Michał Ciałkowski

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.

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.

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.

Iterative algorithm for solving the inverse heat conduction problems with the unknown source function

In this study, the Cauchy problem for the Laplace equation in a multiply connected region was solved. Solving this problem was replaced by solving the Poisson equation in a simply connected region with an unknown source function different from zero in the adjoined region. To determine the power of the sources, a convergent iterative process has been developed based on polyharmonic functions, including the tests on the effect of the polyharmonic function order on the solution accuracy.

Regularization of the inverse heat conduction problem by the discrete Fourier transform

Two methods of solving the inverse heat conduction problem with employment of the discrete Fournier transform are presented in this article. The first one operates similarly to the SVD algorithm and consists in reducing the number of components of the discrete Fournier transform which are taken into account to determine the solution to the inverse problem. The second method is related to the regularization of the solution to the inverse problem in the discrete Fournier transform domain. Those methods were illustrated by numerical examples. In the first example, an influence of the boundary conditions disturbance by a random error on the solution to the inverse problem (its stability) was examined. In the second example, the temperature distribution on the inner boundary of the multiply connected domain was determined. Results of calculations made in both ways brought very good outcomes and confirm the usefulness of applying the discrete Fournier transform to solving inverse problems.

Microfluidic device for a rapid immobilization of Zebrafish larvae in environmental scanning electron microscopy

Small vertebrate model organisms have recently gained popularity as attractive experimental models that enhance our understanding of human tissue and organ development. Despite a large body of evidence using optical spectroscopy for the characterization of small model organism on chip‐based devices, no attempts have been so far made to interface microfabricated technologies with environmental scanning electron microscopy (ESEM). Conventional scanning electron microscopy requires high vacuum environments and biological samples must be, therefore, submitted to many preparative procedures to dehydrate, fix, and subsequently stain the sample with gold–palladium deposition. This process is inherently low‐throughput and can introduce many analytical artifacts. This work describes a proof‐of‐concept microfluidic chip‐based system for immobilizing zebrafish larvae for ESEM imaging that is performed in a gaseous atmosphere, under low vacuum mode and without any need for sample staining protocols. The microfabricated technology provides a user‐friendly and simple interface to perform ESEM imaging on zebrafish larvae. Presented lab‐on‐a‐chip device was fabricated using a high‐speed infrared laser micromachining in a biocompatible poly(methyl methacrylate) thermoplastic. It consisted of a reservoir with multiple semispherical microwells designed to hold the yolk of dechorionated zebrafish larvae. Immobilization of the larvae was achieved by a gentle suction generated during blotting of the medium. Trapping region allowed for multiple specimens to be conveniently positioned on the chip‐based device within few minutes for ESEM imaging.

Application of discrete Fourier transform to inverse heat conduction problem regularization

Purpose This paper aims to present the Cauchy problem for the Laplace’s equation for profiles of gas turbine blades with one and three cooling channels. The distribution of heat transfer coefficient and temperature on the outer boundary of the blade are known. On this basis, the temperature on inner surfaces of the blade (the walls of cooling channels) is determined. Design/methodology/approach Such posed inverse problem was solved using the finite element method in the domain of the discrete Fourier transform (DFT). Findings Calculations indicate that the regularization in the domain of the DFT enables obtaining a stable solution to the inverse problem. In the example under consideration, problems with reconstruction constant temperature, assumed on the outer boundary of the blade, in the vicinity of the trailing and leading edges occurred. Originality/value The application of DFT in connection with regularization is an original achievement presented in this study.

Solution of Cauchy problem to stationary heat conduction equation by modified method of elementary balances with interpolation of the solution in physical plane

This article presents a solution of stationary direct and inverse problems (Cauchy problem) of cooling a circular ring with the modified method of elementary balances. The idea of the method itself relies on the division of the considered range into elements and interpolation of the solution within the elements with the help of linear combination of base functions. In the vicinity of every mesh node, the control regions are created in which the energy is balanced. The regions include or interpenetrate with each other. The numerical calculation has been carried out with the use of a quadrilateral mesh with four nodes and a triangular mesh with six nodes. The presented solution to the inverse problem with randomly perturbed boundary conditions of the flux and temperature at the outer boundary of the ring with maximal error reaching up to 10% gave very good results. The solution of the inverse problem has been obtained in the sense of singular value decomposition algorithm.