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Applications of the addition theorem in engineering problems
|Authors: ||Ying-Te Lee|
|Contributors: ||NTOU:Department of Harbor and River Engineering|
addition theorem;degenerate kernel;multipole expansion;Fourier series;spherical harmonics;null-field integral equation;image method;Trefftz method;method of fundamental solutions
|Issue Date: ||2011-06-30T07:47:35Z
|Abstract: ||本文發展一套系統性的解法來求解含圓形、橢圓形或球形邊界值問題。本法使用零場積分方程搭配退化核及傅立葉級數或球型諧合函數來求解工程問題。基於用圓形、橢圓形或球形幾何條件，閉合型基本解以退化核型式展開。而邊界密度函數則利用傅立葉級數或球型諧合函數展開。針對橢圓形邊界問題，我們發現存在於退化核、邊界未知密度或者是邊界路徑積分中的Jacobian 項，將會自然地互相抵消掉。本法可被視為一套半解析法。因為在實際運算過程中，其誤差是由於傅立葉級數或球型諧合函數擷取有限項所產生的。配合使用選點法使其滿足邊界條件後，我們可經由一個線性代數系統輕易地求得未知的傅立葉或球型諧合函數係數。我們將更進一步把單心推展到快速運算的雙心展開，其中退化核函數被多極展開的核函數所取代。本法有五個好處：(1)免除主值的計算，(2)指數收斂，(3)無須建構網格，(4)邊界層效應的消除，(5)良態代數系統。除了用於求解含孔洞問題外，我們亦將此法延伸到處理含置入物問題。此外，退化核或稱加法定理也被應用來證明同心圓環格林函數Trefftz法與基本解法求得解析解的數學等效性。我們也發現映像法可被視為最佳點源佈點位置及若干給定強度之基本解法的特例。最後，我們將針對含任意數目、不同大小與任意分佈之圓形、橢圓形或球形邊界值問題，發展一套系統性的求解方法。我們也會舉幾個數值算例，包含扭轉桿、水波、地表振動、多體輻射與散射以及彈力等問題，來驗證我們這套方法的可行性與正確性。|
A systematic approach is developed to deal with engineering problems containing circular, elliptic or spherical boundaries. The proposed approach using the null-field integral formulation in conjunction with degenerate kernel and Fourier series (spherical harmonics) is employed to solve engineering problems. Owing to the circular or elliptic (spherical) geometry, the fundamental solutions and the boundary densities are expanded by using degenerate kernels and Fourier series (spherical harmonics), respectively. For the elliptic boundary, it is found that a Jacobian term may exist in the degenerate kernel, boundary density or contour integral and can cancel out to each other. This approach can be seen as one kind of semi-analytical methods, since the error stems from the truncation of Fourier series (spherical harmonics) in the implementation. The unknown Fourier coefficients (spherical harmonics) are easily determined by solving a linear algebraic system after using collocation method and matching the boundary conditions. Further, we extend the single-center to bi-center expansion for fast algorithm. The fundamental functions are extended from degenerate kernels to multipole expansion. The approach has five advantages: (1) free of calculating principal value, (2) convergence rate of exponential order, (3) free of mesh generation, (4) elimination of the boundary-layer effect and (5) well-posed algebraic system. The proposed approach is also extended to deal with the problems including multiple circular inclusions. Besides, degenerate kernel (or so-called addition theorem) is applied to prove the mathematical equivalence of two analytical solutions of the annular Green’s function derived by using the Trefftz method and the method of fundamental solutions. It is worthy of noting that the image method can be seen as a special case of method of fundamental solutions with optimal location of source point and some specified strengths. Finally, a general-purpose program is developed for solving BVPs containing arbitrary number, different sizes and various locations of circular, elliptic or spherical boundaries. Several examples including the torsion bar, water wave, ground motion, multiple radiation and scattering and elasticity problems were given to demonstrate the validity of the present approach.
|Appears in Collections:||[河海工程學系] 博碩士論文|
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