Study on Solving the Green's Function and Boundary Value Problems by Using the Image Method and the Null-Field Integral Equation

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Project title

Code/計畫編號

NSC98-2221-E019-017-MY3

Translated Name/計畫中文名

利用映像法與零場積分方程求解格林函數與邊界值問題

Project Coordinator/計畫主持人

Jeng-Tzong Chen

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=2009925

Year

2010

Start date/計畫起

01-08-2010

Expected Completion/計畫迄

01-07-2011

Bugetid/研究經費

715千元

ResearchField/研究領域

土木水利工程
海洋科學

"基於先前國科會計畫案「利用零場積分方程求解含圓形邊界的邊界值問題」的成功經驗，本計劃將延伸到求解含橢圓邊界的邊界值問題與三維問題。而點源與螺旋差排所產生的格林函數也是我們所關心的議題。我們將考慮下述這幾個方法，如映像法，直接利用格林第三恆等式所推導的零場積分方程與配合疊加法的零場積分方程等方法來求解格林函數問題。對於Trefftz 法、映像法與基本解法這幾個方法之間的關連性，我們也將在此一併討論。進而，我們將利用映像法來說明基本解法中點源的最佳佈點位置。而這些方法主要的關鍵點在於退化核函數的使用，亦可稱它為加法定理。此三年計畫的架構圖，如次頁所示（請參見英文摘要後之附件）。在第一年計畫中，我們主要鎖定在推導含圓形邊界問題的格林函數。除了拉普拉斯運算子外亦將赫姆茲與雙諧和運算子問題納入本計劃討論範圍。並以偏心圓、半平面含一個圓洞與無限域包含兩個圓洞等問題作為算例。而格林函數使用映像法時的最後映像點與雙極座標的焦點位置將會被作一連結。另外，針對求解點聲源與軸對稱聲源在多圓柱體的散射結果，我們會與理論解及實驗資料做比較。在第二年計畫中，我們將嘗試延伸到求解含橢圓邊界的邊界值問題與格林函數。在曲線座標系統下，橢圓與偏心圓的退化核函數需做特別的處理。螺旋差排的角度型基本解將會被展開成分離核型式。最後，在第三年的計畫中，除了將研究三維拉普拉斯方程邊界值問題外，三維偏心圓球的特徵值問題與多球體的輻射與散射問題也將納入討論。並將研究多體輻射與散射的虛擬頻率發生機制及其克服方法。最後，將利用映像法來求解三維空間的格林函數。" "Following the success of previous NSC projects for solving boundary value problems (BVPs) with circular boundaries using null-field integral equations, we will extend to solve the BVPs for the elliptical boundaries and three dimensional problems. The Green’s functions for point source and dislocation are also our concern. Several approaches, null-field integral equations derived by using Green’s third identity and superposition approach as well as the image method will be considered. The relation among the Trefftz method, image method and method of fundamental solution (MFS) will be addressed. Also, the optimal source location of MFS will be examined. The key point of these approaches is the use of the degenerate kernels, namely, the addition theorem. Frame of the three-years project is shown below. The first-year project will focus on deriving the Green’s functions with circular boundaries. Not only Laplace but also Helmholtz and biharmonic operators will be considered. For the eccentric case, a half-plane problem with a hole and infinite plane with two holes, the final successive images for the Green’s function will be linked to the focuses in the bipolar coordinates. Scattering of sound from point and axisymmetry sources by multiple circular cylinders will be solved and the results will be compared with available theoretical solutions and experimental data. Then, it will be extended to solve BVPs and Green’s function with elliptical boundaries in the second-year project. Degenerate kernels in the curvilinear coordinate system for ellipse as well as eccentric annulus should be taken with special care. An angle-type fundamental solution for the dislocation will be first expanded into separable form. Finally, three-dimensional problems will be solved in the third-year project with emphasis on eigenproblems for the eccentric sphere and multiple-spheres radiation or scattering. The fictitious frequencies on multiple-spheres radiation and scattering will be discussed. Also, the image method for solving three-dimensional spatial Green’s functions will be investigated. (See frame of project in the next page)"

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Study on Solving the Green's Function and Boundary Value Problems by Using the Image Method and the Null-Field Integral Equation

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