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Applications of Numerical Global Domain Transformation Method to 2D Elasticity Problems
No full text本研究旨在應用Tsay等人(Tsay and Hsu, 1997; Wang and Tsay, 2005; Tsay, Wang and Huang, 2006; 黃永德，1998)發展出來的「數值全域轉換法」(Numerical Global Domain Transformation Method, NGDTM)，在任意二維幾何區域求解彈性力學的問題(Airy stress function)，並發展出一套通用的數值求解策略。
「數值全域轉換法」為一個將整體解題空間由物理發生的平面轉換至另一個平面求解的方法。利用複變映射原理(Complex mapping)以及邊界元素積分法(Boundary integral element method)求解Laplace方程式，將一個任意二維幾何形狀轉換至一矩形區域，使得有限差分法(Finite difference method)得以在此矩形區域方便地運用。
Airy stress function為求解彈性力學的控制方程式之ㄧ(Timoshenko and Goodier, 1986)，由於定義的關係，traction邊界條件只能提供二階的偏導數值。過去在矩形空間應用FDM求解Airy stress function時，必須透過沿著幾何邊界對traction邊界條件積分獲得一階及零階偏導數的數值邊界條件(Timoshenko and Goodier, 1986)。然而當幾何邊界複雜時，這個積分的動作會變得相當地困難，並且程式不容易撰寫成一般化，因此未被廣泛地應用在數值方法。
本研究透過Tsay等人發展出來的區域數值轉換理論，將任意的幾何邊界條件及traction邊界條件轉換至矩形區域，然後在該矩形區域上將traction邊界條件透過數值積分轉換成運用有限差分法需要之數值邊界條件。控制方程式亦透過此區域數值轉換理論由不規則區域轉換至矩形區域。運用轉換過的控制方程式和邊界條件在矩形區域進行有限差分法之計算。進行控制方程式和邊界條件的轉換時，將透過「柯西-里曼方程式」(Cauchy-Riemann conditions)化簡方程式，並且轉換後的Jacobian， 、 不為常數。在矩形區域進行離散時，本研究採取在區域內(不含邊界)設置以中央差分離散之控制方程式，邊界點上的自由度每個點設置兩條邊界條件，四個角落分別設置三條邊界條件的策略，使得未知數的數目等於方程式的數目，獲得唯一且收歛的解。以一兩端承受彎矩的栱型樑為例，轉換至矩形區域後，在網格點為 ，網格大小為 的情況下，取中間的剖面的數值結果與解析解相較，勢能(Potential)的誤差在 之內，應力 的誤差在 之內，應力 的誤差在 之內，應力 最大誤差為0.004(解析解為0)。
本研究推導出控制方程式、邊界條件及應力式正確的轉換關係，並提出一套通用的數值求解策略，奠定Tsay等人發展出來的「數值全域轉換法」在彈性力學領域應用的基礎。同時經由本研究建立之求解模式，有限差分法可以很方便地運用在二維彈性力學的求解上，提昇了運用有限差分法求解二維彈性力學時適用的廣度。The objective of this research is to develop a general numerical scheme to solve elasticity problems in an arbitrary two dimensional domain by “Numerical global domain transformation method (NGDTM),” a domain transformation theorem developed by Tsay et al. (Tsay and Hsu, 1997; Wang and Tsay, 2005; Tsay, Wang and Huang, 2006; 黃永德，1998).
NGDTM transforms the problem solving domain from an arbitrary two dimensional domain, the physical domain, to a rectangular domain by applying complex mapping theorem and solving Laplace equations by boundary integral element method (BIEM). When the domain is transformed to a rectangle domain, it becomes very convenient to solve the problems with finite difference method (FDM).
Airy stress function is one of the governing equations in elasticity (Timoshenko and Goodier, 1986). Based on the Airy stress function, the traction boundary conditions can only provide the second order derivative of the primary variable. It is thus necessary to integrate the traction boundary condition along the geometry to obtain the numerical boundary condition when one tries to solve the Airy stress function in a rectangular domain (Timoshenko and Goodier, 1986). Such integration becomes relatively difficult when the geometry is complex. Partly due to such shortcoming, FDM is not popular in solving elasticity problems.
In this work, an arbitrary two dimensional domain and traction boundary conditions were transformed to a rectangular domain through the domain transformation theorem. Traction boundary conditions in the rectangular domain were integrated to obtain the numerical boundary condition. The governing equation was also transformed form the arbitrary domain to the rectangular one by the domain transformation theorem. After applying the transformed governing equation and numerical boundary conditions, the finite difference method calculation was performed in a rectangular domain. During the transformation derivative, Cauchy-Riemann conditions were applied in which the Jacobians and , were not constants.
The governing equation and numerical boundary conditions were both expressed in the first-order central difference. The governing equation was discreted to a 13 point formulation and applied on the internal meshes excluding the boundary. Each node on the boundary has two boundary conditions. Each corner has three boundary conditions. This scheme makes the number of unknowns equal to the number of equations and result in an unique and convergent solution.
An arc with a couple of moment applied was used to verify the proposed scheme. The rectangle domain was discreted in nodes and the mesh size was. . In comparison with the analytical solution on the middle of the arc, the error of the potential was less than , the error of the normal stress was less than , the error of the normal stress was less than , and the maximum error of the shear stress was 0.004 (the analytical solution is zero).
A NGDTM for the Airy stress function has been developed in this work. A transformed governing equation, boundary conditions, stress formulation and a general numerical scheme have been derived. Through the machinery of NGDTM developed herein, the FDM can be used conveniently to solve two-dimensional elasticity problems.致謝 ……………………………………………………………………………… i
摘要 ……………………………………………………………………………… v
Abstract …………………………………………………………………………… vii
目錄 ……………………………………………………………………………… ix
圖目錄 …………………………………………………………………………… xi
表目錄 …………………………………………………………………………… xiii
第一章 緒論 …………………………………………………………………… 1
1-1 研究背景與動機 ……………………………………………………… 1
1-2 研究目的 ……………………………………………………………… 2
1-3 文獻回顧與相關研究 ………………………………………………… 3
第二章 二維彈性力學分析 …………………………………………………… 9
2-1控制方程式、邊界條件與應力表示式 ……………………………… 9
2-2 控制方程式、邊界條件與應力表示式之轉換 ……………………… 11
第三章 利用全域轉換法的解析求解 ………………………………………… 23
3-1 推導目的 ……………………………………………………………… 23
3-2 解析求解範例 ………………………………………………………… 23
第四章 數值方法 ……………………………………………………………… 31
4-1 求解流程 ……………………………………………………………… 31
4-2 不規則區域和矩形區域的物理邊界條件轉換關係 ………………… 32
4-3 數值積分求一階及零階偏導數邊界條件 …………………………… 36
4-4 控制方程式與邊界條件的離散 ……………………………………… 39
4-5 自由度的分布 ………………………………………………………… 42
4-6 不規則區域的應力式在矩形區域的表示式 ………………………… 43
4-7 矩形區域計算策略 …………………………………………………… 47
第五章 數值方法之驗證 ……………………………………………………… 57
5-1 解析解 ………………………………………………………………… 57
5-2 本文數值解的前置作業與流程 ……………………………………… 59
5-3 數值解之驗證 ………………………………………………………… 61
第六章 結論與建議 …………………………………………………………… 67
6-1 結論 …………………………………………………………………… 67
6-4 建議 …………………………………………………………………… 68
參考文獻 ………………………………………………………………………… 69
附錄A 偏導數的相關推導 ……………………………………………………… 73
附錄B 本文極座標下偏導數的轉換推導 ……………………………………… 79
附錄C 本文環狀區域之位移 …………………………………………………… 83
附錄D 控制方程式的差分式 …………………………………………………… 87
作者簡曆……………………………………………………9
Study of Effects of Friction Coefficients to Granular Flow Dynamics by Discrete Element Simulation
No full text微觀顆粒的互動行為與巨觀顆粒流的物理現象之間的關係一直是重要的研究議題。近年來，由於數值模擬愈趨成熟，許多過去單憑實驗很難觀察到的微觀顆粒行為，例如：顆粒之間的接觸力與相對速度，都可以透過數值模擬得到合理的結果，進而建立和實驗觀察到的巨觀現象之間的關係。在顆粒流的研究中，離散元素法(discrete element method, DEM)是受到許多學者採用的數值模擬方法，以其能夠進行大量且快速的模擬並得到和實驗相符的結果。 慣性數I是一個能夠捕捉顆粒流特徵的無因次參數，根據I值的大小，可以得知顆粒流的行為比較接近固體還是流體。近年來有學者拿I和有效摩擦係數μ之間的關係來推導顆粒流的組成律，使得I和μ成為研究顆粒流非常重要的巨觀參數。透過DEM的模擬，有學者指出顆粒流的巨觀有效摩擦係數μ與微觀摩擦係數f的關係不大。然而，亦有學者發現當f≤0.4，μ值隨著f呈線性成長。為了釐清f和μ之間的關係，本研究透過DEM進行剪力槽(shear cell)實驗的模擬，發現在相近的慣性數I之下，剪力槽內顆粒流的μ隨著f的增加而增加。這是因為在剪力槽內，f對壁體承受的正向力的影響相對不大。同時，f的增加會提高發生庫倫滑動的門檻，讓顆粒和顆粒以及顆粒和壁體之間產生較大的切向力，造成μ值的上升。此外，本研究發現由於和邊界接觸的顆粒會發生「鐘擺效應」機制，顆粒流和邊界之間的接觸切向力方向不會一致，造成穩態顆粒流的μ明顯低於f。另一方面，當剪力槽的轉速夠快，暫態顆粒流和邊界的切向接觸力有機會完全進入庫倫摩擦。此時所有和邊界接觸的顆粒受到方向一致的摩擦力作用，本研究稱之為「完全同步」狀態。「完全同步」下的顆粒流的μ值幾乎和f一樣。 微觀摩擦係數f和旋轉鼓內速度剖面曲線的關係也是本研究探討的議題。當旋轉鼓內的顆粒流達到穩態，該顆粒系統會因為具有較高的f而形成較大的動態安息角β，具有較高的位能。然而，旋轉鼓內速度剖面曲線並沒有因為顆粒系統的f不同而有顯著的變化，暗示具有不同f的穩態顆粒系統之間動能的差異遠不及彼此位能的差異，也代表微觀摩擦係數f會影響顆粒系統內部顆粒所作的功。本研究透過「功能原理」的應用以及使用DEM進行微觀能量的計算，發現不同f的顆粒系統從靜止到「動能初峰期」這段時間系統內接觸力所做的功Π*會決定顆粒系統在穩態期間的位能。換句話說，不同f的穩態顆粒系統彼此之間的位能差異，主要來自於彼此Π*的差值。本研究進一步分析每種接觸力所作的功隨著f的變化，發現從靜止到「動能初峰期」這段時間內具有較大f的顆粒系統並沒有消散比較多的能量，而是透過切向彈簧力獲得較多的能量。The relation between the interactions of microscopic particles and phenomena of macroscopic granular flow have attracted great interests recently. As the numerical simulation techniques improve, many quantities of microscopic particles that are difficult to be observed or measured by experiments can be ready to computed. For example, the contact forces and relative velocities between particles can be calculated by simulation schemes. Thus, the relations between microscopic behaviors and macroscopic phenomena can be found. For granular flows, one of the most popular numerical simulation schemes is the discrete element method (DEM), because it can process large-scale particle simulations and produce numerical results in a good agreement with experimental measurements. The inertial number, I, is a non-dimensional parameter to quantify the flowing regime of granular flows. Recently, constitutive laws of granular flows have been derived by using the relations between I and the macroscopic effective friction coefficient, μ. Through DEM simulation, some studies suggested that the macroscopic effective friction coefficient was independent of the microscopic friction coefficient, f. However, there were studies claimed that μ was linearly dependent on f when f≤0.4. In order to improve the understanding of the relation between μ and f, shear cell simulations are performed in this research. The macroscopic effective friction coefficients of the granular flows in the shear cell are found to monotonically grow with microscopic friction coefficients under similar values of I, because the influence of f on normal forces is much less than tangential forces on the cell wall. At the same time, the increasing f raises the criterion of Coulomb friction and allows larger tangential forces between particles and cell wall, thus resulting in the increasing of μ. Besides, because the pendulum effect occurs on the particles on the cell wall, the directions of tangential forces between particles and cell wall are not identical, thus resulting in the values of μ from steady state granular flows much less than f. On the other hand, when the rotating speeds of the shear cell are fast enough, all tangential forces between the particles and cell wall of transient granular flows reach the criterion of Coulomb friction. At this moment, all particles on the cell are driven by the friction forces with the same direction, named complete synchronization herein. The granular flows under the complete synchronization state have the same values of μ and f. The relation between the microscopic friction coefficients and velocity profiles of particle systems in rotating drums is the other research issue in this thesis. When the particle systems, half-filled the drum, reach a steady state, the ones with higher f will form higher dynamic repose angles, thus resulting in higher potential energies. However, the velocity profiles of the particle systems in the rotating drum are not obviously influenced by f. These similar velocity profiles suggest that the particle systems with different f have similar kinetic energy. Although the particle systems with higher f have higher potential energies, the kinetic energies are almost identical, which indicates f has significant effects on the work of internal forces between particles. In this research, through the application of principle of work and energy and the calculation of microscopic energy using the DEM simulation, the potential energies of steady state particle systems with different f are found determined by the work, Π*, done by contact forces during the first kinetic peak period. In other words, the difference of potential energies of particle systems with different f comes from the difference of their Π*. Furthermore, to analyze the components of Π* with different f, the particle systems with higher f are found not dissipating more energy, but obtaining more energy from tangential contact force
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Get PDFWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
koamabayili/VECTRON-author-checklist: VECTRON author checklist
No full textWe have done our best to complete the author checklist relating to the use of animals in the hut study. Note that the objective for the hut study was to evaluate the IRS treatment applications for residual efficacy against Anopheles mosquitoes, including the local An. coluzzii mosquito population. Cows were only used to attract mosquitoes into the huts and no tests were carried out directly on the cows. The author checklist is intended for use with studies where experiments are carried out on animals, which is why we have had such difficulty in completing this for the hut study, as many of the questions do not relate to how the cows were used
Author-wise bibliometric analysis based on entropy.
No full textAuthor-wise bibliometric analysis based on entropy.</p
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