IC thermal simulation and modeling via efficient multigrid-based approaches
Citations
82 citations
Cites background from "IC thermal simulation and modeling ..."
...There has been abundant work on designtime full-chip thermal models, such as [11, 16, 17, 18]....
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75 citations
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41 citations
Cites methods from "IC thermal simulation and modeling ..."
...Another enhanced FDM-based IC thermal analyzer is proposed in [9] based on the multigrid technique for a large sparse system of linear equations, which is suitable for a large number of meshes in detailed thermal analysis....
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40 citations
Cites background or methods from "IC thermal simulation and modeling ..."
...Moreover, when calculating the steady state temperature, the input power profile is usually set to the steady power profile (the average power profile for a very long time period estimation) [1], [2], [4], [6], [7], therefore, can be reasonably viewed as a step function with the magnitude being equal to its average power density for a long time period....
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...[4] applied the multi-grid method to speed up the convergence rate of iterative methods, and developed an order reduction scheme to save the runtime of dynamic thermal simulation....
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...However, the dynamic thermal analysis is also necessary while performing the dynamic thermal management and runtime thermal analysis [1], [4], [7]....
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...Based on this heat conduction mechanism, the temperature of die can be governed by the following heat transfer equations [2], [4], [5], [7]:...
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References
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2,505 citations
"IC thermal simulation and modeling ..." refers background or methods in this paper
...They are optimal in a sense that the time complexity is linear in the number of unknowns (h-independent convergence) with a small constant factor [ 16 ], [17]....
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...As a result, IC power density has been rapidly increasing with Moore’s law, and is now projected as a potential show stopper for future performance improvements [ 1 ]....
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...The standard MG V-cycles correspond to m =1 and W-cycles to m =2 . More details on MG methods and definitions can be found in [ 16 ], [17], [20], and [21]....
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...The multigrid (MG) method, as a multilevel iterative scheme, has become increasingly popular for solving a class of PDE problems due to its superior efficiency in the computational fluidic dynamics (CFD) community [ 16 ], [17]....
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1,465 citations
"IC thermal simulation and modeling ..." refers background or methods in this paper
...One benefit of using the passive model order reduction technique of [ 23 ] for SIMO-based reduction is that even though the passivity is not guaranteed, the reduced SIMO models, however, are always stable....
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...To use the Krylov subspace technique of [ 23 ], one needs to compute an orthonormal basis V of the Krylov subspace defined as colspan{r, A −1 r, A −2 r, ... }, where A = G −1 C, r = G −1 b. The reduced SIMO model is defined using the reduced system matrices with smaller dimensions: G = V T GV , C = V T CV, b = V T b. The major computation involved in model construction is due to the multiple linear system solutions defined by matrix G. ......
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...Then, a SPICE-like simulation technique can be applied to the equivalent RC circuit to provide the thermal transient response [ 2 ], [4], [6]....
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...In [ 2 ], [4], and [6], the authors simulate the full-chip temperature profile by discretizing the partial differential equation (PDE) of heat conduction using finite difference and finite element methods....
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...First, high temperature and hot spots degrade the reliability of interconnects and transistors [ 2 ], [3]....
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1,252 citations
574 citations
"IC thermal simulation and modeling ..." refers background in this paper
...…· ◦C)], k is the thermal conductivity of the material (W/m2 · ◦C), g is the power density of the heat sources (W/m3), and ni, hi, and fi are the outward direction normal to the boundary surface i, heat transfer coefficient [W/(m2 · ◦C)], and an arbitrary function at the surface i, respectively....
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...Since the matrix corresponding to (3) can be formulated as symmetric positive definite (SPD), MG iterative methods are applicable and become a good choice....
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