3-D transient analytical solution based on Green’s function to temperature field in friction stir welding

Friction stir welding (FSW) is a relatively modern welding process, which not only provides the advantages offered by fusion welding methods, but also improves mechanical properties as well as metallurgical transformations due to the pure solid-state joining of metals. The FSW process is composed of three main stages; penetrating or preheating stage, welding stage and cooling stage. The thermal history and cooling rate during and after the FSW process are decisive factors, which dictate the weld characteristics. In the current paper, a novel transient analytical solution based on the Green’s function method is established to obtain the three-dimensional temperature field in the welding stage by considering the FSW tool as a circular heat source moving in a finite rectangular plate with cooling surface and non-uniform and non-homogeneous boundary and initial conditions. The effect of penetrating/preheating stage is also taken into account by considering the temperature field induced by the preheating stage to be the non-uniform initial condition for the welding stage. Similarly, cooling rate can be calculated in the cooling stage. Furthermore, the simulation of the FSW process via FEM commercial software showed that the analytical and the numerical results are in good agreement, which validates the accuracy of the developed analytical solution.     

Use of 3D-transient analytical solution based on Green’s function to reduce computational time in inverse heat conduction problems

Inverse problems can be found in many areas of science and engineering and can be applied in different ways. Two examples can be cited: thermal properties estimation or heat flux function estimation in some engineering thermal process. Different techniques for the solution of inverse heat conduction problem (IHCP) can be found in literature. However, any inverse or optimization technique has a basic and common characteristic: the need to solve the direct problem solution several times. This characteristic is the cause of the great computational time consumed. In heat conduction problem, the time consumed is, usually, due to the use of numerical solutions of multidimensional models with refined mesh. In this case, if analytical solutions are available the computational time can be reduced drastically. This study presents the development and application of a 3D-transient analytical solution based on Green’s function. The inverse problem is due to the thermal properties estimation of conductors. The method is based on experimental determination of thermal conductivity and diffusivity using partially heated surface method without heat flux transducer. Originally developed to use numerical solution, this technique can, using analytical solution, estimate thermal properties faster and with better accuracy.     

Solving two-point boundary value problems using combined homotopy perturbation method and Green’s function method

In this paper, an algorithm is proposed for the solution of second-order boundary value problems with two-point boundary conditions. The Green’s function method is applied first to transform the ordinary differential equation into an equivalent integral one, which has already satisfied the boundary conditions. And then, the homotopy perturbation method is used to the resulting equation to construct the numerical solution for such problems. Numerical examples demonstrate the efficiency and reliability of the algorithm developed, it is quite accurate and readily implemented for both linear and nonlinear differential equations with homogeneous and nonhomogeneous boundary conditions. Furthermore, the lower order approximation is of higher accuracy for most cases. Some other extended applications of this algorithm are also exhibited.     

Nonlinear variable separation solutions of some mKdV-type equations and the unified construction method

From the viewpoint of inverse problem research, by using the Hirota’s bilinear transformation method and direct variable separation ansatz, some mKdV-type equations with nonlinear variable separation solutions are unified constructed. These exact special solutions include several lower-dimensional arbitrary functions which can be used to express various soliton structures.     

Mathematical models and solution algorithms for computational design of RC piles under structural effects

The paper presents mathematical models and solution algorithms for RC pile design, through scanning soil stratums from top to downwards with an interactive scanner band. The equilibrium of transferred loads from the superstructure, friction forces and tip bearing forces are considered for the design, which leads to optimum pile length. The most important contribution of this research for designers is supplying an efficient tool to obtain optimum pile length and reinforced concrete design of pile foundation systems. A program package has been developed in MATLAB depending on the proposed algorithm. Soil behaviors depending on external effects, active and passive zone distributions are considered. All possible effects in all freedom degrees are taken into account in design process. Stress and strain distributions due to axial loads, bending moments, shear forces and torsional moments may be monitored. The optimum pile length, cross section dimension and reinforcement details may be found by using developed algorithm.     

Comparison of the Adomian decomposition method with homotopy perturbation method for the solutions of seventh order boundary value problems

The aim of this paper is to compare the Adomian decomposition method and the homotopy perturbation method for solving the linear and nonlinear seventh order boundary value problems. The approximate solutions of the problems obtained with a small amount of computation in both methods. Two numerical examples have been considered to illustrate the accuracy and implementation of the methods.     

Using spectral element method for analyzing continuous beams and bridges subjected to a moving load

Spectral element method in frequency domain is employed to analyze continuous beams and bridges subjected to a moving load. The formulation is developed for an Euler beam under a moving load with an arbitrary amplitude and velocity. It is shown that the procedure is simplified for a moving load with a constant amplitude and velocity. Static Green’s function is used as a modifying function to improve the moment and shear force results. It is further shown that while modifying function is used in conjunction with spectral element method, fewer elements will be required to achieve proper results. The numerical examples show the accuracy of the method.     

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S _j for j=1,…,K at unknown spot locations x _j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S _j and x _j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S _j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.      

A short note on the paper “Convergence of the TAGE iterative method for the system arisen from the cubic spline approximation for the solution of two-point BVPs with forcing function in integral form”, by Mohanty, Jain and Dhall

In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036-3047] and then correct it.     

Painlevé property, soliton-like solutions and complexitons for a coupled variable-coefficient modified Korteweg-de Vries system in a two-layer fluid model

As a model derived from a two-layer fluid system which describes the atmospheric and oceanic phenomena, a coupled variable-coefficient modified Korteweg-de Vries system is concerned in this paper. With the help of symbolic computation, its integrability in the Painlevé sense is investigated. Furthermore, Hirota’s bilinear method is employed to construct the bilinear forms through the dependent variable transformations, and soliton-like solutions and complexitons are derived. Finally, effects of variable coefficients are discussed graphically, and it is concluded that the variable coefficients control the propagation trajectories of solitons and complexitons.     

Monotone method for singular BVP in the presence of upper and lower solutions

In this paper we examine existence of monotone approximations of solutions of singular boundary value problem -(p(x)y^′(x))^′=q(x)f(x,y,py^′) for 0<x?b and lim_x→0⁺p(x)y^′(x)=0,α₁y(b)+β₁p(b)y^′(b)=γ₁. Under quite general conditions on f(x,y,py^′) we show that solution of the singular two point boundary value problem is unique. Here is allowed to have integrable singularity at x=0 and we do not assume .     

Harmonic and Refined Extraction Methods for the Singular Value Problem,with Applications in Least Squares Problems

For the accurate approximation of the minimal singular triple (singular value and left and right singular vector) of a large sparse matrix, we may use two separate search spaces, one for the left, and one for the right singular vector. In Lanczos bidiagonalization, for example, such search spaces are constructed. In SIAM J. Sci. Comput., 23(2) (2002), pp. 606–628, the author proposes a Jacobi–Davidson type method for the singular value problem, where solutions to certain correction equations are used to expand the search spaces. As noted in the mentioned paper, the standard Galerkin subspace extraction works well for the computation of large singular triples, but may lead to unsatisfactory approximations to small and interior triples. To overcome this problem for the smallest triples, we propose three harmonic and a refined approach. All methods are derived in a number of different ways. Some of these methods can also be applied when we are interested in the largest or interior singular triples. Theoretical results as well as numerical experiments indicate that the results of the alternative extraction processes are often better than the standard approach. We show that when Lanczos bidiagonalization is used for the subspace expansion, the standard, harmonic, and refined extraction methods are all essentially equivalent. This gives more insight in the success of Lanczos bidiagonalization to find the smallest singular triples. Finally, we show that the extraction processes for the smallest singular values may give an approximation to a least squares problem at low additional costs. The truncated SVD is also discussed in this context. AMS subject classification (2000) 65F15, 65F50, (65F35, 93E24).Submitted December 2002. Accepted October 2004. Communicated by Haesun Park.M. E. Hochstenbach: The research of this author was supported in part by NSF grant DMS-0405387. Part of this work was done when the author was at Utrecht University.     

A cubic spline approximation and application of TAGE iterative method for the solution of two point boundary value problems with forcing function in integral form

In this article, we report an efficient high order numerical method based on cubic spline approximation and application of alternating group explicit method for the solution of two point non-linear boundary value problems, whose forcing functions are in integral form, on a non-uniform mesh. The proposed method is applicable when the internal grid points of solution interval are odd in number. The proposed cubic spline method is also applicable to integro-differential equations having singularities. Computational results are given to demonstrate the utility of the method.