Abstract In order to efficiently solve nonlinear transient heat conduction problem, a method combining the radial integration boundary element method (RIBEM) and the proper orthogonal decomposition (POD) is proposed to establish the nonlinear reduced-order model, and the implementation of the reduced-order model makes fast numerical simulation possible in engineering field. Transient heat conduction problems with temperature-dependent and temperature-independent thermal conductivity are firstly solved by the RIBEM respectively, and two types of snapshots are composed of these solved transient temperature fields. Subsequently, the POD is applied in the analysis of the snapshots, and then a set of optimal orthogonal basis can be obtained. In the procedure of establishing the reduced-order model, we need to redistribute the nonlinear equations. The strategy is to separate the boundary nodes defined as Dirichlet condition, and then the orthogonal basis is used to derive a reduced-order expression for the remaining unknowns. Therefore, the degree of freedom that needs to be solved can be greatly reduced. Numerical examples show that different reduced-order models based on nonlinear basis and linear basis are all consistent with the full-order model. What is more, the computational efficiency is greatly improved due to the introduction of the reduced-order model.

中文翻译：

---

非线性瞬态热传导问题中基于适当正交分解的径向积分边界元法快速降阶模型

摘要 为有效解决非线性瞬态热传导问题，提出了一种将径向积分边界元法（RIBEM）与适当正交分解（POD）相结合的方法，建立非线性降阶模型，并实现了降阶模型。阶模型使工程领域的快速数值模拟成为可能。RIBEM首先分别解决了热导率与温度相关和温度无关的瞬态热传导问题，并由这些求解的瞬态温度场组成了两种类型的快照。随后，将POD应用于快照的分析，从而得到一组最优正交基。在建立降阶模型的过程中，我们需要重新分配非线性方程组。该策略是将定义为 Dirichlet 条件的边界节点分开，然后使用正交基导出剩余未知量的降阶表达式。因此，需要求解的自由度可以大大降低。数值算例表明，基于非线性基和线性基的不同降阶模型均与全阶模型一致。更重要的是，由于引入了降阶模型，计算效率大大提高。数值算例表明，基于非线性基和线性基的不同降阶模型均与全阶模型一致。更重要的是，由于引入了降阶模型，计算效率大大提高。数值算例表明，基于非线性基和线性基的不同降阶模型均与全阶模型一致。更重要的是，由于引入了降阶模型，计算效率大大提高。