Finite Integral Transformations Research Articles

Analysis of Varying Temperature Regimes in a Conductive Strip during Induction Heating under a Quasi-Steady Electromagnetic Field

Transition processes in a steel conductive strip are analyzed during its induction heating under a quasi-steady electromagnetic field. In particular, the temperature field in the strip is studied. A method of solving corresponding initial boundary problems in a two-dimensional mathematical model for differential equations of electrodynamics and heat conduction is developed. The Joule heat and the temperature are determined with a high level of accuracy. The defining functions are the temperature and component of the magnetic field intensity vector tangent to the bases and end planes of the strip. To find them, we use cubic approximation of the defining functions’ distribution along the thickness coordinate. The original two-dimensional initial boundary value problems for the defining functions are reduced to one-dimensional initial boundary value problems on their integral characteristics. General solutions for these problems are obtained using the finite integral transformation by the transverse variable and the Laplace transform of the integral by time. Integral characteristics’ expressions are represented as convolutions for functions that describe homogeneous solutions of one-dimensional initial boundary value problems and limiting values of defining functions on the bases and end planes of the strip. The change of temperature under a varying regime in the dimensionless Fourier time and temperature distribution over the strip cross-section in a steady state depending on the parameters of induction heating and the Biot number are numerically analyzed. Varying and constant temperature regimes of the strip under conditions of the near-surface and continuous induction heating are studied.

Finite integral transform with homogenized boundary conditions solution of transient heat conduction applied to thermal plug and abandonment of oil wells

After the end of productive life of a geological oil reservoir, a set of operations for well Plug and Abandonment (P&A) is performed to recover the soil layers to its natural state. As traditional P&A techniques are expensive activities, alternative technologies are gaining renewed interest for capital expenditure reduction and leak risks mitigation. Hence, the Thermal Plug and Abandonment (TP&A) procedure is here investigated by focusing on the fulfillment of two intermediate milestones, combining new technology with conventional P&A practices. The first milestone consists in applying a thermite exothermic reaction to generate enough heat for melting the entire thickness of the production tube. The second milestone aims to determine if the resulting molten section has sufficient dimensions to allow for the passage of the cement pumped downhole, thus ensuring the complete sealing of the oil well's cross-section. The thermal investigation was conducted by applying a homogenized form of the Finite Integral Transform (FIT) analytical framework to solve the transient heat conduction equation in 2-D polar coordinates. The well assembly was geometrically discretized as a multilayered circular domain. The thermite reaction was modeled as a theoretical volumetric heat source profile dependent on both time and space, placed in the innermost layer. While a pure FIT approach may only be used to solve Neumann boundary conditions, the combined scheme applied here copes with any type of boundary condition. Hence, the integration of FIT with homogenization satisfies the Dirichlet condition requirement of the TP&A procedure. The methodology was verified through an equivalent Finite Volume Method (FVM) solution obtained using a commercial code. The results evidenced that within the 5 min duration of thermite reaction, the entire circumferential section of the production tube exceeds the steel melting temperature by at least 227 °C, thus fulfilling both milestones set. The research outcomes are part of a series of investigation steps to thoroughly analyze the new TP&A technology with regard to its compliance with the regulatory norms established to P&A operations.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

BOUNDARY VALUE PROBLEM FOR THE EQUATION OF UNSTEADY THERMAL CONDUCTIVITY IN A NON-CYLINDRICAL REGION

The application of the method of decomposition by eigenfunctions of a self-adjoint differential operator to solving a non-stationary heat transfer problem with a phase transition in a non-automatic formulation under special initial conditions is presented for the example of the solidification process in a continuous medium. The one-dimensional problem is solved in spherical coordinates. Solving of the problem begins with its transformation to a problem in a domain with fixed boundaries, then a finite integral transformation with an unknown kernel is constructed to solve the transformed problem, the finding of which is associated with the formulation and solving of the corresponding spectral problem through degenerate hypergeometric functions. The eigenvalues and eigenfunctions are found, as well as the inversion formula for the introduced integral transformation, which makes it possible to write out an analytical solution to the problem. In the course of solving the problem, the parabolic law of motion of the interface of the two phases is established. Problems of this type arise in the mathematical modeling of heat transfer processes in construction, especially in permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.

BOUNDARY VALUE PROBLEM FOR THE EQUATION OF UNSTEADY THERMAL CONDUCTIVITY IN A NON-CYLINDRICAL REGION

The application of the method of decomposition by eigenfunctions of a self-adjoint differential operator to solving a non-stationary heat transfer problem with a phase transition in a non-automatic formulation under special initial conditions is presented for the example of the solidification process in a continuous medium. The one-dimensional problem is solved in spherical coordinates. Solving of the problem begins with its transformation to a problem in a domain with fixed boundaries, then a finite integral transformation with an unknown kernel is constructed to solve the transformed problem, the finding of which is associated with the formulation and solving of the corresponding spectral problem through degenerate hypergeometric functions. The eigenvalues and eigenfunctions are found, as well as the inversion formula for the introduced integral transformation, which makes it possible to write out an analytical solution to the problem. In the course of solving the problem, the parabolic law of motion of the interface of the two phases is established. Problems of this type arise in the mathematical modeling of heat transfer processes in construction, especially in permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.

Finite integral transform solution of unsteady heat conduction applied to thermal plug and abandonment of oil wells

The end of the production phase of an offshore oil well represents a remarkable shift in field operations from extraction to plug and abandonment. The international normative requirements for permanent well plugging demand a series of technical maneuvers to avoid structural failures and the formation of leakage paths during or after the sealing process. The current closure technique requires a complete or partial removal of the production column before the borehole may be plugged with Portland cement. However, the tube removal process frequently results in an increase in the involved time and costs. Hence, the current research was aimed at investigating the prototype approach of Thermal Plug and Abandonment of wells by applying a thermite heat emitter device to melt the production column's steel. Here, the Finite Integral Transform analytical method was computationally implemented through an inhouse code to calculate the resultant transient temperature fields at the multilayered medium. The results were compared to a Finite Volume Method numerical solution to enhance the study’s reliability. The research outcomes provided insight that even in the event of a partial thermite reaction failure and a highly nonuniform heat pattern, the resultant molten azimuthal length of the production column may still allow enough room for cement flow. It was estimated through the temperatures achieved that the heat emitter is capable of melting at least nearly three-quarters of the production tube’s azimuthal length, thus eliminating the need for its removal and significantly reducing the sealing process operational costs.

Straightforward Free Vibration Solutions of Open Cylindrical Shells by the Finite Integral Transform Method

Straightforward analytical free vibration solutions of non-Lévy-type open cylindrical shells are obtained in this study by the finite integral transform method, which was not achieved by conventional semi-inverse methods. Three double finite integral transforms are imposed on the high-order partial differential equations (PDEs) for the problems, with some boundary conditions input, by which the relations between the transformed quantities and specific unknowns are given. Substituting the corresponding inverse transforms into the other boundary conditions provides systems of linear algebraic equations for determining the final analytical solutions. Comprehensive benchmark results for representative open cylindrical shells under different boundary conditions are shown. The results are well verified by other solution methods. With the present solutions, the effects of geometric parameters and boundary conditions on the vibration behaviors are quantitatively revealed by a parametric study. Due to its rigorous and straightforward solution procedure, this study presents a solid easy-to-implement way to explore new analytical solutions of similar problems.

Generalized Method of Finite Integral Transformations in Linear and Nonlinear Static Problems for Shallow Shells

Generalized Method of Finite Integral Transformations in Linear and Nonlinear Static Problems for Shallow Shells

Неосесимметричная нестационарная задача термоэлектроупругости для длинного пьезокерамического цилиндра

A new closed solution to the uncoupled non-axisymmetric problem of thermoelectroelasticity was constructed for a long hollow piezo-ceramic cylinder for the case of non-stationary temperature alteration on its inner surface, taking into account the convective heat transfer between the outer surface and the environment. Cylindrical surfaces were electroded and connected to a measurement device with large input resistance (idle mode), while the internal surface was grounded. The Fourier—Kirchhoff non-stationary heat conduction equation was considered without taking into account the influence of alteration in the body dimensions and the electric field on the temperature field. The closed solution to the heat conduction problem was constructed by the finite integral transformations (FIT) method. The quasi-static coupled electroelasticity problem at a certain temperature field was solved without taking into account the cylinder inertial properties by the FIT method. The calculated relations obtained made it possible to determine the temperature field, the stress-strain state, as well as the electric field in the long piezo-ceramic cylinder under non-stationary non-axisymmetric action in the form of a function of the temperature alteration. Numerical analysis of the results made it possible to determine the cylinder wall thickness and the region of the temperature effect alteration, where deformation could most efficiently transform into the electrical pulse.

Dynamic response of a thin-walled curved beam with a mono-symmetric cross-section under a moving mass

This paper presents analytical solutions for the dynamic response of thin-walled curved beams induced by moving vehicles. The solutions encompass vertical, torsional, radial, and axial motions. A comprehensive set of governing equations for the dynamic response of thin-walled curved beams is established by considering the mass inertia of moving vehicles, rotary inertia, and warping resistance. The solutions are obtained for the four-directional response of curved beams, involving vertical, torsional, radial, and axial motions, based on the Fourier finite integral transformation, the Laplace–Carson transformation, their inverse transformations, and the Galerkin approach. A comparison of the results calculated by the analytical solutions in this study are compared with those calculated by the moving-force solutions using the Galerkin approach reported in a related literature, demonstrates the reliability and superiority of the analytical solutions. Hence, an extensive parametric study is carried out to investigate the effect of the mass inertia of moving vehicles, velocities, higher-order modes of vibrations, and radii of curvature on the dynamic responses of beams. As per the results, the mass and velocity of the moving vehicles play important roles in the dynamic response of curved beams. Furthermore, the dynamic response for thin-walled curved beams increases significantly with heavier vehicles and higher velocities, confirming that it is necessary to consider the mass inertia effect to study the dynamic response of curved beams.

Finite Integral Transform Solutions for Free Vibrations of Rectangular Thin Plates With Mixed Boundary Constraints

Finite Integral Transform Solutions for Free Vibrations of Rectangular Thin Plates With Mixed Boundary Constraints

Theoretical evolution of thermal behaviour of Ti-6Al-4V subjected to selective laser melting: A powder free approach

Additive manufacturing (AM) processes are considered to be the pillar of the next industrial revolution due to their inherent qualities such as design flexibility, the ability to produce complex parts and prototypes, lower cost due to the reduced requirement of materials and curtailed lead time for manufacturing. Selective laser melting (SLM) is one of the most popular metallic AM technologies since it enables accurate control over part dimensions and fabrication of high resolution features. This research paper is aimed to develop an exact analytical model of three-dimensional thermal response captured in SLM of Ti based alloy (Ti-6Al-4V). A physical model has been proposed to predict the temperature profile during the metal additive manufacturing process with consideration of the effect of thermal history developed during moving laser heat source interaction. The corresponding mathematical solution is developed by employing an amalgamation of ‘Duhamel’s theorem’ and ‘Finite Integral Transform method’. The parametric laser-substrate interaction phenomenon is the prime deciding factor for the successful accomplishment of the manufacturing process. This research paper theoretically investigates the thermal characteristics (peak temperature, temperature distribution curvature, pulse time, optical penetration depth, time of laser exposure, laser absorption radius, and so on) by employing Fourier’s heat conduction model with a moving laser heat source. The theoretical estimation has been validated by the existing mathematical as well as experimental research outcomes. Present work might be an asset for deciding the design of process variables and protocols in terms of laser based additive manufacturing specifically the selective laser melting process.

Analytical free vibration solutions of rectangular edge-cracked plates by the finite integral transform method

Analytical free vibration solutions of rectangular edge-cracked thin plates are obtained by the finite integral transform (FIT) method. While the previous studies focus on the mechanical behaviors of intact plates, this study establishes an FIT-based new scheme for analyzing the free vibration characteristics of rectangular thin plates with an edge crack. The plates may have any combined conditions of simply supported, clamped, and free boundaries. In specific solution scheme, an edge-cracked plate is decomposed into four subdomains, and a double cosine FIT is imposed on the fourth-order partial differential governing equations of each subdomain, which yields the relationship between the transformed modal deflections and naturally defined unknowns. By substituting the exterior boundary conditions of the plate, continuity conditions across the subdomains, and free conditions at the crack into the inverse transforms, the natural frequencies and unknowns are successively determined via the established homogeneous linear algebraic equations. The modal deflections are provided by further substituting the obtained unknowns into the inverse transforms. Comprehensive benchmark results for the edge-cracked plates under representative boundary conditions are shown. The results are well verified by other solution methods. A parametric study quantitatively reveals the effects of boundary conditions and crack length on the natural frequencies of edge-cracked plates. Due to the rigorous but easy-to-implement mathematical derivations, this study presents a novel solid way for exploring analytical solutions of vibration problems.

Mathematical modeling of temperature fields of spacecraft antenna reflectors with a new inte-gral transformation

Prediction of the mechanical properties of the reflector, and above all, the deviation of the highly accurate shape of the reflecting surface (RSS) from the given one is the main goal of designing spacecraft antennas. Distortion of the RSS is determined by the stress-deformed state of the elements of the reflector structures under the conditions of orbital operation. At the same time, the main factor determining the distortion of reflectors with RSS in open space is temperature deformation due to the uneven distribution of solar heat fluxes among structural elements. Therefore, the development of methods and models for calculating temperature fields in reflectors during heat flows on the surface is relevant. In the article, for the first time, a new finite integral transformation for the Laplace equation in a cylindrical coordinate system is constructed for a region bounded by several closed piecewise smooth contours. The inverse transformation formula is given. In the article, for the first time, a mathematical model for the calculation of temperature fields in a paraboloid rotating with a constant angular velocity is constructed, taking into account the finite speed of heat propagation in the form of a boundary value problem of mathematical physics for the hyperbolic equation of heat conduction with Dirichlet boundary conditions. With the help of the developed integral transformation, the temperature fields in the paraboloid were found in the form of convergent series according to the Fourier functions. The found solution to the generalized boundary value problem of the heat transfer of the paraboloid of rotation can find application in modulating the temperature fields that arise in the antenna reflectors of space vehicles. The developed integral transformation makes it possible to obtain solutions to complex boundary value problems of mathematical physics.

The nonstationary thermoelectric elasticity problem for a long piezoceramic cylinder

А new closed-loop solution for the coupled nonstationary problem of thermoelectric elasticity is designed for a long piezoceramic radially polarized cylinder. The case of the nonstationary load acting on its inner cylindrical surface is considered as a function of temperature change at a given law of the convection heat exchange on the outer face wall (boundary conditions of heat conductivity of the 1st and 3rd types). Electrodynamic cylinder surfaces are connected to a measuring device with a high input resistance (electric idling). We investigate the problem where the rate of the temperature load changes does not affect the inertial characteristics of the elastic system. It makes it possible to expand the initial linear computational relations with the equilibrium, electrostatics and heat conductivity equations with respect to the radial component of the displacement vector, electric potential as well as the function of temperature field changes. Hyperbolic LS-theory of the thermal conductivity is used in the computations. The problem is solved with a generalized method of biorthogonal finite integral transformation based on a multicomponent ratio of eigen functions of two homogeneous boundary value problems. The structural algorithm of this approach allows identifying a conjugated operator, without which it is impossible to solve non-self-conjugated linear problems in mathematical physics. The resulted computational relations make it possible to determine the stress-strain state, temperature and electric fields induced in the piezoceramic element under an arbitrary external temperature effect. By connecting the electroelastic system to the measuring tool, we can find voltage. Firstly, the analysis of the numerical results allows identifying the rate of the temperature load changes, at which it is necessary to use the hyperbolic theory of thermal conductivity. Secondly, it allows determining the physical characteristics of the piezoceramic material for the case when the rate of changing the body volume leads to a redistribution of the temperature field. The developed computational algorithm can be used to design non-resonant piezoelectric temperature sensors.

Analytical Investigation of Non-Fourier Bioheat Transfer in the Axisymmetric Living Tissue Exposed to Pulsed Laser Heating Using Finite Integral Transform Technique

Abstract This article proposes the closed-form solution of the generalized non-Fourier model-based bioheat transfer equation (BHTE) in Cylindrical coordinates to understand the thermal behavior of living tissue heated by a pulsed laser. The axisymmetric living tissue exposed to the non-Gaussian temporal profile of laser heating has been considered to investigate the non-Fourier bioheat transfer phenomena. The closed-form solution of the generalized non-Fourier model-based BHTE with time-dependent thermal energy generation has been obtained through the finite integral transform (FIT) technique. The analytical solution was juxtaposed to the corresponding numerical solution in order to determine its reliability. The numerical solution of the aforementioned governing equation has been obtained by the finite volume method (FVM). The results of both analytical and numerical solutions have been verified using results given in published literature. Subsequently, the dual-phase-lag (DPL) model's findings were juxtaposed to those obtained using the hyperbolic and traditional Fourier models. The effect of different parameters like relaxation times corresponding to the temperature gradient and heat flux, metabolic energy generation, and blood perfusion on the resultant temperature distribution inside the axisymmetric living tissue exposed to pulsed laser heating has been discussed. The importance of this study might be found in various applications such as laser-based-photothermal therapy, melting of the surface of metal and alloys by laser heating.

Thermoelastic non-stationary fields in a rigidly fixed plate

In this article, a new closed solution of the axisymmetric dynamic problem of the theory of thermoelasticity is constructed for a rigidly fixed circular isotropic plate in the case of temperature changes on its face surfaces. The mathematical formulation of the problem includes linear equations of motion and thermoelasticity in the spatial formulation with respect to the components of the displacement vector, as well as the function of temperature change. The study of non-self-adjoint equations is carried out in an unrelated statement. Initially, we consider the initial boundary value problem of thermoelasticity without taking into account the deformation of the plate, and at the next stage, we study the problem of elasticity theory under the action of a given (defined) temperature change function. To solve the problems, we use a mathematical apparatus for separating variables in the form of finite integral transformations: Fourier, Hankel, and generalized integral transformation (CIP). In this case, at each stage of the study, the procedure is performed to bring the boundary conditions to the form that allows you to apply the appropriate transformation.

Generalization of Finite Integral Transforms for Treating Nonlinear Problems in Heat Diffusion. Part II: Application to a nonlinear case.(Dept.P)

A solution methodology, based on the finite integral transform technique, appropriate for solving nonlinear problems of heat diffusion was developed by the author in previous work [1,2]. In this paper, the method is applied to solve the heat diffusion problem in a finite region subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The results obtained form this analytic Solution are compared with those obtained from a numerical solution developed using an explicit finite difference method.

Bending and torsion vibrations of a beam excited by a moving load

The linear dynamic bending- and torsion vibrations of a simply supported Bernoulli-Euler beam under a moving load with constant velocity are analysed. The main target is to derive an analytical solution of the governing partial differential equations. The deformations of the beam are computed based on the eigenmode expansion. For the computation of the solution first the method of generalized finite integral transformation is used followed by a transformation using the Laplace-Carson integral transformation. The resulting algebraic equation then is rearranged considering the boundary and initial conditions. The inverse transformation has to consider the position of the poles and after a further transformation the solution in the time domain results. The convergence and the necessary number of eigenmodes of this procedure is analysed. For the pure bending load the analysis is performed and the beam vibrations are calculated for various longitudinal speeds. For the special case of static deflections the solution is also given. The resulting deformations are plotted over time and space and the occurring phenomena are discussed. For the pure torsion deformation the Saint-Venant torsion theory is used. The described methods for solving the equations are used again. Based on the boundary and initial conditions the solution shows a similar structure like the pure bending solution. Finally the analytic solutions are compared to a numerical calculation of a finite element model which shows good agreement of the two solutions.

The generalized finite integral transformations method in linear and nonlinear static problems for shallow shells

The generalized finite integral transformations method in linear and nonlinear static problems for shallow shells