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Des. Autom. Electron. Syst."],"published-print":{"date-parts":[[2015,9,28]]},"abstract":"<jats:p>\n            In this article, we propose an efficient finite-element-based (FE-based) method for both steady and transient thermal analyses of high-performance integrated circuits based on the hierarchical matrix (\n            <jats:italic>H<\/jats:italic>\n            -matrix) representation.\n            <jats:italic>H<\/jats:italic>\n            -matrix has been shown to provide a data-sparse way to approximate the matrices and their inverses with almost linear-space and time complexities. In this work, we apply the\n            <jats:italic>H<\/jats:italic>\n            -matrix concept for solving heating diffusion problems modeled by parabolic partial differential equations (PDEs) based on the finite element method. We show that the matrix from a FE-based steady and transient thermal analysis can be represented by\n            <jats:italic>H<\/jats:italic>\n            -matrix without any approximation, and its inverse and Cholesky factors can be evaluated by\n            <jats:italic>H<\/jats:italic>\n            -matrix with controlled accuracy. We then show and prove that the memory and time complexities of the solver are bounded by\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>k<\/jats:italic>\n            <jats:sub>1<\/jats:sub>\n            <jats:italic>N<\/jats:italic>\n            log\n            <jats:italic>N<\/jats:italic>\n            ) and\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>k<\/jats:italic>\n            <jats:sub>1<\/jats:sub>\n            <jats:sup>2<\/jats:sup>\n            <jats:italic>N<\/jats:italic>\n            log\n            <jats:sup>2<\/jats:sup>\n            <jats:italic>N<\/jats:italic>\n            ), respectively, where\n            <jats:italic>k<\/jats:italic>\n            <jats:sub>1<\/jats:sub>\n            is a small quantity determined by accuracy requirements and\n            <jats:italic>N<\/jats:italic>\n            is the number of unknowns in the system. The comparison with existing product-quality LU solvers, CSPARSE and UMFPACK, on a number of 3D IC thermal matrices, shows that the new method is much more memory efficient than these methods, which however prevents CPU time comparison with those methods on large examples. But the proposed method can solve all the given thermal circuits with decent scalabilities, which shows good agreement with the predicted theoretical results.\n          <\/jats:p>","DOI":"10.1145\/2714563","type":"journal-article","created":{"date-parts":[[2015,9,29]],"date-time":"2015-09-29T19:22:29Z","timestamp":1443554549000},"page":"1-25","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["<i>H<\/i>\n            -Matrix-Based Finite-Element-Based Thermal Analysis for 3D ICs"],"prefix":"10.1145","volume":"20","author":[{"given":"Hai-Bao","family":"Chen","sequence":"first","affiliation":[{"name":"Shanghai Jiao Tong University"}]},{"given":"Ying-Chi","family":"Li","sequence":"additional","affiliation":[{"name":"The University of Hong Kong"}]},{"given":"Sheldon X.-D.","family":"Tan","sequence":"additional","affiliation":[{"name":"University of California, Riverside"}]},{"given":"Xin","family":"Huang","sequence":"additional","affiliation":[{"name":"University of California, Riverside"}]},{"given":"Hai","family":"Wang","sequence":"additional","affiliation":[{"name":"University of Electronic Science and Technology of China"}]},{"given":"Ngai","family":"Wong","sequence":"additional","affiliation":[{"name":"The University of Hong Kong"}]}],"member":"320","published-online":{"date-parts":[[2015,9,28]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-002-0445-6"},{"volume-title":"Fundamentals of Heat and Mass Transfer","author":"Bergman T.","key":"e_1_2_1_2_1","unstructured":"T. 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