A computational cost-effective algorithm is proposed to solve steady-state heat conduction problems with uncertain thermal conductivity which appears locally at some part of structures. Such local uncertainty is assumed to be induced by a crack or notch, and modelled by probability or interval models. The deterministic steady-state heat conduction problem is formulated by the Scaled Boundary Finite Element Method (SBFEM), facilitating to treat heat flux singularity around the crack conveniently and efficiently, and generate SBFE mesh flexibly, and the uncertain analysis is conducted via the Monte Carlo simulation (MCS). To alleviate the computational burden of repeated updating deterministic solutions in MCS, a Woodbury formula-based tactic is presented, such that the update of inverse heat conduction matrix can be carried out with a small solution scale, and substantial benefit can be achieved in term of computational cost. The proposed algorithm is applicable for the cases with single and multiple local uncertainties. Numerical examples are employed to illustrate the effectiveness and efficiency.

中文翻译：

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一种解决具有局部不确定性的稳态热传导问题的有效数值算法

提出了一种计算成本有效的算法来解决结构某些部分局部出现的导热系数不确定的稳态热传导问题。假设这种局部不确定性是由裂纹或缺口引起的，并通过概率或区间模型进行建模。采用尺度边界有限元法(SBFEM)来表述确定性稳态热传导问题，便于方便高效地处理裂纹周围的热流奇异性，并灵活地生成SBFE网格，并通过蒙特卡罗进行不确定性分析模拟（MCS）。为了减轻MCS中重复更新确定性解的计算负担，提出了一种基于Woodbury公式的策略，使得逆热传导矩阵的更新可以在较小的解规模下进行，并且可以在以下方面获得显着的好处：计算成本。该算法适用于具有单个和多个局部不确定性的情况。采用数值例子来说明有效性和效率。