Parallel Multiscale Finite Element with Adaptive Bubble Enrichment Method

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姓名

李孟哲(Meng-Zhe Li) 查詢紙本館藏

畢業系所

數學系

論文名稱

(Parallel Multiscale Finite Element with Adaptive Bubble Enrichment Method)

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摘要(中)

模擬在多孔介質中的流體運動，通常使用達西定律（Darcy′s law）計算流體的流場，它是一個在非均值滲透透性下的橢圓問題，而多尺度的方法是一類適合解這樣非均質問題的方法。多尺度有限元素法（multiscale finite element method）可以將細網格（fine-grid）的大問題分解成多個在粗網格（coarse-grid）與子網格（sub-grid）上較小的問題，且在非均質（heterogeneous）的問題上較不會遺漏細微的資訊，但是近似解的精確度容易受到多尺度基底（multiscale basis）的影響，所以我們從多重網格法（multigrid method）引進smoothing的概念進而發展自適應多尺度有限元素法（iApMsFEM），透過迭代過程自動調整多尺度基底來得到近似解，再根據多尺度有限元素法的特性將它平行化來達到加速的效果。但是，自適應多尺度有限元素法因為多尺度基底的改變必須重複的作一些計算量較大的運算，為了改善自適應多尺度有限元素法的這個缺點，我們發展multiscale finite element with adaptive bubble enrichment method (MsFEM_bub)，不再透過迭代過程改善多尺度基底來修正數值解，而是透過迭代過程改善氣泡函數（bubble function）來修正數值解，因為多尺度基底不再隨著迭代而改變，所以可以大幅減少那些重複的計算。
　　在實驗階段，透過均質（homogeneous）與非均質的橢圓問題來尋找 MsFEM_bub 的最佳參數設定以及探討其延展性（scability），也透過這些的問題來比較 MsFEM_bub 與多尺度自適應有限元素法和多重網格法的差異。這份論文的實驗結果皆是在國立中央大學數學系的叢集電腦與國家高速網路與計算中心的ALPS上實驗而得到的。

摘要(英)

To simulate some physical behavior of fluid in a porous media, we usually compute the flow field by Darcy′s law which is an elliptic problem with heterogeneous permeability. And the multiscale methods are suitable to solve the heterogeneous problem. Multiscale finite element method (MsFEM) can separate the huge fine-grid problem into several small sub-grid problems and coarse-grid problem, and it less lost important fine-grid information for the heterogeneous problem when it does information exchange between fine-grid and coarse-grid. But the accuracy of MsFEM is easily affected by the multiscale basis. Therefore, we develop the iteratively adaptive multiscale finite element method (iApMsFEM) which introduces the conception of smoothing from the multi-grid method (MG). iApMsFEM can update the multiscale basis by iteration and then it can improve the approximation. We also can use some properties of MsFEM to parallel and accelerate iApMsFEM. But iApMsFEM must do some repeated computations because that the multiscale basis is updated as the method iterates. To improve this disadvantage, we develop multi-scale finite element with adaptive bubble enrichment method (MsFEM_bub) which no longer update the multiscale basis but update the bubble function in the iteration process to modify the approximation. Because of the fixed multiscale basis, we can avoid those repeated computations.

In the numerical experiment, we use the homogeneous and heterogeneous elliptic problems to find the best parameter of MsFEM_bub and test its scalability. And we also use these problems to compare the difference between MsFEM_bub, iApMsFEM and MG. In this thesis, all of the numerical results are produced by the cluster Leopard in National Central University Department of Mathematics and ALPS in National Center for High-performance Computing (NCHC, NARLabs).

關鍵字(中)

★ 多尺度有限元素法
★ 自適應氣泡函式
★ 多孔介質
★ 非均質問題
★ 橢圓問題

關鍵字(英)

★ Multiscale finite element method
★ Adaptive bubble function
★ Porous media
★ Heterogeneous problem
★ Elliptic problem

論文目次

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 2D elliptic partial differential equation problem . . . . . . . . . . . . . . . . 3
2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Galerkin finite element method [11,16] . . . . . . . . . . . . . . . . . . . 3
2.3 Galerkin finite element discretization . . . . . . . . . . . . . . . . . . . . 4
3 A review of iteratively adaptive multiscale finite element method . . . . . . 7
3.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Iteratively adaptive multiscale finite element method (iApMsFEM) . . . . 7
3.3 The local problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.5 Flowchart and algorithm of iApMsFEM . . . . . . . . . . . . . . . . . . 13
4 Multiscale finite element with adaptive bubble enrichment method . . . . . 16
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Fundamental idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3 Multiscale finite element with adaptive bubble enrichment method (Ms-
FEM bub) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4 The local problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.5 Flowchart and algorithm of MsFEM bub . . . . . . . . . . . . . . . . . . 22
5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1 Parallel tool - PETSc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2.1 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2.2 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.3 The local problem . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.1 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.1.1 Homogeneous elliptic boundary value problems . . . . . . . . . . 31
6.1.2 Heterogeneous elliptic boundary value problems . . . . . . . . . 32
6.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.2.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.2.3 Case 3 (r1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2.4 Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2.5 Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.3 Parameter study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.3.1 The selection of stopping condition . . . . . . . . . . . . . . . . 45
6.3.2 The selection of smoother . . . . . . . . . . . . . . . . . . . . . 46
6.4 Comparison with other methods . . . . . . . . . . . . . . . . . . . . . . 48
6.4.1 Comparison with other methods for the convergence history on
homogeneous problem . . . . . . . . . . . . . . . . . . . . . . . 48
6.4.2 Comparison with other methods for the convergence history on
heterogeneous problem . . . . . . . . . . . . . . . . . . . . . . . 48
6.4.3 Comparison with iApMsFEM . . . . . . . . . . . . . . . . . . . 50
6.4.4 Comparison with GMRES+ASM(LU) . . . . . . . . . . . . . . . 54
6.4.5 Comparison with multigrid method . . . . . . . . . . . . . . . . 56
7 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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指導教授

黃楓南(Feng-Nan Hwang)

審核日期

2016-7-25

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