A Two—Streamfunction—Coordinate Method for the Numerical Solution of 3—D Aero—Thermoldynamics Direct and Inverse Problems

This paper describes a new method which has been developed for the solution of direct and inverse problemsof 3-D compressible flows in turbomachinery.Two types of streamfunctions are proposed in the paper andthe streamfunction-coordinate system is applied in numerical computations.The algorithm is applied to statorblades and the results are compared with experimental data,It is shown that the comparisons are very satis-factory.     

On the inverse heat convection problem of the flow over a cascade of rectangular blades

In this study, an inverse algorithm based on the conjugate gradient method is applied to a steady flow over a cascade of rectangular blades to estimate the inlet flow temperature. The objective is to study the difficulties associated with inverse heat convection problems. Therefore, the measurement quantity has been deliberately placed at five different locations over the domain, each of them covered by a unique flow feature. The computation shows that at very low Reynolds number, the accuracy of the inverse method is not affected by the relative position between the estimated and measurement quantities. At a higher Reynolds number, however, the accuracy of the inverse method strongly depends on the relative position between the two quantities. The inverse method only returns satisfactory estimation for some cases but not others.     

The Method of Particular Solutions for Solving Inverse Problems of a Nonhomogeneous Convection-Diffusion Equation with Variable Coefficients

A new version of the method of particular solutions (MPS) has been proposed for solving inverse problems for nonhomogeneous convection-diffusion equations with variable coefficients (IPCD). Coupled with the time discretization and MPS, the proposed method is a truly meshless method which requires neither domain or boundary discretization. Even though the final temperature is almost undetectable or is disturbed by significant noise, the proposed method can still recover the initial temperature very well. The effectiveness of the proposed inverse scheme using radial basis functions is demonstrated by several examples in 2-D and 3-D.     

Application of grey prediction to inverse nonlinear heat conduction problem

The purpose of this research is to estimate the thermal conductivity with the inverse method which is modified by grey prediction; herein the thermal conductivity is a nonlinear function. When the thermal conductivity is the function of position and temperature, if one would try to obtain the thermal conductivity with the inverse method, then the measuring points of the temperature shall be distributed in whole object, consequently there would be a large number of measuring points for the relevant temperatures. The method of grey prediction will be able to dramatically decrease the number of measuring points for the temperature accordingly. However, the method of grey prediction should be accompanied with the prediction errors, thus the estimation of inverse method will produce a major deviation. This paper adopts the methods of the “rolling grey prediction” and the “comparison of temperature measurement” to correct the errors of grey prediction, and then proceed the inverse method to estimate the thermal conductivity. The estimated value obtained by the proposed method and the actual value compares very well.     

Solving Direct and Inverse Thermoelasticity Problems by Means of Trefftz Base Functions for Finite Element Method

The Trefftz functions method has been developed very quickly. The paper presents the application of this method to solving direct and inverse problems of elasticity and thermoelasticity. The system of equations for displacements is reduced to a system of wave equations. Then the wave polynomials (Trefftz functions for wave equation) as base functions for several variants of Finite Element Method are used. In the paper, continuous FEMT and substructuring are considered. In the case of thermoelasticity, the temperature field occurs as inhomogeneity in one of the wave equations. It is shown how to get the particular solution in 2D and 3D. When using FEMT, the difference of solutions between the elements has to be minimized. The mechanical energy of the body depends on the velocity of the displacements. Therefore, the difference of the velocities between the elements is also minimized – it is a kind of physical regularization. The quality of the approximate solutions of direct and inverse problem was verified on the test example.     

Virtual boundary element method in conjunction with conjugate gradient algorithm for three-dimensional inverse heat conduction problems

In this article, the virtual boundary element method (VBEM) in conjunction with conjugate gradient algorithm (CGA) is employed to treat three-dimensional inverse problems of steady-state heat conduction. On the one hand, the VBEM may face numerical instability if a virtual boundary is improperly selected. The numerical accuracy is very sensitive to the choice of the virtual boundary. The condition number of the system matrix is high for the larger distance between the physical boundary and the fictitious boundary. On the contrary, it is difficult to remove the source singularity. On the other hand, the VBEM will encounter ill-conditioned problem when this method is used to analyze inverse problems. This study combines the VBEM and the CGA to model three-dimensional heat conduction inverse problem. The introduction of the CGA effectively overcomes the above shortcomings, and makes the location of the virtual boundary more free. Furthermore, the CGA, as a regularization method, successfully solves the ill-conditioned equation of three-dimensional heat conduction inverse problem. Numerical examples demonstrate the validity and accuracy of the proposed method.     

What is the most suitable fixed grid solidification method for handling time-varying inverse Stefan problems in high temperature industrial furnaces?

This paper presents a simple inverse heat transfer method for predicting the time-varying bank thickness of the phase change protective layer inside a high temperature furnace. The direct problem is handled with the single phase method which is neglecting the latent heat. The inverse method rests on the Adjoint Problem and the Conjugate Gradient Method. It is shown that the iterative inverse procedure based on the single phase method predicts a virtual incident heat flux in the liquid zone that yields accurate time-varying bank predictions. Results also indicate that the benefits of the virtual iterative approach are twofold: the CPU time required for solving the inverse problem is reduced and the lagging effect inherent to the inversion is diminished for non isothermal phase change processes. For typical melting furnace operating conditions, it is shown that the virtual approach doubles the allowable diagnostic frequency for predicting the time-varying bank thickness.     

Reconstruction of a space and time dependent heat source from finite measurement data

This paper deals with an inverse problem of determining a heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable x, but also depends on the time t. On the basis of the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established. The convergence of the minimizer as the mesh parameters tend to zero is also proved. The conjugate gradient method is applied to the inverse problem and some typical numerical experiments are performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well.     

EFFICIENCY OF AN INVERSE METHOD TO DETERMINE NATURAL-CONVECTION HEAT

The final aim of this study is to identify the value of convective heat transfer generated by a local perturbation analog to a sun patch. A first approach to this work consisted of developing a means of evaluating the local heat transfer based on the use of an inverse method. We first chose the well-known academic case of a vertical heated plane plate to generate natural heat convection transfer. Then, for several temperatures of the plate, a first validation of the chosen inverse method was obtained in the steady state. It consisted of a comparison between the results provided by the inverse method and those obtained by the use of thermocouples to measure the temperature inside the boundary layer. These two types of results are also compared with relations published in the literature. In conclusion, the authors observe that the inverse method is pertinent and gives satisfactory results.     

A simplex search method for a conductive–convective fin with variable conductivity

An inverse problem is solved for simultaneously estimating the convection–conduction parameter and the variable thermal conductivity parameter in a conductive–convective fin with temperature dependent thermal conductivity. Initially, the temperature field is obtained from a direct method using an analytical approach based on decomposition scheme and then using a simplex search minimization algorithm an inverse problem is solved for estimating the unknowns. The objective function to be minimized is represented by the sum of square of the error between the measured temperature field and an initially guessed value which is updated in an iterative manner. The estimation accuracy is studied for the effect of measurement errors, initial guess and number of measurement points. It is observed that although very good estimation accuracy is possible with more number of measurement points, reasonably well estimation is obtained even with fewer number of measurement points without measurement error. Subject to selection of a proper initial guess, it is seen that the number of iterations could be significantly reduced. The relative sensitiveness of the estimated parameters is studied and is observed from the present work that the estimated convection–conduction parameter contributes more to the temperature distribution than the variable conductivity parameter.     

A heat-flux based “building block” approach for solving heat conduction problems

A numerical approximation of the Green’s function equation based on a heat-flux formulation is given. It is derived by assuming as a functional form of the surface heat flux a stepwise variation with space and time. The obtained approximation is very important in investigation of the inverse heat conduction problems (IHCPs) because it gives a convenient expression for the temperature in terms of the heat flux components. Additionally, it is very important for the unsteady surface element (USE) method which is a modern boundary discretization method. Green’s function approximate solution equation (GFASE) also creates ‘naturally’ fixed groups or modules of work elements called “building blocks” that may be added together to obtain space and time values of temperature. In the current case, they are subject to a partial heating by an applied surface heat flux. The “building block” solution can be derived by using the various analytical and numerical approaches available in heat conduction literature though the exact analysis is preferable, as discussed in the text. Poorly-convergent series deriving from Green’s functions approach are replaced by closed-form algebraic solutions.     

A rapid method for solving Laplace equation in a doubly connected region

A new methodology is developed for solving 2-D Laplace equation (heat conduction problem) within a doubly connected region with prescribed temperature and heat flux distribution along outer or inner prescribed boundary. Both direct problem (with specified geometry of the other boundary) and inverse problem (with prescribed temperature or heat flux distribution along the other boundary) can be solved by this method. The computation work needed is very simple and can be programed with a very small computer such as Sharp PC-1500. This method can be extended to a 3-D one also.     

Fast inverse prediction of phase change banks in high temperature furnaces with a Kalman filter coupled with a recursive least-square estimator

An inverse heat transfer procedure for predicting the time-varying thickness of phase-change banks on the inside surface of the walls of high temperature furnaces is presented. The main feature of the inverse method is its unique capability of making fast predictions so that it can be easily integrated to existing real-time control systems of industrial facilities. The method rests on fast computing state-space models (direct model) that are designed to mimic the response of a full finite-difference model of the phase change problem. A Kalman filter coupled with a recursive least-square estimator (inverse method) is employed to estimate the time-varying phase front position from the data collected by a temperature and/or heat flux sensor located in the furnace wall. The inverse heat transfer procedure is thoroughly tested for typical phase change conditions that prevail inside industrial facilities. The effect of the sensor type (temperature sensor or heat flux sensor), of its location and of the measurement noise on the accuracy and stability of the predicted bank thickness is investigated. It is shown that the proposed inverse heat transfer procedure becomes increasingly reliable and accurate for predicting the bank thickness as it shrinks. This feature is of the utmost interest for preventing the sudden and accidental loss of the protective banks of industrial furnaces filled with molten material. Recommendations are also made concerning the type and location of sensors.     

Inverse problem of unsteady conjugated forced convection in parallel plate channels

The heat transfer phenomena of the unsteady laminar forced convection in parallel plate channels with wall conduction effects are still not very well understood. An inverse algorithm based on the conjugate gradient method is proposed to estimate the boundary conditions of these problems, and the minimization of object function is used to reduce the estimated error. The estimation of applied heat flux is found to be highly dependent of temperature sensor location and uncertainty, plate thickness, and heating way. The results show that the predicted boundary conditions by the present inverse method are consistent with the initially specified ones.     

神经网络非线性逆控制方法实现PEMFC分布式发电系统的负荷响应

质子交换膜燃料电池(PEMFC)分布式发电系统的负载典型特点是时时有瞬间的大的峰值功率。因此,系统必须找到实时的最佳工作点才能响应负载的功率需求。以往采用的优化搜索方法需要在负载连续变化和峰值现象之间权衡搜索步长,影响控制的速度和精度。分析PEMFC分布式发电系统的工作特点以及各工作参数与净输出功率之间的关系,提出利用BP神经网络实现PEMFC输出功率的复杂非线性逆模型。在此基础上,根据非线性逆控制的思想设计了系统实时功率响应的逆控制器。经分析和仿真验证,利用神经网络非线性逆控制方法,PEMFC系统在稳定运行过程中能够满足系统内部功率损耗的同时良好的响应系统负载的实时功率需求。与优化搜索方法比较,该实时控制设计在实现精度和速度上都有所改进。图5表1参6     

A method for magnetizing curve identification in rotor fluxoriented induction machines

Operation of an indirect rotor flux oriented induction machine in the field weakening region is usually realized by varying the rotor flux reference in inverse proportion to the speed of rotation. In order to provide the correct stator d-axis current reference at all speeds, it is necessary to incorporate the inverse magnetizing curve of the machine in the controller. The paper proposes an experimental method for identifying the inverse magnetizing curve, specifically developed for the type of vector controlled drives described. The method utilizes the same indirect vector controller and PWM inverter that are used in subsequent normal operation of the drive. It requires that the machine can run under no-load conditions and that the fundamental component of the stator voltage can be measured. The simplicity and accuracy of the method make it well suited for use during commissioning of the drive. The method is verified by extensive experimentation     

逆算法对涡轮特性柯特略尔估算的改进

以柯特略尔法为例,对在级叠加式涡轮特性计算时使用逆算法的适用性进行了讨论。证明了末级首先临界的假定在各折合级设计压比相同的条件下可自然满足;当各折合级设计压比不全相同时,给出了寻找首先临界级的判据,并提出采用顺逆结合处理临界问题的方法,突破了末级首先临界的简化假定,优化了计算步骤,提高了估算精度。     

Solution of inverse heat conduction problem using the Tikhonov regularization method

It is hard to solve ill-posed problems, as calculated temperatures are very sensitive to errors made while calculating “measured” temperatures or performing real-time measurements. The errors can create temperature oscillation, which can be the cause of an unstable solution. In order to overcome such difficulties, a variety of techniques have been proposed in literature, including regularization, future time steps and smoothing digital filters. In this paper, the Tikhonov regularization is applied to stabilize the solution of the inverse heat conduction problem. The impact on the inverse solution stability and accuracy is demonstrated.     

An inverse problem in estimating the reaction functions and solute concentration simultaneously in a reversible process

An inverse algorithm basing on the Iterative Regularization Method (IRM) is applied in this study in determining the unknown time-dependent reaction functions and solute concentration in the solution, i.e. three unknown time-dependent functions, simultaneously in a reversible process by using measurements of concentration components. It is assumed that no prior information is available on the functional form of the unknown functions in the present study, it can thus be classified as function estimation for the inverse calculations. The accuracy of this inverse problem is examined by using the simulated exact and inexact concentration measurements in the numerical experiments. Results show that the estimation of the time-dependent reaction functions and solute concentration in the solution can be obtained in a very short CPU time on a HP d2000 2.66 GHz personal computer. Moreover, the sensors should be placed as close to the boundary as possible to obtain better estimations.     

A modified genetic algorithm for solving the inverse heat transfer problem of estimating plan heat source

This work considers a new approach for solving the inverse heat conduction problem of estimating unknown plan heat source. It is shown that the physical heat transfer problem can be formulated as an optimization problem with differential equation constraints. A modified genetic algorithm is developed for solving the resulting optimization problem. The proposed algorithm provides a global optimum instead of a local optimum of the inverse heat transfer problem with highly-improved convergence performance. Some numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.