IC thermal analyzer for versatile 3-D structures using multigrid preconditioned krylov methods

Abstract

Thermal analysis is crucial for determining the propagation of heat and to track the formation of hot spots in advanced integrated circuit technologies. At the core of the thermal analysis for the integrated circuits is the numerical solution of the heat equation. Prior academic thermal analysis tools typically compute temperature by applying finite difference methods on uniform grids with time integration methods having fixed time step size. Additionally, the linear systems arising from the discretized heat equation are solved using direct methods based on matrix factorizations. Direct methods, however, do not scale well as the problem size increases. Moreover, most of the tools support only 2-D or a limited number of 3-D technologies. To address these issues, this paper presents a novel thermal analyzer with the ability to model both 2-D and 3-D circuit technologies. The analyzer solves the heat equation using the finite element method for the spatial discretization coupled with implicit time integration methods for advancing the solution in time. It also offers fully adaptive spatio-temporal refinement features for improved accuracy and computational efficiency. The resulting linear systems are solved by a multigrid preconditioned Krylov subspace iterative method, which gives superior performance for 3-D transient analyses. The analyzer is shown to accurately capture the propagation of heat in both the horizontal and vertical directions of integrated systems.

The MTA: An Advanced and Versatile Thermal Simulator for Integrated Systems

“…The power density dissipated in the active components of the circuit is supplied from an external power trace file that is generated either by architecture level simulators with a power modeling framework such as [24] or commercial power analysis tools. Further details on the XML and power trace files are given in [25]. In the remainder of this section, the governing heat equation is reviewed, followed by the details of the mesh generation process, the spatial discretization, and the time integration methods.…”

“…The (nonlinear) mass and stiffness matrices are denoted by M (u h ), K(u h ) ∈ R n×n . For further details of the discretization process see [25]. The length n (dimension) of the above vectors (matrices) is equal to the number of unknown nodal temperatures and is commonly referred to as the number of degrees of freedom (DOF).…”

“…The numerical solution of the heat equation (1) in the current version of the simulator is implemented with the open-source finite element library deal.II [36], [37] for improved computational speed and parallel efficiency. As a result of this new implementation, the MTA supports larger and faster thermal simulations compared to [25].…”

“…As described in [25], the proposed solution methodology allows the execution time to roughly scale linearly in the number of DOFs. This scaling trend can be seen for each phase of the simulation in Table X, which reports the breakdown of times (in secs) for a set of increasingly larger transient simulations conducted in mode M1 with Q1 elements and a fixed time step size ∆t = 0.01 sec.…”

A Preconditioned Iterative Approach for Efficient Full Chip Thermal Analysis on Massively Parallel Platforms

“…Furthermore, in [11,12], the FDM approach and the RCequivalent were used along with modeling of the fluids for micro-cooling 3D structures. In [13], FEM was adopted for 2D and 3D geometries along with a multigrid preconditioning method and automatic mesh generation for chip geometries. Finally, Green's functions were used in [14] with discrete cosine transform and its inversion in order to accelerate the numerical computation of the homogeneous and inhomogeneous solution.…”