Krzysztof Grysa

An inverse temperature field problem of the theory of thermal stresses

Abstract Unlike the investigations reported in the literature, which are restricted to the inverse problems of heat conduction, the current study presents an analytical approximate solution to an inverse problem of temperature field in the thermal stresses theory. Based upon the so-called internal responses of a different kind, an approximate form of surface temperature for a slab is predicted. To get the approximate solution—and also the exact one — the Laplace transform techniques are used. The theoretical considerations are carried out in the following way: (1) solution of a direct — initial-boundary — problem of the thermal stresses theory is exploited to obtain the transformed form of a formal solution of the problem, (2) conditions for the admissible internal responses are settled, (3) functions describing the internal responses are constructed, and (4) approximate solution of the considered problem is found and discussed. It appears that a step function is admissible for description of an internal response. The form of solution is convenient to numerical evaluation. Several numerical examples are presented as an indication of the accuracy of the theoretical results.

On a certain inverse problem of temperature and thermal stress fields

SummaryThe paper discusses the method of solution of an inverse problem of one-dimensional temperature and stress fields for a sphere, a circular cylinder and an infinite plate. The inverse problem describes the dependance of the boundary conditions of different types on the prescribed temperature state or stress state within the body under consideration, in contrast with the direct problem which relates the temperature and stress states to known boundary conditions. To obtain a function describing the temperature of a heating medium and/or the Biot number in a simple form use has been made of the Laplace transformation. The numerical examples for both types of the inverse problems are presented.ZusammenfassungDie Arbeit Antersucht die Lösungsmethode des inversen Problems eindimensionaler Temperatur-und Spannungsfelder für eine Kugel, einen Kreiszylinder und eine unendliche Platte. uas inverse Problem beschreibt die Abhängigkeit der Randbedingungen verschiedener Drt vom vorgegebenen Temperatur- oder Spannungszustand innerhalb des betrachteten Körpers im Vergleich zum direkten Problem, welches den Temperatur- und Spannungszustand zu bekannten Randbedingungen in Beziehung setzt. Zum Erhalt einer Funktion, die die Temperatur des erwärmten Mediums und/oder die Biot-Zahl in einer einfachen Form beschreiben, wurde die Laplace-Transformation verwendet. Numerische Beispiele für beide Arten der inversen Probleme werden angegeben.

Different finite element approaches for inverse heat conduction problems

In this study, three different versions of the finite element method (FEM) based on Trefftz functions are presented for solving transient two-dimensional inverse heat conduction problems. The following cases are considered: (a) FEM with the condition of continuity of temperature in the common nodes of elements, (b) no temperature continuity at any point between elements and (c) nodeless FEM. Instead, in each finite element the temperature is approximated by a linear combination of Trefftz functions. These investigations are carried out using temperature data obtained from numerical simulations, exact and disturbed. For the inaccurate data some efficient method of smoothing is presented.

Physical regularization for inverse problems of stationary heat conduction

In the paper, physical foundation of a functional for solving discontinuous stationary heat conduction problems with a kind of regularisation parameter has been presented. Then, the noncontinuous FEM with Trefftz base functions and a wide range of the regularisation parameter values has been applied to solving direct and inverse problem of linear heat conduction. 2D test problem solution has been discussed. Then a problem of finding heat transfer coefficient and temperature on the inner boundary of a turbine blade has been considered. To solve the problem the numerical entropy production intensity and energy dissipation function have been minimized on the boundary of the blade cross-section. In order to simplify the numerical calculation the Bernstein polynomials have been used to approximate the boundary temperature. Increasing the number of base functions in the finite element substantially decreases the inaccuracies of direct and inverse problem solution. It seems that the only disadvantage of minimising the functionals of entropy production intensity or energy dissipation lead to nonlinear system of algebraic equation.

One-dimensional problems of temperature and heat flux determination at the surfaces of a thermoelastic slab: Part I: The analytical solutions

Abstract In this paper the approximate analytical and the exact solutions of the problems of heat flux and temperature determination on slabs are presented. The problems are considered on the ground of the theory of thermoelasticity. The interior thermal and mechanical responses are used in determining the unknown functions describing heat flux and temperature of the surface of a slab. The approximate forms of the solutions are simplified significantly if the time step is sufficiently large.

Trefftz method in solving the inverse problems

Abstract In this article we present different types of Trefftz functions for stationary and non-stationary linear differential equations. The Trefftz methods are then defined and briefly described. One of them, namely the LSM applied to the region divided on finite elements, is discussed. Examples of the T-functions usage in solving inverse problems and smoothing inaccurate input data are presented. Moreover, some remarks concerning nonlinear non-stationary problem of heat conduction is briefly discussed and some open problems are formulated.

One-dimensional problems of temperature and heat flux determination at the surfaces of a thermoelastic slab: Part II: Numerical analysis

Abstract In this paper a thorough numerical analysis of an identification process, that has been analytically described in the preceding paper in this issue, is presented. The conclusions have shown some new aspects of the problem and are confirmed by brief additional analytical considerations.

Inverse Heat Conduction Problems

In the heat conduction problems if the heat flux and/or temperature histories at the surface of a solid body are known as functions of time, then the temperature distribution can be found. This is termed as a direct problem. However in many heat transfer situations, the surface heat flux and temperature histories must be determined from transient temperature measurements at one or more interior locations. This is an inverse problem. Briefly speaking one might say the inverse problems are concerned with determining causes for a desired or an observed effect. The concept of an inverse problem have gained widespread acceptance in modern applied mathematics, although it is unlikely that any rigorous formal definition of this concept exists. Most commonly, by inverse problem is meant a problem of determining various quantitative characteristics of a medium such as density, thermal conductivity, surface loading, shape of a solid body etc. , by observation over physical fields in the medium or – in other words a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in. The fields may be of natural appearance or specially induced, stationary or depending on time, (Bakushinsky & Kokurin, 2004). Within the class of inverse problems, it is the subclass of indirect measurement problems that characterize the nature of inverse problems that arise in applications. Usually measurements only record some indirect aspect of the phenomenon of interest. Even if the direct information is measured, it is measured as a correlation against a standard and this correlation can be quite indirect. The inverse problems are difficult because they ussually are extremely sensitive to measurement errors. The difficulties are particularly pronounced as one tries to obtain the maximum of information from the input data. A formal mathematical model of an inverse problem can be derived with relative ease. However, the process of solving the inverse problem is extremely difficult and the so-called exact solution practically does not exist. Therefore, when solving an inverse problem the approximate methods like iterative procedures, regularization techniques, stochastic and system identification methods, methods based on searching an approximate solution in a subspace of the space of solutions (if the one is known), combined techniques or straight numerical methods are used.

Stability Investigation For The Inverse HeatConduction Problems

Discretisation with respect to time changes the heat conduction equation into a system of Helmholtz equations. An error introduced by such a simplification is investigated. For the inverse problems the constrains for a time step are considered in the case of two inaccurate internal responses. Nomographs helping to choose a proper time step for inverse problems with two internal responses are presented.