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In this paper, an approach to improve the method of angular basis function (MABF) proposed by Young et al. (2015) is proposed. Instead of using lnr in the method of fundamental solution (MFS), the MABF adopts θ to construct the solution. However, since the nature of θ introduces multiple values along the branch cut such that to avoid the branch cut passing through the domain is an important issue (Li et al., 2018). Noticing this difficulty, Alves et al. (2018) first proposed a remedy which used a pair of two points to restrict the discontinuity appearing only along the line segment between two points, and they named this approach as cracklets. In this article, the two-point angular basis function (cracklets) is investigated deeply. We explain why for a multiply connected domain with a logarithm singular solution the cracklets will encounter failure. To resolve this difficulty, one can adopt the proposed method (cracklets) with the MFS or one can use domain decomposition method to separate the domain into several singly connected domains. Seven numerical examples are provided to show the validity of this method, and examples for dealing with the multiply connected domain are focused to support our claims.

An almost exact solution is derived for the forced vibration of a composite beam with periodically varying non-smooth interface through a moderate weak-form formulation. The material property of a non-uniform beam is characterized by its flexural rigidity function R(x). In the novel method, R(x) is relaxed to be an integrable function rather than a $${\mathcal{C}}^2$$ smooth function in the usual approach. The R(x)-orthogonal bases in the linear span of all boundary functions are derived such that the second-order derivatives of the bases elements are orthogonal with respect to the weight function R(x). When the deflection of the beam is expressed in terms of the bases, the expansion coefficients can be determined exactly in closed form owing to the R(x)-orthogonality of the bases. The solution obtained is almost exact, since its accuracy can be up to the order $$10^{-15}$$ . This powerful method is used to analyze the forced vibration behavior of composite beams with three different periodic interfaces.

In this paper, after deriving certain adjoint eigenfunctions as test functions, as well as the bases of unknown pollutant source, the numerical algorithm based on the weak-form integral equation method is developed, of which the expansion coefficients can be found exactly in closed-form, without resorting to numerical solution. The closed-form expansion coefficients method is effective for the inverse pollutant source problem (IPSP), which can recover the unknown pollutant source by means of an extra final time condition. Numerical examples of the IPSP demonstrate that the closed-form expansion coefficients method is stable against large noises up to 10%–50%.

In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome t...

The polynomial Trefftz method consists of the polynomial type solutions as bases, providing a cheap boundary-type meshless method to solve the heat conduction equation, since the bases automatically satisfy the governing equation. In order to stably solve the backward heat conduction problem (BHCP), and the inverse heat source problem (IHSP) together with the boundary condition recovery problem by a polynomial Trefftz method, which are both known to be highly ill-posed, we introduce a multiple-scale post-conditioner in the resultant linear system to reduce the condition number. Then the conjugate gradient method (CGM) is used to solve the post-conditioned linear system to determine the unknown expansion coefficients. In the multiple-scale polynomial Trefftz method (MSPTM) the scales are determined a priori by the collocation points on space–time boundary, which can retrieve the missing initial data, the unknown time-dependent heat source as well as the boundary condition rather well. Several numerical examples of the inverse heat conduction problems demonstrate that the MSPTM is effective and accurate, even for those of severely ill-posed inverse problems under very large noises.

In this paper, we recover unknown space–time dependent forces imposed on the vibrating Euler–Bernoulli beams under different boundary conditions, including simply supported, clamped-hinged, cantilevered, and two-end fixed conditions. The data overspecified to recover the external force are the final time displacement and the right-side strain or right-side moment of the beam. We develop a family of boundary shape functions, which automatically satisfy the initial conditions, final time condition, and homogeneous boundary conditions for each type of beam. When the solution is obtained using the method of superimposing boundary shape functions and solving a small-scale linear system to satisfy an extra right-side boundary condition, the unknown force can be recovered through back-substitution of the solution into the Euler–Bernoulli beam equation. The accuracy and robustness of the proposed methods are confirmed by comparing the recovered results of seven examples to the exact forces, even though considerable noise is present in the overspecified data.

The power series solution is a cheap and effective method to solve nonlinear problems, like the Duffing-van der Pol oscillator, the Volterra population model and the nonlinear boundary value problems. A novel power series method by considering the multiple scales $R_k$ in the power term $(t/R_k)^k$ is developed, which are derived explicitly to reduce the ill-conditioned behavior in the data interpolation. In the method a huge value times a tiny value is avoided, such that we can decrease the numerical instability and which is the main reason to cause the failure of the conventional power series method. The multiple scales derived from an integral can be used in the power series expansion, which provide very accurate numerical solutions of the problems considered in this paper.

The polynomial expansion method is a useful tool to solve partial differential equations (PDEs). However, the researchers seldom use it as a major medium to solve PDEs due to its highly ill-conditioned behavior. We propose a single-scale and a multiple-scale Pascal triangle formulations to solve the linear elliptic PDEs in a simply connected domain equipped with complex boundary shape. For the former method a constant parameter R 0 is required, while in the latter one all introduced scales are automatically determined by the collocation points. Then we use the multiple-scale method to solve the inverse Cauchy problems, which is very accurate and very stable against large noise to 20%. Numerical results confirm the validity of the present multiple-scale Pascal polynomial expansion method.

In this paper, the modified polynomial expansion method is developed to solve problems of identifying the time dependent heat source, in which an inverse problem is encountered. Aimed at this problem, the variation of variables is adopted to eliminate the unknown heat source and obtain a six-line boundary value problem. As compared with the conventional four-line boundary value problem, the six-line boundary value problem is quite hard to be dealt with. After the unknown non-homogeneous term being eliminated, the polynomial expansion method is introduced to discretize the time and space fields, respectively. Then, the distribution of temperature is expressed as a linear superposition of polynomial functions. After that, a characteristic length concept is adopted to resolve the ill-posed matrix problems arising in those conventional polynomial expansion methods. The desired heat source function can be obtained by putting the solution of the six-line boundary value problem into differential operations. Several numerical experiments with designed examples are included to validate the accuracy and effectiveness of the proposed approach.

In this article, the solutions for the Helmholtz equation for forward problems with high wave number and ill-posed inverse problems using the multiple scales Trefftz collocation method are investigated. The resulting linear algebraic systems for these problems are ill-posed and therefore require special treatments. The equilibrated matrix concept is adopted to determine the scales and to construct an equivalent linear algebraic problem with a leading matrix less ill-posed such that standard solver like the conjugate gradient method (CGM) can be adopted. Five examples, including two forward problems with the high wave number and three inverse Cauchy problems, are given to show the validity for the approach. Results show that the equilibrated matrix concept can yield a less ill-posed leading matrix such that the conventional linear algebraic solver like CGM can be successfully adopted. This approach has a very good noise resistance.

This paper proposes the numerical solution of three-dimensional Laplacian problems based on the multiple scale Trefftz method with the incorporation of the dynamical Jacobian-inverse free method. A numerical solution for three-dimensional Laplacian problems was approximated by superpositioning T-complete functions formulated from 18 independent functions satisfying the governing equation in the cylindrical coordinate system. To mitigate a severely ill-conditioned system of linear equations, this study adopted the newly developed multiple scale Trefftz method and the dynamical Jacobian-inverse free method. Numerical solutions were conducted for problems involving three-dimensional groundwater flow problems enclosed by a cuboid-type domain, a peanut-type domain, a sphere domain, and a cylindrical domain. The results revealed that the proposed method can obtain accurate numerical solutions for three-dimensional Laplacian problems, yielding a superior convergence in numerical stability to that of the conventional Trefftz method.

We consider an inverse problem of a nonlinear heat conduction equation for recovering unknown space-dependent heat source and initial condition under Cauchy-type boundary conditions, which is known as a sideways heat equation. With the aid of two extra measurements of temperature and heat flux which are being polluted by noisy disturbances, we can develop a Lie-group differential algebraic equations (LGDAE) method to solve the resulting differential algebraic equations, and to quickly recover the unknown heat source and initial condition simultaneously. Also, we provide a simple LGDAE method, without needing extra measurement of heat flux, to recover the above two unknown functions. The estimated results are quite promising and robust enough against large random noise.

This article is aimed to reconstruct a time-dependent heat source for a one-dimensional heat conduction equation. The extra measurement data are used to transform the original equation into a homogeneous equation with three-point boundary conditions. Then the modified polynomial expansion method is developed to deal with the resulting three-point boundary-value problem. By considering the characteristic length, the modified polynomial expansion method can obtain a convergent series solution and improve the stability of the algorithm. The accuracy and efficiency of the present method are validated by comparing the estimating results with those of designed examples even under noisy measurement data.

In this paper, the boundary detection problem, which is governed by the Laplace equation, is analyzed by the modified collocation Trefftz method (MCTM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the boundary detection problem, the Cauchy data is given on part of the boundary and the Dirichlet boundary condition on the other part of the boundary, whose spatial position is unknown a priori. By adopting the MCTM, which is meshless and integral-free, the numerical solution is expressed by a linear combination of the T-complete functions of the Laplace equation. The use of a characteristic length in MCTM can stabilize the numerical procedure and ensure highly accurate solutions. Since the coefficients of MCTM and the position of part of the boundary are unknown, to collocate the boundary conditions will yield a system of nonlinear algebraic equations; the ECSHA, which is exponentially convergent, is adopted to solve the system of nonlinear algebraic equations. Several numerical examples are provided to demonstrate the ability and accuracy of the proposed meshless scheme. In addition, the consistency of the proposed scheme is validated by adding noise into the boundary conditions.

We propose a simple extension of the two-dimensional method of fundamental solutions (MFS) to a two-dimensional like MFS for the numerical solution of the three-dimensional Laplace equation in an arbitrary interior domain. In the directional MFS (DMFS) the directors are planar orientations, which can take the geometric anisotropy of the problem domain into account, and more importantly the order of the logarithmic singularity with $\ln R$ of the new fundamental solution is reduced than that of the conventional three-dimensional fundamental solution with singularity $1/r$. Some numerical examples are used to validate the performance of the DMFS.

This study presents the numerical predictions of the fluid flow characteristics within the recirculation zone for a backward-facing step with uniform normal mass bleed. The turbulent governing equations are solved by a control-volume-based finite-difference method with power-law scheme. A new turbulence model is proposed to describe the turbulent structure. Non-uniform staggered grids are used. The parameters studied include entrance Reynolds number (Re), and the velocity of the normal mass bleed (Vs). The channel expansion ratio ER = 1.3, and the working medium is air. The numerical results show the uniform normal mass bleed suppresses the reverse horizontal velocity, turbulence intensity, and Reynolds shear stress within the recirculation zone. The attachment point extends to downstream. Better computational predictions are obtained with the new turbulence model by the introduction of the Kolmogorov velocity scale, Ue = (ve) 1 4 instead of friction velocity Uv.

This study investigates the application of Fuzzy ART neural network to the part-machine grouping problem in group technology. Fuzzy ART neural network provides a framework for both binary and continuous values, and offers several advantages, particularly the reduction in computational complexity and ability to handle large scale industrial problems. In this study, a modified Fuzzy ART neural network has been developed to improve the solution quality of part-machine grouping problem. The performance of the modified Fuzzy ART was evaluated by grouping efficiency and the number of exceptional elements.

Carbon nanotubes (CNTs) directly grown on aluminum foil or nanoporous alumina templates by chemical vapor deposition are treated with acid for different times or annealed at different temperatures. Then, the CNTs are immersed into a manganese nitrate aqueous solution and then manganese oxides are formed by thermally decomposing the manganese nitrate at different temperatures. The longer the time that the CNTs are treated with the acid, the slower the decreasing rate of the specific capacitance. Also, the higher the annealing temperature of the CNTs, the slower the decreasing rate of the specific capacitance. Furthermore, the electrochemical stability for thermally decomposing manganese nitrate at 400°C is better than that at 200 or 300°C and the operational stability with nanoporous alumina templates as a substrate is better than that with Al foil as a substrate.

In this paper, the two-dimensional Stokes problem is analyzed by the modified collocation Trefftz method (MCTM) and the Laplacian decomposition. The coupled Stokes equations are converted to three Laplace equations by utilizing the Laplacian decomposition and then the boundary-type meshless MCTM is adopted to solve the resultant Laplace equations. The MCTM, free from mesh and numerical quadrature, is derived from the conventional Trefftz method by considering the characteristic length of the domain, which stabilize the numerical scheme and obtain highly accurate results. Besides, the solutions are expressed as the linear combination of T-complete functions and the velocity as well as pressure are coupled by collocating the boundary conditions. Several numerical examples are provided to demonstrate the efficacy and accuracy of the proposed meshless scheme. In addition, the numerical results demonstrates that the proposed meshless scheme can solve the Stokes problems accurately in simplyand doubly-connected domains.

The Newton algorithm based on the ”continuation” method may be written as being governed by the equation x(subscript j)(t)+B(superscript -1 subscript ij) F(subscript i)(x(subscript j)) =0, where F(subscript i) (x(subscript j))=0, i, j =1, ...n are nonlinear algebraic equations (NAEs) to be solved, and B(subscript ij)=Əf(subscript i)/Əx(subscript j) is the corresponding Jacobian matrix. It is known that the Newton's algorithm is quadratically convergent; however, it has some drawbacks, such as being sensitive to the initial guess of solution, and being expensive in the computation of the inverse of B(subscript ij) at each iterative step. How to preserve the convergence speed, and to remove the drawbacks is a very important issue in the solutions of NAEs. In this paper we discretize the above equation being written as B(subscript ij) X(subscript j)(t) +F(subscript i) (x(subscript j)) =0, by a backward difference scheme in a new time scale of s=1-e^(-1), and an ODEs system is derived by introducing a fictitious time-like variable. The new algorithm is obtained by applying a numerical integration scheme to the resultant ODEs. The new algorithm does not need the inverse of B(subscript ij), and is thus resu1ting in a significant reduction in computational time than the Newton's algorithm. A similar technique is also used to modify the homotopy method. Numerical examples given confirm that the modified Newton method is highly efficient, insensitive to the initial condition, to find the solutions with a very small the residual error.

In the era of mass customization and ever-changing market, it is important for enterprises to develop the dynamic and quick-response planning techniques in their flexible manufacturing systems (FMSs). This research paper presents an agent-based inference system, called Intelligent Decision Inference System (IDIS), to support existing FMS for production dispatching. When system integration is required between two separate operating systems, there are many issues to be taken into consideration, including system communication method, data flow design, and application program control. In the past, the most universal method was to integrate databases with two or more systems. However, this method had a main weakness of “data confusion and inconsistent”. Therefore, this research combines the agent, JESS, and related IT technologies to systematically integrate the IDIS with FMS. The agents can easy communicate and deliver the messages, data, or information among the different systems and automatically scan the signals of the programmable logic controller (PLC)/sensors periodically. The method applies web service techniques as a standard information exchange mechanism and the agent communication language (ACL) of the Foundation for Intelligent Physical Agents (FIPA) for multi-agents to communicate/coordinate with each other. The agents, implemented in Java Agent Development Framework (JADE), are designed to decide whether a pallet should be moved in the shop floor of the FMS. Finally, one explicit example about how the cooperative multi-agent supporting the IDIS and FMS integration is presented. The unique contribution of this research is in the integration of expert systems and an agent framework to apply in real-time shop floor control such as FMS.