Estimation of front surface temperature and heat flux of a locally heated plate from distributed sensor data on the back surface

We present a new method of solving the three-dimensional inverse heat conduction (3D IHC) problem with the special geometry of a thin sheet. The 3D heat equation is first simplified to a 1D equation through modal expansions. Through a Laplace transform, algebraic relationships are obtained that express the front surface temperature and heat flux in terms of those same thermal quantities on the back surface. We expand the transfer functions as infinite products of simple polynomials using the Hadamard Factorization Theorem. The straightforward inverse Laplace transforms of these simple polynomials lead to relationships for each mode in the time domain. The time domain operations are implemented through iterative procedures to calculate the front surface quantities from the data on the back surface. The iterative procedures require numerical differentiation of noisy sensor data, which is accomplished by the Savitzky–Golay method. To handle the case when part of the back surface is not accessible to sensors, we used the least squares fit to obtain the modal temperature from the sensor data. The results from the proposed method are compared with an analytical solution and with the numerical solution of a 3D heat conduction problem with a constant net heat flux distribution on the front surface.     

Analytical closed-form solution of a space-vector modulated VSIfeeding an induction motor drive

The paper presents the analysis of the time response of the stator and rotor currents in induction motor fed by space-vector pulse-width modulated voltage source inverter. This mathematical model uses the Laplace and modified Z-transform. The solution is made in two steps: (a) finding the Laplace transform of the stator voltage space vectors; and (b) finding the inverse transform of the load currents (original function) using the modified Z-transform. The solution is not dependent on the number of the pulses of the PWM pattern. All the analytical waveforms were visualized from the derived relations with the program MATHCAD. Experimental results prove the feasibility of the proposed mathematical model     

Three-dimensional generalized thermoelastic diffusion and application for a thermoelastic half-space subjected to rectangular thermal pulse

In this article, a model of three-dimensional generalized thermo-diffusion in a half-space thermoelastic medium subjected to permeating gas and the rectangular thermal pulse has been constructed. The half-space is considered to be made of an isotropic homogeneous thermoelastic material. The chemical potential is also assumed to be known on the bounding plane. Laplace transform techniques have been applied, and the solution is obtained in the Laplace transform domain using a direct approach. The solution of the problem in the physical domain is obtained numerically using a numerical method based on a Riemann-sum approximation for the inversion of Laplace transform. The temperature increment, stress, strain, diffusion concentration, and chemical potential distributions are represented graphically. The nonzero value of the relaxation time parameter predicts the finite speed of thermal, mechanical, diffusion waves.     

Temperature and heat flux estimation from sampled transient sensor measurements

Laplace transform is used to solve the problem of heat conduction over a finite slab. The temperature and heat flux on the two surfaces of a slab are related by the transfer functions. These relationships can be used to calculate the front surface heat input (temperature and heat flux) from the back surface measurements (temperature and/or heat flux) when the front surface measurements are not feasible to obtain. This paper demonstrates that the front surface inputs can be obtained from the sensor data without resorting to inverse Laplace transform. Through Hadamard Factorization Theorem, the transfer functions are represented as infinite products of simple polynomials. Consequently, the relationships between the front and back surfaces are translated to the time-domain without inverse Laplace transforms. These time-domain relationships are used to obtain approximate solutions through iterative procedures. We select a numerical method that can smooth the data to filter out noise and at the same time obtain the time derivatives of the data. The smoothed data and time derivatives are then used to calculate the front surface inputs.     

Real-time solution of heat conduction in a finite slab for inverse analysis

Laplace transform is used to solve the problem of heat conduction over a finite slab. The transfer functions relating the temperature and heat flux on the front and back surfaces of the finite slab are developed. Although there are many competing methods for constructing the inverse Laplace transform, we use polynomial approximation of the transfer function. Therefore, transient solutions for given boundary conditions are easily obtained using SIMULINK. This process is much simpler than other numerical solution methods for the heat equation. Most importantly, our method of solution allows us to obtain, in real-time, the front surface temperature and heat flux based on the thermodynamic measurements on the back surface. We also demonstrate the feasibility of reconstructing the front surface temperature when sensor noise is incorporated to the back surface measurements.     

Study of Dynamical Response in a Two-Dimensional Transversely Isotropic Thick Plate Due to Heat Source

This paper deals with a two dimensional problem for a transversely isotropic thick plate having heat source. The upper surface of the plate is stress free with prescribed surface temperature while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The study is carried out in the context of three-phase-lag thermoelastic model, GN model II (TEWOED) and GN model III (TEWED). The governing equations for displacement and temperature fields are obtained in Laplace–Fourier transform domain by applying Laplace and Fourier transform techniques. The inversion of double transform has been done numerically. The numerical inversion of Laplace transform is done by using a method based on Fourier Series expansion technique. Numerical computations have been done for magnesium (Mg) and a comparison of the results for different theories (three-phase-lag model, GN model II, GN model III) are presented graphically. The results for an isotropic material (Cu) have been deduced numerically and presented graphically to compare with those of transversely isotropic material (Mg).     

海上升压站Laplace域振动响应预报方法研究

针对传统时域预报方法的准确性依赖计算时间步长、频域法只能得到稳态响应的问题,基于Laplace变换提出一种新的海上升压站动力振动响应预报方法。通过解耦得到模态坐标系振动方程,在Laplace域求解海上升压站结构振动响应的极值和留数,进而得到时域响应。该方法考虑海上升压站的初始条件,可避免时域积分引起的误差。分别采用悬臂梁模型和导管架式海上升压站模型对该方法进行验证,结果表明了其振动响应预报的准确性。     

THERMOELASTIC TRANSIENT RESPONSE OF MULTILAYERED HOLLOW CYLINDER WITH INITIAL INTERFACE PRESSURE

This article deals with the transient response of one-dimensional axisymmetric quasi-static coupled thermoelastic problems with initial interface pressure. The initial interface pressure in a multilayered cylinder caused by the heat-assembling method is considered as an initial condition for the thermoelastic equilibrium problem. The Laplace transform and finite difference methods are used to analyze problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. The solution is obtained using the matrix similarity transformation and inverse Laplace transform. We obtained solutions for the temperature and thermal stress distributions in a transient state. Moreover, the computational procedures established in this article can solve the generalized thermoelasticity problem for a multilayered hollow cylinder.     

Generalized Coupled Thermoelasticity of a Layer

The generalized thermoelasticity based on the Lord-Shulman (LS), Green-Lindsay (GL), and Green-Naghdi (GN) theories admit the second sound effect. By introducing some parameters all these theories are combined and a unified set of equations is rendered. These equations are then solved for a layer of isotropic and homogeneous material to study the thermal and mechanical wave propagations. The disturbances are generated by a sudden application of temperature to the boundary. The non-dimensionalized form of the governing equations are solved utilizing the Laplace transform method in time domain. Closed form solutions are obtained for the layer in Laplace transform domain, and a numerical inverse Laplace transform method is used to obtain the temperature, displacement, and stress fields in the physical time domain. The thermo-mechanical wave propagations and reflections from the layer boundaries are investigated.     

Two-Temperature Generalized Thermoelasticity with Variable Thermal Conductivity

In this work, the consideration of variable thermal conductivity as a linear function of temperature has been taken into account in the context of two-temperature generalized thermoelasticity (Youssef's model). The governing equations have been derived and used to solve the one-dimensional problems of an elastic half-space. The governing equations have been cast into a matrix form by using Bahar–Hetnarski method, and Laplace transform is used to get the general solution for any set of boundary conditions. The solution has been applied for a thermally shocked medium that has no strain on its bounding plane. The numerical inversion of the Laplace transform has been calculated by using the Riemann-sum approximation method. The distribution of the conductive temperature, the thermo-dynamical temperature, the strain, the displacement, and the stress have been shown graphically with some comparisons.     

Generalized coupled transient response of 3-D multilayered hollow cylinder

This paper deals three-dimensional axisymmetric quasi-static coupled thermoelastic problems. Laplace transform and finite difference methods are used to analyze problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal deformation distributions in a transient and steady state. Moreover, the computational procedures established in this thesis, can solve the generalized thermoelasticity problem for different length hollow cylinder with nonhomogeneous materials.     

Hybrid Numerical Method Applied to Multilayered Hollow Cylinder with Time-Dependent Boundary Conditions

ABSTRACT

This article deals with one-dimensional axisymmetric quasi-static coupled thermoelastic problems with time-dependent boundary conditions. Laplace transform and finite difference methods are used to analyze the problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal deformation distributions for a transient and steady state. It is demonstrated that the computational procedures established in this article are capable of solving the generalized thermoelasticity problem of a hollow cylinder with nonhomogeneous layers.     

NUMERICAL ANALYSIS OF NON-FICKIAN DIFFUSION PROBLEMS IN A POTENTIAL FIELD

The primary difficulty encountered in the numerical solution of non-Fickian diffusion problems is numerical oscillations in the vicinity of sharp discontinuities. The present study applies a hybrid numerical scheme of the Laplace transform technique and the controlvolume method in conjunction with the hyperbolic shape functions to investigate the one-dimensional non-Fickian diffusion problems in the presence of a potential field for finite or semi-infinite geometry. The Laplace transform method used to remove the time-dependent terms in the governing differential equation and boundary conditions, and then the transformed equations are discretized by the control-volume scheme. To show the accuracy of the present numerical method, a comparison of the mass concentration distribution between the present numerical results and the analytic solutions is made. Results show that the present numerical results agree well with the analytic solutions and do not exhibit numerical oscillations in the vicinity of the jump discontinuity for various potential values. The potential gradient dV/dx has a great effect on the mass concentration distribution. The strength of the jump discontinuity is reduced as the value of the dimensionless potential gradient is increased.     

BOUNDARY ELEMENT ANALYSIS OF GREEN AND LINDSAY THEORY UNDER THERMAL AND MECHANICAL SHOCK IN A FINITE DOMAIN

A boundary element formulation based on the Laplace transform method is developed for transient coupled thermoelasticity problems with relaxation times in a two-dimensional finite domain. The dynamic thermoelastic model of Green and Lindsay is selected for the present study. The Laplace transform method is applied to the time domain, and the resulting equations in the transformed field are discretized using the boundary element method. The nodal dimensionless temperature and displacements in the transformed domain are inverted to obtain the actual physical quantities using the numerical inversion of the Laplace transform method. This work considers the Green and Lindsay theory of thermoelasticity with the thermal and mechanical loading in a finite domain. The creation and propagation of elastic and thermoelastic waves in a finite domain and their effects on each other are investigated. Different relaxation times are chosen to briefly illustrate the events that take place in GL theory. Details of the formulation and numerical implementation are also presented.     

Thermoelastic Analysis of an Annulus Using the Green-Naghdi Model

ABSTRACT

The linear theory of thermoelasticity of Green-Naghdi (GN) types II and III for homogeneous and isotropic materials are employed to study the thermal and mechanical waves in an annulus domain. The disturbances are generated by sudden application of temperature to the boundary. The nondimensional form of the governing equations are solved utilizing the Laplace transform method. Locally transversal linearization (LTL) technique, and a numerical inverse Laplace transform method are used to obtain the temperature, displacement, and stress fields in the physical time domain. The thermomechanical wave propagation and reflection from the boundary are investigated and the influence of the damping parameter on temperature, displacement, and stress fields in the Green-Naghdi type III is discussed.     

A MODE-I CRACK PROBLEM FOR AN INFINITE SPACE IN GENERALIZED THERMOELASTICITY

ABSTRACT

In this work, we solve a dynamical problem of an infinite space with a finite linear crack inside the medium. The Fourier and Laplace transform techniques are used. The problem is reduced to the solution of a system of four dual integral equations. The solution of these equations is shown to be equivalent to the solution of a Fredholm integral equation of the first kind. This integral equation is solved numerically using the method of regularization. The inverse Laplace transforms are obtained numerically using a method based on Fourier expansion techniques. Numerical values for the temperature, stress, displacement, and the stress intensity factor are obtained and represented graphically.     

Application of inverse solution in two‐dimensional heat conduction problem using Laplace transformation

An inverse solution has been explicitly derived for two‐dimensional heat conduction in cylindrical coordinates using the Laplace transformation. The applicability of the inverse solution is checked using the numerical temperatures with a normal random error calculated from the corresponding direct solution. For a gradual temperature change in a solid, the surface heat flux and temperature can be satisfactorily predicted, while for a rapid change in the temperature this method needs the help of a time partition method, in which the entire measurement time is split into several partitions. The solution with the time partitions is found to make an improvement in the prediction of the surface heat flux and temperature. It is found that the solution can be applied to experimental data, leading to good prediction. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 602–617, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10115     

Interfacial stresses and fracture analysis of a two-phase composite under time-dependent heat flux

An analysis of a two-phase composite component under time-dependent heat flux is presented. The fundamental thermoelastic solution is obtained in terms of complex potentials via the technique of the analytical continuation in order to satisfy the continuous conditions on the interface. The hereditary integral associated with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking inverse Laplace transform. Some typical examples of interface stresses induced by various time-dependent heat flux are discussed. Finally, the solution of a crack embedded in the bi-material subjected to a uniform heat flux is also discussed.     

DISCONTINUITIES IN GENERALIZED THERMO-VISCOELASTICITY UNDER FOUR THEORIES

A model of the equations of generalized thermoviscoelasticity for isotropic media is given. The formulation is applied to the generalized thermoelasticity theories—Lord–Shulman, Green–Lindsay, and Chandrasekharaiah and Tzou—as well as to the dynamic coupled theory. The state-space approach is adopted for the solution of the one-dimensional problem of plane distribution of heat sources. The Laplace transform technique is used. The expansions of the stress component, the temperature increment, and the displacement, in Laplace transform domain, in power series, and the exact inversions for arbitrary time, are given. The jump discontinuities are calculated for the four theories and the kinematic conditions of compatibility are verified. Numerical results are given and illustrated graphically by employing the numerical method for the inversion of the Laplace transforms. Comparisons are made with the results predicted by the four theories.     

GENERALIZED COUPLED THERMOELASTICITY OF DISKS BASED ON THE LORD–SHULMAN MODEL

A one-dimensional generalized thermoelasticity model of a disk based on the Lord–Shulman theory is presented. The dynamic thermoelastic response of the disk under axisymmetric thermal shock loading is studied. The effects of the relaxation time and coupling coefficient are studied. The Laplace transform method is used to transform the coupled governing equations into the space domain, where the Galerkin finite element method is employed to solve the resulting equations in the transformed domain. The dimensionless temperature, displacement, and stresses in the transformed domain are inverted to obtain the actual physical quantities using the numerical inversion of the Laplace transform method.