Iterative Finite-Difference Solution of Heat Transfer and Fluid Flow With Truncated Modal Acceleration and Imaginary Time (Algorithm, Modeling, Steady-State)
Mech, Andrew Raymond
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https://hdl.handle.net/2142/70139
Description
Title
Iterative Finite-Difference Solution of Heat Transfer and Fluid Flow With Truncated Modal Acceleration and Imaginary Time (Algorithm, Modeling, Steady-State)
Author(s)
Mech, Andrew Raymond
Issue Date
1986
Department of Study
Mechanical Engineering
Discipline
Mechanical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Mechanical
Abstract
This work discusses acceleration of the convergence of some classic point-iterative methods; Jacobi, simultaneous overrelaxation (JOR), Gauss-Seidel and successive overrelaxation (SOR). Each of these methods is diffusive in nature. Such methods do not distribute calculated changes of mass, momentum, and energy properly. As a result local imbalances diffuse and dissipate quickly while global imbalances control the convergence rate.
The proposed acceleration technique is used in conjunction with an iteration of one of the four point-iterative methods. The residual fields are decomposed into orthogonal modes with individual convergence rates. Each mode in residual space is mapped to a corresponding mode in error space which in turn is subtracted from the existing approximation of the solution. Rather than calculating each mode, the expansion is truncated after the lowest (slowest-converging) modes have been calculated.
An "imaginary" time parameter, quantifying the time-link nature of the algorithms, is used to estimate the convergence rate of each mode for steady-state one-dimensional heat conduction. In particular, elimination of the lowest mode for the JOR scheme was predicted to improve the convergence rate by a factor of four. Numerical tests indicate the actual factor is 3.7. Other numerical tests illustrate applicability to one- and two-dimensional conduction and convection. The results of fluid flow tests were somewhat disappointing.
Imaginary time was found to have an additional application to the modeling of fluid flow problems. Analysis using imaginary time shows the momentum equations are not coordinated with the pressure corrector when upwinding is used. Chorin's artificial compressibility and Hirt's pressure corrector are examined in particular. It is shown that the proper coefficient for that method should vary spatially. Numerical results are provided and other work applying this technique is cited.
Recently modes were developed which adjust to the mode dominating convergence. The modes are not sensitive to geometry. Furthermore, mappings between residual and error space are unnecessary. Their flexibility shows great promise.
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