Assessment of Multigrid Schemes with Fixed Patterns for a Two-Dimensional Heat Transfer Problem

Year 2024, , 322 - 338, 01.11.2024

Mustafa Serdar Tekçe , Kürşad Melih Güleren

https://doi.org/10.47480/isibted.1403905

Abstract

Heat transfer problems and their solutions are of critical importance in almost all areas of engineering and technology, while many real-world problems are inherently three-dimensional. Simplifying them to 2D models offers practical advantages with reasonable models. With being the fundamental class of problem of heat transfer, the 2D thermal diffusion problem was selected for the study. Multigrid methods were referred to as standing out in terms of cost reduction while keeping solution accuracy. The effectiveness of multigrid methods employing fixed pattern schemes was subject to investigation. To set up numerical experimentation, an authentic code generation effort was given that implements a basic finite volume method, intergrid operations and iterative solvers. A reference case with an analytical Laplace solution was selected and properly validated by the results. A variety of multigrid schemes with fixed patterns were explored around parameters, iterations per sweep and maximum coarsening level. Results were compiled on the focal points of cost and performance. A comparison of the direct iterative methods with multigrid schemes proved the effectiveness of multigrid schemes. With the assessment of the cost and performance outcomes, it is concluded that any multigrid scheme should visit the maximum coarsest level possible while keeping a minimum number of iterations on each grid resolution.

Keywords

Finite Volume Method, Multigrid Schemes, Thermal Diffusion, Computational Cost, Convergence Rate

Isı Transferi Problemi için Sabit Modelli Çoklu Ağ Şemalarının Değerlendirilmesi

Abstract

Isı transferi problemleri ve çözümleri mühendislik ve teknolojinin hemen hemen tüm alanlarında kritik öneme sahipken, gerçek problemlerinin çoğu doğası gereği üç boyutludur. Bunları 2 boyutlu modellere basitleştirmek, makul modellerle pratik avantajlar sunar. Isı transferi probleminin temel sınıfı olan 2 boyutlu termal difüzyon problemi çalışma için seçilmiştir. Çoklu ağ yöntemleri, çözüm doğruluğunu korurken maliyet düşürme açısından da öne çıkan yöntemler olarak bilinmektedir. Sabit model şemaları kullanan çoklu ağ yöntemlerinin etkinliği incelenmiştir. Sayısal deney oluşturmak için temel sonlu hacim yöntemini, ağlar arası işlemleri ve yinelemeli çözücüleri uygulayan özgün bir kod oluşturulmuştur. Analitik Laplace çözümüne sahip bir referans durum seçilmiş ve sonuçlarla uygun şekilde doğrulanmıştır. Sabit modellere sahip çeşitli çoklu şebeke şemaları parametreler, tarama başına yinelemeler ve maksimum seyreltme düzeyinde araştırılmıştır. Sonuçlar maliyet ve performansın odak noktalarına göre derlenmiştir. Doğrudan yinelemeli yöntemlerin çoklu ağ şemalarıyla karşılaştırılması, çoklu ağ şemalarının etkinliğini kanıtlanmıştır. Maliyet ve performans sonuçlarının değerlendirilmesiyle, herhangi bir çoklu ağ şemasının, her ağ çözünürlüğünde minimum sayıda yinelemeyi korurken, mümkün olan en düşük seviyeye gelmesi gerektiği sonucuna varılmıştır.

Keywords

Sonlu Hacim Yöntemi, Termal Difüzyon, Yakınsama Oranı, Çoklu Ağ Şemaları, Hesaplama Maliyeti

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