Microelectronics Reliability xxx (2017) xxx–xxx ARTICLE IN PRESS MR-12476; No of Pages 10 Contents lists available at ScienceDirect Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel Partition-based approach to parametric dynamic compact thermal modeling Lorenzo Codecasa a, * , Vincenzo d’Alessandro b , Alessandro Magnani b , Niccolò Rinaldi b , Andre G. Metzger c , Robin Bornoff d , John Parry d a Department of Electronics, Information and Bioengineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan I-20133, Italy b Department of Electrical Engineering and Information Technology, University Federico II, via Claudio 21, Naples 80125, Italy c Skyworks Solutions Inc., Newbury Park, CA 91320, USA d Mentor Graphics, Mechanical Analysis Division, 81 Bridge Road, Hampton Court, Surrey, UK ARTICLE INFO Article history: Received 20 February 2017 Received in revised form 16 June 2017 Accepted 25 June 2017 Available online xxxx Keywords: Dynamic Compact Thermal Model (DCTM) Parametric Model-Order Reduction (MOR) Sparse matrices ABSTRACT This paper presents three procedures for the extraction of parametric Dynamic Compact Thermal Models (DCTMs) with controlled and user-chosen accuracy, namely, (i) a DCTM with dense matrices obtained by a direct conventional method, (ii) a partition-based approach leading to a sparse DCTM suited for heat conduc- tion problems suffering from a massive number of independent heat sources and/or parameters, for which extracting conventional dense DCTMs may be too resource-demanding or even unviable, and (iii) a novel algorithm that quickly translates a sparse DCTM into a dense one, which allows reducing the simulation time. The proposed methodologies are validated through the application to two state-of-the-art electronics systems. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction During the last decades, various approaches have been proposed in literature for constructing Dynamic Compact Thermal Models (DCTMs) of electronic components and packages [1–4]. In this con- text, the Authors have proposed effective Model-Order Reduction (MOR) methods for extracting DCTMs for the detailed heat conduc- tion models of electronic components and packages through the Multi-Point Moment Matching (MPMM) algorithm [5–8] and its suc- cessive improvements [9–21]. MOR-based DCTMs are an extremely efficient alternative to standard numerical simulations since they allow quickly and accurately evaluating both the key temperatures (point or surface and/or volumetric average) directly as output, and the whole space-time temperature evolution with a subsequent projection [17]. Prominent among the aforementioned models are the parametric DCTMs, which explicitly include the dependence on a subset of boundary conditions, material thermal properties, and geometry details, thereby enabling the fast study of the influence of parameter * Corresponding author. E-mail address: lorenzo.codecasa@polimi.it (L. Codecasa). variations in a wide range [11,15,19-21]. In this case, the heat con- duction problem to be reduced is preliminarily reformulated in an equivalent way by considering a reference domain with fixed geom- etry and variable material properties (i.e., a single mesh has to be constructed). Bases obtained with a suitable expansion are then introduced for approximating the temperature field, and the dis- cretized equations describing the parametric heat conduction are projected onto the space spanned by the previously-computed bases. A conventional parametric MOR technique operates on the whole spatial domain describing the electronic component/system, directly extracting dense (full) DCTMs [11,15]. In the case of low number of independent heat sources (IHSs) and parameters, the achieved DCTMs have small dimensions and can be numerically solved at neg- ligible computational cost. However, if the number of IHSs and/or parameters is high, the dimension of the bases can become large. This leads to the following drawbacks: the number of matrix elements defining the dense DCTMs quadratically increases with the dimension of the bases, thus reducing its advantages in terms of both extrac- tion and – to a lesser extent – simulation times; in severe cases, the orthonormalization of the bases and the projection of the discretized equations onto the space spanned by these bases may even become unviable due to RAM shortage. In [20], a novel MOR approach was proposed for overcoming the above drawbacks for the case of linear parametric DCTMs, while http://dx.doi.org/10.1016/j.microrel.2017.06.059 0026-2714/© 2017 Elsevier Ltd. All rights reserved. Please cite this article as: L. Codecasa et al., Partition-based approach to parametric dynamic compact thermal modeling, Microelectronics Reliability (2017), http://dx.doi.org/10.1016/j.microrel.2017.06.059