Isogeometric analysis in BIE for 3-D potential problem

The isogeometric analysis is introduced in the Boundary Integral Equation (BIE) for solution of 3-D potential problems. In the solution, B-spline basis functions are employed not only to construct the exact geometric model but also to approximate the boundary variables. And a new kind of B-spline function, i.e., local bivariate B-spline function, is deducted, which is further applied to reduce the computation cost for analysis of some special geometric models, such as a sphere, where large number of nearly singular and singular integrals will appear. Numerical tests show that the new method has good performance in both exactness and convergence.     

Multi-dimensional clearing functions for aggregate capacity modelling in multi-stage production systems

Nonlinear clearing functions have been proposed in the literature as metamodels to represent the behaviour of production resources that can be embedded in optimisation models for production planning. However, most clearing functions tested to date use a single-state variable to represent aggregate system workload over all products, which performs poorly when product mix affects system throughput. Clearing functions using multiple-state variables have shown promise, but require significant computational effort to fit the functions and to solve the resulting optimisation models. This paper examines the impact of aggregation in state variables on solution time and quality in multi-item multi-stage production systems with differing degrees of manufacturing flexibility. We propose multi-dimensional clearing functions using alternative aggregations of state variables, and evaluate their performance in computational experiments. We find that at low utilisation, aggregation of state variables has little effect on system performance; multi-dimensional clearing functions outperform single-dimensional ones in general; and increasing manufacturing flexibility allows the use of aggregate clearing functions with little loss of solution quality.     

The B-spline finite element method applied to axi-symmetrical andnonlinear field problems

A B-spline FEM (finite-element method) using the B-spline functions for rectangular elements as shape functions is presented. It is very effective for solving two-dimensional electromagnetic field problems in regular regions. Compared with the conventional FEM, it gives more accurate potential values and due to the inherent properties of B-spline functions yields much closer field values. Both the computing time and the storage capacity are greatly reduced as well     

Dual reciprocity BEM: local versus global approximation functions for diffusion, convection and other problems

The use of the global approximation functions (elements of Pascal's triangle, sine expansions and others) in the dual reciprocity boundary element method is compared to the better known local radial basis functions for convection, diffusion and other problems in which the volume integrals considered contain first and second derivatives of the problem variables, time derivatives and sums and products of functions, including nonlinear terms. It will be shown that whilst it is possible to obtain accurate solutions to the problems considered using the global functions, a successful solution to a given problem depends very much on the function chosen, as well as other factors.     

A Trefftz method in space and time using exponential basis functions: Application to direct and inverse heat conduction problems

In this paper we present a Trefftz method based on using exponential basis functions (EBFs) to solve one (1D) and two (2D) dimensional transient problems. We focus on direct and inverse heat conduction problems, the latter being the more challenging ones, to show the capabilities of the method. A summation of exponential basis functions (EBFs), satisfying the governing equation in time and space, with unknown coefficients is considered for the solution. The unknown coefficients are determined by the satisfaction of the prescribed time dependent boundary and initial conditions through a collocation method. Several 1D and 2D direct and inverse heat conduction problems are solved. Some numerical evidence is provided for the convergence and sensitivity of the method with respect to the noise levels of the measured data and time steps.     

A new 48 d.o.f. quadrilateral shell element with variable-order polynomial and rational B-spline geometries with rigid body modes

A 48 degrees-of-freedom (d.o.f.) quadrilateral thin elastic shell finite element using variable-order polynomial functions, B-spline functions and rational B-spline functions to model the shell surface is developed. This development may allow the stiffness formulation of the shell element to be linked to the geometry data bases created by computer aided design systems. The displacement functions are that of bicubic Hermitian polynomials. The displacement functions and d.o.f. are expressed and investigated in both the curvilinear and Cartesian forms. The cuivilinear form is simpler and can provide the proper solution for a certain class of shell problems. For certain highly curved shells such as bellows, however, the curvilinear form fails to properly model some rigid body modes even with either the explicit inclusion of rigid body terms or the high order displacement functions. It is suggested in this study that such difficulty can be circumvented and the rigid body modes can be properly included if a Cartesian form is used for displacement functions. The strain–displacement equations are expressed in curvilinear co-ordinates. Thus, the Cartesian displacement functions require a transformation to curvilinear displacement at each numerical integration point. Examples include a pinched cylinder, a translational shell under central load, a uniformly loaded hypar shell, a pressurized ovel shell, a semi-toroidal bellows and a U-shaped bellows. For the first four examples, geometric modellings consist of polynomials of second-order (subparametric), third-order (isoparametric), and fourth and fifth-order (both superparametric) as well as B-spline functions of fourth- and fifth-order. The geometries of the pinched cylinder, the semi-toroidal bellows, and the U-shaped bellows were modelled exactly using rational B-spline functions. All the results obtained are in good agreement with alternative existing solutions.     

Dynamic solution of unbounded domains using finite element method: discrete Green's functions in frequency domain

In this paper we present a new approach for finite element solution of time‐harmonic wave problems on unbounded domains. As representatives of the wave problems, discrete Green's functions are evaluated in finite element sense. The finite element mesh is considered to be of repeatable pattern (cell) constructed in rectangular co‐ordinates. The system of FE equations is therefore reduced to a set of well‐known dispersion equations by using a spectral solution approach. The spectral wave bases are constructed directly from the FE dispersion equations. Radiation condition is satisfied by selecting the wave bases so that the wave information is transmitted in appropriate directions at the cell level. Dirichlet/Neumann boundary conditions are defined at the edges of a quadrant of the main domain while using the axes of symmetry of the problem. A new discrete transformation method, recently proposed by the authors, is used to satisfy the boundary conditions. Comprehensive studies are made for showing the validity, accuracy and convergence of the solutions. The results of the benchmark problems indicate that the proposed method can be used to evaluate discrete Green's functions whose analytical forms are not available. Copyright © 2006 John Wiley & Sons, Ltd.     

Radial basis functions methods for solving Fokker-Planck equation

In this paper two numerical meshless methods for solving the Fokker-Planck equation are considered. Two methods based on radial basis functions to approximate the solution of Fokker-Planck equation by using collocation method are applied. The first is based on the Kansa's approach and the other one is based on the Hermite interpolation. In addition, to conquer the ill-conditioning of the problem for big number of collocation nodes, two time domain Discretizing schemes are applied. Numerical examples are included to demonstrate the reliability and efficiency of these methods. Also root mean square and N_e errors are obtained to show the convergence of the methods. The errors show that the proposed Hermite collocation approach results obtained by the new time-Discretizing scheme are more accurate than the Kansa's approach.     

Reliability Forecasting of a Load‐Haul‐Dump Machine: A Comparative Study of ARIMA and Neural Networks

Both the autoregressive integrated moving average (ARIMA or the Box–Jenkins technique) and artificial neural networks (ANNs) are viable alternatives to the traditional reliability analysis methods (e.g., Weibull analysis, Poisson processes, non‐homogeneous Poisson processes, and Markov methods). Time series analysis of the times between failures (TBFs) via ARIMA or ANNs does not have the limitations of the traditional methods such as requirements/assumptions of a priori postulation and/or statistically independent and identically distributed observations for TBFs. The reliability of an LHD unit was investigated by analysis of TBFs. Seasonal autoregressive integrated moving average (SARIMA) was employed for both modeling and forecasting the failures. The results were compared with a genetic algorithm‐based (ANNs) model. An optimal ARIMA model, after a Box–Cox transformation of the cumulative TBFs, outperformed ANNs in forecasting the LHD's TBFs. Copyright © 2015 John Wiley & Sons, Ltd.     

A robust time‐space formulation for large‐scale scalar wave problems using exponential basis functions

In this paper, a new effective boundary node method is presented for the solution of acoustic problems, directly in time domain, using exponential basis functions. Unlike many other methods using boundary information, the final coefficient matrix is sparse. The formulation is well suited for domains whose extent is relatively larger than the distance traveled by the acoustic wave in an increment of time. The exponential basis functions used satisfy the time‐space governing equation. This helps to choose a relatively large time increment and a moderate number of boundary points, which leads to reduction of computation time. The computation is performed incrementally using a weighted residual in time. Through a series of numerical examples, it is shown that the method, when combined with a domain decomposition strategy, is effectively capable of solving various 1‐ to 3‐dimensional acoustic problems.     

Implementation of a generalized exponential basis functions method for linear and non‐linear problems

In this paper, we address shortcomings of the method of exponential basis functions by extending it to general linear and non‐linear problems. In linear problems, the solution is approximated using a linear combination of exponential functions. The coefficients are calculated such that the homogenous form of equation is satisfied on some grid. To solve non‐linear problems, they are converted to into a succession of linear ones using a Newton–Kantorovich approach. The generalized exponential basis functions (GEBF) method developed can be implemented with greater ease compared with exponential basis functions, as all calculations can be performed using real numbers and no characteristic equation is needed. The details of an optimized implementation are described. We compare GEBF on some benchmark problems with methods in the literature, such as variants of the boundary element method, where GEBF shows a good performance. Also, in a 3D problem, we report the run time of the proposed method compared with that of Kratos, a parallel, highly optimized finite element code. The results show that in this example, to obtain the same level of error, much less computational effort is needed in the proposed method. Practical limitations might be encountered, however, for large problems because of dense matrix operations involved. Copyright © 2015 John Wiley & Sons, Ltd.     

Exponential basis functions in solution of static and time harmonic elastic problems in a meshless style

In this paper, exponential basis functions (EBFs) are used in a boundary collocation style to solve engineering problems whose governing partial differential equations (PDEs) are of constant coefficient type. Complex‐valued exponents are considered for the EBFs. Two‐dimensional elasto‐static and time harmonic elasto‐dynamic problems are chosen in this paper. The solution procedure begins with first finding a set of appropriate EBFs and then considering the solution as a summation of such EBFs with unknown coefficients. The unknown coefficients are determined by the satisfaction of the boundary conditions through a collocation method with the aid of a consistent and complex discrete transformation technique. The basis and various forms of the transformation have been addressed and discussed. We shall propose several strategies for selection of EBFs with the aid of the basis explained for the transformation. While using the transformation, the number of EBFs should not necessarily be equal to (or less than) the number of boundary information data. A library of EBFs has also been presented for further use. The effect of body forces is included in the solution via construction of particular solution by the use of the discrete transformation and another series of EBFs. A number of sample problems are solved to demonstrate the capabilities of the method. It has been shown that the time harmonic problems with high wave number can be solved without much effort. The method, categorized in meshless methods, can be applied to many other problems in engineering mechanics and general physics since EBFs can easily be found for almost all problems with constant coefficient PDEs. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical method for heat transfer problems using collocation and radial basis functions

Simple, mesh/grid free, numerical schemes for the solution of heat transfer problems are developed and validated. Unlike the mesh or grid-based methods, these schemes use well-distributed quasi-random collocation points and approximate the solution using radial basis functions. The schemes work in a similar fashion as finite differences but with random points instead of a regular grid system. This allows the computation of problems with complex-shaped boundaries in higher dimensions with no extra difficulty. © 1998 John Wiley & Sons, Ltd.     

Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

This study proposes a new uncertain analysis method for multibody dynamics of mechanical systems based on Chebyshev inclusion functions The interval model accounts for the uncertainties in multibody mechanical systems comprising uncertain‐but‐bounded parameters, which only requires lower and upper bounds of uncertain parameters, without having to know probability distributions. A Chebyshev inclusion function based on the truncated Chebyshev series, rather than the Taylor inclusion function, is proposed to achieve sharper and tighter bounds for meaningful solutions of interval functions, to effectively handle the overestimation caused by the wrapping effect, intrinsic to interval computations. The Mehler integral is used to evaluate the coefficients of Chebyshev polynomials in the numerical implementation. The multibody dynamics of mechanical systems are governed by index‐3 differential algebraic equations (DAEs), including a combination of differential equations and algebraic equations, responsible for the dynamics of the system subject to certain constraints. The proposed interval method with Chebyshev inclusion functions is applied to solve the DAEs in association with appropriate numerical solvers. This study employs HHT‐I3 as the numerical solver to transform the DAEs into a series of nonlinear algebraic equations at each integration time step, which are solved further by using the Newton–Raphson iterative method at the current time step. Two typical multibody dynamic systems with interval parameters, the slider crank and double pendulum mechanisms, are employed to demonstrate the effectiveness of the proposed methodology. The results show that the proposed methodology can supply sufficient numerical accuracy with a reasonable computational cost and is able to effectively handle the wrapping effect, as cosine functions are incorporated to sharpen the range of non‐monotonic interval functions. Copyright © 2013 John Wiley & Sons, Ltd.     

B-spline approximation in boundary face method for three-dimensional linear elasticity

In this paper, basis functions generated from B-spline or Non-Uniform Rational B-spline (NURBS), are used for approximating the boundary variables to solve the 3D linear elasticity Boundary Integral Equations (BIEs). The implementation is based on the BFM framework in which both boundary integration and variable approximation are performed in the parametric spaces of the boundary surfaces to keep the exact geometric information in the BIEs. In order to reduce the influence of tensor product of B-spline and make the discretization of a body surface easier, the basis functions defined in global intervals are translated into local form. B-spline fitting function built with the local basis functions is converted into an interpolation type of function in which the nodal values of the boundary variables are used for control points. Numerical tests for 3D linear elasticity problems show that the BFM with B-spline basis functions outperforms that with the well-known Moving Least Square (MLS) approximation.     

密码学中的布尔函数研究综述

概述了目前密码学中布尔函数的研究现状和重要研究方向上的新成果，并对布尔函数的研究进行了展望，指出了一些重要的研究热点问题．     

Transient behavior of time‐between‐failures of complex repairable systems

It is well known for complex repairable systems (with as few as four components), regardless of the time‐to‐failure (TTF) distribution of each component, that the time‐between‐failures (TBFs) tends toward the exponential. This is a long‐term or ‘steady‐state’ property. Aware of this property, many of those modeling such systems tend to base spares provisioning, maintenance personnel availability and other decisions on an exponential TBFs distribution. Such a policy may suffer serious drawbacks. A non‐homogeneous Poisson process (NHPP) accounts for these intervals for some time prior to ‘steady‐state’. Using computer simulation, the nature of transient TBF behavior is examined. The number of system failures until the exponential TBF assumption is valid is of particular interest. We show, using a number of system configurations and failure and repair distributions, that the transient behavior quickly drives the TBF distribution to the exponential. We feel comfortable with achieving exponential results for the TBF with 30 system failures. This number may be smaller for configurations with more components. However, at this point, we recommend 30 as the systems failure threshold for using the exponential assumption. Copyright © 2002 John Wiley & Sons, Ltd.     

Assessment of safety effects for widening urban roadways in developing crash modification functions using nonlinearizing link functions

Since a crash modification factor (CMF) represents the overall safety performance of specific treatments in a single fixed value, there is a need to explore the variation of CMFs with different roadway characteristics among treated sites over time. Therefore, in this study, we (1) evaluate the safety performance of a sample of urban four-lane roadway segments that have been widened with one through lane in each direction and (2) determine the relationship between the safety effects and different roadway characteristics over time. Observational before–after analysis with the empirical Bayes (EB) method was assessed in this study to evaluate the safety effects of widening urban four-lane roadways to six-lanes. Moreover, the nonlinearizing link functions were utilized to achieve better performance of crash modification functions (CMFunctions). The CMFunctions were developed using a Bayesian regression method including the estimated nonlinearizing link function to incorporate the changes in safety effects of the treatment over time. Data was collected for urban arterials in Florida, and the Florida-specific full SPFs were developed and used for EB estimation.     

Elastostatic Green’s functions for an arbitrary internal load in a transversely isotropic bi-material full-space

The problem of a full-space which is composed of two half-spaces with different transversely isotropic materials with an internal load at an arbitrary distance from the interface is considered. By virtue of Hu-Nowacki-Lekhnitskii potentials, the equations of equilibrium are uncoupled and solved with the aid of Hankel transform and Fourier decompositions. With the use of the transformed displacement- and stress-potential relations, all responses of the bi-material medium are derived in the form of line integrals. By appropriate limit processes, the solution can be shown to encompass the cases of (i) a homogeneous transversely isotropic full-space, and (ii) a homogeneous transversely isotropic half-space under arbitrary surface load. As the integrals for the displacement- and stress-Green’s functions, for an arbitrary point load can be evaluated explicitly, illustrative results are presented for the fundamental solution under different material anisotropy and relative moduli of the half-spaces and compared with existing solutions.