bluuebunny/crossref_metadata_2025 huggingface dataset:
Generalized finite element analysis with T‐complete boundary solution functions
Authors: Zielinski A. P. , Zienkiewicz O. C.
Published: August 2005

The use of a complete and nonsingular set of Trefftz functions in the solution of quasi‐harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary‐type approximation. Procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated.
Finite prism analysis of plates and shells
Authors: Wong Carlos C. K. , Vardy Alan E.
Published: August 2005

The finite prism technique introduced by Zienkiewicz and Too4 is extended to include 12‐node prism elements and, more importantly, a novel offset beam element. The use of 12‐node prism elements enables parabolic strain distributions to be simulated, this being useful for structures which have strongly tapered cross‐sections. The offset beam element is used to simulate flexure and torsion of a beam whose centroid is offset from the main structure. The element is specified completely at the nodes of adjacent prism elements and so is not really an ‘element’ in the usual sense.

The analysis is applied to thin and thick plates and to shells, with and without edge beams. It is shown to be more versatile than the finite strip method and it requires smaller computer resources than the finite element method. Experimental verification of the analysis is obtained by comparison with measurements for a double‐T bridge deck tested by Loo14.
Analytical representation of a surface defined only at a discrete set of points
Authors: Gharghoury Emmanuel
Published: August 2005

Interpolation of a curved triangular surface element by a complete cubic polynomial that minimizes C1discontinuity is described.
The boundary integral equation method for the solution of problems involving elastic slabs
Authors: Clements David L. , Haselgrove M.
Published: August 2005

Three boundary integral equations for the solution of an important class of elastic slab problems are considered. Some numerical examples are examined in order to illustrate the application of the integral equations to particular boundary‐value problems.
On the Galerkin finite difference method: A multi‐bases approach
Authors: Prochnow Dieter
Published: August 2005

At present there are many general methods of approximating a possibly nonlinear operator equation by a finite equation system. The most commonly applied methods are the finite element and the finite difference. Numerous papers (e.g. References 9 and 10) have dealt with a comparison of these methods.

In Reference 12 the Galerkin finite difference method (GFDM) is developed. The GFDM is a special finite element method designed t o solve nonlinear and possibly coupled partial differential equations numerically. It consists of a finite difference scheme derived from a Galerkin finite element method through the use of special local basis functions and a special grid.

In this paper, we are concerned with extending the GFDM to derive ‘normal’ difference schemes. e.g. five‐point schemes on two‐dimensional domains for a general class of operators, even in nonlinear cases.

Using GFDM or the finite element method on two‐dimensional regions generally leads to at least 7‐ or 9‐point schemes as well as expensive approximations of the nonlinear terms. Often, numerical integration is necessary. These computational costs are due to the non‐orthogonality of the continuous and differentiable local basis functions that are needed in this case.

The basic idea of the multi‐bases approaches, which are the major concern in this paper11, is to reduce the smoothnessproperties ofthelocal basis functions in favour oftheir orthogonality. Thiscan beachieved with the help of so‐called transfer operators which map the non‐orthogonal and differentiable basis onto an only bounded but orthogonal basis.
INACOF—A computer program for the inelastic analysis of reinforced concrete framed structures
Authors: Gurujee C. S. , Agashe S. N.
Published: August 2005

The paper presents a general‐purpose computer program for the inelastic analysis of reinforced concrete frames based on the method of imposed rotations. The program is capable of calculating the response of the structure to a given loading and also capable of performing collapse load analysis of the given structure. The program has been tested for a variety of problems and has been found to be extremely efficient compared to the other existing routines with much more limited capabilities. Complete listing and information for using the program has been presented.
Discrete orthogonalization of parameter lines on curved surfaces for finite element analysis of thin shells
Authors: Eberhardsteiner J. , Mang H. A.
Published: August 2005

Orthogonal curvilinear co‐ordinates represent the logical choice for description of the middle surface of the majority of shells in engineering practice. By means of orthogonalization of originally skew parameter lines at integration points for shell analysis by the finite element method (FEM), the remaining minority can be treated without having to resort to skew curvilinear co‐ordinates. Therefore, it does not seem to be worth while to extend existing computer codes for shell analysis by the FEM, restricted to orthogonal curvilinear co‐ordinates, to general curvilinear co‐ordinates. Depending on the structure of such a code, this extension may require a substantial amount of recoding. (However, it is advisable to consider general curvilinear co‐ordinates if a new computer code for shell analysis by the FEM, based on thin‐ or thick‐shell theory, is written).

Based on these considerations, the concept of ‘discrete orthogonalization of parameter lines’ has been developed. It is presented in this paper and applied to the analysis of a hyperbolic paraboloid groined vault subjected to dead load. In the numerical investigation it is demonstrated that treating skew parameter lines incorrectly as orthogonal has a significant effect on the results for the displacements and the internal forces.
A comparison of Galerkin and streamline techniques for integrating strains from an Eulerian flow field
Authors: Agrawal Ajay , Dawson Paul R.
Published: August 2005

Two methods are compared for integrating the strains that can arise in finite element solutions for Eulerian velocity fields associated with large strain material forming processes. With the Galerkin formulation, partial differential equations for the deformation gradient are solved over the entire domain based on a weighted residual; with the streamline integrated technique, the corresponding ordinary differential equations are integrated along characteristic lines. Both methods have yielded accurate integrations for the radial flow and planar rolling problems studied. A finite element technique is also presented for ensuring that the free surfaces of the fluid flow are streamlines. This technique has been used for ensuring proper boundary conditions in the rolling analysis.
A study of mathematical programmingmethods for structural optimization. Part II: Numerical results
Authors: Belegundu Ashok D. , Arora Jasbir S.
Published: August 2005

Various mathematical programming methods for structural optimization are studied. In a companion paper, these methods have been studied based on certain theoretical considerations. In this paper, the methods are studied based on solving a set of test problems. The methods that are studied include recursive QP, feasible directions, gradient projection, SUMT and multiplier methods. Various computer codes have been developed, and are studied together with some existing programs such as CONMIN and OPTDYN. The test problems considered have 3–47 design variables and 3–252 constraints. The evaluation criteria consist of studying the accuracy, reliability and efficiency of a code. It turns out that globally convergent algorithms (multiplier methods, in particular) are very reliable but not efficient. Primal algorithms (like CONMIN), which are not proved to be globally convergent, are efficient but not reliable.
A finite element method for analysing skin effect in conductors with unknown values of surface boundary conditions
Authors: Gasiorski A. K.
Published: August 2005

This paper presents a method of analysing the distribution of a two‐dimensional electromagnetic field in a conducting medium of known cross‐section carrying sinusoidal current and of unknown numerical values of boundary conditions on conductors surfaces. The method combines the finite element and separation of variables methods. An example is given of application of the method to calculation of impedance and electrodynamic forces of two thick, full and round conductors carrying oppositely directed sinusoidal currents. On the basis of numerical computation, graphs of resistance, reactance and electrodynamic forces are plotted.
Comparison of some single‐step methods for the numerical solution of the structural dynamic equation
Authors: Penry S. N. , Wood W. L.
Published: August 2005

This paper compares the performance of the SS22 and SS23 (References 1 and 2) single step algorithms for the numerical solution of the second‐order structural dynamic equation and a related new algorithm SS32B applied to the equivalent first‐order system, with sine and step forcing functions. Various aspects of stability relevant to these equations are discussed.
A consistent finite element technique for recovery of distributed reactions and surface tractions
Authors: Haber Robert B.
Published: August 2005

An effective technique is presented for recovering surface tractions and distributed reaction forces from finite element displacement solutions of structural mechanics problems. Nodal values of surface tractions and assumed trial functions are used to represent the distribution of surface stresses. Equations obtained from the familiar Galerkin formulation are used to compute the nodal traction values. The method is demonstrated to be efficient and highly accurate, even in the presence of jump conditions or singularities. Extensions of the method are possible for recovering secondary solution variables in non‐structural finite element applications.
Continuously deforming finite elements
Authors: Mueller Alan C. , Carey Graham F.
Published: August 2005

A general class of time‐dependent co‐ordinate transformations is introduced in a variational formulation for evolution problems. The variational problem is posed with respect to both solution and transformation field variables. An approximate analysis using finite elements is developed from the continuous variational form. Modified forms of the variational functional are considered to ensure the deforming mesh is not top irregular. ODE system integrators are utilized to integrate the resulting semidiscrete systems. In Part I we consider the formulation for problems in one spatial dimension and time, including, in particular, convection‐dominated flows described by the convection‐diffusion, Burgers' and Buckley–Leverett equations. In Part II the extension of the method to two dimensions and supporting numerical experiments are presented.
A generalized displacement method for the finite element analysis of thin shells
Authors: Kui Li Xi , Liu Guo Qiang , Zienkiewicz O. C.
Published: August 2005

The shear locking problem for the bilinear degenerated thick shell elements, when used in the context of thin shell structures, can be overcome by a generalized displacement method presented in this paper. The transverse shear energy in the degenerated thick shell elements is totally suppressed by introducing discrete Kirchhoff constraints in each element. The constrained variational problem based on the nodal displacement space is transformed into an unconstrained one based on a so‐called generalized displacement subspace. It is shown that shear looking phenomena completely disappear and no degradation of results is observed as the ratio of thickness to span approaches zero.
On the implementation of iterated defect correction to finite difference methods for elliptic boundary‐value problems
Authors: Hertling JÖRg
Published: August 2005

We present a method which makes it possible to apply the idea of iterated defect correction to finite difference methods for the numerical treatment of partial differential equations. The method yields numerical approximations of very high accuracy for the solution, while the corresponding algebraic systems of equations still have ‘reasonable’ size.
An approach for undrained geotechnical problems accounting for the development of partial saturation
Authors: Genna Francesco , Gioda Giancarlo
Published: August 2005

A mathematical programming approach is discussed for the finite element‐analysis of two‐phase undrained problems in the field of geomechanics. This approach enables us to account for the possible development of partially saturated zones in the geotechnical medium, due to the attainment of a given lower limit in the pore pressure distribution. Two alternative formulations are presented, both of them in terms of linear complementarity and quadratic programming problems, and the computational advantages of one of them are illustrated. The main characteristics of a step‐by‐step solution technique are presented. Finally, the solutions of some test problems are discussed and compared with those deriving from the traditional analyses in which the development of partial saturation is neglected.
Unilateral buckling of thin elastic plates by the boundary integral equation method
Authors: Bezine G. , Cimetiere A. , Gelbert J. P.
Published: August 2005

The unilateral buckling of thin elastic plates, according to Kirchhoff's theory, is studied by using a boundary integral method. A representation for the second member of the equation is given. In the matrix formulatiea, boundary unknowns are eliminated; therefore, the unilateral buckling problem reduces to compute the eigenvalues and the eigenvectors of a matrix depending on the contact zone with the rigid foundation. An iterative process allows this zone and the buckling load to be computed. The capacities of the proposed method are illustrated by four examples.
Analysis of laminated composite plates using a higher‐order shear deformation theory
Authors: Phan N. D. , Reddy J. N.
Published: August 2005

A higher‐order shear deformation theory is used to analyse laminated anisotropic composite plates for deflections, stresses, natural frequencies and buckling loads. The theory accounts for parabolic distribution of the transverse shear stresses, and requires no shear correction coefficients. A displacement finite element model of the theory is developed, and applications of the element to bending, Vibration and stability of laminated plates are discussed. The present solutions are compared with those obtained using the classical plate theory and the three‐dimensional elasticity theory.
Nonlinear subincremental method for determination of elastic–plastic–creep behaviour
Authors: Ottosen Niels Saabye , Gunneskov Ole
Published: August 2005

The frequently used subincremental method has so far been based on a linear interpolation of the total strain path within each main step. This method has proven successful when elastic–plastic behaviour and secondary creep is involved. The present paper proposes a nonlinear subincremental method applicable to general elastic–plastic–creep behaviour including problems with a highly nonlinear total strain path caused by the occurrence of creep hardening. This nonlinear method degenerates to the linear‐approach for elastic–plastic behaviour and when secondary creep is present. It is also linear during step loadings and it becomes increasingly more nonlinear, the more creep hardening deformations dominate the behaviour. A wide range of structures are analysed and the results from both subincremental methods are compared; the nonlinear strategy increases the accuracy by a factor of typically 3–5 without affecting the computer time. Moreover, the implementation of the nonlinear method is extremely simple. The optimum number of substeps in each main step is found to be around 5. For such a choice, the advantage of using the subincremental method as compared to the more conventional solution technique, where only one type of time step is used, is a significant reduction in computer time without, in practice, affecting the accuracy.
On the heat‐flow distribution from and the temperature profile on an equally heated rectangle, calculated by series development
Authors: Nieuwenhuizen J. K.
Published: August 2005

A method is proposed to approximate the solutions of a certain class of differential equations, linear or nonlinear, in two or three, dimensions, provided that the boundary conditions are given on a rectangle or a parallelepiped, respectively. For other boundary shapes the co‐ordinate system must be transformed to meet that requirement. As in illustration, an example is given for the solution of Poisson's equation in two dimensions with a constant heat source, giving the temperatures on the rectangle together with the heat‐flow distribution along its edges. The basis of the method is a Taylor‐series development around one point; the result is given in terms of as many partial derivatives in that point as is desired. A similar method has already been described by Small2 for the ‘heat equation’ ∂θ/∂t = ∂2θ/∂x22. Compared with finite difference and finite element solutions, these methods have the advantage that the solution is continuous, whereas first and second derivatives such as heat fluxes are available at hardly any effort. The results are compared with those of the exact solution. Even if the size of the determinant is limited to 4 × 4, the accuracy is already better than 98.98 per cent. When more effort is spent to solve a system of 10 equations, the accuracy is better than 99–95 per cent.
Exact Bernoulli–Euler dynamic stiffness matrix for a range of tapered beams
Authors: Banerjee J. R. , Williams F. W.
Published: August 2005

Bernoulli‐Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as yn and GJ and I both vary as y(n + 2), where A, GJ and I have their usual meanings; y = (cx/L) + 1; c is a constant such that c > − 1; L is the length of the beam; and x is the distance from one end of the beam. Numerical checks give better than seven‐figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements. A procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped. This enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members.
Implicit modelling and mesh refinement for complex structures
Authors: Taig Ian C.
Published: August 2005

When analysing complex structures it is rarely feasible to represent explicitly all the structural features which may contribute significantly to local or global structural performance. This paper presents a variety of methods for representing features which are below the economic analysis scale. Two approaches are described—referred to as implicit stiffness calculation and implicit mesh refinement. The former is essentially approximate and uses relatively simple formulae and computations. The latter involves quite extensive subsidiary calculation, but can lead to numerically exact simulation of fine mesh behaviour in so far as it affects neighbouring structure. Stiffness equivalence principles are stated and a number of computation schemes are presented and illustrated with simple two‐dimensional examples.
The patch test for elements
Authors: Razzaque Abdur
Published: August 2005

The main purpose of this paper is to emphasize the necessity of convergence tests in general and the patch test in particular. Following a brief introduction, the patch test is described with examples. This, it is hoped, will help to remove any lingering misunderstanding of this very useful convergence test. Some positive contributions of the test in the formulation and improvement of elements are also discussed.
TRUNC for shells—an element possibly to the taste of Bruce Irons
Authors: Argyris J. , Haase M. , Mlejnek H.‐P. , Schmolz P. K.
Published: August 2005

In his delightful and original textbook,18 Bruce Irons introduces the chapter of plate bending with the remark ‘History of disasters, but the pioneering days are past’. This is certainly true. On the other hand, we learn in the same context from Irons: ‘We believe that the world has not yet exhausted the older formulations.’ In this paper we present a review of theoretical aspects of the plate‐bending triangle TRUNC (Reference 4), together with extensive tests in linear statics, dynamics, buckling and nonlinear large deformations both for plates and shells. The unconventional TRUNC model is based on the patch test, which was originated by Irons.1 In spite of its limited theoretical foundation, the performance of this element on a fairly broad field of applications appears at least from the practical point of view very satisfactory.
A study of the influence of hierarchical nodes on the performance of selected parametric elements
Authors: Delpak R. , Peshkam V.
Published: August 2005

A number of programs have been developed and tested to enable the routine addition of elements with internal nodes into most finite element packages. A subsequent examination was carried out to assess the effect of those nodes on (a) the accuracy of the displacements and stresses using parametric elements with varying formulation complexities, and (b) the possible additional cost in processing, owing to adequacy of representation. It was found in general that, by including the internal nodes, the representation of both the displacement and the stress fields improved dramatically. It was also found that having hierarchical nodes could lead to a reduction in the processing cost.
Optimal design in elasticity and plasticity
Authors: Strang Gilbert , Kohn Robert V.
Published: August 2005

A direct weight minimization subject to compliance constraints or plastic yielding constraints leads to a non‐convex variational problem. Both the theoretical and the numerical analysis are unsatisfactory: the minimum weight is not achieved by any design, and the approximate designs oscillate as the element mesh is refined. We look for equivalent ‘relaxed problems’ with the same minima. They come from allowing composite materials constructed in an optimal way from the original materials. The constructions are different for elasticity and plasticity, but surprisingly the final relaxed problems are in some cases the same.
Ultimate load behaviour of reinforced concrete plates and shells under dynamic transient loading
Authors: Liu G. Q. , Owen D. R. J.
Published: August 2005

This paper considers a layered thick shell finite element procedure for determining the dynamic transient nonlinear response of plates and shells. The degenerated three‐dimensional isoparametric shell element with independent rotational and translational degrees‐of‐freedom is employed and a layered formulation is adopted to represent the steel reinforcement and to simulate progressive concrete cracking through the thickness. The dynamic yielding function is assumed to be a function of the current strain rate, in addition to being total plastic strain or work dependent. The concrete model also simulates both compressive crushing and tensile cracking behaviours and an implicit Newmark algorithm is employed for time integration of the equations of motion. Several numerical examples are presented for both slab and shell structures and the results obtained compared with those from other sources wherever available.
Continuous stress fields by the finite element–difference method
Authors: Cook Robert D. , Huang Xianrui
Published: August 2005

Standard finite elements yield a stress field that is discontinuous across interelement boundaries. In this paper, results given by standard finite elements are postprocessed by an efficient iterative scheme that makes use of nodal strains, finite difference operations and interpolations that augment (and may differ from) the shape functions used to generate the element stiffness matrix. Various types of interpolation are considered. Each leads to a continuous stress field that is more accurate than the unprocessed stress field.
Penalty‐function iterative procedures for mixed finite element formulations
Authors: Felippa C. A.
Published: August 2005

Iterative methods for solving mixed finite element equations that correct displacement and stress unknowns in ‘staggered’ fashion are attracting increased attention. This paper looks at the problem from the standpoint of allowing fairly arbitrary approximations to be made on both the stiffness and compliance matrices used in solving for the corrections. The resulting iterative processes usually diverge unless stabilized with Courant penalty terms. An iterative procedure previously constructed for equality‐constrained displacement models is recast to fit the mixed finite element formulation in which displacements play the role of Lagrange multipliers. The penalty function iteration is shown to reduce to an ordinary staggered stress‐displacement iteration if the weight is set to zero. Convergence conditions for these procedures are stated and the potentially troublesome effect of prestress modes noted.
The design of transonic aerofoils by a well‐posed inverse method
Authors: Volpe G. , Melnik R. E.
Published: August 2005

This paper describes a numerical method for designing closed aerofoils that correspond to given surface speed distributions including flows with shock waves. The prescribed surface speed distribution in general contains three parameters whose values are found as part of the solution. The inverse aerofoil design problem is not well posed unless the prescribed speed satisfies three constraints that can be used to fix the value of the three adjustable parameters. The computational method is based on the numerical solution of the full potential equation in conservation form by an approximate factorization–multigrid scheme. The nature of the constraints and several strategies for introducing the necessary freedom in the prescribed speed are discussed along with typical solutions illustrating the method.
Application of elliptic continuation method for transonic aerofoil design and experimental verification
Authors: Redeker G. , Wichmann G. , Oelker H.‐Chr.
Published: August 2005

With the aid of the elliptic continuation method, transonic aerofoils for two different purposes have been optimized. The design process is described and it is shown that the optimization is more difficult for higher design Mach numbers. The results of experimental investigations confirmed the good aerodynamic performances of the designed aerofoils, although no real shock‐free pressure distributions could be achieved in the experiments and the highest aerodynamic efficiencies occurred always at higher lift coefficients, with pressure distributions already having shock waves.
Shallow water problems: A general explicit formulation
Authors: Peraire J. , Zienkiewicz O. C. , Morgan K.
Published: August 2005

We present in this paper a general formulation for shallow water equations which is capable of solving problems of induced harbour oscillation, tidal flow, tsunami propagation, dispersion of pollutant, etc.

The shallow water equations are written in a novel fully conservative form. The solution algorithm is based on an explicit time integration procedure which exploits the conservative properties of the governing equations and incorporates accurate treatment of convective terms without introduction of artificial damping. Its accuracy and low cost of computation make it attractive vis‐à‐vis existing procedures.
Variational analysis of a slender fluid–structure system: The elasto‐acoustic beam—a new symmetric formulation
Authors: Ohayon R.
Published: August 2005

We present a new finite element analysis of the linear dynamic responses of a slender fluid–structure system, namely the elasto‐acoustic beam, neglecting flow and viscosity effects. Using one unknown field in the fluid, namely the ‘mass–flow’ corresponding to a cross‐section mean value of the longitudinal displacement field component, an original symmetric formulation is derived which does not exhibit the usual spurious modes associated with the irrotationality constraint occurring in displacements formulations of fluid–structure problems.
A return mapping algorithm for plane stress elastoplasticity
Authors: Simo J. C. , Taylor R. L.
Published: August 2005

An unconditionally stable algorithm for plane stress elastoplasticity is developed, based upon the notion of elastic predictor‐return mapping (plastic corrector). Enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation. Consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm. Use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of Newton methods. The accuracy of the algorithm is assessed by means of iso‐error maps. The excellent performance of the algorithm for large time steps is illustrated in numerical experiments.
Solution of non‐linear thermal transient problems by a reduction method
Authors: Cardona Alberto , Idelsohn Sergio
Published: August 2005

A new algorithm for solving transient thermal problems in a reduced subspace of the original space of discretization is described. The basis of the subspace is formed by using the system response at the first time step (or an approximation to it) and a set of orthogonal vectors obtained by the algorithm of Lanczos. Derivatives of these vectors are included when treating non‐linear cases. The method allows one to handle the sharp gradients that appear in thermally loaded structures, and the response is accurately predicted by using only a small number of degrees of freedom in the reduced system. The algorithm is specially well suited for treating large‐scale problems. Examples dealing with one‐, two‐ and three‐dimensional cases of linear and non‐linear conduction problems are presented.
A vectorized version of a sparse matrix‐vector multiply
Authors: Hayes Linda J. , Devloo Phillippe
Published: August 2005

A fast vectorized algorithm is presented for a sparse matrix‐vector multiply. It can be used when the matrix, A, can be represented as a multiplitting, A = ∑Ae. In particular, it can be applied to a matrix‐vector multiply arising in finite element techniques where the matrices Ae are associated with the individual element contributions to the global matrix A. The algorithm presented here uses a data structure which is based on the individual matrices Ae and can be applied both to symmetric and to non‐symmetric matrices. This algorithm would be attractive for vector architecture similar to either the CYBER 205 or the CRAY and has been implemented for both regular and irregular finite element grids on the CYBER 205. Execution times and storage requirements are compared to standard sparse and band matrix‐vector multiply algorithms.
Free vibration of coil springs of arbitrary shape
Authors: Nagaya Kosuke , Takeda Sadahiko , Nakata Yoshitaka
Published: August 2005

This paper is concerned with the method for solving the vibration problem of coil springs of arbitrary shape. In the analysis the exact field transfer matrix is derived from the solution of the fundamental equation of the three‐dimensional curved beam theory. The transfer matrix is given by combining this field transfer matrix and the point transfer matrix. The analysis derives the frequency equation which is applicable to coil springs of general shape. Numerical calculations have been carried out for hyperboloidal springs and barrel springs. Experimental tests also have been carried out to verify the present results.
A simple algorithm for the analysis of axisymmetric thin shell metal forming
Authors: Gavriushin S. S. , Zienkiewicz O. C.
Published: August 2005

A numerical method for analysing large elasto‐visco‐plastic deformations of thin axisymmetric metal shells is presented. The formulation is based on membrane shell theory, using the total Lagrangian description and simple constitutive relations, which include elastic, plastic and creep effects.

The method of lines is used in the computational algorithm to separate variables. The algorithm involves two steps: first the initial value problem in load‐time is considered, then the boundary problem is solved by finite differences. A numerical procedure to study the inverse metal forming problem is developed.

The effectiveness of the proposed algorithm is illustrated by several examples.
A field‐consistent two‐noded curved axisymmetric shell element
Authors: Babu C. Ramesh , Prathap G.
Published: August 2005

The efficiency of a field‐consistent two‐noded linear curved axisymmetric shear‐flexible shell element is shown to be due to the removal of both shear and membrane locking. Typical applications illustrate how the field‐consistent representation dramatically improves performance, and allows greater flexibility in tailoring element design to satisfy specific problem needs.
Mixed element approach for two‐dimensional field problems with open boundaries
Authors: Sanavullah M. Y. , Dharmalingam K.
Published: August 2005

A new mixed finite element mesh division and its recursion algorithm are proposed for the solution of two‐dimensional field problems with open boundaries. The scheme is applied to a pin insulator in open space. The results are in close agreement with those of an established technique for modelling field problems. The advantage of the mixed element mesh division over the star point radial ray division of the ballooning technique and its application to a three‐dimensional mesh division are brought out.
Dynamic optimization of framed structures with variable layout
Authors: Sadek Edward A.
Published: August 2005

The paper presents a solution to the problem of dynamic optimization of framed structures for which the design variables are not only the sizes of the members but also the positions of the joints. The optimal design obtained is the least weight design which satisfies a specific frequency requirement plus upper and lower bounds on the design variables. The design algorithm is an iterative solution of the Kuhn‐Tucker optimality criterion. The procedure is to modify an initial design to first obtain the desired fundamental frequency and then, while this frequency is held constant, to minimize the weight.

The most important features of the algorithm are (a) a small number of design iterations is needed to reach an optimal or near‐optimal design, (b) structural elements with a wide variety of size‐stiffness may be used.

The design procedure is illustrated with some numerical examples. The results show that the method is rapidly convergent and the optimized configurations are determined in a small number of redesign cycles. The procedure has been completely automated in a computer program on the HP 45 desk‐top computer.
Reduction method for the non‐linear analysis of symmetric anisotropic panels
Authors: Noor Ahmed K.
Published: August 2005

Reduction method and computational procedures are presented for reducing the size of the analysis model and the number of degrees of freedom used in predicting the non‐linear response of symmetric anisotropic panels. The two key elements of the method are (a) operator splitting, or decomposition of the characteristic arrays of the finite element model into sums of orthotropic and non‐orthotropic contributions, (b) application of a reduction method through the successive use of the finite element method and the classical Rayleigh‐Ritz technique. The finite element method is first used to generate a small number of global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the classical Rayleigh‐Ritz technique.

The global approximation vectors are selected to be those commonly used in single (or multiple) parameter perturbation techniques, namely a non‐linear solution corresponding to zero non‐orthotropic arrays and a number of its derivatives with respect to an anisotropic tracing parameter (and possibly, to a load or arc‐length parameter in the solution space). The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding orthotropic structure.

The effectiveness of the proposed reduction method is demonstrated by means of a numerical example, and its potential for solving quasi‐symmetric non‐linear problems of anisotropic panels is discussed.
Stress oscillations in plane stress modelling of flexure—a field‐consistency interpretation
Authors: Prathap G. , Subramanian G. , Babu C. Ramesh
Published: August 2005

Exactly integrated isoparametric plane stress elements behave poorly in flexure. The 4‐noded element ‘locks’, with errors that progress indefinitely as element aspect ratio increases. Reduced integration of the shear strain energy eliminates this locking entirely. The 8‐noded element does not lock, but improves in performance with reduced integration of shear strain energy. Both elements, with their original shape functions, show severe shear stress oscillations in flexure. In this paper we attribute these oscillations to the lack of ‘consistency’ of shear strain fields derived directly from independent field‐variable interpolations. We derive error models for specific tractable examples which can confirm the accuracy of this conceptual scheme through digital computation using the finite element models. A field‐consistent redistribution strategy for the shear strain field is offered as an elegant procedure to free the elements of spurious oscillations and give a ‘lock’‐free performance.
Generalized geometric programming with many equality constraints
Authors: Burns Scott A.
Published: August 2005

A monomial treatment for solving mixed equality/inequality constrained generalized geometric programs is presented. The method is shown to be equivalent to a Newton–Raphson procedure carried out in log space on the equality constraints at the same time that a cutting plane procedure is performed on a convex region constrained within the feasible region of the inequality constraints. Known limitations of this monomial treatment are outlined and a technique for improving the efficiency of the GGP method is suggested. A discussion of how other methods of solving mixed equality/inequality constrained GGP problems compare to the monomial treatment is presented and two example problems are solved.
Semiconductor device modelling from the numerical point of view
Authors: Polak S. J. , Den Heijer C. , Schilders W. H. A. , Markowich P.
Published: August 2005

In this paper mathematical and computational aspects of device modelling are treated. Four main subjects are discussed; the analytical model, the discretizations, the non‐linear and the linear systems of equations. On the one hand the most commonly used numerical techniques are described; on the other hand several new techniques are presented. Noteworthy new techniques are the generalized upwind box scheme and the mixed variable approach. Aspects such as the choice of variables and the singularly perturbed nature of the problem are treated. Properties of the matrices involved are presented in a systematic way. Throughout the paper a standard one‐dimensional diode is used to illustrate the concepts involved. Finally several realistic two‐dimensional examples are given.
On the preconditioned conjugate gradient method for solving (A – λB)X =0
Authors: Papadrakakis Manolis , Yakoumidakis Michalis
Published: August 2005

Classical iterative methods when applied to the partial solution of the generalized eigenvalue problem Ax =λBx, may yield very poor convergence rates particularly when ill‐conditioned problems are considered. In this paper the preconditioned conjugate gradient (CG) method via the minimization of the Rayleigh quotient and the reverse power method is employed for the partial eigenproblem. The triangular splitting preconditioners employed are obtained from an incomplete Choleski factorization and a partial Evans preconditioner. This approach can dramatically improve the convergence rate of the basic CG method and is applicable to any symmetric eigenproblem in which one of the matrices A, B is positive definite. Because of the renewed interest in CG techniques for FE work on microprocessors and parallel computers, it is believed that this improved approach to the generalized eigenvalue problem is likely to be very promising.
Hybrid‐Trefftz plate bending elements with p‐method capabilities
Authors: Jirousek J.
Published: August 2005

The p‐method capabilities have thus far been used only in connection with the assumed displacement model elements. When such elements are used, singular points have to be isolated by one or two layers of small elements graded in a suitable geometrical progression towards singularity. This paper presents an alternative formulation which circumvents this drawback and enables excellent solution results to be obtained with uniform FE‐grids. The formulation is based on the recently presented hybrid‐Trefftz FE‐model and makes use of optional expansion sets for various singularities or stress concentrations. Stress concentrations due to concentrated loads are also properly accounted for. It is shown that by augmenting the order of approximation over a fixed grid of such elements rapid convergence towards the accurate solution is obtained in the most efficient way.

This paper summarizes the results of a preliminary study that had been undertaken to critically evaluate the potential of the new approach as a suitable basis for subsequent developments of a fully automated adaptive process.
Computer algorithms for calculating efficient initial vectors for subspace iteration method
Authors: Cheu Tsu‐Chien , Johnson C. Philip , Craig Roy R.
Published: August 2005

In this paper, the effects of selecting initial vectors on computation efficiency for a subspace iteration method are investigated. Four algorithms are used for selecting the initial vectors. First, arbitrary starting iteration vectors are chosen according to Bathe and Wilson's algorithm.1 In the other algorithms, the initial vectors are the retrieved eigenvectors from the Guyan and quadratic reduction methods. Improvement of the eigenvalue approximations of the subspace iteration method over reduction methods is presented. The computation effort is examined for the various algorithms used for initial iteration vectors.
Basis random variables in finite element analysis
Authors: Lawrence Mark A.
Published: August 2005

Galerkin's method is applied to random operator equations. Appropriate Hilbert spaces are defined for random functions and solutions are projected into these spaces, allowing the first‐ and second‐moment properties of the solution to be calculated. An equivalent energy‐based approach similar to the Rayleigh–Ritz method is developed, from which a stochastic finite element technique is derived. Several one‐ and two‐dimensional example problems are solved and the results discussed.
A numerical study of steady plane granular chute flows using the Jenkins–Savage model and its extension
Authors: Szidarovszky F. , Hutter K. , Yakowitz S.
Published: August 2005

The granular flow model proposed by Jenkins and Savage and extended by us is used here to construct numerical solutions of steady chute flows thought to be typical of granular flow behaviour.

We present the governing differential equations and discuss the boundary conditions for two flow cases: (i) a fully fluidized layer of granules moving steadily under rapid shear and (ii) a fluidized bottom‐near bed covered by a rigid slab of gravel in steady motion under its own weight. The boundary value problem is transformed into a dimensionless form and the emerging system of non‐linear ordinary differential equations is numerically integrated. Singularities at the free surface and (in one case) also at an unknown point inside the solution interval make the problem unusual. Since the non‐dimensionalization is performed with the maximum particle concentration and the maximum velocity, which are both unknown, these two parameters also enter the formulation of the problem through algebraic equations. The two‐point boundary value problem is solved with the aid of the shooting method by satisfying the boundary conditions at the end of the soluton interval and these normalizing conditions by means of a minimization procedure. We outline the numerical scheme and report selective numerical results. The computations are the first performed with the exact equations of the Jenkins–Savage model; they permit delineation of the conditions of applicability of the model and thus prove to be a useful tool for the granular flow modeller.
Field‐consistency and violent stress oscillations in the finite element method
Authors: Prathap G. , Babu C. Ramesh
Published: August 2005

The fact that finite element models can give rise to violent stress oscillations and that there are optimal locations where stresses can be correctly sampled in spite of the presence of these violent stress fluctuations has been known for some time. However, it is less well known that these oscillations arise in a specific class of problems—where there are multiple strainfields arising from one or more field‐variables and where one or more of these strain‐fields must be constrained in particular physical limits. In this paper, we show that unless the interpolations for these constrained strain‐fields are ‘field‐consistent’, violent oscillations would set in. These oscillations represent spurious self‐equilibrating stress‐fields generating spurious energy terms that lead to ‘locking’.

The field‐consistency interpretation offers a conceptual scheme to delineate these problems and an operational procedure called the functional reconstitution technique allows the errors resulting from field‐inconsistency to be anticipated a priori. We demonstrate the power of this approach through an interesting example of a multi‐strain‐field problem—the inextensional/nearly inextensional deformation of a shear flexible curved beam.
Design sensitivity analysis of non‐linear structural systems part I: Theory
Authors: Choi Kyung K. , Santos Jose L. T.
Published: August 2005

A unified approach is presented for design sensitivity analysis of non‐linear structural systems that include truss, beam, plane elastic solid and plate components. Both geometric and material non‐linearities are treated. Sizing design variables, such as thickness and cross‐sectional areas of components of individual members and built‐up structures, are considered. A distributed parameter structural design sensitivity analysis approach is used that retains the continuum elasticity formulation throughout the derivation of design sensitivity analysis results. Using this approach and an adjoint variable method, expressions for design sensitivity in terms of design variations are derived in the continuous setting which can be evaluated numerically using analysis results of finite element analysis. Both total Lagrangian and updated Lagrangian formulations in non‐linear analysis of solid mechanics are used for design sensitivity analysis. Numerical implementation of design sensitivity analysis results using existing finite element code will be presented in Part II of this paper.