Multifrontal incomplete factorization for indefinite and complex symmetric systems

A new class of preconditioners based on the adaptive threshold incomplete multifrontal factorization for indefinite and complex symmetric systems is developed. Numerical experiments consisting of the 3D Helmholtz equations, fluid–structure interaction and localization problems demonstrate the excellent performance of the preconditioner. Copyright © 2001 John Wiley & Sons, Ltd.     

Stability analysis of hyper symmetric skeletal structures using group theory

The main objective of this article is to develop a methodology for the efficient calculation of buckling loads for frame structures having high-order symmetry properties in order to reduce the size of their associated eigenvalue problems. This is achieved by decomposing the second-order stiffness matrix of a symmetric model into submatrices using a representation of its symmetry group, via a step-by-step approach. The physical interpretation of the resulting submatrices is shown as substructures (factors), and the possibility of further decomposition is then investigated for each of the constructed submodels. Due to the similarity in transformation, the constructed submatrices contain the eigenvalues of the main structural matrix. The buckling load of the entire structure is obtained by calculating the buckling loads of its factors. The methods of the present paper provide a mathematical foundation and a logical means to deal with symmetry instead of looking for various boundary conditions to be imposed for symmetric structures, as in the traditional methods. Examples are provided to illustrate the simplicity and efficiency of the present method.     

Buckling load of symmetric plane frames using canonical forms and group theory

Summary The main objective of this article is to develop a methodology for an efficient calculation of buckling loads for symmetric frame structures in order to reduce the size of the eigensolution problems involved. This is achieved by decomposing a symmetric model into two submodels followed by their healing to obtain the factors of the model. The buckling load of the entire structure is then obtained by calculating the buckling loads of its factors. For each case, the results are approved using group theoretical approaches. The methods of the present paper provide a mathematical foundation and a logical means for dealing with symmetry, in place of looking for various boundary conditions to be imposed for symmetric structures, as in the traditional methods. Examples are provided to illustrate the simplicity and efficiency of the present method.     

Block diagonalization of Laplacian matrices of symmetric graphs via group theory

In this article, group theory is employed for block diagonalization of Laplacian matrices of symmetric graphs. The inter‐relation between group diagonalization methods and algebraic‐graph methods developed in recent years are established. Efficient methods are presented for calculating the eigenvalues and eigenvectors of matrices having canonical patterns. This is achieved by using concepts from group theory, linear algebra, and graph theory. These methods, which can be viewed as extensions to the previously developed approaches, are illustrated by applying to the eigensolution of the Laplacian matrices of symmetric graphs. The methods of this paper can be applied to combinatorial optimization problems such as nodal and element ordering and graph partitioning by calculating the second eigenvalue for the Laplacian matrices of the models and the formation of their Fiedler vectors. Considering the graphs as the topological models of skeletal structures, the present methods become applicable to the calculation of the buckling loads and the natural frequencies and natural modes of skeletal structures. Copyright © 2006 John Wiley & Sons, Ltd.     

Analysis of the dynamic behavior of two parallel symmetric cracks using the non-local theory

In this paper, the dynamic behavior of two parallel symmetric cracks under harmonic anti-plane shear waves is studied using the non-local theory. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the problem to obtain the stress occurs near the crack tips. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the lattice parameter and the distance between two parallel cracks, respectively.     

Graph theory and networks in Biology

A survey of the use of graph theoretical techniques in Biology is presented. In particular, recent work on identifying and modelling the structure of bio-molecular networks is discussed, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronisation and disease propagation.     

Block preconditioners for symmetric indefinite linear systems

This paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a Krylov subspace method for solving symmetric indefinite linear systems. The focus is on large‐scale real world problems where block approximations are a practical necessity. The main illustration is the performance of the block diagonal, constrained, and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large‐scale finite element discretization of Biot's consolidation equations. This system of equations is of fundamental importance to geomechanics. Numerical studies show that simple diagonal approximations to the (1,1) block K and inexpensive approximations to the Schur complement matrix S may not always produce the most spectacular time savings when K is explicitly available, but is able to deliver reasonably good results on a consistent basis. In addition, the block diagonal preconditioner with a negative (2,2) block appears to be reasonably competitive when compared to the more complicated ones. These observation are expected to remain valid for coefficient matrices whereby the (1,1) block is sparse, diagonally significant (a notion weaker than diagonal dominance), moderately well‐conditioned, and has a much larger block size than the (2,2) block. Copyright © 2004 John Wiley & Sons, Ltd.     

Reliability of dynamic systems in random environment by extreme value theory

A practical method is developed for estimating the performance of highly reliable dynamic systems in random environment. The method uses concepts of univariate extreme value theory and a relatively small set of simulated samples of system states. Generalized extreme value distributions are fitted to state observations and used to extrapolate Monte Carlo estimates of reliability and failure probability beyond data. There is no need to postulate functional forms of extreme value distributions since they are selected by the estimation procedure. Our approach can be viewed as an alternative implementation of the method in [7] for estimating system reliability. Numerical examples involving Gaussian and non-Gaussian system states are used to illustrate the implementation of the proposed method and assess its accuracy.     

An approximate nonlinear dynamic theory for plates

Summary In this paper an approximate theory for the moderately large motion of transversely isotropic (orthotropic) plates is presented. Unlike other methods used in the past, the method used here involves no artificial assumptions. The method of approach begins with the equations of a partially non-linear elasticity theory and utilizes a method of asymptotic integration to arrive at successive two-dimensional approximations of increasing accuracy. The first approximation theory obtained here is the dynamic counterpart of Karman's plate theory. It includes the effect of rotatory inertia.     

Adaptive observers for linear dynamic systems

A synthesis method is given for an adaptive observer for a linear dynamic system with a variable gain in the adaptation loop. If the condition for bandwidth at the input to the system is not met, an adaptive observer is proposed with a multiplicative vector for the adjustable parameters.     

Condensate theory versus pairing theory for degenerate Bose systems

The statistical variational principle for the thermodynamic potential is used to investigate the quantum statistically degenerate phase of an interacting Bose system. Starting with a general mean-field trial Hamiltonian, we find a set of integral equations that has two different types of solution: (a) the condensate sssssstheory with a finite c-number part for the lowest momentum state operator; (b) the pairing theory with zero c-number part. The condensate theory is identical to the self-consistent Hartree-Fock-Bogoliubov approximation of the Beliaev equations, whereas the pairing theory is identical to the theory proposed by Evans and Imry. We solve the resulting integral equations for both theories numerically and show that—independent of the special form of interacting potentials—the condensate theory gives the lower free energy and the higher transition temperature. Therefore we conclude that the condensate theory describes the thermodynamically stable state rather than the pairing theory.Project of the Sonderforschungsbereich Frankfurt/Darmstadt financed by special funds of the Deutsche Forschungsgemeinschaft.     

Graph theory and stability analysis of protein complex interaction networks

Protein complexes play an essential role in many biological processes. Complexes can interact with other complexes to form protein complex interaction network (PCIN) that involves in important cellular processes. There are relatively few studies on examining the interaction topology among protein complexes; and little is known about the stability of PCIN under perturbations. We employed graph theoretical approach to reveal hidden properties and features of four species PCINs. Two main issues are addressed, (i) the global and local network topological properties, and (ii) the stability of the networks under 12 types of perturbations. According to the topological parameter classification, we identified some critical protein complexes and validated that the topological analysis approach could provide meaningful biological interpretations of the protein complex systems. Through the Kolmogorov–Smimov test, we showed that local topological parameters are good indicators to characterise the structure of PCINs. We further demonstrated the effectiveness of the current approach by performing the scalability and data normalization tests. To measure the robustness of PCINs, we proposed to consider eight topological‐based perturbations, which are specifically applicable in scenarios of targeted, sustained attacks. We found that the degree‐based, betweenness‐based and brokering‐coefficient‐based perturbations have the largest effect on network stability.Inspec keywords: graph theory, perturbation theory, proteins, molecular configurations, molecular biophysics, pattern clustering, pattern classification, cellular biophysics, biology computingOther keywords: stability analysis, protein complex interaction networks, biological processes, protein complex interaction network, cellular processes, interaction topology, graph theoretical approach, local network topological properties, topological parameter classification, Kolmogorov‐Smirnov test, network structures, data normalisation tests, topological‐based perturbations, highly clustered protein complexes, brokering‐coefficient‐based perturbations, betweenness‐based perturbations     

Generic kanban systems for dynamic environments

Kanbans have shown successful results in lowering inventory and shortening lead time in repetitive production systems. Unfortunately, such systems are not applicable to production environments with dynamic characteristics. Here a modified kanban system, the generic kanban system, is proposed for such dynamic environments. The generic kanban system behaves similarly to the push system except that it is more flexible with respect to system performance and more robust as to the location of the bottleneck. The simulation results that the generic kanban system is dominant over the dedicated kanban system and the CONWIP system. The adaptability of such a system to dynamic environments is justified.     

Optimum group designs for random-effects nonlinear dynamic processes

We give the theoretical background and an algorithm for calculating optimum group experimental designs for regression models with random model parameters. The theory is applicable to those practical situations in which a dynamical system is sensitive to sampling or gives a different response at each run of the experiment. This difference in response is due both to the inherent nature of the system and to random noise in the observations. We treat all observations as independent. Together with the definition of the group designs that we introduce, this structure leads to a practical and numerically tractable representation of optimum designs for estimation of the mean values of the parameters. We present two different examples of chemical reactions where such modelling of the random structure is particularly relevant, one for a single response, the other for multivariate data.     

Variational principles for linear coupled dynamic theory of thermoviscoelasticity

Variational principles for linear coupled dynamic theory of thermoviscoelasticity are constructed using variational theory of potential operators. The functional derived herein gives, when varied, all the governing equations, including the boundary and initial conditions, as the Euler equations. The procedure shown herein does not require, in contrast to Gurtin's method, the transformation of field equations into an equivalent set of integro-differential equations, and includes the initial conditions of the problem explicitly. Gurtin's variational principle for dynamic theory of thermoviscoelasticity is also derived and compared with the present one. Variational principles for elastodynamics, visco-elasticity, etc. are derived as special cases of the variational principle derived herein.     

Capacitated dynamic lotsizing heuristics for serial systems

This paper deals with multi-item capacitated lotsizing in a serial production environment under dynamic demand conditions. The primary objective of this research is to investigate the performance of simple level-by-level heuristics. A problem classification is proposed to distinguish between easy problems and harder ones for which no simple heuristics can be developed. It is shown how simple algorithms can be constructed for the easier problems using a level-by-level approach. These heuristics are the first of their kind for this problem. They are compared with the optimal solution for a set of test problems. The results show that for some problem classes strategies which are frequently used in practice may result in rather poor schedules.     

A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations

The standard SSOR preconditioner is ineffective for the iterative solution of the symmetric indefinite linear systems arising from finite element discretization of the Biot's consolidation equations. In this paper, a modified block SSOR preconditioner combined with the Eisenstat‐trick implementation is proposed. For actual implementation, a pointwise variant of this modified block SSOR preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness. Numerical experiments show that the proposed modified SSOR preconditioned symmetric QMR solver can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime. Moreover, the proposed modified SSOR preconditioners can be generalized to non‐symmetric Biot's systems. Copyright © 2005 John Wiley & Sons, Ltd.