Sequential Function Specification Research Articles

An algorithm for solving multidimensional inverse heat conduction problem

The unknown time-dependent boundary heat flux of a multidimensional body is determined from temperature measurements inside the body or on its boundary. The sequential function specification method with the assumption that future-boundary heat flux varies linearly with time is used to solve the inverse problem. The method of discretization used by the proposed algorithm is the boundary element method. The effectiveness of the algorithm is illustrated by solving a sample two-dimensional problem.

Estimation of heat flux and temperature in a tool during turning

The heat flux and the temperature at the tip of a tool used in a turning process are estimated from temperature measurements in an interior point of the tool insert. Instead of using a classical direct model of the transient thermal behaviour of the tool, two models that express the heat flux and the temperature at the tip of the insert according to the interior temperature are identified. These models are expressed in a recursive transfer function form. The identification of the parameters in the two models has been performed for a specific apparatus that permits controlling the heat flux and the temperature at the tip of the insert. In a second stage, the inverse heat conduction problem in the tool is solved both using an exact matching method, a sequential function specification method and a sequential regularization method in order to identify the heat flux and the temperature during a turning process.

Inverse heat conduction problem of determining time-dependent heat transfer coefficient

The time-dependent Biot number in a one-dimensional linear heat conduction problem is obtained from the solutions of the inverse heat conduction problems of determining boundary heat flux and boundary temperature. The sequential function specification method with the linear basis function and the assumption of linearly varying future boundary heat flux or temperature components is used to solve the inverse problem. The expression for Biot number is found to be a nonlinear function of measured temperatures. The variance in input data is shown to cause variance and nonlinear bias in estimated Biot number. The method presented offers three tunable parameters that may be used to improve the quality of the solution.

Comparison of three sequential function specification algorithms for the inverse heat conduction problem

Three algorithms for implementing the sequential function specification method of estimating boundary heat flux in the inverse heat conduction problem are compared. They differ from one another in the type of piecewise function used to describe the heat flux and the assumed variation of heat flux over future time. The results of the comparison show that the algorithm that makes use of linear piecewise function for the heat flux and assumes linearly varying heat flux over future time performs slightly better than the other two algorithms.

Time-varying heat transfer coefficients between tube-shaped casting and metal mold

An experimental and numerical study were made on the time-varying heat transfer coefficient h( t) between a tube-shaped casting and metal molds. One dimensional treatment was adopted in analyzing the heat flows between the casting and the inner and the outer mold. The sequential function specification method was employed to solve the nonlinear inverse heat conduction problem. In order to investigate the different behavior of h( t) for different alloys, casting experiments were carried out with three Al-base alloys and pure Al having different types of solidification behavior. It was found that the temperature change of the outer mold showed a normal heating and cooling curve. However, that of the inner mold was unusual especially for the alloys with a wide solidification range, i.e. the temperature increases first rapidly, then halts for a while and then increases again showing finally a regular heating and cooing curve. The resulting heat transfer coefficient at the interface to the inner mold h i( t) decreases temporarily and then increases, while the one at the interface to the outer mold h o( t) decreases monotonously to a quasi steady state. The abnormal heat transfer phenomenon at the inner interface for the alloys with a wide solidification range was concluded to be caused by a slight movement of the semi-solid inner wall at the inside of the tube-shaped casting due to the solidification contraction of the casting freezing in a mushy type.

An inverse thermal field problem based on noisy measurements: comparison of a genetic algorithm and the sequential function specification method

Draws a comparison between the use of a genetic algorithm and the sequential function specification method for the solution of a one‐dimensional linear inverse thermal field problem, based on the use of noisy measurements. In solving this problem aims to estimate the value of a single constant convective heat transfer coefficient. Documents the findings that both approaches can provide estimates within 1 per cent of the target solution and that the sensitivity and robustness of each approach to measurement location, time step size and measurement errors are markedly different.

Redundant data and future times in the inverse heat conduction problem

This paper considers a case study focusing on the use of redundant data and the trade between data density and number of future time steps used in Beck's Sequential Function Specification Method (SFSM) [1] for solving the Inverse Heat Conduction Problem (IHCP). The effect of the number of future time steps (r) and size of data time difference (Δt) have been observed and are presented in this paper. The results indicate that use of redundant sensors can improve heat flux estimates in the one-dimensional IHCP. Furthermore, it is demonstrated that the number of future time steps used r and the size of the time step in the data Δt should be combined into a new parameter, the “look-ahead” time period, p = rΔt. Different applications of the method with the same value of p are seen to give correspondingly similar results. A subsequent numerical experiment is used to give guidelines on the selection of the parameter “p” for the method.

A Stability Analysis on Beck's Procedure for Inverse Heat Conduction Problems

The sequential function specification method proposed first by Beck is considered as one of the most efficient methods for the inverse heat conduction problem (IHCP) which is extremely ill-posed and time-dependent. This method determines an “inverse solution” advancing in a sequential fashion in time. The values estimated at any given time depend on the solution obtained previously. The main question connected with this method is the stability;i.e.,the cumulative error in the solution must remain bounded at all time. Since the first paper of Beck in 1970, few theoretical stability analyses have been studied in the literature. The aim of this paper is to find the conditions under which this method is stable irrespective of the data measurements. For a 1D linear IHCP, we try to construct a sequenced (1)X1= 1,Xj= Σj−1l=1αj−l+1Xl,j≥ 2, such that the coefficients αiare independent of the data measured and the convergence of the series Σi=1 |Xi| guarantees the stability of the method. In other words, we need to find an adequate condition on αisuch that Σ∞i=1|Xi| is convergent, implying that the method is stable. The values of αidepend on the discretization sizehof the function to be determinedq(t) and the sliding time horizon (or future time interval) τ of the method. The range of values ofhand τ which give the values of αisuch that the series Σ∞i=1 |Xi| is convergent is established numerically. Under the stability condition, an error estimation of the Beck's method is derived. The approach presented could be also applied to multidimensional IHCPs, in which the coefficients αiandXiare no longer scalar but become square matrices.

A NUMERICAL SOLUTION FOR THE INVERSE NATURAL-CONVECTION PROBLEM

A solution algorithm, based on the sequential function specification method, is presented for the inverse natural-convection problem. Sensitivity coefficients are computed from an explicit difference equation for temperature. Results are presented for a square cavity with a time-varying uniform heat flux on one face at various Rayleigh numbers. It is found that the stability of the method is improved by using multiple future temperatures, but remains sensitive to the location of the sensors, particularly when convection is strong.

Application of inverse techniques to determine heat- transfer coefficients in heat- treating operations

An existing sequential function specification algorithm for the solution of the inverse heat conduction problem (IHCP) has been applied to determine the response of both the surface heat flux and the surface temperature of flat stainless steel samples subjected to water quenching under controlled laboratory conditions that ensured one-dimensional heat flow. From this information, combined convective and radiative heat-transfer coefficients have been obtained as a function of steel surface temperature. The computer code was subsequently modified to solve the IHCP for air-cooled cylindrical carbon steel samples. In the algorithm, the problem is linearized by assuming the thermophysical properties of the steel to be fixed at values from the previous time step while estimating the current surface heat flux, which results in a more efficient code without a severe loss of accuracy. When compared with iterative (“brute force”) methods commonly used in the past, techniques like sequential function specification offer a more robust strategy for solving the IHCP. By including information on future measurements, while solving for the unknown surface heat flux at a particular time, the sequential function specification algorithm effectively prevents over-responses to measured temperatures, and large variations in calculated heat-transfer coefficients, observed when sequential matching is applied, can be reduced. Sensitivity coefficients, a measure of the response of temperature to changes in the unknown surface heat flux which are calculated with this algorithm, can be used to design experiments involving the IHCP effectively.

An Inverse Convection Problem

The nature of inverse problems in convective environments is investigated. The ill-posed quality inherent in inverse problems is verified for free convection laminar flow in a vertical channel. A sequential function specification algorithm is adapted for the semiparabolic system of equations that governs the flow and heat transfer in the channel. The procedure works very well in alleviating the ill-posed symptoms of inverse problems. The performance of a simple smoothing routine is also tested for the prescribed conditions.