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 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.

中文翻译：

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一种针对热问题的基于通量的弱公式，它发展了毕奥变分原理


我们提出了固体瞬态热方程的弱形式，作为热位移矢量场的时间相关空间变化，其时间导数是热通量。这发展了比奥最初提出的变分原理，因为傅立叶定律被嵌入为完整约束，而能量守恒则由变分产生（比奥反之亦然）。这是一个简洁的公式，因为变分方程中只出现热位移，而毕奥形式还涉及未知的温度场：傅立叶定律仅用于恢复温度。由于热位移通常比温度场更规则，因此它代表了材料不均匀性、辐射不均匀、热冲击等问题的自然变量。三维分析装置与 Biot 的分析装置进行了比较，用于考虑辐射和对流的边界条件。对于一维情况，建议使用具有粘性约束的弹性杆的平衡进行机械类比。变分方程以有限元代码实现。涉及高温梯度的基准问题的数值实验证实了所提出的方法在许多结构问题中的效率。