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Title: 含隧道之半圓形或半橢圓形山峰SH波散射問題之零場積分方程解法
Scattering Problem of Semi-Circular And/Or Semi-Elliptical Hills Containing Tunnels Subject to Sh Wave by Using the Null-Field Boundary Integral Equation Method
Authors: 陳正宗
Contributors: NTOU:Department of Harbor and River Engineering
國立臺灣海洋大學:河海工程學系
Keywords: SH波;繞射;半圓山峰;零場邊界積分方程法;退
SH-wave;diffraction;semi-circular hill;null-field BIEM;degenerate kernel
Date: 2012-08
Issue Date: 2013-10-07T02:25:50Z
Publisher: 行政院國家科學委員會
Abstract: 摘要:地形效應對於地震波所產生的地表位移的影響是很重要的議題,因此地震波作用到山峰所產生的 放大效應是結構安全的重要因素。根據申請人NTOU/MSV 研究團隊求解半圓形山谷問題成功的經驗 (國科會計畫NSC95-2115-M-019-003-MY2 與SDEE2008 論文),本計畫第一年擬用零場積分方程法 搭配退化核處理含圓形隧道之半圓形或半橢圓形山峰受SH 波入射的散射問題,為了充分應用零場積 分方程搭配退化核的解析特性,求解此問題需將凸起的山峰部分先取自由體圖,將求解的問題領域拆 成兩個次領域,第一個次領域為取出的圓形山峰含隧道部份,可將之視為一個同心圓問題;另一次領 域可視為一個半圓形山谷問題,將其埋入無窮域後成為一個無限域含圓形人工邊界的問題。針對其圓 形邊界基本解可使用退化核來展開。邊界密度可使用傅立葉係數的自然基底展開。透過這種方式配置 點即可真實座落在問題的邊界上並且無須計算柯西及阿達馬主值;滿足各種邊界條件、界面連續與平 衡後產生六條限制方程式可建立一套線性代數系統。在第二年中,擬推廣到含橢圓隧道之半橢圓形山 峰問題,為了充分利用橢圓幾何特性,我們使用橢圓座標系統將基本解展成含Mathieu 函數的退化核 形式,邊界物理量亦使用Mathieu 函數為基底展開。最後一年我們將嘗試使用這套線性系統求解半圓 形或半橢圓形山峰中含夾雜問題,尺寸效應、入射波數與入射角的參數研究均是我們有興趣的議題。 本計劃將提供很多測試例題的半解析法,另外,我們將一倂探討地震波聚焦效應以及SH 波在山峰散 射和水波在港池共振兩者看似不同的問題間之相似性。此國科會計畫書的組織架構如圖一所示。
abstarct:The influence of a hill on the ground motion due to earthquake is an important issue for engineers. The amplification factor is a key for structural safety. The scattering problem of SH wave by hills containing tunnels will be point the main focus of this proposal. Following the successful experiences of solving the scattering problem subject to the SH wave for a semi-circular canyon by the PI’s group (NSC95-2115-M-019-003-MY2 and “Surface motion of multiple alluvial valleys for incident plane SH-waves by using a semi-analytical approach,” SDEE, Vol. 28, pp. 58-72, 2008 [1]), we will extend the null-field boundary integral equation method (BIEM) to deal with the hill problem instead of canyon problem. To fully utilize the analytical property of the null-field boundary integral equations in conjunction with the degenerate kernels for solving the hill problem, the original problem is divided into two subdomains by using the concept of taking free body. The one region is a semi-circular hill containing tunnels and the other is a half-plane problem containing a semi-circular cut. The half-plane problem can be imbedded to an infinite domain with an artificial boundary of a full circle such that the degenerate kernel can be fully adopted. Next, the fundamental solution is expanded to the degenerate kernel in the polar coordinates for the problem containing circular boundaries. Boundary densities are expanded by using the natural base of the Fourier series. By this way, the field point can be exactly located on the real boundary and is free from calculating Cauchy and Hadamard principal values. Then, we will consider the displacement continuity and equilibrium of force on the artificial interface. Finally, the semi-analytical formulation of the null-field BIE in companion with matching boundary conditions can yield six constraint equations. In the second year, we will extend the proposed approach to deal with scattering problems of semi-elliptical hills containing circular or elliptical tunnels subject to the SH wave. For fully utilizing the elliptic geometry, the fundamental solution expanded into the degenerate form in terms of the elliptic coordinate system by using the Mathieu functions and modified Mathieu functions. Boundary densities are represented by using the Mathieu functions, too. In the third year, we attempt to solve the problem of semi-circular or semi-elliptical hills containing circular or elliptical inclusions. We will provide semi-analytical solutions for several benchmark examples. Parameter studies of the tunnel size, wave number and incident angle of SH wave will be further investigated. The focusing effect will be discussed. Moreover, similarity between the SH wave impinging on the hill and harbor resonance caused by water-waves will be also addressed. The frame of this NSC proposal is shown in Fig. 1.
Relation: NSC100-2221-E019-040-MY3
URI: http://ntour.ntou.edu.tw/handle/987654321/34327
Appears in Collections:[Department of Harbor and River Engineering] Research Reports

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