【摘要】：帶純Neumann邊界條件三維線彈性模型是描述固體力學(xué)問題和計算材料力學(xué)問題的重要模型,有限元法是數(shù)值求解三維線彈性問題最常用的離散方法,并行代數(shù)多層網(wǎng)格(AMG)法是快速求解三維線彈性問題離散系統(tǒng)的最為有效的方法之一。本文針對一種滿足適定性條件的純Neumann邊界的三維線彈性問題的二次有限元離散代數(shù)系統(tǒng),研究其并行快速求解算法及解法器。首先,在文[33]的基礎(chǔ)上,討論了二次有限元離散化代數(shù)系統(tǒng)的適定性,數(shù)值實(shí)驗(yàn)表明二次有限元誤差函數(shù)在L2(Ω)和H1(Ω)范數(shù)下均具有飽和誤差階。接著,針對二次有限元代數(shù)系統(tǒng)給出了兩類求解算法,其中重點(diǎn)研究了兩種基于AMG法和高斯賽德爾迭代法(GS)的組合型預(yù)條件子,并研制了相應(yīng)的預(yù)條件GMRES(k)解法器(BAG-GMRES(k)和BvAG-GMRES(k))。數(shù)值實(shí)驗(yàn)驗(yàn)證了基于AMG的組合型預(yù)條件GMRES(k)法的迭代次數(shù)和求解時間均優(yōu)于常用求解方法ILU(0)-GMRES(k)。相比較而言,BvAG-GMRES(k)解法器更穩(wěn)定高效。最后,在上述串行預(yù)條件子算法解法器的基礎(chǔ)上,設(shè)計了相應(yīng)的具有極小化數(shù)據(jù)通信的并行AMG法和GS的組合型預(yù)條件算法,數(shù)值試驗(yàn)驗(yàn)證了該算法具有良好的擴(kuò)展性。
[Abstract]:The three-dimensional linear elastic model with pure Neumann boundary conditions is an important model for describing the problems of solid mechanics and computational material mechanics. The finite element method is the most commonly used discrete method for numerical solution of three-dimensional linear elastic problems. The parallel algebraic multi-layer grid (AMG) method is one of the most effective methods for fast solving three-dimensional linear elastic problem discrete systems. In this paper, a quadratic finite element discrete algebraic system for three-dimensional linear elastic problem with pure Neumann boundary satisfying the well-posed condition is studied, and its parallel fast algorithm and solver are studied. Firstly, on the basis of [33], the well-posed quadratic finite element discrete algebraic system is discussed. Numerical experiments show that the quadratic finite element error function has saturation error order in L2 (惟) and H1 (惟) norm. Then, two kinds of solving algorithms are given for quadratic finite element algebraic system, in which two kinds of combinatorial preforms based on AMG method and Gao Si Seidel iterative method (GS) are studied emphatically. The corresponding preconditioned GMRES (k) solver (BAG-GMRES (k) and BvAG-GMRES (k).) is developed. Numerical experiments show that the iteration times and solution time of the combined preconditioned GMRES (k) method based on AMG are better than that of ILU (0)-GMRES (k). Method. In contrast, the BvAG-GMRES (k) solver is more stable and efficient. Finally, on the basis of the subalgorithm solver of serial prebar, the parallel AMG method with minimized data communication and the combined preconditioned algorithm of GS are designed. Numerical experiments show that the algorithm has good expansibility.
【學(xué)位授予單位】：湘潭大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O343.2;O241.82

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本文編號：2461984