An analytical method for magnetothermoelastic analysis of functionally graded hollow cylinders

This paper considers magnetothermoelastic behavior of a functionally graded material (FGM) hollow cylinder, placed in a uniform magnetic field, subjected to thermal and mechanical loads. Exact solutions for stresses and perturbations of the magnetic field vector in FGM hollow cylinders is determined by using the infinitesimal theory of magnetothermoelasticity. Numerical results indicate that the inhomogeneous constants presented in the present study are useful parameters from a design point of view in that it can be tailored for specific applications to control the stress and perturbation of magnetic field vector distributions. This research is helpful for the optimum design annular cylindrical FGM sensors/actuators.     

Nonlinear transient thermal stress and elastic wave propagation analyses of thick temperature-dependent FGM cylinders,using a second-order point-collocation method

Nonlinear transient thermal stress and elastic wave propagation analyses are developed for hollow thick temperature-dependent FGM cylinders subjected to dynamic thermomechanical loads. Stress wave propagation, wave shape distortion, and speed variation under impulsive mechanical loads in thermal environments are also investigated. In contrast to researches accomplished so far, a second-order formulation rather than a first-order one is employed to improve the accuracy. The FDM method (as a point-collocation FEM method) is used. It is known that other FEM methods cannot show the actual trend jumps due to distributing the abrupt changes in the quantities as the numerical errors and the residuals of the governing equations among the nodal results. Furthermore, the required computational time and allocated computer memory are much reduced by the present solution algorithm. The cylinder is not divided into isotropic sub-cylinders. Therefore, artificial wave reflections from the hard interfaces are avoided. Time variations of the temperatures, displacements, and stresses due to the dynamic or impulsive loads are determined by solving the resulted highly nonlinear governing equations using an iterative updating solution scheme. A sensitivity analysis includes effects of the volume fraction indices, dimensions, and temperature-dependency of the material properties is performed. Results reveal the significant effect of the temperature-dependency of the material properties on the thermoelastic stresses and present some interesting characteristics of the thermoelastic and wave propagation behaviors.     

A remark on an integral equation via the method of moving spheres

In this paper, we study the following integral equation:

     

Structural elastic and creep models of a UD composite in longitudinal shear

The transient creep of a UD composite with a quadratic arrangement of elastic fibers of quadratic cross section is investigated. The deformational properties of the composite are determined from the known properties of its constituents. A structural model of the UD composite is developed, whose minimal elementary cell contains four elements. The stress-strain state of the elements is assumed homogeneous. Two types of basic and resolving governing equations of transient creep are deduced, which are based on static or kinematic assumptions. In each of the cases, a formula for the longitudinal elastic shear modulus of the composite is found. The stationary solutions of creep equations allow one to obtain formulas of the steady-state creep of the composite in a form similar to Norton’s law. Numerical calculations are also performed, and a comparison of the results with data given in the literature bears witness to the efficiency of the models developed and the solutions obtained. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 437–448, July–August, 2007.     

Constitutive modeling of void-growth-based tensile ductile failures with stress triaxiality effects

In most metals and alloys, the evolution of voids has been generally recognized as the basic failure mechanism. Furthermore, stress triaxiality has been found to influence void growth dramatically. Besides strain intensity, it is understood to be the most important factor that controls the initiation of ductile fracture. We include sensitivity of stress triaxiality in a variational porous plasticity model, which was originally derived from hydrostatic expansion. Under loading conditions rather than hydrostatic deformation, we allow the critical pressure for voids to be exceeded so that the growth due to plasticity becomes dependent on the stress triaxiality. The limitations of the spherical void growth assumption are investigated. Our improved constitutive model is validated through good agreements with experimental data. Its capacity for reproducing realistic failure patterns is also indicated by a numerical simulation of a compact tensile (CT) test.     

Numerical solution of thermal convection problems using the multidomain boundary element method

The multidomain dual reciprocity method (MD‐DRM) has been effectively applied to the solution of two‐dimensional thermal convection problems where the momentum and energy equations govern the motion of a viscous fluid. In the proposed boundary integral method the domain integrals are transformed into equivalent boundary integrals by the dual reciprocity approach applied in a subdomain basis. On each subregion or domain element the integral representation formulas for the velocity and temperature are applied and discretised using linear continuous boundary elements, and the equations from adjacent subregions are matched by additional continuity conditions. Some examples showing the accuracy, the efficiency and flexibility of the proposed method are presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 469–489, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10016     

Optimized Dropping and Rolling (ODR) method for packing of poly-disperse spheres

Powder-based microstructures are traditionally modelled using densely packed spherical particles. In this letter, a new dropping and rolling packing algorithm is employed by means of analytical equations, to investigate the random packing of spherical particles. The three-dimensional randomly packed microstructures are quantified in terms of packing density and computational time.     

Numerical solution of the one phase 1D fractional Stefan problem using the front fixing method

In this paper, we present a novel numerical scheme to solve a 1D, one‐phase extended Stefan problem with fractional Caputo derivative with respect to time. The proposed method is based on a suitable choice of the new space coordinate for the subdiffusion equation and extends the front‐fixing method to the subdiffusion case. In the final part, examples of numerical results are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

The solution of the two-dimensional sine-Gordon equation using the method of lines

The method of lines is used to transform the initial/boundary-value problem associated with the two-dimensional sine-Gordon equation in two space variables into a second-order initial-value problem. The finite-difference methods are developed by replacing the matrix-exponential term in a recurrence relation with rational approximants. The resulting finite-difference methods are analyzed for local truncation error, stability and convergence. To avoid solving the nonlinear system a predictor–corrector scheme using the explicit method as predictor and the implicit as corrector is applied. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given.     

A solution for the local plane strain problems using elliptic integrals of Cauchy type

We consider a parallel fiber‐reinforced periodic elastic composite that present an imperfect contact of spring type between the fiber and the matrix. We use the elliptic integral of Cauchy type for solving the plane strain local problems that arise from the asymptotic homogenization method. Several general conditions are assumed, which include that the fibers are disposed of arbitrary manner in the local cell, that all fibers present contact perfect with different constants of imperfection, and that their cross section is a smooth closed arbitrary curve. We find that there are infinity solutions for these problems, and we find relations between these solutions and effective coefficients of the composite. Finally, we obtain analytic formulae for the circular fiber case and show some numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

A hybrid legendre tau method for the solution of a class of nonlinear wave equations with nonlinear dissipative terms

This article presents a technique based on the hybrid Legendre tau‐finite difference method to solve the fourth order wave equation which arises in the elasto‐plastic‐microstructure models for longitudinal motion of an elasto‐plastic bar. Illustrative examples and numerical results obtained using new technique demonstrate that the proposed approach is efficient in treating longitudinal equation of ealsto‐plastic bar. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1055–1071, 2011     

High performance solution of partial differential equations discretized using a Chebyshev spectral collocation method

When a Chebyshev spectral collocation method is applied to a flow problem in a rectangularly decomposable domain it leads to the solution of a structured linear system. Since the linear system is solved at each step of a Newton-type iterative process an efficient method to solve it needs to be provided. A level 3 BLAS-based algorithm is presented and its efficiency is studied.     

Algebraic formulations for the solution of the nullspace‐free eigenvalue problem using the inexact Shift‐and‐Invert Lanczos method

Given the generalized symmetric eigenvalue problem Ax=λMx, with A semidefinite and M definite, we analyse some algebraic formulations for the approximation of the smallest non‐zero eigenpairs, assuming that a sparse basis for the null space is available. In particular, we consider the inexact version of the Shift‐and‐Invert Lanczos method, and we show that apparently different algebraic formulations provide the same approximation iterates, under some natural hypotheses. Our results suggest that alternative strategies need to be explored to really take advantage of the special problem setting, other than reformulating the algebraic problem. Experiments on a real application problem corroborate our theoretical findings. Copyright © 2002 John Wiley & Sons, Ltd.     

Post processing of solution and flux for the nodal mimetic finite difference method

We develop and analyze a post processing technique for the family of low‐order mimetic discretizations based on vertex unknowns for the numerical treatment of diffusion problems on unstructured polygonal and polyhedral meshes. The post processing works in two steps. First, from the nodal degrees of freedom, we reconstruct an elemental‐based vector field that approximates the gradient of the exact solution. Second, we solve a local problem for each mesh vertex associated with a scheme degree of freedom to determine a post processed normal flux that is conservative and divergence preserving. Theoretical results and numerical experiments for two‐dimensional (2D) and 3D benchmark problems show optimal convergence rates. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 336–363, 2015     

Global optimization method using adaptive and parallel ensemble of surrogates for engineering design optimization

As a robust and efficient technique for global optimization, surrogate-based optimization method has been widely used in dealing with the complicated and computation-intensive engineering design optimization problems. It’s hard to select an appropriate surrogate model without knowing the behaviour of the real system a priori in most cases. To overcome this difficulty, a global optimization method using an adaptive and parallel ensemble of surrogates combining three representative surrogate models with optimized weight factors has been proposed. The selection of weight factors is treated as an optimization problem with the desired solution being one that minimizes the generalized mean square cross-validation error. The proposed optimization method is tested by considering several well-known numerical examples and one industrial problem compared with other optimization methods. The results show that the proposed optimization method can be a robust and efficient approach in surrogate-based optimization for locating the global optimum.     

High‐order numerical solution of second‐order one‐dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method

We present a high‐order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second‐order one‐dimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well‐conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 307–349, 2016     

A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation

This paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations. An algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed. Several illustrative examples, with linear and nonlinear fractional differential equations, are given to demonstrate the effectiveness of the method.     

A study about the solution of convection–diffusion–reaction equation with Danckwerts boundary conditions by analytical,method of lines and Crank–Nicholson techniques

We compare different solutions of the convection–diffusion–reaction problem with Danckwerts boundary conditions. Analytical solution is found, and method of lines and Crank–Nicholson method are described, applied, and compared for different values of Péclet and Damköhler numbers. The eigenvalues and eigenfunctions have been obtained for all the possible values of the dimensionless parameters. And the analytical expression of the concentration has been calculated with the optimum number of terms in the Fourier expansion.