This article presents a finite-element-based domain decomposition method (DDM) for thermal analysis of 3-D large-scale and geometrically complicated integrated systems. The presented DDM decomposes the entire problem domain into many nonoverlapping subdomains, and a novel transmission condition (TC) based on a fictitious thermal contact resistance is introduced and then imposed through an interior penalty (IP) Galerkin procedure to enforce continuity at the interfaces between subdomains. The major advantage of the novel TC combined with the IP Galerkin procedure is that no extra unknowns are introduced and the system matrix arising from the finite element method will be symmetric and positive definite. Furthermore, the solving procedure of matrix equation adopts the preconditioned conjugate gradient method with a two-level preconditioner. To verify the convergence and accuracy, this article calculates the transient temperature profile of a brick and compares the computed numerical approximation with the exact solution. Meanwhile, the solving advantage of the proposed DDM is shown by comparing with Robin-TC DDM. Finally, numerical results for a 3-D integration package are shown to demonstrate the accuracy and superior performance of the proposed DDM.

中文翻译：

---

3D集成系统热分析的内部罚域分解方法

本文提出了一种基于有限元的区域分解方法（DDM），用于对3D大型且几何复杂的集成系统进行热分析。提出的DDM将整个问题域分解为许多不重叠的子域，并引入了基于虚拟热接触电阻的新型传输条件（TC），然后通过内部惩罚（IP）Galerkin程序强加了该条件，以在子域之间的接口处实现连续性。新型TC与IP Galerkin程序相结合的主要优势在于，不会引入额外的未知数，并且由有限元方法产生的系统矩阵将是对称且为正定的。此外，矩阵方程的求解过程采用带两级前置条件的前置共轭梯度法。为了验证收敛性和准确性，本文计算了砖的瞬态温度曲线，并将计算出的数值近似值与精确解进行了比较。同时，通过与Robin-TC DDM进行比较，显示了所提出DDM的解决优势。最后，显示了3-D集成软件包的数值结果，以证明所提出的DDM的准确性和优越的性能。