Physics-based models for thermal simulation that involve numerical solution of the heat equation are well placed to accurately capture the heterogeneity of materials and structures in modern 3-D integrated circuits (ICs). The introduction of non-linear effects such as leakage power significantly improves their accuracy. However, this non-linearity increases considerably the complexity and computational time of the analysis. In this paper, we introduce a linearised thermal model by demonstrating that the weak temperature dependence of the specific heat and the thermal conductivity of IC related materials has only minor effect to computed temperature profiles. Thus, these parameters can be considered constant for the operating temperature ranges of modern ICs. The non-linearity in leakage power is approximated by a piecewise linear least square fit and the resulting model is linearised by exact Newton's method. The method is applied to transient thermal analysis with adaptive time step selection, where we demonstrate the importance of applying Newton corrections to obtain the right time step size selection. The resulting method is typically 2 - 3× faster than a full non-linear method with a global relative error of less than 1%.

The verification of integrated Circuits (ICs) in deep submicron technologies requires that all mutual inductive effects are taken into account to properly validate the performance and reliable operation of the chip. However, the inclusion of all mutual inductive couplings results in a fully dense inductance matrix that renders the circuit simulation computationally prohibitive. In this paper, we present efficient techniques for the solution of the linear systems arising in transient analysis of large mutually inductive circuits. These techniques involve the compression of the dense inductance matrix block by low-rank products in hierarchical matrix format, as well as the development of a Schur-complement preconditioner for the iterative solution of the transient linear system (which comprises sparse blocks alongside the dense inductance block). Experimental results indicate that substantial compression rates of the inductance matrix can be achieved without compromising accuracy, along with considerable reduction in iteration counts and execution time of iterative solution methods.

Fast and accurate thermal analysis is crucial for determining the propagation of heat and tracking the formation of hotspots in integrated circuits (ICs). Existing academic thermal analysis tools primarily use compact models to accelerate thermal simulations but are limited to linear problems on relatively simple circuit geometries. The Manchester Thermal Analyzer (MTA) is a comprehensive tool that allows for fast and highly accurate linear and nonlinear thermal simulations of complex physical structures including the IC, the package, and the heatsink. The MTA is targeted for 2.5/3-D IC designs but also handles standard planar ICs. The MTA discretizes the heat equation in space using the finite element method and performs the time integration with unconditionally stable implicit time stepping methods. To improve the computational efficiency without sacrificing accuracy, the MTA features adaptive spatiotemporal refinement. The large-scale linear systems that arise during the simulation are solved with fast preconditioned Krylov subspace methods. The MTA supports thermal analysis of realistic integrated systems and surpasses the computational capabilities and performance of existing academic thermal simulators. For example, the simulation of a processor in a package attached to a heat sink, modeled by a computational grid consisting of over 3 million nodes, takes less than 3 min. The MTA is fully parallel and publicly available.1

Thermal analysis of integrated circuits (IC) is a high performance computing problem because the nanoscale spatiotemporal features of the problem result in a large discrete problem. In previous works, compact models of ICs were introduced to speed up the modeling process. However, such methods have limited accuracy as they approximate the underlying physics. They are also ill-suited to simulate the thermal characteristics of an IC at the cell-level. The finite element method (FEM) is an appropriate computational technique for providing both fast and accurate thermal analyses. Considering that the number of cells in modern ICs is on the order of millions, thermal analysis at this abstraction level is a formidable task. Consequently, handling the computational meshes and computing thermal profiles of an IC at the cell-level requires substantial computing power. In order to provide accurate cell-level thermal simulations at a lower computational cost, this work introduces advanced techniques that judiciously trade off mesh granularity with simulation accuracy which allows fast analysis of cell-level floorplans. The proposed cell-homogenization techniques start with a flat cell-level floorplan and a related power trace and produce reduced order meshes that accelerate thermal simulations with a negligible loss in accuracy. Results show that the proposed techniques achieve up to a 90% reduction in the number of nodes in the mesh with less than 5% error in the temperature compared to the full scale mesh. The simulation time is also reduced by an order of magnitude.

Thermal analysis is crucial for determining the propagation of heat and tracking the formation of hot spots in advanced integrated circuit technologies. At the core of the thermal analysis for 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.