Inverse finite element formulations for transient heat conduction problems

Transient heat conduction problems are normally simulated by the conventional consistent and lumped finite element methods. The discretization error and the physical reality violation in such problems are noticeable and unwanted responses are observed in the results when using the consistent formulations. Although in utilizing the lumped formulations, the violation of physical reality becomes reduced; however, emerging the discretization error would also become obvious to the degree of being quantifiable. In using the inverse finite element method without considering the element shape functions, the element matrices will be obtained by minimizing the governing equation and its generalized discretized corresponding formula. The results obtained by using this method indicates that the reduction in both the discretization error as well as the violation of physical reality would be realized.     

瞬态热传导问题的加权最小二乘无网格法

加权最小二乘无网格法是一种基于节点信息的纯无网格法,该方法使用最小二乘法建立系统的变分原理,通过移动最小二乘法构造近似函数,控制方程在节点处的残量使用最小二乘法予以消除,边界条件通过罚函数法引入。本文推导了瞬态热传导问题的加权最小二乘无网格计算格式,编制了相应的计算程序,算例结果表明,该方法具有精度高、前后处理简单的优点,是一种高效的的新型无网格法。     

二阶非定常热传导反问题的多宗量辨识

引入Bregman距离构造同伦函数,建立了二阶非定常多宗量热传导反问题的一种求解模式,可对导热系数和边界条件等宗量进行识别。时域上采用精细算法,建立了便于敏度分析的有限元正/反演模型,对各宗量进行有效的组合识别。对信息测量误差和初值选取作了初步探讨,数值验证取得了满意的结果。     

A space marching method for multidimensional transient inverse heat conduction problems

A new method for the solution of multidimensional heat conduction problems is formulated. The developed space marching method allows to determine quickly and exactly unsteady temperature distributions in the construction elements of irregular geometry. The method is especially appropriate for determining transient temperature distribution in thick-wall pressure components based on temperature measurements at the outer surface. Two examples are included to demonstrate the capabilities of the new approach. Received on 28 October 1998     

PRECISE INTEGRAL ALGORITHM BASED SOLUTION FOR TRANSIENT INVERSE HEAT CONDUCTION PROBLEMS WITH MULTI-VARIABLES

IntroductionIHCPs (InverseHeatConductionProblems)arecloselyassociatedwithmanyengineeringaspects,andwelldocumentedintheliteratures,coveringtheidentificationsofthermalparameters[1,2 ],boundaryshapes[3],boundaryconditions[4 ]andsource_relatedterms[5 ,6 ]etc .Howeveritseemsthatonlylittleworkisdirectlyconcernedwithmulti_variablesidentificationsbyauthors’knowledge.Tsengetal.presentedanapproachtodeterminingtwokindsofvariables[7],butonlygavefewnumericalexamplestodeterminethemsimultaneously .Thesol…     

基于物理信息神经网络的功能梯度材料稳态/瞬态热传导分析

采用物理信息神经网络PINN(Physics-informed Neural Networks)求解稳态和瞬态功能梯度材料(FGMs)热传导问题。该方法利用控制方程、边界及初始条件的残差构造损失函数,在无任何响应数据的情况下得到了更具泛化能力的神经网络模型,同时避免了传统数值方法在求解计算力学问题时所需的微分、积分公式推导以及繁重的建模和划分网格等前处理工作。本文探究了PINN及其域分解的扩展物理信息神经网络XPINN(eXtended Physics-informed Neural Networks)在求解稳态和瞬态FGMs热传导问题时的适用性,讨论了网络结构对预测结果的影响。研究结果表明,PINN/XPINN在解决几何复杂的稳态和瞬态FGMs热传导问题时仍具有较高的可靠性和简洁的求解流程,同时,为极端环境下求解复杂多场耦合和夹杂等问题提供了新思路。     

Critical parameters for thermal conduction equations

Thermal stability of two models, whose thermal conductivity is a function of temperature, is investigated. Studies are made for the cylindrical geometry for both exponetial and power forms for the temperature dependence of conductivity. The maximum value of a certain parameter, δ_cr, defines criticality and this is the condition for explosion. The purpose of this paper is to ascertain the way in which explosion is affected by boundary conditions and how changes in the values of a certain constant, β, affect the critical parameter δ_cr.     

New analytic solutions to 2D transient heat conduction problems with/without heat sources in the symplectic space

The two-dimensional (2D) transient heat conduction problems with/without heat sources in a rectangular domain under different combinations of temperature and heat flux boundary conditions are studied by a novel symplectic superposition method (SSM). The solution process is within the Hamiltonian system framework such that the mathematical procedures in the symplectic space can be implemented, which provides an exceptional direct rigorous derivation without any assumptions or predetermination of the solution forms compared with the conventional inverse/semi-inverse methods. The distinctive advantage of the SSM offers an access to new analytic heat conduction solutions. The results obtained by the SSM agree well with those obtained from the finite element method (FEM), which confirms the accuracy of the SSM.     

Stationary and transient heat conduction in a non homogeneous material

An apparent thermal conductivity for inhomogeneous materials is widely used. In this paper it is demonstrated that the apparent thermal conductivity for stationary heat conduction is not sufficient to describe the transient heat response of an inhomogeneous medium. In the geometry we used the heat transfer is estimated too high when the stationary thermal conductivity is employed. A numerical solution of the equation of thermal diffusion has been used to check several approximations. For short and for long times a separate approximate analytic expression can be used.

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Stationäre und Übergangswärmeleitung in nichthomogenen Materialien

Zusammenfassung Oft wird eine scheinbare Wärmeleitfähigkeit für inhomogene Materialien verwendet. In diesem Artikel wird gezeigt, daß es im allgemeinen nicht genügt, die instationäre Übergangswärmeleitung mit der stationären Wärmeleitfähigkeit zu beschreiben. In unserer Geometrie gibt dies eine Überschätzung der Wärmeleitung. Ein numerisches Modell für die Wärmediffusions-gleichung ist entwickelt worden, um mehrere Schätzungen der scheinbaren Wärmeleitfähigkeit zu kontrollieren. Für kurze und lange Zeiten stehen unterschiedliche analytische Beziehungen zur Verfügung.

     

Initial and boundary value problems of hyperbolic heat conduction

This is a study on the initial and boundary value problem of a symmetric hyperbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density e and a balance equation for the heat flux , where e and are the four basic fields of the theory. The initial and boundary value problem that uses exclusively prescribed boundary data for the energy density e is solved by a new kinetic approach that was introduced and evaluated by Dreyer and Kunik in [1], [2] and Pertame [3]. This method includes the formation of shock fronts and the broadening of heat pulses. These effects cannot be observed in the linearized theory, as it is described in [4]. The kinetic representations of the initial and boundary value problem reveal a peculiar phenomenon. To the solution there contribute integrals containing the initial fields as well as integrals that need knowledge on energy and heat flux at a boundary. However, only one of these quantities can be controlled in an experiment. When this ambiguity is removed by continuity conditions, it turns out that after some very short time the energy density and heat flux are related to the initial data according to the Rankine Hugoniot relation. Received October 6, 1998     

Approximate solution of non-linear transient heat conduction in cylindrical geometry

The applicability of the Heat Balance Integral method (HBI) using hybrid profiles (HP _n ) for heat diffusion problems in cylindrical geometry is studied. Three test problems are taken to validate the method, namely, TP1 — long hollow cylinder, inner boundary is supplied with constant heat flux and the outer boundary is at infinity, TP2 — the same geometry with the inner boundary at constant temperature, TP3 — the same geometry with convection at the boundary. It is found that HBI in combination with HP _n gives much better solutions as compared to the HBI-polynomial (P _n ) combinations. This method is applied to non-linear cases like temperature dependent thermal conductivity, boiling or condensation at the boundary. For assessment of accuracy Root Mean Square Residual (E _rms) is defined. The residual minimization technique is used to select the parameter in the HP _n profile. It is shown that this procedure yields solutions which are in excellent agreement with the available exact solutions in the case of the three test problems. In the case of nonlinear problems, the value ofE _rms which is comparable to the values in the test cases, shows that the HBI-HP _n is a very good combination for all problems.In dieser Studie wurde die Verwendbarkeit des Wärmebilanz-Integrationsverfahrens (HBI) unter Benutzung von hybriden Profilen (HP _n ) für Wärmeausbreitungsprobleme in zylindrischen Geometrien untersucht. Um das Verfahren zu veranschaulichen wurden drei Testprobleme genommen; wobei für alle drei Fälle die gleiche Geometrie benützt wurde, und zwar ein langer, ausgehöhlter Zylinder. Beim TP-1 ist die innere Grenze mit konstantem Wärmestrom versorgt und die äußere Grenze ist im Unendlichen angenommen worden. Beim TP-2 herrscht an der inneren Grenze eine konstante Temperatur. Beim TP-3 herrscht Konvektion an den Grenzen. Hierbei wurde festgestellt, daß HBI in Verbindung mit HP _n sehr viel bessere Lösungen liefert, als die HBI-Polynom-Kombinationen (P _n ). Dieses Verfahren ist für nicht lineare Fälle, wie temperaturabhängige Wärmeleitfähigkeit, Sieden oder Kondensation and den Grenzen verwendet worden. Für die Abschätzung der Genauigkeit ist das restliche quadratische Mittel definiert worden. Die Residuen-Minimisierungstechnik ist für die Auswahl des Parameters \ benützt worden. Es ist gezeigt worden, daß dieses Verfahren Lösungen liefert, welche mit den benützten genauen Lösungen der drei Testfälle hervorragend übereinstimmen. Im Fall der nicht linearen Probleme zeigt der Wert vonE _rms, der mit den Testfällen vergleichbar ist, daß die HBI-HP _n eine sehr gute Kombination für alle Probleme ist.     

Approximate solution of nonlinear transient heat conduction in one dimension

Three approximate methods, viz. the Heat Balance Integral method (HBI), the Modified Heat Balance Integral method (MHBI) and the Double Integral Method (DIM) in combination with hybrid profiles (HP - consisting of an exponential and polynomial) are evaluated for relative merits in solving the onedimensional heat diffusion equation. Applications to two linear test problems (TP1 - semi-infinite solid with constant heat flux at its boundary, TP 2 - semi-infinite solid with step change in its surface temperature) yield the HP-MHBI as the most desirable combination. Applications to nonlinear problems show that these combinations are ideally suited for obtaing reliable and accurate appoximate solutions.

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Näherungslösung von eindimensionaler nicht linearer transienter Wärmeleitung

Zusammenfassung Es wurden drei Näherungsmethoden, die Heat Balance Integral-Methode (HBI), die Modified Heat Balance IntegralMethode (MHBI) und die Double Integral-Methode (DIM) in Verbindung mit hybriden Profilen (HP - bestehend aus Exponentialund Polynomfunktionen) ausgewählt, um die eindimensionale Wärmediffusionsgleichung zu lösen. Anwendungen auf zwei lineare Testprobleme (TP 1 - halb-unendlicher Festkörper mit konstantem Wärmestrom an dessen Grenze, TP2 - halb-unendlicher Festkörper mit sprunghaftem Wechsel der Oberflächentemperatur) ergaben, daß die HP-MHBI-Methode die erwünschte Kombination darstellt. Anwendungen auf nichtlineare Probleme zeigen, daß diese Kombination ideal für das Erzielen zuverlässiger und aussagekräftiger Näherungslösungen ist.

     

非均质材料等效瞬态热传导时域分析

基于时域自适应算法,结合均匀化技术和有限元方法,从而提出一种瞬态温度场求解模型,用来预测非均质材料等效性能并评估宏观温度场的等效行为.在整个计算中,通过自适应算法保证计算精度和稳定性,避免时间段尺寸变化可能引起的计算误差.在数值算例中,等效分析的结果与利用ANSYS求解的非均质有限元解比较,从计算效率和计算精度综合效果而言,结果是令人满意的.     

On the thermal rectification factor in steady heat conduction

Thermal rectification in heat conduction problems has been extensively studied in planar slabs. Here we consider the rectification problem in planar, cylindrical and spherical geometries involving two layers one of which has a temperature variable heat conductivity. The rectification factor is analytically calculated. It is shown that a maximum theoretical value of 1.618 is obtained.     

Propagation of blast waves with exponential heat release and internal heat conduction and thermal radiation

The problem of reactive blast waves in a combustible gas mixture, where the heat release at the detonation front decays exponentially with the distance from the center, is analyzed. The central theme of the paper is on the propagation of reactive blast into a uniform, quiescent, counterpressure atmosphere of a perfect gas with constant specific heats. The limiting cases of Chapman-Jouguet detonation waves are considered in the phenomenon of point explosion. In order to deal with this problem, the governing equations including thermal radiation and heat conduction were solved by the method of characteristics using a problem-specific grid and a series expansion as start solution. Numerical results for the distribution of the gas-dynamic parameters inside the flow field are shown and discussed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Estimate of the transient conduction of heat in materials with linear thermal properties based on the solution for constant properties

A method of analysis is described which yields quasianalytical solutions for one and multidimensional unsteady heat conduction problems with linearly dependent thermal properties, such as thermal conductivity and volumetric specific heat. The method accomodates rather general thermal boundary conditions including arbitrary variations in surface temperature or in surface heat flux or a convective exchange with a fluid having even varying temperature. Once the solution for the identical problem but with constant properties has been developed, its practical realization is rather direct, being facilitated by a reduced number of iterations. The four applied examples given in this work show that a wide variety of nonlinear heat conduction problems can be tackled by this procedure without much difficulty. These simple solutions compare favorably with more laborious results reported in the archival heat transfer literature.

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Berechnung nichtstationärer Wärmeleitvorgänge mit linear temperaturabhängigen Stoffwerten aus der Lösung für konstante Stoffwerte

Zusammenfassung Es werden quasi-analytische Lösungen für ein- und mehrdimensionale nichtstationäre Wärmeleitprobleme mit linear temperaturabhängigen Stoffwerten, wie Wärmeleitfähigkeit und volumetrische Wärmekapazität, mitgeteilt. Die Methode gilt für recht allgemeine Randbedingungen wie beliebige Veränderungen der Oberflächentemperatur, der Wärmestromdichte oder auch konvektiven Wärmeaustausch mit veränderlicher Fluidtemperatur. Ist die Lösung für das identische Problem mit konstanten Stoffwerten bekannt, kann die Methode direkt mit einer begrenzten Zahl von Iterationen angewandt werden. Die vier hier mitgeteilten Beispiele zeigen, daß eine große Zahl nichtlinearer Wärmeleitprobleme auf diese Weise ohne Schwierigkeit angepackt werden können. Die einfachen Lösungen stimmen befriedigend mit komplizierteren Ergebnissen aus der Literatur überein.

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Nomenclature a side of square bar - B _i0 reference Biot number,hR/k₀ - B _i0 ^T transformed Biot number, equation (16) - c geometric parameter, equation (8) - h convective coefficient - k thermal conductivity - k ₀ value ofk atT ₀ - K dimensionless thermal conductivity,k/k ₀ - K _i value ofK at _i - K _i+1 value ofK at _i+1 - m _k slope of theK- line, equation (3) - m _s slope of theS- line, equation (4) - R characteristic length - s volumetric specific heat - s ₀ value of s at T₀ - S dimensionless volumetric specific heat, s/s₀ - S _i value ofS at _i - S _i+1 value of S at _i+1 - t time - T temperature - T ₀ reference temperature - x, y cartesian coordinates - X, Y dimensionless cartesian coordinates,x/a andy/a - thermal diffusivity - _k transformed time, equation (11) - _s transformed time, equation (37) - _k dimensionless time for variable conductivity, equation (8) - _s dimensionless time for variable specific heat, equation (34) - dimensionless temperature,T/T ₀ - dimensionless coordinate,r/R - ₀ value of at T₀ - _i lower value of the interval (_i, _i+1) - _i+1 upper value of the interval (_i, _i+1