Zienkiewicz動態多孔彈性力學模型之穩定性探討

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姓名

李子立(Tsu-Li Li) 查詢紙本館藏

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機械工程學系

論文名稱

Zienkiewicz動態多孔彈性力學模型之穩定性探討

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★ 冠狀動脈三維重建之初步架構

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摘要(中)

Zienkiewicz等人基於了Biot提出之多孔彈性動力學模型重新推導並額外考慮了一固體位移慣性項和流體加速度項，為了滿足位移–壓力型多孔彈性力學，Zienkiewicz等人在實際物理機制允許的情況下將流體加速度項省略了，以至於後來鮮少人能針對Zienkiewicz等人提出之多孔彈性動力學模型進行額外的分析及模擬，吾人通過馮諾依曼穩定性分析針對該多孔彈性力學模型分析之，並發現該模型存在無條件不穩定的區域，雖然該多孔彈性力學模型並不穩定，吾人仍可找出兩個條件收斂的穩定性條件，而且這兩條穩定性條件也分別對應了統御方程式中的質量方程式與動量方程式。另一方面，吾人於馮諾依曼穩定性分析中考慮不同變數(如：位移、孔壓)擁有不同之波數，相較於前人提出之馮諾依曼穩定性分析，此不同波數的假設更能全面的了解變數對穩定性的影響，甚至能得到假設相同波數的穩定性分析無法得到的結果。

摘要(英)

Zienkiewicz et al. consider an additional inertial term of displacement and acceleration term of fluid in the poroelastodynamics model established by Biot. To satisfy the displacement-pressure formula. They consider neglecting acceleration terms of fluid with some physical condition. After they received his study, there were rarely researches also studying on theirs. Because of that, this study forces on his poroelastodynamic model using von Neumann stability analysis. However, this study points out that their poroelastodynamic model is unconditionally instability, finding out unconditionally instability region. Although this model is unstable, this paper finds the other two conditionally stable conditions, wave-like and diffusion-like conditions. On the other hand, this study assumes the different wavenumbers for each unknown, e.g. u (displacement) and p (pore pressure). This assumption is not the same as previous studies, but it can analyse this model more generally. In addition, utilizing this assumption can get more results than assuming the same wavenumbers for each unknown.

關鍵字(中)

★ 穩定性
★ 穩定區
★ 多孔彈性

關鍵字(英)

★ stability
★ stable region
★ poroelasticity

論文目次

摘要 i
Abstract ii
致謝 iii
目錄 iv
表目錄 v
圖目錄 vi
一、緒論 1
1-1 研究動機 1
1-2 文獻回顧 2
二、 Zienkiewicz動態多孔彈性力學模型基於有限差分法之穩定性分析 3
2-1 馮諾依曼穩定性分析 5
2-2 類波動穩定性條件 10
2-3 類擴散穩定性條件 12
2-4 類波動穩定性條件與類擴散穩定性條件之臨界點與交集 16
三、數值模擬與討論 18
3-1 無條件不穩定之波數區域的數值驗證與討論 19
3-2 穩定性條件的數值驗證與討論 20
四、結論 26
五、未來展望 27
參考文獻 28

參考文獻

[1] Terzaghi, K., Principles of soil mechanics. Engineering News-Record, 1925. 95(19-27): p. 19-32.
[2] Biot, M.A., General theory of three‐dimensional consolidation. Journal of applied physics, 1941. 12(2): p. 155-164.
[3] Biot, M.A., Mechanics of deformation and acoustic propagation in porous media. Journal of applied physics, 1962. 33(4): p. 1482-1498.
[4] Biot, M.A., Generalized theory of acoustic propagation in porous dissipative media. The Journal of the Acoustical Society of America, 1962. 34(9A): p. 1254-1264.
[5] Biot, M.A., Theory of propagation of elastic waves in a fluid‐saturated porous solid. II. Higher frequency range. The Journal of the acoustical Society of america, 1956. 28(2): p. 179-191.
[6] Bonnet, G., Basic singular solutions for a poroelastic medium in the dynamic range. The Journal of the acoustical Society of america, 1987. 82(5): p. 1758-1762.
[7] Zienkiewicz, O. and T. Shiomi, Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution. International journal for numerical and analytical methods in geomechanics, 1984. 8(1): p. 71-96.
[8] Zienkiewicz, O.C., et al., Static and dynamic behaviour of soils: a rational approach to quantitative solutions. I. Fully saturated problems. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 1990. 429(1877): p. 285-309.
[9] Zienkiewicz, O., C. Chang, and P. Bettess, Drained, undrained, consolidating and dynamic behaviour assumptions in soils. Geotechnique, 1980. 30(4): p. 385-395.
[10] Miga, M.I., K.D. Paulsen, and F.E. Kennedy, Von Neumann stability analysis of Biot′s general two-dimensional theory of consolidation. International Journal for Numerical Methods in Engineering, 1998. 43(5): p. 955-974.
[11] Hwang, C.T., N.R. Morgenstern, and D.W. Murray, On Solutions of Plane Strain Consolidation Problems by Finite Element Methods. Canadian Geotechnical Journal, 1971. 8(1): p. 109-118.
[12] Masson, Y.J., S. Pride, and K. Nihei, Finite difference modeling of Biot′s poroelastic equations at seismic frequencies. Journal of Geophysical Research: Solid Earth, 2006. 111(B10).
[13] Masson, Y.J. and S. Pride, Finite-difference modeling of Biot’s poroelastic equations across all frequencies. Geophysics, 2010. 75(2): p. N33-N41.
[14] O’Brien, G.S., 3D rotated and standard staggered finite-difference solutions to Biot’s poroelastic wave equations: Stability condition and dispersion analysis. Geophysics, 2010. 75(4): p. T111-T119.
[15] Chi, S.W., T. Siriaksorn, and S.P. Lin, Von Neumann stability analysis of the u-p reproducing kernel formulation for saturated porous media. Computational Mechanics, 2017. 59(2): p. 335-357.
[16] Zienkiewicz, O., Soils and other saturated porous media under transient, dynamic conditions. General formulation and the validity of various simplifying assumptions. Soil mechanics-transient and cyclic loads, 1982.
[17] Zienkiewicz, O., et al. Staggered, time marching schemes in dynamic soil analysis and selective explicit extrapolation algorithms. in Second International Symposium on Innovative Numerical Analysis in Applied Engineering Sciences, Canada. 1980.

指導教授

鍾禎元周鼎贏(Chen-Yuan Chung Dean Chou)

審核日期

2021-10-8

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