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系統識別號 U0026-2107202023440800
論文名稱(中文) 腦多孔彈性模型的混合解析和數值方法
論文名稱(英文) A Mixed Analytical and Numerical Method for Brain Poroelastic Models
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 108
學期 2
出版年 109
研究生(中文) 廖士綱
研究生(英文) Shih-Kang Liao
電子信箱 L18021010@mail.ncku.edu.tw
學號 L18021010
學位類別 博士
語文別 英文
論文頁數 64頁
口試委員 口試委員-陳宜良
口試委員-侯世章
口試委員-胡偉帆
口試委員-陳旻宏
指導教授-舒宇宸
中文關鍵字 多室多孔彈性模型  腦力學  解析解 
英文關鍵字 multicompartmental poroelastic model  Cerebral Poromechanics  Analytical Solution 
學科別分類
中文摘要 在這項研究中,我們為腦多孔彈性模型提出了一種混合分析和數值方法。單網絡和多室多孔彈性模型被用來描述大腦組織位移和各種腦脊液壓力的相互作用。通過匹配邊界條件,導出了解析的穩態通解,並透過邊界條件對係數進行了數值求解。對於動力學問題,採用交錯網格的有限差分法消除了非物理振動。 數值實驗表明穩態解的二階收斂性。得到當在邊界上施加衝擊時,腦質裡波傳輸的模擬。結果表明,該波被吸收並收斂到穩態解。
英文摘要 In this study, we proposed a mixed analytical and numerical method for brain poroelastic models. Single-network and multicompartmental poroelastic models are applied to describe the interaction between the displacement of the brain tissue and the pressure of various cerebrospinal fluid. General solutions of the steady state are derived analytically, and the coefficients are solved numerically to agree with the boundary conditions. For dynamical problems, non-physical oscillation is eliminated by finite difference method with staggered grid. The numerical experiments show the second-order convergence for the steady state solutions. Wave transmission in brain is observed when an impact is applied to the boundary. The results show that the wave decays in time, and the displacement eventually converges to the steady state.
論文目次 Contents
1 Introduction 1
2 Modeling 4
2.1 Modeling PoroElasticity Theory 4
2.1.1 Solid Structure of Brain 4
2.1.2 Circulatory of CSF 4
2.1.3 Blood circulation and interaction 5
2.1.4 Governing Equations 6
2.1.5 Boundary Conditions 9
3 Analytical Solutions 11
3.1 Single-network PoroElasticity Theory (SPET) 11
3.1.1 Analytical Steady State Solution of SPET 12
3.1.2 Duhamal’s Principle in SPET 16
3.2 MPET 19
3.2.1 MPET Boundary Conditions 19
3.2.2 Steady State Solution in MPET 21
4 Numerical Solutions 26
4.1 Numerical Solution of SPET 26
4.1.1 Uniform Mesh 26
4.1.2 Computing Process in Uniformly Mesh 27
4.1.3 Dual Mesh 30
4.1.4 Computing Process in Dual Mesh 34
4.2 Numerical Steady State Solution of SPET 35
5 Numerical experiments 40
5.1 Impact Experiment in SPET 40
5.2 Neumann/Dirichlet Condition in Capillary (MPET) 45
6 Conclusions 46
References 47
Appendix A Gradient, Divergence, Laplacian in Spherical Coordinates 50
Appendix B Effect of Coefficient in MPET 51
Appendix C Difference in two type of grids 62
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