A semi-Markovian model allowing for inhomogenities with respect to process time

A non-homogeneous semi-Markov process is considered as an approach to model reliability characteristics of components or small systems with complex test resp. maintenance strategies. This approach generalizes previous results achieved for ordinary inhomogeneous Markov processes. This paper focuses on the following topics to make the application of semi-Markovian models feasible: rather than transition probabilities Q_ij(t), which are used in normal mathematical text books to define semi-Markov processes, transition rates λ_ij( ) are used, as is usual for ordinary Markov processes. These transition rates may depend on two types of time in general: on process time and on sojourn time in state i. Such transition rates can be followed from failure and repair rates of the underlying technical components, in much the same way, as this is known for ordinary Markov processes. Rather than immediately starting to solve the Kolmogorov equations, which would result in N² integral equations, a system of N integral equations for frequency densities of reaching states is considered. Once this system is solved, the initial value problem for state probabilities can be solved by straightforward integration. An example involving 14 states has been solved as an illustration using the approach.     

Semi-analytical solution for orthotropic piezoelectric laminates in cylindrical bending with interfacial imperfections

The static and dynamic problems of an imperfectly bonded, orthotropic, piezoelectric laminate in cylindrical bending are investigated based on the equations of piezoelasticity. A general spring layer is adopted to model the bonding imperfection at the interfaces of the laminate. A recently proposed semi-analytical approach, which makes a hybrid use of the state-space method (SSM) and the differential quadrature method (DQM), is employed. This approach allows us to efficiently analyze the laminate with a large number of plies and with arbitrary boundary conditions at the two ends. Numerical examples are considered and discussed to show the efficiency of the present semi-analytical solution and the effects of relevant parameters on the behavior of the laminate.     

Effect of Friction on Edge Singularities in Slip Bands

Resistance to dislocation glide in slip bands due to the presence of other dislocations may be represented by an additional term in the integral equation formulation of the equilibrium state. As a consequence, the asymptotic behaviour of dislocation density at the edge of interval is modified from the conventional square root singularity. We propose an efficient and fast numerical method that is suitable for solving the resulting singular equations of the second kind based on the Gauss-Jacobi quadrature. The quadrature formulae involve fixed nodal points and provide exact results for polynomials up to degree 2n-1, where n is the number of nodes. A numerical example of the application of the Gauss-Jacobi rule to a slip band problem is provided as a demonstration of the validity of the method.     

A wideband fast multipole accelerated boundary integral equation method for time‐harmonic elastodynamics in two dimensions

This article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two‐dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low‐frequency FMM and the high‐frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura's group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low‐frequency FMM and the quadrature order for the high‐frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton–Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, Nlog N, where N is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite‐size phononic crystal was demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.     

Two-dimensional version of Sternberg and Al-Khozaie fundamental solution for viscoelastic analysis using the boundary element method

In this work a general and concise two-dimensional fundamental solution is obtained for quasi-static linear viscoelastic problems using the boundary element method. For this purpose, the three-dimensional fundamental displacement, derived by Sternberg and Al-Khozaie from the generalization of Navier equation, is integrated with respect to z-coordinate. A time formulation is constructed from the viscoelastic Reciprocity Principle, defined in terms of the Stieltjes integral and the material functions are acquired by means of Boltzmann's rheological model. The collocation method and a semi-analytical procedure for the singular boundary integral are employed to the numerical analysis of the boundary integral. The Gaussian quadrature, the analytical method and an incremental approach are used to deal with the convolution integral. As the latter has presented the best performance, it is employed in most analyses of the examples. Finally, numerical results of problems, found in the literature, are presented in order to validate the formulation and the two-dimensional fundamental solution.     

A general method for analyzing moderately large deflections of a non-uniform beam: an infinite Bernoulli–Euler–von Kármán beam on a nonlinear elastic foundation

The present paper concerns a semi-analytical procedure for moderately large deflections of an infinite non-uniform static beam resting on a nonlinear elastic foundation. To construct the procedure, we first derive a nonlinear differential equation of a Bernoulli–Euler–von Kármán “non-uniform” beam on a “nonlinear” elastic foundation, where geometrical nonlinearities due to moderately large deflection and beam non-uniformity are effectively taken into account. The nonlinear differential equation is transformed into an equivalent system of nonlinear integral equations by a canonical representation. Based on the equivalent system, we propose a method for the moderately large deflection analysis as a general approach to an infinite non-uniform beam having a variable flexural rigidity and a variable axial rigidity. The method proposed here is a functional iterative procedure, not only fairly simple but straightforward to apply. Here, a parameter, called a base point of the method, is also newly introduced, which controls its convergence rate. An illustrative example is presented to investigate the validity of the method, which shows that just a few iterations are only demanded for a reasonable solution.     

An analytical framework for reliability growth of one-shot systems

In this paper, we introduce a new reliability growth methodology for one-shot systems that is applicable to the case where all corrective actions are implemented at the end of the current test phase. The methodology consists of four model equations for assessing: expected reliability, the expected number of failure modes observed in testing, the expected probability of discovering new failure modes, and the expected portion of system unreliability associated with repeat failure modes. These model equations provide an analytical framework for which reliability practitioners can estimate reliability improvement, address goodness-of-fit concerns, quantify programmatic risk, and assess reliability maturity of one-shot systems. A numerical example is given to illustrate the value and utility of the presented approach. This methodology is useful to program managers and reliability practitioners interested in applying the techniques above in their reliability growth program.     

Cauchy-type integrals and integral equations with logarithmic singularities

Summary The Gauss-type quadrature methods with a logarithmic weight function can be extended to the evaluation of Cauchy-type integrals and to the solution of Cauchy-type integral equations by reduction of the latter to a linear system of algebraic equations. This system is obtained by applying the integral equation at properly selected collocation points. The poles of the integrand lying in the integration interval were treated as lying outside this interval. The efficiency of the method, both in evaluating integrals and solving integral equations, is exhibited by a numerical example. Finally, an application of the method to a crack problem of plane elasticity is made.     

Transient analysis of reliability with and without repair for K-out-of-N:G systems with two failure modes

This paper presents Markov models for transient analysis of reliability with and without repair for K-out-of-N:G systems subject to two failure modes. The reliability of repairable systems can be calculated as a result of the numerical solution of a simultaneous set of linear differential equations. Closed form solutions of the transient probabilities are used to obtain the reliability for nonrepairable systems.     

A semi-analytical solution for multiple circular inhomogeneities in one of two joined isotropic elastic half-planes

The paper presents a semi-analytical method for solving the problem of two joined, dissimilar isotropic elastic half-planes, one of which contains a large number of arbitrary located, non-overlapping, perfectly bonded circular elastic inhomogeneities. In general, the inhomogeneities may have different elastic properties and sizes. The analysis is based on a solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using a Taylor series expansion. Apart from round-off, the only errors introduced into the solution are due to truncation of the Fourier series. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the inhomogeneities. Numerical examples are included to demonstrate the effectiveness of the approach.     

Transient analysis of reliability with and without repair for K-out-of-N :G systems with M failure modes

This paper represents Markov models for transient analysis of reliability with and without repair for K-out-of-N :G systems subject to M failure modes. The reliability and the mean time between failures of repairable systems can be calculated as a result of numerical solution of simultaneous set of linear differential equations. Closed form solutions of the transient probabilities are used to obtain the reliability and the mean time to failure for nonrepairable systems.     

Stokes flow around cylinders in a bounded two‐dimensional domain using multipole‐accelerated boundary element methods

The multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from O(N²) (iterative solvers) or O(N³) (gaussian elimination) to O(N log N) or O(N), where N is the number of nodes in the discretized system. Such a reduction in the growth of the calculation time and data storage is crucial in applications where N is large, such as when modelling the macroscopic behaviour of suspensions of particles. In such cases, a minimum of 1000 particles is needed to obtain statistically meaningful results, leading to systems with N of the order of 10 000 for the smallest problems. When only boundary velocities are known, the indirect boundary element formulation for Stokes flow results in Fredholm equations of the second kind, which generally produce a well‐posed set of equations when discretized, a necessary requirement for iterative solution methods. The direct boundary element formulation, on the other hand, results in Fredholm equations of the first kind, which, upon discretization, produce ill‐conditioned systems of equations. The model system here is a two‐dimensional wide‐gap couette viscometer, where particles are suspended in the fluid between the cylinders. This is a typical system that is efficiently modelled using boundary element method simulations. The multipolar technique is applied to both direct and indirect formulations. It is found that the indirect approach is sufficiently well‐conditioned to allow the use of fast multipole methods. The direct approach results in severe ill‐conditioning, to a point where application of the multipole method leads to non‐convergence of the solution iteration. Copyright © 1999 John Wiley & Sons, Ltd.     

Complex hypersingular integrals and integral equations in plane elasticity

Summary Complex hypersingular (finite-part) integrals and integral equations are considered in the functional class of N. Muskhelishvili. The appropriate definition is given. Three regularization (equivalence) formulae follow from this definition. They reduce hypersingular integrals to singular ones and allow to derive hypersingular analogues for Sokhotsky-Plemelj's formulae and for conditions that are necessary and sufficient for the function to be piecewise holomorphic. Two approaches to get and investigate complex hypersingular equations follow from these results: one of them is based on the equivalence formulae; as to the other, it is based on above-mentioned conditions. As an example, authors' equation for plane elasticity is studied. The existence of a unique solution is stated and some advantages over singular equations are outlined. To solve hypersingular equations the quadrature rules are presented. The accuracy of different quadrature formulae is compared, the examples being used. They confirm the need to take into account asymptotics and to carry out a thorough analytical investigation to get safe numerical results.     

Static solutions for functionally graded simply supported plates

In this article mixed semi-analytical and analytical solutions are presented for a rectangular plate made of functionally graded (FG) material. All edges of a plate are under simply supported (diaphragm) end conditions and general stress boundary conditions can be applied on both top and bottom surface of a plate during solution. A mixed semi-analytical model consists in defining a two-point boundary value problem governed by a set of first-order ordinary differential equations in the plate thickness direction. Analytical solutions based on shear-normal deformation theories are also established to show the accuracy, simplicity and effectiveness of mixed semi-analytical model. The FG material is assumed to be exponential in the thickness direction and Poisson’s ratio is assumed to be constant.     

Analytical solutions to the planar non-axisymmetric elasticity and thermoelasticity problems for homogeneous and inhomogeneous annular domains

This paper presents an analytical approach to solving the plane non-axisymmetric elasticity and thermoelasticity problems in terms of stresses for isotropic, homogeneous or inhomogeneous annular domains. The key feature of this approach is integration of the equilibrium equations in order to: a) express all the stress-tensor components in terms of a governing stress; b) deduce the integral equilibrium conditions, which are vital for the solution. Because the equilibrium equations are insensitive of material properties, the obtained expressions and integral conditions fit both homogeneous and inhomogeneous cases. The governing stress is derived out of the compatibility equation. Regarding complete construction of the solution, the integral compatibility conditions are deduced by integrating the strain-displacement relations. In the case of inhomogeneous material, the governing compatibility equation is reduced to Volterra type integral equation which then is solved by simple iteration method. The rapid convergence of the iterative procedure is established.     

Mathematical Modeling of Scattering of Electromagnetic Waves by a Thin Penetrable Defect

We propose a new approximate system of singular integral equations for the mathematical modeling of scattering of electromagnetic waves by a thin elongated defect. The system consists of two singular integral equations of the second kind given in the median line of the cross section of the scatterer. Its form is identical for the TM and TE polarizations of exciting waves. The effective direct (without preliminary analytic regularization) numerical algorithm used for the solution of the problem is based on special quadrature formulas for singular integrals. The applicability of the proposed mathematical model is checked for a closed cylindrical dielectric shell.     

A multiwavelet Galerkin method for Stokes problems using boundary integral equations

A multiwavelet Galerkin method for the simulation of two-dimensional incompressible viscous fluid flow in primitive variables is developed in this paper. This method combines a Galerkin variational formulation of the boundary integral equation based on single layer potential to the Stokes equation with the trial and test functions constructed by Alpert multiwavelets. Due to the use of multiwavelets and compression strategies, the present method reduces the computational complexity of system matrix from O(N²) to almost O(N) (where N is the number of unknowns). In order to compute the double integrals with logarithmic kernel more efficiently, the analytical formulae to calculate the inner integrals are presented in this paper and then the Gaussian quadrature is used for the outer integrals. Some numerical examples are given to demonstrate the availability of the method.     

Stress intensity factors in two bonded elastic layers containing cracks perpendicular to and on the interface—I. Analysis

In this paper the basic crack problem which is essential for the study of subcritical crack propagation and fracture of layered structural materials is considered. Because of the apparent analytical difficulties, the problem is idealized as one of plane strain or plane stress. An additional simplifying assumption is made by restricting the formulation of the problem to crack geometries and loading conditions which have a plane of symmetry perpendicular to the interface. The general problem is formulated in terms of a coupled system of four integral equations. For each relevant crack configuration of practical interest the singular behavior of the solution near and at the ends and points of intersection of the cracks is investigated and the related characteristic equations are obtained. The edge crack terminating at and crossing the interface, the T-shaped crack consisting of a broken layer and a delamination crack, the cross-shaped crack which consists of delamination crack intersecting a crack which is perpendicular to the interface and a delamination crack initiating from a stress-free boundary of the bonded layers are some of the practical crack geometries considered as examples. The formulation of the problem is given in Part I of the paper. Part II deals with the solution of the integral equations and presentation of the results.     

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Summary A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.With 2 Figures     

Estimation of heat flux and mechanical loads on laminated functionally graded plate

The spatially and temporally varying heat flux and mechanical load on the top (bottom) surface of laminated plates with functionally graded layers are estimated using an inverse algorithm. The temperature and strains at a number of points on the bottom (top) surface of the plate are the only measured input data. The solution of corresponding direct problem is used to simulate the measured temperatures and strains, which are obtained based on a three-dimensional layerwise thermoelastic analysis. The conjugate gradient method as a powerful technique for optimization in conjunction with the discrepancy principle is employed to develop the inverse solution procedure. A semi-analytical approach composed of the layerwise-differential quadrature method and series solution is adopted to discretize the governing differential equations subjected to the related boundary and initial conditions. The influence of measurement errors on the accuracy of the estimated heat flux and mechanical load is investigated. The good accuracy of the results validates the presented inverse approach.