Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method

Abstract

The multipole Trefftz method is proposed to solve the scattering of flexural wave by multiple circular inclusions in an infinite thin plate. The near-field dynamic moment concentration factor (DMCF) and the far-field scattering amplitude are determined theoretically. Owing to the addition theorem, the solution represented by multiple coordinates centered at each circle can be transformed into one coordinate centered at one circle where continuity conditions are required. In this way, a coupled infinite linear algebraic system is derived as an analytical model for an infinite thin plate with multiple inclusions subject to incident flexural wave. The formulation is general and is applicable to dealing with the problem containing multiple circular inclusions. Some numerical results are presented in the truncated finite system. The effects of the incident wave number, the thickness of inclusion and the central distance between inclusions on the DMCF and the far-field scattering amplitude are examined. Numerical results show that the DMCF of two inclusions is larger than that of one, when two inclusions are close to each other. The effect of the space between inclusions on the near-field DMCF is different from that on the far-field scattering amplitude. 本文提出多極Trefftz法以求解含多圖形置入物之無限域薄板彎曲波散射問題。動應力集中係數(Dynamic Moment Concentration Factor)與遠場散射強度將利用解析推導並以數值計算求得。由於引入加法定理，由多個座標系統所表示的解可轉換成由一個座標系統表示，而該座標系統所在的圖邊界必須滿足連續條件。依此方式可求得一無限耦合線性代數系統作為含多圖形置入物之無限域薄板彎受入射彎曲波作用的解析模式。本文理論具有一般化特性可推廣置多圖形置入物問題。在捨去高次項的有限性代數系統中，本文提出數個數值算例，以探討入射波數、置入物厚度與置入物間距等因素對動應力集中係數與遠場散射強度的影響。數值結果顯示，當置入物彼此相當接近時，雙置入物的應力集中係數大於單置入物，置入物間距對近場動應力集中係數與遠場散射強度有不同的影響。