Implicit numerical multidimensional heat-conduction algorithm parallelization and acceleration on a graphics card

Abstract

Analytical solutions are much less computationally intensive than numerical ones, and moreover, they are more accurate because they do not contain numerical errors; however, they can only describe a small group of simple heat-conduction problems. A numerical simulation of heat conduction is often used as it is able to describe complex problems, but its computational time is much longer, especially for unsteady multidimensional models with temperature-dependent material properties. After a discretization using the implicit scheme, the heat-conduction problem can be described with N non-linear equations, where N is the large number of the elements of the discretized model. This set of equations can be efficiently solved with an iteration of the line-by-line method, based on the heat-flux superposition, although the computational procedure is strictly serial. This means that no parallel computation can be done, which is strictly required when a graphics card is used to accelerate the computation. This paper describes a multidimensional numerical model of unsteady heat conduction solved with the line-by-line method and a modification of this method for a highly parallel computation. An enormous increase in the speed is demonstrated for the modified line-by-line method accelerated on the graphics card, and the durations of the computations for various mesh sizes are compared with the original line-by-line method. Keywords: heat conduction, numerical simulation, multidimensional numerical model algorithm, acceleration, parallelization, graphics card Analiti~ne re{itve so mnogo manj ra~unsko intenzivne kot numeri~ne, poleg tega pa so bolj natan~ne, ker ne vsebujejo numeri~nih napak, vendar pa lahko opisujejo samo majhno skupino enostavnih problemov prevajanja toplote. Numeri~na simulacija prevajanja toplote se pogosto uporablja, ker je sposobna opisati kompleksne probleme, vendar pa je~as izra~una mnogo dalj{i, {e posebno pri nestabilnih ve~dimenzijskih modelih, z lastnostmi materiala, odvisnimi od temperature. Po diskretizaciji z uporabo implicitne sheme je mogo~e problem prevajanja toplote opisati z N nelinearnimi ena~bami, kjer je N veliko {tevilo elementov diskretiziranega modela. Ta sklop ena~b je mogo~e u~inkovito re{iti s pribli`kom metode vrsta za vrsto, ki temelji na predpostavki toka toplote,~eprav izra~un poteka serijsko. To pomeni, da ni mogo~vzporedni izra~un, kar je striktna zahteva, kadar se uporablja grafi~no kartico za pohitritev izra~una. Ta~lanek opisuje ve~dimenzijski numeri~ni model nestabilnega prevajanja toplote, kar je bilo re{eno z metodo vrsta za vrsto in s pospe{itvijo modifikacije te metode na grafi~ni kartici. Trajanje izra~una je primerjano z osnovno metodo vrsta za vrsto pri razli~nih dimenzijah mre`e.

Sequential Inverse Heat Conduction Problem in OpenFOAM

“…At present, the sequential IHCP algorithm is commonly used with a direct minimization (the quadratic minimization) in the Heat Transfer and Fluid Flow Laboratory [71] [72]. The direct part of the in-house IHCP solver is computed using the line-by-line algorithm [73], hereafter referred to as the line-by-line IHCP solver. The proposed IHCP solver in OpenFOAM is compared to the line-by-line solver in the following text.…”

Heat treatment of thin sheets, research methodology to design a cooling system

“…Heat transfer coefficient dependence on the position and surface temperature are obtained from these tests using inverse task (Figure 5). Detail information about inverse task could be found in [11][12][13][14][15]. Cooling intensity and homogeneity is evaluated by comparison of obtained heat transfer coefficient dependences on the surface temperature for selected position interval.…”