Linear Inverse Heat Conduction Problem Research Articles

Stable numerical solution to linear inverse heat conduction problems by the conjugate gradient method

Article Stable numerical solution to linear inverse heat conduction problems by the conjugate gradient method was published on January 1, 1995 in the journal Journal of Inverse and Ill-posed Problems (volume 3, issue 6).

ANALYSIS OF SEQUENTIAL METHODS OF SOLVING THE INVERSE HEAT CONDUCTION PROBLEM

The emphasis of this article is on an error and stability analysis of sequential methods solving the linear inverse heat conduction problem. The well-known method of J. V. Beck is studied as well as modifications thereof. At the end, our results are discussed by means of numerical examples. The following aspects are most important in the article. A fundamental relation is established that shows how the heat fluxes determined by Beck's method depend explicitly on the previous heat fluxes and on the data. On the one hand, this presents a new way of computing the Beck method and, on the other hand, leads to various modifications of the method for which analogous relations and computational procedures are available. Moreover, for all sequential methods under consideration, an error analysis can be established.

Two-dimensional linear transient inverse heat conduction problem - Boundary condition identification

This article deals with the identification of unknown time- and space-dependent boundary conditions for systems driven by the heat equation. We first consider a one-dimensiona l and single-input problem, dealing with the identification of the time-dependent heat flux on one side of a one-dimensional linear thermal wall, from temperature and heat flux measurements on the other side. We then focus on a quenching process; our interest is to identify the time- and space-dependent heat flux on the boundary of a metal piece from temperature measurements performed inside the material (two-dimensional geometry). Those inverse problems are solved by use of a regularization method, and the solution is obtained by minimization of a quadratic criterion. Because of the linearity of the input-output relationship, the solution of this minimization is derived from the linear quadratic optimal control theory (resolution of a nonstationary Riccati equation). The robustness of the method for very small signal-to-noise ratio is shown. In the two-dimensional multi-input problem, the identification sensitivity to the localization of the measurement points is analyzed. In both cases, we consider input that are discontinuous in time, in order to show the method accuracy in the high-frequency domain.

WELL-CONDITIONED NUMERICAL METHOD FOR THE NONLINEAR INVERSE HEAT CONDUCTION PROBLEM

A numerical procedure recently presented by the authors for linear inverse heat conduction problems is extended to nonlinear cases. The method is well conditioned in the sense that it always generates bounded solutions and never generates heat fluxes oscillating with increasing amplitude. The convergence of the algorithm is studied, and a set of examples shows convergence for a wide range of nonlinear problems. In the case of perfect data the accuracy of the estimated heat fluxes increases as the time step is decreased without need for additional stabilization or regularization. Filters allow perturbed data to be treated and facilitate attaining a balance between accuracy and resolving power.

A stable space marching finite differences algorithm for the inverse heat conduction problem with no initial filtering procedure

We consider the one-dimensional semi-infinite linear inverse heat conduction problem (IHCP) and present a new solution algorithm based on singular perturbation techniques. The new method successfully incorporates the needed initial filtering procedure of the noisy data into the marching scheme. The concept of formal stability for the IHCP is analyzed and its relationship with the high frequency components of the measured data funciton is further studied by comparing the new method with the mollification method and the method of Weber. A fully explicit space marching finite differences scheme is developed and several numerical experiments of interest, showing the accuracy and stability properties of the algorithm in the presence of noisy data and for small dimensionless time steps, are investigated.

Noise reconstruction for the inverse heat conduction problem

A new automatic procedure to numerically recover the sample root mean square norm of the data error for the linear inverse heat conduction problem (IHCP)—when this information is not readily available—is presented. Numerical results are described which illustrate the accuracy of the algorithm.

An integral solution for the inverse heat conduction problem after the method of Weber

We consider the linear inverse heat conduction problem (IHCP) in the one-dimensional semi-infinite slab, and present a new solution algorithm and error bounds, based on the integral representation of Weber's hyperbolic approximation and a suitable filtered version of the noisy data. We discuss the determination of the hyperbolic parameter and describe several numerical examples of interest, showing the accuracy and stability of the method.

Periodic B-spline basis for quasi-steady periodic inverse heat conduction

A method based on the use of periodic B-splines and the integral transform technique is proposed for the solution of quasi-steady periodic linear inverse heat conduction problems. Previous approaches based on a finite Fourier series representation of the unknown surface condition are best suited to smooth time variations of the surface condition. Now, using a B-spline representation, problems with discontinuities or abrupt variations in the surface condition can be handled readily. The versatility of the B-spline basis allows prior information concerning the general functional behavior of the surface condition to be better incorporated into the model.

A direct analytic approach for solving two-dimensional linear inverse heat conduction problems

A stable and accurate analytic approach is developed for solving two-dimensional inverse heat conduction problems where the front surface conditions are calculated from the knowledge of the time variation of the temperature at an interior point in the region. A combination of a splitting-up procedure and the least square technique is utilized for the solution of this inverse problem.

Whole domain function-specification method for linear inverse heat conduction

A generalized function-specification approach in the full time domain is presented for solving linear inverse heat conduction problems involving an unknown time-varying uniform applied condition at one of the boundary surfaces. The formalism allows unified treatment of such problems in the 1-D,2-D and 3-D domains in rectangular, cylindrical and spherical coordinates. A wide variety of smooth time variations of the boundary condition as well as variations with an abrupt change can readily be accommodated with the present method. A statistical analysis is performed to establish confidence bounds for the error involved in the determination of boundary quantities. The application of the method is illustrated with specific examples in cylindrical geometry.

On the solution of three-dimensional inverse heat conduction in finite media

An accurate and stable analytic approach is developed for solving three-dimensional linear inverse heat conduction problems involving the determination of the surface temperature from the knowledge of the time variation of the temperature at the opposite boundary surface. The least-square technique is utilized to compute the unknown parameters associated with solution. The numerical results show that the current method of analysis: is insensitive to measurement errors; remains stable with measurements involving large number of data points taken with extremely small time steps; and can accommodate, with high degree of accuracy, abrupt changes with time in the unknown surface temperature.

Solution of the linear one-dimensional inverse heat-conduction problem on the basis of a hyperbolic equation

The linear problem of determining the temperature in an infinite one-dimensional plate with a stationary heat sensor and a stationary boundary is solved. The allowable approximation time steps in the calculations can be diminished by using a hyperbolic heat-conduction equation with a suitably chosen hyperbolicity parameter.

Hyperbolic equation of thermal conductivity. Solution of the direct and inverse problems for a semiinfinite bar

The first and second boundary-value problems as well as the linear boundary-value inverse heat-conduction problem with fixed heat collector and boundary have been solved. Account of the finite heat-propagation velocity increases the boundary-value inverse heat-conduction problem stability.