Hybrid Spline Difference Method Research Articles

A higher‐order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions

AbstractWe propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin‐boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature‐based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first‐order in time and second‐order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.

Multiobjective Optimization of Annular Fin Array Subject to Thermal Load

The life of an annular fin is shortened by the thermal stress developed due to the temperature difference along the length of the fin. However, the thermal stress in fin arrays is yet to be analyzed in detail. Because of that fact, the design of an annular fin array is formulated here as a multiobjective optimization problem, primarily for maximizing the heat transfer rate and minimizing the developed maximum thermal stress. For further assessment, minimization of the total fin volume of the array and maximization of the surface efficiency are considered as two additional objective functions. Considering an array of identical fins having a plane profile, the half-thickness at the base, the half-thickness at the tip, and the outer radius of an individual fin along with the number of fins in the array are taken as four design variables. Evaluating through the hybrid spline difference method, the four objective functions are optimized in different combinations by employing nondominated sorting genetic algorithm II. The proposed procedure should help a designer to choose a compromise fin array configuration from a set of tradeoff optima based upon the availability of resources.

Multi-objective optimization of annular fin array with B-spline curve based fin profiles

ABSTRACTThe design of annular fin array with variable thickness fin profiles defined by B-spline curves is studied as a multi-objective optimization problem for simultaneously maximizing heat transfer rate and minimizing thermal stress. Maximization of surface efficiency and augmentation factor as well as minimization of fin volume are considered as additional objective functions for further assessment of fin array performance. Evaluating the objective values through hybrid spline difference method, different cases are investigated by solving the optimization model by non-dominated sorting genetic algorithm II. The proposed scheme should aid designers in selecting compromise optimal solutions for practical problems.

Geometric Size Optimization of Annular Step Fin Using Multi-Objective Genetic Algorithm

Although an annular stepped fin can produce better cooling effect in comparison to an annular disk fin, it is yet to be studied in detail. In the present work, one-dimensional heat transfer in a two-stepped rectangular cross-sectional annular fin with constant base temperature and variable thermal conductivity is modeled as a multi-objective optimization problem. Taking cross-sectional half-thicknesses and outer radii of the two fin steps as design variables, an attempt is made to obtain the efficient fin geometry primarily by simultaneously maximizing the heat transfer rate and minimizing the fin volume. For further assessment of the fin performance, three more objective functions are studied, which are minimization of the fin surface area and maximization of the fin efficiency and effectiveness. Evaluating the heat transfer rate through the hybrid spline difference method, the well-known multi-objective genetic algorithm, namely, nondominated sorting genetic algorithm II (NSGA-II), is employed for approximating the Pareto-optimal front containing a set of tradeoff solutions in terms of different combinations of the considered five objective functions. The Pareto-optimal sensitivity is also analyzed for studying the influences of the design variables on the objective functions. As an outcome, it can be concluded that the proposed procedure would give an open choice to designers to lead to a practical stepped fin configuration.

B-spline curve based optimum profile of annular fins using multiobjective genetic algorithm

ABSTRACTThermal stress in annular fins, which influences the life of a fin, still needs its detail analysis. Heat transfer in annular fins is studied here as a multiobjective optimization problem. Representing fin profiles by B-spline curves, fin geometries are obtained primarily by maximizing heat transfer rate and minimizing thermal stress. Fin performance is further assessed by minimizing fin volume and maximizing fin efficiency and effectiveness. Evaluating temperature and thermal stress by hybrid spline difference method, non-dominated sorting genetic algorithm II is applied to approximate the Pareto-optimal front. The proposed procedure would be helpful for designers to adopt suitable fin configurations from the Pareto front.

A BFGS and simple step method for estimating the interface temperature in 2D ultrasonic seam welding

An inverse method for estimating unknown interface heat generation in an ultrasonic seam welding process based on a combination of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) and Simple Step Method is presented. In this study, a nonlinear direct problem is solved by using the Hybrid Spline Difference Method (HSDM). Heat generation and interface temperature, which are extremely difficult to directly measure, can be determined inversely through the Direct Problem, Adjoint Problem, and the Simple Step Method without sensitivity analysis. Special features of proposed method are stable, accurate, and efficient for estimating moving heat sources and interface temperature even including measurement error. The predicted interface heat generation and temperatures are in very good agreement with exact solutions for different heat sources profiles. Furthermore, the influence of the measurement location on the accuracy of estimated solutions with or without error of measurement was also discussed. These findings indicate that the estimated heat generation is more sensitive than the temperature to different measured errors and locations. According to analysis of results, the occurrence of thermal defects can be avoided during the welding process. In addition, the proposed method strongly provides good information for optimization of the welding process.

HSDM and BFGS method for determining the heat generation and range of heat distribution in 2-D ultrasonic seam welding problems

ABSTRACTAn inverse method based on the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method and the simple step method is proposed to estimate unknown heat generation at the welding interface during actual ultrasonic seam welding with knowledge of temperatures taken in a workpiece. The efficiency of the hybrid spline difference method (HSDM) compared to the finite-difference method (FDM) is presented. In addition, a comparison of the BFGS method and the conjugate gradient method (CGM) is made. Results show that the estimated temperature is in very good agreement with the measurement temperature. In addition, the heat generation and its distributed range are predicted rapidly. According to the analysis results, the proposed method is an accurate, robust, and efficient method for the prediction of temperature and moving heat generation in an ultrasonic welding process.

An inverse method for estimating the electromechanical parameters of moving-coil loudspeakers

This article presents an inverse method for estimating the electromechanical parameters of a moving-coil loudspeaker with or without the eddy current and suspension creep effects. With known voice-coil displacement, voice-coil current, and stimulus signal as inputs, four calculation procedures for the direct problem, adjoint problem, sensitivity problem, and conjugate gradient method are involved in inversely solving the unknown electromechanical parameters. The proposed method features high efficiency in solving the direct problem through a hybrid spline difference method. It requires a small number of iterations for the computational algorithm, while offering excellent accuracy in parameter estimations. Analysis results demonstrate small differences between the estimated and measured electromechanical parameters under a variety of stimulus signals, excitation times, and initial guesses. The results are also confirmed by experimental measurements. These results indicate that the proposed method has a strong potential for estimating the electromechanical parameters of moving-coil loudspeakers.

A study of nonlinear harmonic distortion in a balanced armature actuator with asymmetrical magnetic flux

This study investigates the nonlinear harmonic distortion caused by the asymmetrical magnetic flux in a balanced armature actuator (BAA). To account for asymmetrical magnetic flux, a set of transduction equations with nonlinear parameters for a BAA is first established. Next, the nonlinear transduction equations are solved correctly and with great efficiency by utilizing the hybrid spline difference method. Unlike symmetrical magnetic flux that leads to odd-order harmonic distortion only, analysis results demonstrate that asymmetrical magnetic flux in the electromagnetic field leads to the simultaneous appearance of even- and odd-order harmonic distortions. Meanwhile, the present study also indicates that the nonlinear harmonic distortion of the BAA can be reduced by adjusting the eccentric position of the armature and the cross-sectional area of the air gap between the permanent magnets and armature. Results presented herein provide valuable information not only on the desired design of a BAA but also on the diagnosis and improvement of its performance.

Hybrid Spline Difference Method for Heat Transfer and Thermal Stresses in Annular Fins

This study proposes a high-accuracy numerical method—the hybrid spline difference method—which is as simple as the finite-difference method, and discusses its feasibility by analyzing the internal temperature and thermal stresses of rectangular annular fins changing the heat transfer coefficient. The hybrid spline difference method can simplify the complex computational process of the traditional spline method, and obtain the high accuracy of O(ξ 4) regardless of whether the equation and boundary conditions have a first differential term. As compared with other numerical methods, the numerical accuracy of temperature and thermal stresses obtained by internal arithmetic is improved greatly considering the effects of heat transfer, convection, and radiation. The results prove that the proposed method can the replace traditional cubic spline method and finite-difference method directly, as a new simple and high-accuracy numerical method.

Analysis of Convective-Radiative Fins by Using Hybrid Spline Difference Method

This study used rectangular fins with constant heat transfer coefficient as material to discuss convective and radiative heat transfer, so as to prove that the hybrid spline difference method proposed in this study is an easy to operate method with high accuracy. According to the computational process described in this paper, the hybrid spline difference method is as simple as finite difference method and is easy to use. The complex computational process of traditional spline method can be simplified by using this method, but the numerical accuracy can be increased to second order. Therefore, the high accuracy numerical method of hybrid spline difference method replacing traditional spline method for future heat transfer analyses is expectable.

Hybrid spline difference method for the Burgers’ equation

This study developed a high accuracy hybrid spline difference method, and deduced a difference equation that is very similar to finite difference method from the concept of spline difference. As validated by nonlinear Burgers’ equation, the concept of difference made the computational process as simple as the finite difference method, and easy to be implemented. The free parameters α→1/12 and α→1/6 were combined in the concept of spline hybrid, in order to increase the accuracy of the first and second derivatives of space from O(h2) of finite difference method to the O(h4). The accuracy was improved, and the numerical oscillation with the increase in parameter Re was improved greatly.

Hybrid Spline Difference Method (HSDM) for Transient Heat Conduction

This study develops a high-accuracy numerical method named the hybrid spline difference method (HSDM), to replace the finite-difference method. Verification by transient heat transfer finds that the concept of the spline finite-difference method can greatly simplify the complexity of the traditional spline calculation, while having a solution procedure as in the finite-difference method. In addition, the hybrid spline technique can prevent the problem of selection of the optimal parameter, and increase the numerical accuracy to O(Δt2, Δx4, Δt2Δx2). This method features high accuracy and similarity with the concept of the finite-difference method, and could replace the traditional finite-difference method.

Hybrid Spline Difference Method for Steady-State Heat Conduction

This work proposes a simple and high-precision hybrid spline difference method, and finds that the computing method of spline difference is as simple as that of the finite-difference method. It also discusses steady-state heat conduction problems to overcome the complex computation and time-consuming problems of the previous spline method. In addition, due to the concept of hybrid parameters, the truncation error of numerical solutions is enhanced to o(h4). This method is expected to be a simple and high-precision numerical method that can replace the finite-difference method.