A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low‐rank damping

The low‐rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low‐rank damping property, we propose a Padé approximate linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only n + ?m, which is generally substantially smaller than the dimension 2n of the linear eigenvalue problem produced by a direct linearization approach, where n is the dimension of the quadratic eigenvalue problem, and ? and m are the rank of the damping matrix and the order of a Padé approximant, respectively. Numerical examples show that by exploiting the low‐rank damping property, the PAL algorithm runs 33–47% faster than the direct linearization approach for solving modest size quadratic eigenvalue problems. Copyright © 2015 John Wiley & Sons, Ltd.     

An exact method for solving the manufacturing cell formation problem

The cell formation problem is extensively studied in the literature, but very few authors have proposed exact methods. In this paper a linear binary programming formulation is introduced to generate a solution for the cell formation problem. To verify the behaviour of the proposed model, a set of 35 benchmark problems is solved using the branch and cut method implemented in the IBM ILOG CPLEX 10.11 Optimiser. Moreover, these results allow us to validate the quality of the solution generated with heuristic methods proposed in the literature. This experimentation indicates that, for the smaller problems, the best-known solutions are the same as those generated with CPLEX 10.11 Optimiser. These results indicate a fair confidence in the optimality of the best-known solutions generated by the heuristic methods. Furthermore, our approach is the first exact method providing results of this quality.     

An algorithm for the solution of a third-order eigenvalue problem

A simple, rapidly convergent procedure is described for solving a third-order symmetric eigenvalue problem Au = λ Bu typically arising in vibration analysis. The eigenvalue problem is represented in terms of its variational dual, the Rayleigh quotient, and the eigenosolution is obtained through a topographical search for points of quotient stationarity. The associated computer routine is compact and can easily be incorporated within the calling program. Degenerate eigensolutions cause no difficulty. An example FORTRAN routine is given.     

An efficient method for determining the chemical rank of three-way fluorescence data arrays

An efficient method is proposed for determining the chemical rank of three-way fluorescence data arrays. At first, the original three-way fluorescence data arrays are preprocessed by Monte Carlo simulation and a new set of data arrays is generated. The new set of data arrays obtained does not only keep all the useful information, but the noises from the common background are largely removed, which results in the improvement of the signal to noise ratio of the data and is beneficial for the later frequency analysis. Then, we perform singular value decomposition over the new data and frequency analysis on the subsequent eigenvectors, with which it is very easy to distinguish the spectra from the noises. Furthermore, a new quantity frequency localization is introduced to quantify the frequency characteristics of the eigenvectors. With this quantity, we can easily and accurately select out the spectra from the mess of data. The feasibility of the method is verified by determining the chemical rank of two-component mixtures with simple calculation procedures and high efficiency. Finally, the efficiency of our method is further illustrated by comparison with the core consistency diagnostic (CORCONDIA) method in the analysis of mixtures with different concentration and different number of components.     

An improved algorithm for solving the economic lot size problem (ELSP)

A bounded enumeration combining non-linear optimization, heuristic procedure, and mixed integer programming is proposed to solve the capacitated multi-item lot scheduling problem. The combined effect is demonstrated by a better solution to problem data which has been the subject of inquiry and investigation for over 20 years. In addition, the approach avoids the escape from infeasibilily deficiencies of prior solution procedures. The attractiveness of the solution found can be confirmed by the close proximity of the production run frequency to that obtained from the independent solution. Since the independent solution reflects the optimum economic tradeoff between incurring setup costs and inventory carrying costs for each product individually, the advantages of deriving a multi-product production schedule where the frequency of setups conform to these cost relationships are obvious.     

An error indicator for the generalized eigenvalue problem using the hierarchical finite element method

The hierarchical concept applied to eigenvalue problems is considered. An error indicator is derived from the pertinent Rayleigh quotient. The indicator serves as an estimation ‘a posteriori’ of the relative change in an eigenvalue for a hierarchical refinement. A numerical example is included.     

An efficient method of solving the Navier–Stokes equations for flow control

A new method of solving the Navier–Stokes equations efficiently by reducing their number of modes is proposed in the present paper. It is based on the Karhunen–Loève decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena, and consequently reduce the Navier–Stokes equation defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom. The present algorithm is well suited for the problems of flow control or optimization, where one has to compute the flow field repeatedly using the Navier–Stokes equation but one can also estimate the approximate solution space of the flow field based on the range of control variables. The low-dimensional dynamic model of viscous fluid flow derived by the present method is shown to produce accurate flow fields at a drastically reduced computational cost when compared with the finite difference solution of the Navier–Stokes equation. © 1998 John Wiley & Sons, Ltd.     

Design or problem solving — a critical choice for the engineering profession

There are two distinct interpretations of engineering design. The problem-solving approach, popular in many tertiary institutions and with an emphasis on solving structured, welldefined problems using standardized techniques, may be traced to ‘hard’ systems thinking. The creative design approach, on the other hand, combines analytical and systems thinking with human factors in engineering design to create and take advantage of opportunities to serve society. This paper discusses the limitations of the problem-solving approach in dealing with many realworld tasks. It explores the nature of those tasks and seeks to develop an understanding of the intellectual resources needed for design. This paper concludes with a discussion of the choices open to the profession; an acceptance of a ‘fix it’ role deriving from a problem-solving mentality or belief in a creative, revolutionary role in shaping future society. Although emphasizing engineering design, the arguments developed in the paper are relevant to all designers and design educators.     

Parametric method of solving a nonlinear heat-conduction problem for a semiinfinite body

A new method is proposed for solving heat-conduction problems with nonlinear boundary conditions.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 3, pp. 510–519, September, 1978.     

An optimal method for the solution of the constrained eigenvalue/response problem for large structures comprising rigidly-connected substructures

Dynamic substructuring or modal synthesis is established as a powerful means by which the computational effort required to perform dynamic analyses of large structures comprising many degrees of freedom can be reduced. For the case where the substructures are rigidly connected a method of synthesis is proposed which gives very substantial improvements in efficiency over existing methods both in terms of the number of floating-point operations which must take place and the on-board computer memory required. The proposed method makes use of a stable co-ordinate transformation which reduces the mass matrix for the constrained composite-system to the identity matrix and makes the stiffness matrix banded. The bandwidth is exactly equal to twice the total number of constraints plus 1. This is usually very small in comparison with the total number of degrees of freedom in the constrained composite-system. The proposed method is ideal for combining a number of substructures into a single new substructure in an efficient way without necessarily disposing of any degrees of freedom. Approximate formulae are presented which give the ratios of the number of operations required for established methods compared with the number needed for the proposed method. These formulae determine when it is appropriate to use the method proposed here.     

A proof for the quasisteady method of solving the Stefan problem

The location limit for the interphase boundary is found in the region exterior to a sphere with a finite radius. It is shown that the solution to the Stefan problem for this region by the method of quasisteady states approaches the same limit as t .Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 24, No. 1, pp. 135–137, January, 1973.     

Finite-difference method for solving a one-dimensional nonstationary problem of radiative-conductive heat transfer

An algorithm and examples of the solution of problems of complex heat transfer are given.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 51, No. 5, pp. 840–847, November, 1986.     

Numerical-analytical method of solving the nonlinear heat-conduction problem for a doubly connected variable-thickness plate

The method permits construction of an approximate solution in an analytically closed form on each of the radial rays into which the plate domain is separated.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 53, No. 4, pp. 654–658, October, 1987.     

A method of solving the heat-conduction problem for simple laminated bodies

The temperature is expressed as the sum of integrals of a particular form. Expressions for the required time functions are determined in accordance with prescribed boundary and contact conditions by means of the Laplace transformation. Sample solutions are given.     

Difference method of solving an inverse problem for the convective diffusion equation

An inverse problem related to the recovery of the initial and boundary conditions for the one-dimensional equation of convective diffusion that describes the process of the physicochemical method of stimulation of the oil bed is considered. A numerical method based on the use of the difference scheme whose differential approximation is a hyperbolic-type equation is suggested for solving the problem.     

Theoretical basis of the finite element iterative method for the eigenvalue problem in stationary cracks

In a recently published paper a finite element based iterative method was introduced for the solution of the eigenvalue problem of stationary cracks.¹ In this paper we give the theoretical basis of this iterative method and we show why it converges and how it could be extended to more complex fracture problems. The cases of cracks at interfaces are illustrated.     

An improved computational technique for perturbations of the generalized symmetric linear algebraic eigenvalue problem

A technique is developed and presented which improves the accuracy of the predictions for the eigenvalues of the usual first order perturbation scheme for the generalized symmetric linear algebraic eigenvalue problem. The improved method very simply replaces the usual B -norms appearing in the denominator of the expressions for the first order changes in the eigenvalues by B ^P-norms where B ^P is the perturbed B matrix. With increasing sizes for the perturbations the improved technique produces results whose accuracy approaches those of the usual second order predictions. Numerical studies are presented to support the analysis.