Inverse heat conduction model for the resistance spot welding of aluminum alloy

ABSTRACT

During the resistance spot welding (RSW), the thermal process plays a crucial role on nugget formation, especially the temperature field at the workpiece–workpiece interface, since it dominates the nugget diameter which is acknowledged generally as the quality criterion for welds, whereas it could hardly be measured experimentally. This work developed a solution for the RSW process of aluminum alloy according to inverse heat conduction problems. A direct transient heat conduction model was first established considering the variation of contact resistance with temperature and electrode force, the temperature-dependent thermophysical material properties, etc. A developed inverse model was then solved via the conjugate gradient method combined with the direct model based on the experimental temperature measurements at the workpiece surface by infrared thermometry. The calculated temperature distributions at the interface of workpieces were examined by the resulting development of the nugget diameter, which agrees well with the experimental results.     

THERMALLY INDUCED VIBRATION IN A THIN PLATE UNDER THE WAVE HEAT CONDUCTION MODEL

Thermally induced vibration in a thin plate under a thermal excitation is investigated. The excitation is in the form of a suddenly applied laser pulse (thermal shock). The resulting transient variations of temperature are predicted using the wave heat conduction model (hyperbolic model), which accounts for the phase lag between the heat flux and the temperature gradient. The resulting heat conduction equation is solved semianalytically using the Laplace transformation and the Riemann sum approximation to calculate the temperature distribution within the plate. The equation of motion of the plate is solved numerically using the finite difference technique to calculate the transient variations in deflections.     

Weld Pool Shape Identification by Using Bezier Surfaces

This paper deals with the heat transfer analysis in a welding process: A method is developed to determine the shape of the three-dimensional (3-D) phase change front and to estimate the temperature field within the solid part of the work piece. The problem is formulated and solved as an inverse phase-change problem by using an optimization method. The direct problem is solved in the torch frame and so formulated as an Eulerian approach. The interface between the weld pool and the solid region is parameterized by Bezier surfaces. The most important feature of the presented approach is that the liquid–solid interface as well as the temperature distribution within the solid region can be obtained from additional temperature data available in the solid region, without considering heat transfer and fluid flow in a molten zone. The estimate of these thermal characteristics then allows a thermomechanical calculation of the welded joint (calculation of the deformations and residual stresses). The validity of the numerical solution of the inverse problem is checked by comparing the results with the direct solution of the problem.     

Estimation of internal fuel cell temperatures from surface temperature measurements

In this paper, a method to monitor PEM fuel cells internal temperature from surface measurements is presented. The aim of this work is to monitor fuel cells to prevent damages due to internal overheating. The measurements are taken at the side of the bipolar plate, and heat flux and temperature at the border of the active zone are estimated. The method is based on sensitivity analysis and inverse problem algorithms. The mathematical formulation and algorithm are described. The model is a transient heat conduction model in two dimensions, the inverse problem is solved with an optimization method using adjoint equation. Numerical test cases are presented for graphite and steel bipolar plates. The results show that internal temperature can be correctly estimated. The response time of the method is limited by the heat transfer rate in the material. Therefore, the method is particularly appropriate to fuel cells made of steel bipolar plates.     

ANALYSIS OF TRANSIENT THERMAL BENDING MOMENTS AND STRESSES OF THE WORKPIECE DURING SURFACE GRINDING

Geometrical inaccuracy is often induced by heat generated during grinding. Furthermore, the transient thermal process is the main cause for the residual stresses on theground surface. The objective of this article is to investigate the three-dimensional transient temperature distribution of the workpiece using the finite difference method,and based on the acquired temperature and beam theory, the thermal moment and thermoelastic stress as calculated using Simpson's multiple numerical integral method. The energypartition is the key factor in accurately predicting the temperature distribution, on which the solution of the thermal moment and stress rely. As the heat conductivity of the workpiece decreases, the stress and moment increase near the wheel-workpiece contact zone and the peaks move closer to the contact position. A smaller thickness results in higher thermal stress and lower thermal moment. Enhancing cooling in grinding effectively reduces temperature and the induced stress.     

An inverse problem of finding the time-dependent thermal conductivity from boundary data

We consider the inverse problem of determining the time-dependent thermal conductivity and the transient temperature satisfying the heat equation with initial data, Dirichlet boundary conditions, and the heat flux as overdetermination condition. This formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data cause large errors in the output solution. The finite difference method is employed as a direct solver for the inverse problem. The inverse problem is recast as a nonlinear least-squares minimization subject to physical positivity bound on the unknown thermal conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. We investigate the accuracy and stability of results on a few test numerical examples.     

辐射腔体热源反向优化设计

介绍的待设计热力系统通过辐射加热使目标从初始已知状态按照设定温度进程升温到最终的热稳定状态。对于热源就是寻求满足整个加热过程的热流输入函数,在空间结构和腔体表面性质已知的条件下,热流输入函数可以通过目标的步时确定,问题转变成非稳态问题。非稳态问题的引入使问题更具有实际应用的价值,采用截断奇异值分解法来求解。     

Inverse Hyperbolic Heat Conduction in Fins with Arbitrary Profiles

This study aims to estimate unknown base temperature distribution in different non-Fourier fins. The Cattaneo–Vernotte (CV) heat model is used to predict the heat conduction behavior in these fins. This inverse problem is solved by the function-estimation version of the Adjoint conjugate gradient method (ACGM) based on boundary temperature measurements. The ACGM includes direct, sensitivity, and adjoint problems. For each of these problems, a one-dimensional general formulation of the non-Fourier model for longitudinal fins with arbitrary profile is driven and solved by an implicit finite difference method. In this study, three different profiles are considered: triangular, convex parabolic, and concave parabolic. For each of them, two different base temperature distributions are estimated using an inverse method. Moreover, the effects of sensor positions at the fin tip and a specific place in-between are considered on the base temperature estimation. A close agreement between the exact values and the estimated results is found, confirming the validity and accuracy of the proposed method. The results show that the ACGM is an accurate and stable method to determine the thermal boundary conditions in different non-Fourier fin problems.     

Neural Computation to Estimate Heat Flux in an Atmospheric Plasma Spray Process

Temperature is a key parameter in the thermal spray process and is a consequence of the heat flux experienced by the workpiece. This paper deals with the estimation of the heat flux transmitted to a workpiece from an atmospheric plasma spray torch during the preheating process often implemented in thermal spraying. An inverse heat conduction problem solution using a conjugate gradient method was considered to determine the heat flux starting from a known temperature distribution. Results from the later method were used to train an artificial neural network to discover correlations between selected processing parameters and heat flux.     

On the inverse transient heat-transfer problem in structural elements exposed to solar radiation

The inverse transient heat-transfer problem in walls, whether or not exposed to solar radiation, i.e. the estimation of the thermal properties of a wall if the transient temperature and/or heat flow fields are known, is interesting both from the theoretical and practical point of view. An attempt is made to analyse and solve this problem. Methods are developed for estimating the thermal properties of structural elements which are already parts of existing buildings, i.e. under real transient, non-periodic conditions. The finite-difference and experimental examples presented show that the thermal diffusivity, the thermal conductivity and the overall heat transfer coefficient may be estimated with very good accuracy by taking on-site temperature and heat flow measurements.     

Inverse problem of transpiration cooling for estimating wall heat flux by LTNE model and CGM method

In this work, a transient inverse problem of transpiration cooling is investigated in detail. The heat flux on the wall to be cooled is estimated by single point temperature measurement. The local thermal non-equilibrium (LTNE) model is utilized to describe the energy conservation of transpiration cooling process, and the conjugate gradient method (CGM) is extended to solve the inverse problem. The accuracy of the solutions of the inverse problem is examined through three given heat fluxes with given measurement errors. The examination shows that with the LTNE model and CGM, satisfactory solutions can be obtained. The influences of the variation in thermal properties, compressibility and the location of sensor on the accuracy of the solutions are analyzed. The analysis indicates that the variation in thermal properties and compressibility should be considered when a large temperature gradient exists, and the sensor location should be as close as possible to the hot wall. The inverse solutions obtained by the measurements of solid and fluid temperatures are compared. Through the comparison, it is found that using the solid temperature measurement as the input of the inverse problem is better than using the fluid temperature measurement.     

Inverse problem on the identification of temperature and thermal stresses in an FGM hollow cylinder by the surface displacements

A problem on the identification of time-dependent temperature on one of the limiting surfaces of a radially inhomogeneous hollow cylinder is formulated and solved under the temperature and radial displacement given on the other limiting surface. The analysis of temperature and thermal stress distribution in the cylinder is performed. The solution has been constructed by the reduction to an inverse thermoelasticity problem. By making use of the finite difference method, a stable solution algorithm is suggested for the analysis of inverse problem. The solution technique is verified numerically by making use of the solution to a relevant direct problem. It is shown that the proposed technique can be e?ciently used for the identification of a heat flux or unknown parameters (the surrounding temperature or the heat-exchange coe?cient) in the third-kind boundary conditions.     

Thermal Stresses in Multilayer Gun Barrel with Interlayer Thermal Contact Resistance

A hybrid numerical method of the Laplace transformation and the finite difference is applied to solve the transient heat transfer problem of a gun barrel, in which the interlayer thermal contact resistance between the steel cylinder and the chrome coating is taken into account in the boundary conditions. The general solutions of the governing equations are first solved in the transform domain. Then the inversion to the real domain is completed by the method of Fourier series technique. The transient distributions of temperature and thermal stresses for the gun barrel in the real domain are calculated numerically.     

The use of non-linear inverse problem and enthalpy method in GTAW process of aluminum

This work presents an alternative approach for the thermal analysis of the GTA (Gas Tungsten Arc) welding process on a 6065 T5 aluminum alloy. For this purpose, a C++ code was developed, based on a transient three-dimensional heat transfer model with phase change and mobile heat source. The thermal model took into account the dependence of thermal properties on temperature and heat losses by convection and radiation. The convection heat transfer coefficient and the emissivity were also considered variables with the temperature. To estimate the amount of heat delivered to the plate, the BFGS (Broydon–Fletcher–Goldfarb–Shanno) technique was used. Moreover, an analysis of the duration time of the electrode in positive (t+) and negative (t−) polarities was carried out. The methodology was validated by accomplishing lab controlled experiments. The aluminum samples lay on four conical head screws and submitted to a heat flux on one surface by the GTAW process torch. The torch displaces along the sample, thus simulating a real process. The numerical results presented low deviation when compared to the experimental results, which in turn, confirm the validation of the methodology for the study of the welding process presented.     

A spectral method for the estimation of a thermomechanical heat source from infrared temperature measurements

The inverse problem of 2D time-dependent heat source reconstruction is solved. The scientific objectives are the quantification of thermal effects associated to the mechanical deformation of materials during tensile tests. The experiment provides infrared measurements of the specimen’s surface temperature and the inverse algorithm aims at providing a volumic heat source that is free of errors due to heat diffusion. This algorithm is based on an analytical solution of the direct problem in the Laplace-Fourier domain. The solution proposed here is compared to a previously used method [1] based on an adjoint formulation and a regularization of Tikhonov type. This allows to check the validity of the results.     

MODELING OF TRANSPORT PHENOMENA IN A GAS METAL ARC WELDING PROCESS

A numerical study of three-dimensional heat transfer and fluid flow in a moving gas metal arc welding (GMAW) process is performed by considering various driving forces of fluid flow such as buoyancy, Lorentz force, and surface tension. The computation of the current density distribution and the resulting Lorentz force field is performed by solving the Maxwell equations numerically in the domain of the workpiece. The phase change process during melting and solidification is modeled using the enthalpy-porosity technique. Mass and energy transports by droplet transfer are also considered through a thermal analysis of the electrode. The droplet heat addition to the molten pool is considered to be a volumetric heat source distributed in an imaginary cylindrical cavity within the weld pool ("cavity" model). This nature of the heat source distributed due to the falling droplets takes into account the momentum and thermal energy of the droplets. The numerical model is able to capture the well-known "finger" penetration commonly observed in the GMAW process. Numerical prediction regarding the weld pool shape and size is compared with the corresponding experimental results, showing good qualitative agreement between the two. The weld pool geometry is also found to be dependant on some key parameters of welding, such as the torch speed and power input to the workpiece.     

Inverse problem of estimating the heat flux at the roller/workpiece interface during a rolling process

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown space-dependent heat flux at the roller/workpiece interface during rolling process from the knowledge of temperature measurements taken within the roller. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. The results show that an excellent estimation on the space-dependent heat flux can be obtained for the test cases considered in this study.     

Inverse Photoacoustic Technique for Parameter and Temperature Estimation in Tissues

The knowledge of tissues' properties and the noninvase monitoring of internal temperature are required in novel medical diagnostic and therapeutic techniques. For example, in the hyperthermia therapy of cancer, local heating must be accurately controlled in order to promote necrosis of the cancerous cells in thermoablation, or to induce apoptosis as an adjuvant treatment to chemotherapy or radiotherapy, without thermally affecting healthy cells. Photoacoustic imaging, also called optoacoustic imaging, is a new biomedical technique based on laser-generated ultrasound, which combines the high contrast of optical imaging with the high spatial resolution of ultrasound. Since parameters appearing in the mathematical formulation of the photoacoustic problem are temperature dependent, their estimation can be used as indirect temperature measurement. In the present work, sound speed, absorption coefficient, and a parameter that includes thermal expansion coefficient, laser energy density, and specific heat, are estimated through inverse analysis aiming at the identification of the tissue temperature. The forward problem is solved analytically using Laplace's transform, while the inverse problem is solved with a Markov chain Monte Carlo method within the Bayesian framework. Results obtained with simulated measurements reveal the capabilities of the proposed technique of parameter and temperature estimation.     

Monitoring of transient 3D temperature distribution and thermal stress in pressure elements based on the wall temperature measurement

Abstract

Two methods for monitoring the thermal stresses in pressure components of thermal power plants are presented. In the first method, the transient temperature distribution in the pressure component is determined by measuring the transient wall temperature at several points located on the outer insulated surface of the component. The transient temperature distribution in the pressure component, including the temperature of the inner surface is determined from the solution of the inverse heat conduction problem (IHCP). In the first method, there is no need to know the temperature of the fluid and the heat transfer coefficient. In the second method, thermal stresses in a pressure component with a complicated shape are computed using the finite element method (FEM) based on experimentally estimated fluid temperature and known heat transfer coefficient. A new thermometer with good dynamic properties has been developed and applied in practice, providing a much more accurate measurement of the temperature of the flowing fluid in comparison with standard thermometers. The heat transfer coefficient on the inner surface of a pressure element can be determined from the empirical relationships available in the literature. A numerical-experimental method of determination of the transient heat transfer coefficient based on the solution of the 3D-inverse heat conduction problem has also been proposed. The heat transfer coefficient on the internal surface of a pressure element is determined based on an experimentally determined local transient temperature distribution on the external surface of the element or the basis of wall temperature measurement at six points located near the internal surface if fluid temperature changes are fast. Examples of determining thermal and pressure stresses in the thick-walled horizontal superheater header and the horizontal header of the steam cooler in a power boiler with the use of real measurement data are presented.