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Research Papers: Heat and Mass Transfer

Application of the Semi-Analytic Complex Variable Method to Computing Sensitivities in Heat Conduction

[+] Author and Article Information
James Grisham, Ashkan Akbariyeh

Department of Mechanical and Aerospace
Engineering,
University of Texas at Arlington,
Arlington, TX 76019

Weiya Jin

Professor
College of Mechanical Engineering,
Zhejiang University of Technology,
Hangzhou 310032, Zhejiang, China

Brian H. Dennis, Bo P. Wang

Professor
Department of Mechanical and Aerospace
Engineering,
University of Texas at Arlington,
Arlington, TX 76019

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received February 27, 2017; final manuscript received February 17, 2018; published online June 29, 2018. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 140(8), 082006 (Jun 29, 2018) (9 pages) Paper No: HT-17-1109; doi: 10.1115/1.4039541 History: Received February 27, 2017; Revised February 17, 2018

Sensitivity information is often of interest in engineering applications (e.g., gradient-based optimization). Heat transfer problems frequently involve complicated geometries for which exact solutions cannot be easily derived. As such, it is common to resort to numerical solution methods such as the finite element method. The semi-analytic complex variable method (SACVM) is an accurate and efficient approach to computing sensitivities within a finite element framework. The method is introduced and a derivation is provided along with a detailed description of the algorithm which requires very minor changes to the analysis code. Three benchmark problems in steady-state heat transfer are studied including a nonlinear problem, an inverse shape determination problem, and a reliability analysis problem. It is shown that the SACVM is superior to the other methods considered in terms of computation time and sensitivity to perturbation size.

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Figures

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Fig. 1

Comparison between exact and numerical solutions for different mesh resolutions (every third point is shown for the medium grid and every sixth point for the fine)

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Fig. 2

Order of accuracy of nonlinear solution

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Fig. 3

Sensitivity comparison between exact solution (solid lines) and the semi-analytic complex variable solution (symbols for which every other point is shown)

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Fig. 4

Schematic of inverse problem parametrization (nz = 5 and nθ = 4)

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Fig. 5

Target and initial results for nz = 5 and nθ = 4 (contours of temperature in Kelvin): (a) target solution and (b) initial guess for inner geometry

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Fig. 6

Variation of OF and design variables' error with respect to step size used in sensitivity computations

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Fig. 7

Effect of mesh size on sensitivities' computation time

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Fig. 8

Diagram of flip-chip BGA package

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Fig. 9

Ball grid array package mesh and boundary conditions

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Fig. 10

Sample BGA temperature field in degrees Kelvin

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