptive analysis ; error estimate ; mesh refinement ; finite elements ; plate bending ; computations ; Engineering ; Engineering General
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    Notes: A new mesh refinement formulation for h-adaptive solutions of linear elliptic problems is presented. The reliability of the methods employed for estimating errors and the convergence rate of the errors during an adaptive process are investigated by means of an asymptotic analysis method. The present adaptive mesh refinement strategy significantly improves adaptivity performance and has higher error convergence rates than those usually used. Numerical examples are shown to illustrate the efficiency of the method shown.
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    Communications in Numerical Methods in Engineering 11 (1995), S. 995-1003 
    ISSN: 1069-8299
    Keywords: bandwidth ; ordering ; finite elements ; graphs ; algorithms ; sparsity ; Engineering ; Engineering General
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    Notes: Graph theoretical algorithms are presented for finite element nodal ordering to reduce the bandwidth of their stiffness matrices. Five different graphs are defined for representing the connectivity of FE models, and the problem of finite element nodal ordering is transformed into that of a graph nodal ordering. The efficiencies of these methods are compared through examples of two- and three-dimensional finite element models.
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    Communications in Numerical Methods in Engineering 13 (1997), S. 583-592 
    ISSN: 1069-8299
    Keywords: RLW equation ; finite elements ; Petrov-Galerkin ; undular bore ; Engineering ; Numerical Methods and Modeling
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    Notes: The equal width equation is solved by a Petrov-Galerkin method using quadratic B-spline spatial finite elements. A linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is obtained via a Crank-Nicolson approach involving a product approximation. The motion of solitary waves is studied to assess the properties of the algorithm. The development of an EW undular bore is investigated and compared with that of the RLW bore. © 1997 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 13 (1997), S. 627-634 
    ISSN: 1069-8299
    Keywords: finite elements ; doubly connected regions ; Poisson's equation ; viscous flow ; Engineering ; Numerical Methods and Modeling
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    Notes: In this paper we give a finite element method and its error estimate for evaluating double integrals over a smooth domain. The results are used to compare rates of flow of a viscous incompressible fluid in a pipe-in-pipe system with different doubly connected cross-sections. These numerical results confirm an earlier conjecture regarding the rates of flow. ©1997 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 13 (1997), S. 909-921 
    ISSN: 1069-8299
    Keywords: surface condensers ; steam ; finite elements ; modelling ; heat exchangers ; Engineering ; Numerical Methods and Modeling
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    Notes: Two numerical procedures are proposed for surface condenser analysis. In the first method a two-noded element is used with a variable heat transfer coefficient for each element. This method is suitable for performance analysis under various operating conditions. A four-noded element is also proposed for studying the axial heat conduction effect in tubes. In the second method, shell side flow and heat transfer are simulated in order to study the flow distribution and pressure drop characteristics. The governing equations are solved in primitive variable form. The finite element method with a Eulerian velocity correction algorithm is used. These two procedures are applied to two different types of surface condensers and the results are found to be in good agreement with experimental data. © 1997 John Wiley & Sons, Ltd.
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    Communications in Numerical Methods in Engineering 13 (1997), S. 963-976 
    ISSN: 1069-8299
    Keywords: plane stress ; finite elements ; finite strain plasticity ; Engineering ; Numerical Methods and Modeling
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    Notes: A two-dimensional finite element method is developed for large deformation plasticity. Principal axes are used for the description of the material behaviour, and the use of principal logarithmic stretches leads to exact formulae for finite deformation problems with large elastic and plastic strains. An efficient return mapping algorithm and the corresponding consistent tangent are derived and applied to plane stress problems. Two examples show the performance of the proposed formulation. © 1997 John Wiley & Sons, Ltd.
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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
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    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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    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
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    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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    Numerical Linear Algebra with Applications 2 (1995), S. 347-361 
    ISSN: 1070-5325
    Keywords: finite elements ; multilevel methods ; optimal order preconditioners ; newest vertex mesh refinement ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The application of some recently proposed algebraic multilevel methods for the solution of two-dimensional finite element problems on nonuniform meshes is studied. The locally refined meshes are created by the newest vertex mesh refinement method. After the introduction of this refinement technique it is shown that, by combining levels of refinement, a preconditioner of optimal order can be constructed for the case of local refinement along a line. Its relative condition number is accurately estimated. Numerical tests demonstrating the performance of the proposed preconditioners will be reported in a forthcoming paper.
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    ISSN: 1572-9265
    Keywords: scale transformations ; convergence ; finite elements ; approximation from rapidly varying data ; spline ; 41A35 ; 41A25 ; 65D10 ; 41A15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Scale transformations are common in approximation. In surface approximation from rapidly varying data, one wants to suppress, or at least dampen the oscillations of the approximation near steep gradients implied by the data. In that case, scale transformations can be used to give some control over overshoot when the surface has large variations of its gradient. Conversely, in image analysis, scale transformations are used in preprocessing to enhance some features present on the image or to increase jumps of grey levels before segmentation of the image. In this paper, we establish the convergence of an approximation method which allows some control over the behavior of the approximation. More precisely, we study the convergence of an approximation from a data set $$\{ x_i ,f(x_i )\} $$ of $$\mathbb{R}^n \times \mathbb{R} $$ , while using scale transformations on the $$f(x_i ) $$ values before and after classical approximation. In addition, the construction of scale transformations is also given. The algorithm is presented with some numerical examples.
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    Applied mathematics and mechanics 17 (1996), S. 333-339 
    ISSN: 1573-2754
    Keywords: finite elements ; subsonic flow ; supersonic flow
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract In this paper we begin with Taylor-Galerkin Finite Elements, then improve it completely and finally construct Modified Taylor-Galerkin Finite Elements. Computation is done by two methods to study two kinds of subsonic and supersonic flow field. The results show that the new one is much better than the old one.
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    Numerical algorithms 21 (1999), S. 311-322 
    ISSN: 1572-9265
    Keywords: finite elements ; adaptivity ; numerical linear algebra ; recursive methods ; 65F50 ; 65M60
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper an Approximate Waves-Bordering algorithm (AWB) is presented. It computes the finite elements linear system solution-update after a refinement/unrefinement step. This is done taking into consideration only the equations that correspond to the nodes whose solution is modified above a certain tolerance and it appears to be very efficient. The algorithm considers an increasing set of equations that updates recursively and stops when the norm of the residual has gone under a user-defined threshold.
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    Numerical algorithms 21 (1999), S. 323-341 
    ISSN: 1572-9265
    Keywords: drift-diffusion equations ; Gummel-like procedure ; nonlinear Poisson equation ; finite elements ; local scaling ; iterative methods ; quasi-Newton algorithm ; 65P05 ; 35Q60 ; 78A55 ; 65H10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The drift-diffusion model can be described by a nonlinear Poisson equation for the electrostatic potential coupled with a system of convection-reaction-diffusion equations for the transport of charge. We use a Gummel-like process [10] to decouple this system. Each of the obtained equations is discretised with the finite element method. We use a local scaling method to avoid breakdown in the numerical algorithm introduced by the use of Slotboom variables. Solution of the discrete nonlinear Poisson equation is accomplished with quasi-Newton methods. The nonsymmetric discrete transport equations are solved using an incomplete LU factorization preconditioner in conjunction with some robust linear solvers, such as (CGS), (BI-CGSTAB) and (GMRES). We investigate the behaviour of these iterative methods to define the effective strategy for this class of problems.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 469-488 
    ISSN: 0029-5981
    Keywords: finite elements ; substructuring ; static condensation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper describes and analyses the implementation of two static condensation algorithms used to exploit the regular patterns created by two schemes of regular mesh substructuring. Computer resources required by the implementation are assessed in terms of the number of multiplications, the total CPU time, the core and the disk storage requirements. It is discovered that, when implemented for the regular mesh substructuring, the newly proposed condensation algorithm is generally more cost effective than the one proposed by Han and Abel, and modified for the regular mesh substructuring by the present authors.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 717-734 
    ISSN: 0029-5981
    Keywords: finite elements ; arbitrary Lagrange-Euler ; free surfaces ; surface tension ; contact angles ; multiphase flow ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The arbitrary Lagrange-Euler (ALE) kinematic description has been implemented in a 3-D transient finite element program so as to simulate multiple fluid flows with surfaces and interfaces of general shapes.The description of fluid interfaces includes continuity of velocity and a discontinuous tangential and normal stress appropriate for fluids of variable interfacial tension.The formulation is particularly relevant to the description of low and intermediate Reynolds number flow dominated by interfacial tension.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 3675-3694 
    ISSN: 0029-5981
    Keywords: machining ; finite elements ; remeshing ; thermo-mechanical coupling ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A Lagrangian finite element model of orthogonal high-speed machining is developed. Continuous remeshing and adaptive meshing are the principal tools which we employ for sidestepping the difficulties associated with deformation-induced element distortion, and for resolving fine-scale features in the solution. The model accounts for dynamic effects, heat conduction, mesh-on-mesh contact with friction, and full thermo-mechanical coupling. In addition, a fracture model has been implemented which allows for arbitrary crack initiation and propagation in the regime of shear localized chips. The model correctly exhibits the observed transition from continuous to segmented chips with increasing tool speed.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 2951-2977 
    ISSN: 0029-5981
    Keywords: finite elements ; plates ; equilibrium ; assumed strain ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Equilibrium constraints are used to improve the stress accuracy of existing assumed natural-co-ordinate strain (ANS) plate elements. This is accomplished by introducing hierarchical degrees of freedom, which are subsequently eliminated by invoking kinematic consistency and element-interior homogeneous equilibrium conditions. The resulting element enrichments are completely contained in the strain-displacement matrix. In addition, the modified displacement shape functions become available for consistent mass and nodal load calculations, a feat not shared by some hybrid-type elements. The performance of the present element (ANS-EC) is compared with other plate bending elements. This performance comparison includes convergence studies for both displacements and stresses.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 433-450 
    ISSN: 0029-5981
    Keywords: mesh partitioning ; finite elements ; parallel computing ; optimization ; heuristics ; frontal method ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We address the problem of automatic partitioning of unstructured finite element meshes in the context of parallel numerical algorithms based on domain decomposition. A two-step approach is proposed, which combines a direct partitioning scheme with a non-deterministic procedure of combinatorial optimization. In contrast with previously published experiments with non-deterministic heuristics, the optimization step is shown to produce high-quality decompositions at a reasonable compute cost. We also show that the optimization approach can accommodate complex topological constraints and minimization objectives. This is illustrated by considering the particular case of topologically one-dimensional partitions, as well as load balancing of frontal subdomain solvers. Finally, the optimization procedure produces, in most cases, decompositions endowed with geometrically smooth interfaces. This contrasts with available partitioning schemes, and is crucial to some modern numerical techniques based on domain decomposition and a Lagrange multiplier treatment of the interface conditions.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 1219-1235 
    ISSN: 0029-5981
    Keywords: plasticity ; numerical integration algorithms ; consistent linearization ; finite elements ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper presents a consistent algorithm, which combines the advantages of the exact time integration of Prandtl-Reuss elastoplastic models and the quadratic asymptotic convergence of Newton-Raphson iteration strategies. The consistent modulus is evaluated by a full linearization of the exact stress update procedure. Numerical tests for a thin wall tube subjected to combined loads of tension and torsion are performed to illustrate the accuracy and efficiency of the consistently linearized exact stress update algorithm described in the paper. For comparison purpose numerical results of the radial return method are also given.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 2603-2617 
    ISSN: 0029-5981
    Keywords: frictionless contact ; large deformations ; finite elements ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two-body contact problem into two simultaneous sub-problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two-dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 1437-1454 
    ISSN: 0029-5981
    Keywords: large elastic deformations ; enhanced strain method ; finite elements ; Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: The numerical simulation of processes undergoing finite deformations requires robust elements. For a broad range of applications these elements should have a good performance in bending dominated situations as well as in the case of incompressibility. The element should be insensitive against mesh distortions which frequently occurs during finite deformations. Furthermore, due to efficiency reasons a good coarse mesh accuracy in required in non-linear analysis. The QS6 element, developed in this paper, tries to fulfil the above-mentioned requirements. The performance is depicted by means of numerical examples.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 1079-1096 
    ISSN: 0029-5981
    Keywords: flexible multibody dynamics ; finite elements ; beam element ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new finite element beam formulation for modelling flexible multibody systems undergoing large rigid-body motion and large deflections is developed. In this formulation, the motion of the ‘nodes’ is referred to a global inertial reference frame. Only Cartesian position co-ordinates are used as degrees of freedom. The beam element is divided into two subelements. The first element is a truss element which gives the axial response. The second element is a torsional spring-like bending element which gives the transverse bending response. D'Alembert principle is directly used to derive the system's equations of motion by invoking the equilibrium, at the nodes, of inertia forces, structural (internal) forces and externally applied forces. Structural forces on a node are calculated from the state of deformation of the elements surrounding that node. Each element has a convected frame which translates and rotates with it. This frame is used to determine the flexible deformations of the element and to extract those deformations from the total element motion. The equations of motion are solved along with constraint equations using a direct iterative integration scheme. Two numerical examples which were presented in earlier literature are solved to demonstrate the features and accuracy of the new method.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 1955-1977 
    ISSN: 0029-5981
    Keywords: structural dynamics ; finite elements ; algorithms ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An overview of new and recent advances towards a VIrtual-Pulse (VIP) time-integral methodology for general linear/non-linear dynamic systems is presented. Attention is focused on providing a brief overview and an indepth evaluation of the developments, methodology, computational issues and implementation aspects for practical problems. Different from the way we have been looking at the developments encompassing existing direct time-integration type methods and mode superposition techniques, the proposed methodology capitalizes on the computational attributes of both and thereby offers new perspectives and several attractive favourable features in terms of stability and accuracy, storage, and computational costs for a wide variety of inertial dynamic problems. Recently, the authors have shown the theoretical developments via the VIP methodology for transient structural problems and for transient thermal problems. The purpose of the present paper is to summarize the theoretical developments, improve upon the computational/implementation aspects for general linear/non-linear dynamic structural problems, and demonstrate the pros and cons via numerous illustrative test cases. The theoretical analysis and results of several test cases show that the VIP methodology has improved accuracy/stability characteristics and computational advantages in comparison to the commonly advocated explicit and implicit methods such as the Newmark family. Overall, an analysis of the theoretical developments, algorithmic study, and the implementation and evaluation of the formulations strongly suggest the proposition that the VIP methodology is a viable alternative for general structural dynamic applications encountered in practical engineering problems.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 2417-2455 
    ISSN: 0029-5981
    Keywords: finite elements ; shell ; shell element ; warped configuration ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A reformulated four-node shell element, based on the analysis of the moment redistribution mechanism development by C0 plate bending and shell elements, is presented. The moment redistribution mechanism of a finite shell element model is shown to be predominantly activated by the membrane flexural action of the shell. This action is triggered through the membrane strain components which participate in the moment equilibrium equations of the finite element assembly system. An equivalent elastic foundation action, along with the activation of the in-plane twisting stiffness of the shell, may also contribute to the moment redistribution mechanism of the finite shell element model. The proposed shell element formulation aims at retaining the non-spurious contribution of the transverse shear/membrane strain energy to the flexural behaviour of the shell, through the activation of the moment redistribution mechanism. Yet, any potentially spurious, whether locking or kinematic, mechanism is rejected. In warped configurations, the element activates appropriate coupling mechanisms of the bending terms to nodal translations. The so-obtained reformulated four-node shell element exhibits an excellent behaviour without experiencing any locking phenomena or zero-energy modes, while its formulation is kept simple, based on physical considerations. The proposed formulation performs equally well in flat as well as in warped shell element applications.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 2803-2820 
    ISSN: 0029-5981
    Keywords: finite rotations ; shells ; finite elements ; Engineering ; Engineering General
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    Topics: Mathematics , Technology
    Notes: The present paper considers a finite rotation formulation for curved shell elements with rotations about the element sides as nodal degrees of freedom. Attention is mainly on the derivation of a consistent finite rotation formulation. Significant simplifications of the governing equations are presented. These simplifications lead to more efficient finite element implementations. Numerical examples demonstrate the differences between the present consistent and previous approximate formulations.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 3731-3755 
    ISSN: 0029-5981
    Keywords: finite elements ; generalized variables ; softening ; localization ; gradient plasticity ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A mesh-independent finite element method for elastoplastic problems with softening is proposed. The regularization of the boundary value problem is achieved introducing in the yield function the second order gradient of the plastic multiplier. The backward-difference integrated finite-step problem enriched with the gradient term is given a variational formulation where the consitutive equations are treated in weak form as well as the other field equations. A predictor-corrector scheme is proposed for the solution of the non-linear algebraic problem resulting from the finite element discretization of the functional. The expression of the consistent tangent matrix is provided and the corrector phase is formulated as a Linear Complementarity Problem. The effectiveness of the proposed methodology is verified by one- and two-dimensional tests.
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    International Journal for Numerical Methods in Engineering 40 (1997), S. 211-232 
    ISSN: 0029-5981
    Keywords: beam theory ; Timoshenko beam ; finite elements ; Engineering ; Numerical Methods and Modeling
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    Topics: Mathematics , Technology
    Notes: A new general beam stiffness matrix which accounts for bending, torsion and shear deformation is derived from an elasticity solution of the beam. The influence of shear and torsion is considered using a 3×3 matrix of deformation coefficients. Numerical examples are presented to demonstrate the behaviour of the deformation coefficients and of the beam stiffness matrix. © 1997 by John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 40 (1997), S. 599-619 
    ISSN: 0029-5981
    Keywords: finite elements ; contact compliance ; orthotropic friction law ; Engineering ; Numerical Methods and Modeling
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    Topics: Mathematics , Technology
    Notes: The present study deals with the solution of the fully three-dimensional contact/friction problem taking into account microstructural characteristics of the surfaces. An incremental non-associated hardening friction law model analogous to the classical theory of plasticity is used. Two different non-linear friction functions in the orthogonal directions are used to account for the orthotropic properties of the contacting bodies. A frontal solver processing unsymmetric matrices is adopted. Two numerical examples have been selected to show applicability of the method proposed. © 1997 by John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 40 (1997), S. 295-317 
    ISSN: 0029-5981
    Keywords: finite elements ; plate ; foundation ; dynamic analysis ; Engineering ; Numerical Methods and Modeling
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    Notes: The main objective of the present work is to give the systematic way for derivation of Kirchhoff plate-elastic foundation interaction by mixed-type formulation using the Gâteaux differential instead of well-known variational principles of Hellinger-Reissner and Hu-Washizu. Foundation is a Pasternak foundation, and as a special case if shear layer is neglected, it converges to Winkler foundation in the formulation. Uniform variation of the thickness of the plate is also included into the mixed finite element formulation of the plate element PLTVE4 which is an isoparametric C0 class conforming element discretization. In the dynamic analysis, the problem reduces to solution of the standard eigenvalue problem and the mixed element is based upon a consistent mass matrix formulation. The element has four nodes and at each node transverse displacement two bending and one torsional moment is the basic unknowns. Proper geometric and dynamic boundary conditions corresponding to the plate and the foundation is given by the functional. Performance of the element for bending and free vibration analysis is verified with a good accuracy on the numerical examples and analytical solutions present in the literature. © 1997 by John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 40 (1997), S. 1667-1685 
    ISSN: 0029-5981
    Keywords: finite elements ; vector elements ; magnetostatics ; tree ; cotree ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Edge-based finite elements have important applications in modelling both quasi-static and high-frequency electromagnetic problems. Recent work has demonstrated that formulations that exploit the graph structure of the edge-based finite element mesh may be extended to higher-order elements. This paper presents the details of how such an extension to a first-order complete finite element is accomplished and illustrates its application in two-dimensional magnetostatics. © 1997 by John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 40 (1997), S. 3229-3245 
    ISSN: 0029-5981
    Keywords: short waves ; diffraction problem ; wave envelope ; phase ; finite elements ; iterative procedure ; Engineering ; Numerical Methods and Modeling
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    Notes: We consider progressive waves such that the time independent potential satisfies the Helmholtz equation, for example, the travelling wave diffracted from a body. In order to model the wave potential using finite elements it is usual to discretize the domain such that there are about ten nodal points per wavelength. However, such a procedure is computationally expensive and impractical if the waves are short. The goal is to be able to model accurately with few elements problems such as sonar and radar. Therefore we seek a new method in which the discretization of the domain is more economical. To do so, we express the complex potential φ in terms of the real wave envelope A and the real phase p such that φ=Aeip, and expect that in most regions the functions A and p vary much more gradually over the domain than does the oscillatory potential φ. Therefore instead of modelling the potential we model the wave envelope and the phase.The usual approach then uses the well known geometrical optics approximation (see p. 109 of Reference 1) : if the wave number k is large then the potential can be expanded in decreasing powers of k. The first two terms give the eikonal equation for the phase and the transport equation for the wave envelope respectively (see p. 149 of Reference 2). However, using the geometrical optics approximation (or ray theory) gives no diffraction effects. This approach shall therefore not be considered. (We note though that Keller's theory of geometrical diffraction, an extension to geometrical optics, does allow for diffraction effects and this may be considered at a later date.)We shall consider a new method which shall be described in the present paper and apply it to two-dimensional problems, although the method is equally valid for arbitary three-dimensional problems. (The method has already been validated for the case of one-dimensional problems.) An iterative procedure is described whereby an estimate of the phase is first given and from the resulting finite element calculation for the wave envelope a better estimate for the phase is obtained. The iterated values for the phase and wave envelope converge to the expected values for the test progressive wave examples considered. Even if a very poor estimate for the phase is first given the iterated values converge to the exact values but very slowly. © 1997 John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 40 (1997), S. 4275-4311 
    ISSN: 0029-5981
    Keywords: friction ; finite elements ; thermomechanical coupling ; thermal contact ; Engineering ; Numerical Methods and Modeling
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    Topics: Mathematics , Technology
    Notes: In this work, a combined theoretical and numerical approach is taken to consider the dynamic behaviour and the thermomechanical interactions of systems in which friction has an important role. Emphasis is placed on the proper numerical description of fully coupled thermomechanical response on frictional interfaces. From a theoretical standpoint, the first and second laws of thermodynamics are carefully formulated in the context of large deformations, giving rise to a numerical description of contact interaction that is consistent with these laws. Physical observations encompassed by the proposed interface description include thermal softening of frictional response, rate dependence, pressure dependent heat conduction across the interface, and heat sinks on the interfaces created by trapped wear debris. The finite element implementation of the framework is discussed and demonstrated, with numerical examples successfully describing such phenomena as fading braking torque in automobiles and the observed temperature and friction evolution at the interface between a gun barrel and a projectile passing through it. © 1997 John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 40 (1997), S. 4119-4136 
    ISSN: 0029-5981
    Keywords: finite elements ; adaptivity ; mesh optimization ; unstructured grids ; Engineering ; Numerical Methods and Modeling
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    Notes: The construction of solution-adapted meshes is addressed within an optimization framework. An approximation of the second spatial derivative of the solution is used to get a suitable metric in the computational domain. A mesh quality is proposed and optimized under this metric, accounting for both the shape and the size of the elements. For this purpose, a topological and geometrical mesh improvement method of high generality is introduced. It is shown that the adaptive algorithm that results recovers optimal convergence rates in singular problems, and that it captures boundary and internal layers in convection-dominated problems. Several important implementation issues are discussed. © 1997 John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 42 (1998), S. 409-442 
    ISSN: 0029-5981
    Keywords: non-linear shell dynamics ; energy-momentum method ; finite elements ; Engineering ; Numerical Methods and Modeling
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    Notes: Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton-Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 42 (1998), S. 127-143 
    ISSN: 0029-5981
    Keywords: unsymmetric sparse matrices ; frontal solver ; direct methods ; finite elements ; BLAS ; computational kernels ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We consider the implementation of a frontal code for the solution of large sparse unsymmetric linear systems on a high-performance computer where data must be in the cache before arithmetic operations can be performed on it. In particular, we show how we can modify the frontal solution algorithm to enhance the proportion of arithmetic operations performed using Level 3 BLAS thus enabling better reuse of data in the cache. We illustrate the effects of this on Silicon Graphics Power Challenge machines using problems which arise in real engineering and industrial applications. © 1998 John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 42 (1998), S. 1279-1305 
    ISSN: 0029-5981
    Keywords: Ogden model ; shell structures ; large elastic strains and finite rotations ; incompressibility ; finite elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The present contribution proposes a variational procedure for the numerical implementation of the Ogden material model. For this purpose the strain energy density originally formulated in terms of the principal stretches is transformed as variational quantities into the invariants of the right Cauchy-Green tensor. This formulation holds for arbitrary three-dimensional deformations and requires neither solving eigenvalue problems nor co-ordinate system transformations. Particular attention is given to the consideration of special cases with coinciding eigenvalues. For the analysis of rubber-like shells this material model is then coupled with a six parametric shells kinematics able to deal with large strains and finite rotations. The incompressibility condition is considered in the strain energy, but it is additionally used as 2-D constraint for the elimination of the stretching parameter at the element level. A four node isoparametric finite element is developed by interpolating the transverse shear strains according to assumed strain concept. Finally, examples are given permitting to discuss the capability of the finite element model developed concerning various aspects. © 1998 John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 42 (1998), S. 1463-1476 
    ISSN: 0029-5981
    Keywords: J-integral ; inhomogeneous ; fracture ; finite elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The three-dimensional interaction of a crack front with particles is investigated under mode-I loading. The J-integral is applied to characterize the crack-inclusion interactions. Numerical examples are presented, using 20 node, isoparametric finite elements, for the compact tension specimen and elastic materials. The J-integral is calculated for various moduli of the particles, distances of the crack from the interface and particle size. The problem of the crack penetrating a cluster of particles is discussed. © 1998 John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 43 (1998), S. 493-506 
    ISSN: 0029-5981
    Keywords: damage ; composites ; unilateral constraints ; finite elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper a flexible model for the description of damage in heterogeneous structural materials is presented. The approach involves solving the equations of equilibrium, with unilateral constraints on the maximum attainable values of selected internal variables. Due to the unilateral constraints, the problem is non-linear. Accordingly, a simple iterative algorithm is developed to solve this problem by (1) computing the internal fields with the initial undamaged microstructure and (2) reducing the material stiffness at locations where the constraints are violated. This process is repeated until a solution, with a corresponding microstructure, that satisfies the equations of equilibrium and the constraints, is found. The corresponding microstructure is the final ‘damaged’ material. As an application, the method is used in an incremental fashion to generate response curves describing the progressive macroscopic damage for a sample of commonly used fibre-reinforced Aluminum/Boron composite. The results are compared to laboratory experiments published by Kyono et al. and computational results using standard numerical methods, published by Brockenbrough et al. © 1998 John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 43 (1998), S. 1029-1052 
    ISSN: 0029-5981
    Keywords: soil-pore fluid interaction ; staggered solution ; finite elements ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new class of unconditionally stable staggered implicit-implicit time-stepping algorithms for coupled soil-pore fluid dynamic problems is presented. The proposed schemes are stabilized with a pressure correction method and the staggered procedure introduced earlier by the second author is a simplified case of the new algorithms. The stabilized procedure is efficient in numerical time domain analysis of soil-pore fluid interaction occurring in earthquake and consolidation problems. Several examples of linear and non-linear problems demonstrate the stability and accuracy of the proposed numerical solution algorithms. © 1998 John Wiley & Sons, Ltd.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 1475-1506 
    ISSN: 0029-5981
    Keywords: finite elements ; incompressible Navier-Stokes ; upwinding schemes ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper examines a new Galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1-D scalar advection-diffusion and that leads to non-oscillatory solutions as the streamline upwinding algorithms for 2-D scalar advection-diffusion and incompressible Navier-Stokes. This method retains the satisfaction of the Babuska-Brezzi condition and, thus, leads to optimal performance in the incompressible limit. This method, when, combined with the recently proposed linear unconditionally stable algorithms of Simo and Armero (1993), yields a method for solution of the incompressible Navier-Stokes equations ideal for either diffusive or advection-dominated flows. Examples from scalar advection-diffusion and the solution of the incompressible Navier-Stokes equations are presented.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 2165-2184 
    ISSN: 0029-5981
    Keywords: finite elements ; CAD ; non-conforming meshes ; constraints ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A methodology to connect two or more non-conforming meshes generated from a geometric model is presented. The problem leads to a variational problem subject to constraints. The emphasis of the work is on constraints construction. Two methods, namely, discrete connection and continuous connection, are developed. The relation with hybrid formulations is considered and two application examples are presented.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 3365-3381 
    ISSN: 0029-5981
    Keywords: non-linear elasticity ; finite elements ; solution algorithms ; three-dimensional membranes ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The treatment of finite hyperelasticity set on the spatial configuration is described. The fully consistent linearization of the corresponding field equations is carried out in detail. It provides the basis of the exact Newton-Raphson procedure for solution of the associated boundary value problem. A membrane formulation is presented in which the incompressibility constraint usually employed in the description of rubber-like materials is satisfied trivially. The excellent performance of the proposed framework is demonstrated by means of 3-D membrane finite element simulations.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 1-32 
    ISSN: 0029-5981
    Keywords: finite volumes ; finite elements ; compressible flows ; unstructured grids ; hybrid grids ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A framework for the construction of node-centred schemes to solve the compressible Euler and Navier-Stokes equations is presented. The metric quantities are derived by exploiting some properties of C0 finite element shape functions. The resulting algorithm allows to implement both artificial diffusion and one-dimensional upwind-type discretizations. The proposed methodology adopts a uniform data structure for diverse grid topologies (structured, unstructured and hybrid) and different element shapes, thus easing code development and maintenance. The final schemes are well suited to run on vector/parallel computer architectures. In the case of linear elements, the equivalence of the proposed method with a particular finite volume formulation is demonstrated.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 1295-1312 
    ISSN: 0029-5981
    Keywords: finite elements ; mixed models ; hybrid models ; plates ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A mixed-hybrid formulation for stress finite elements is presented. The stresses and the displacements in the domain of the element and the displacements on the boundary are simultaneously and independently approximated using orthogonal functions. The stress approximation functions are used as weighting functions in the weighted residual enforcement of the local compatibility and constitutive equations. Similarly, the displacement approximation functions in the domain and on the boundary are used as weighting functions in the weighting residual enforcement of the local equilibrium equation and of the static boundary conditions, respectively. Legendre polynomials and Fourier series are used to illustrate the performance of the finite element formulation when applied to elastostatic problems.
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    International Journal for Numerical Methods in Engineering 38 (1995), S. 3297-3313 
    ISSN: 0029-5981
    Keywords: finite elements ; iterative solutions ; preconditioning ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A class of preconditioners built around a coarse/fine mesh framework is presented. The proposed method involves the reconstruction of the stiffness equations using a coarse/fine mesh idealization with relative degrees-of-freedom derived from the element shape functions. This approach leads naturally to effective preconditioners for iterative solvers which only require a factorization involving coarse mesh variables. A further extension is the application of the proposed method to super-elements in conjunction with substructuring (domain decomposition) techniques. The derivation of the coarse/fine mesh discretization via the use of transformation matrices, allows a straightforward implementation of the proposed techniques (as well as multigrid type procedures) within an existing finite element system.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 1659-1680 
    ISSN: 0029-5981
    Keywords: elasticity ; stress singularity ; finite elements ; optimization ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A semianalytical singular element is proposed to model the stress fields in plane and axisymmetric elastic bodies in the vicinity of singular points. The element generation is performed taking into account the structure of analytical solution in the vicinity of such points. This renders the element well suited for any type of singular points. Introduction of specially generated shape functions in the approximating expressions for displacements enables one to ensure the displacement compatibility with the standard elements. A detailed consideration is given to the convergence of the finite-element procedure. The extensive possibilities of the proposed singular element are exemplified by solving particular elastic problems. The applicability of the elaborated singular element is discussed with respect to optimization of geometry and material properties in the vicinity of singular points. Two optimization problems are presented to illustrate a search for the body geometry providing the optimal stress distribution.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 2249-2263 
    ISSN: 0029-5981
    Keywords: transport problems ; control volumes ; finite elements ; conservation properties ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A physical approach, based on energy balances at element and at node levels, is utilized to derive both control volume and Galerkin finite element formulations of convection-type problems. This direct approach leads to the same systems of differential equations yielded by conventional formulations, but allows us to demonstrate that there are no differences in conservation properties between the two methods. Besides, we provide reliable physical interpretations for all the steps involved in standard mathematical procedures of discretization.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 3101-3121 
    ISSN: 0029-5981
    Keywords: optimal slewing ; finite elements ; flexible beam ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new computational approach to modelling and control of a flexible beam is proposed. The structural modelling and the control design problems are formulated in a unified mathematical framework that allows simultaneous structural and control design iterations that result in an optimal overall system performance. The method employs the space-time spectral elements for simultaneous space and time discretizations of a Timoshenko beam model. Dimensionless equations of motion are derived using Hamilton's principle of variable action and an integral formulation in the framework of space-time spectral elements is introduced. An optimal control problem formulated for the continuum model is transformed by the space-time spectral element formulation into an optimization problem in a finite-dimensional parameter space. Dynamic programming is then used to obtain both open and closed loop control laws. A simulation study shows good performance of the control law applied to the nominal model. It is also demonstrated that proper discretization yields performance robustness of the system with respect to modal truncation.
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    International Journal for Numerical Methods in Engineering 39 (1996), S. 3351-3365 
    ISSN: 0029-5981
    Keywords: force method ; finite elements ; structural analysis ; orthogonalization ; redundants ; natural factor formulation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: If the force method is fully optimized by introducing two appropriate co-ordinate transformations on the internal generalized force unknowns, the self-stress states (or redundant force systems) are determined so as to be an orthonormal set which is also orthogonal to the particular solution. The magnitudes of the redundant forces then become zero for any loading. This eliminates all operations with the redundants. In other words, the particular solution which is found is the solution to the problem. The result is a large decrease in the amount of computation necessary in the force method by eliminating the need to form up and solve the matrices associated with the compatibility equations.Although a different approach, this optimized form of the force method is shown to result in the same numerical procedures as those for the natural factor formulation of the displacement method.The same transformations developed to orthogonalize completely the self-stress states may also be applied directly to the compatibility and equilibrium equations as an alternative procedure. This approach to the solution of the structural equations is designated as the force transformation method.
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