基于通量的热问题弱公式,发展了毕奥特变分原理

IF 5.7 1区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY
Ali Haydar , Laura Galuppi , Gianni Royer-Carfagni
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引用次数: 0

摘要

我们提出了固体体瞬态热方程的弱形式,即热位移矢量场随时间的空间变化,其时间导数为热通量。这发展了最初由 Biot 提出的变分原理,因为傅里叶定律被嵌入为整体约束条件,而能量守恒则来自变分(Biot 则相反)。这是一种简洁的表述,因为只有热位移出现在变分方程中,而毕奥的表述还涉及未知温度场:傅里叶定律仅用于事后恢复温度。由于热位移通常比温度场更有规律,因此在材料不均匀、不均匀辐射、热冲击等问题中,热位移是一个自然变量。在考虑到辐射和对流的边界条件下,与 Biot 的三维分析设置进行了比较。对于一维情况,提出了与具有粘性约束的弹性杆平衡的力学类比。变分方程是在有限元代码中实现的。在涉及高温梯度的基准问题上进行的数值实验证实了所提出的方法在许多结构问题上的效率。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A neat flux-based weak formulation for thermal problems which develops Biot’s variational principle

We propose a weak form of the transient heat equations for solid bodies, as a time-dependent spatial variation of the heat displacement vector field, whose time derivative is the heat flux. This develops the variational principle originally proposed by Biot, inasmuch Fourier’s law is embedded as a holonomic constraint, while energy conservation results from the variation (the vice-versa from Biot). This is a neat formulation because only the heat displacement appears in the variational equations, whereas Biot’s form also involved the unknown temperature field: Fourier’s law is used only a posteriori to recover the temperature. Since the heat displacement is generally more regular than the temperature field, it represents a natural variable in problems with material inhomogeneities, uneven radiation, thermal shocks. The three-dimensional analytical set-up is presented in comparison with Biot’s, for boundary conditions accounting for radiation and convection. A mechanical analogy with the equilibrium of an elastic bar with viscous constraints is suggested for the one-dimensional case. The variational equations are implemented in a finite element code. Numerical experiments on benchmark problems, involving high temperature gradients, confirm the efficiency of the proposed approach in many structural problems.

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来源期刊
International Journal of Engineering Science
International Journal of Engineering Science 工程技术-工程:综合
CiteScore
11.80
自引率
16.70%
发文量
86
审稿时长
45 days
期刊介绍: The International Journal of Engineering Science is not limited to a specific aspect of science and engineering but is instead devoted to a wide range of subfields in the engineering sciences. While it encourages a broad spectrum of contribution in the engineering sciences, its core interest lies in issues concerning material modeling and response. Articles of interdisciplinary nature are particularly welcome. The primary goal of the new editors is to maintain high quality of publications. There will be a commitment to expediting the time taken for the publication of the papers. The articles that are sent for reviews will have names of the authors deleted with a view towards enhancing the objectivity and fairness of the review process. Articles that are devoted to the purely mathematical aspects without a discussion of the physical implications of the results or the consideration of specific examples are discouraged. Articles concerning material science should not be limited merely to a description and recording of observations but should contain theoretical or quantitative discussion of the results.
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