本文選題：能帶結(jié)構(gòu)　切入點(diǎn)：Bloch原理　出處：《廣西民族大學(xué)》2016年碩士論文

【摘要】：自上世紀(jì)90年代以來(lái),在固體力學(xué)中具有廣泛應(yīng)用的能帶理論被拓廣到研究液體表面波在海底周期排列的無(wú)窮陣列結(jié)構(gòu)上的傳播并取得大量成果.然而很遺憾,二十多年來(lái)國(guó)際上所有的相關(guān)研究,幾乎全都局限于簡(jiǎn)單的分段或分片常數(shù)深度的底部結(jié)構(gòu),如二維矩形臺(tái)階系列和三維(出水、水下或底部鉆孔)直立圓柱陣列或直立棱柱陣列等,而周期性排列的變水深無(wú)限結(jié)構(gòu)陣列幾乎從未被研究過(guò),原因在于變水深情形下,若采用淺水波方程為控制方程,則需要求解變系數(shù)偏微分方程或常微分方程,若采用葉氏方程或者緩坡類(lèi)方程,由于線性波色散關(guān)系中波數(shù)為水深函數(shù)的隱函數(shù),則需要求解系數(shù)為隱函數(shù)形式的偏微分方程或常微分方程.本學(xué)位論文研究線性淺水波越過(guò)海底周期排列的無(wú)限圓臺(tái)陣列的能帶結(jié)構(gòu).顯然,在圓臺(tái)之上的水深是連續(xù)變化的,稱(chēng)為變水深,這是以往有關(guān)液體表面波能帶結(jié)構(gòu)研究中極少涉及的.因?yàn)楸疚目紤]的是線性淺水波,所以采用的控制方程為線性淺水波方程(也稱(chēng)線性長(zhǎng)波方程).基于二元周期函數(shù)的傅里葉級(jí)數(shù)展開(kāi)理論,我們將周期變化的水深函數(shù)展開(kāi)成傅里葉級(jí)數(shù),同時(shí)也將所求的自由液面高程函數(shù)的周期性部分展開(kāi)成傅里葉級(jí)數(shù),系數(shù)待定.然后將它們雙雙代入控制方程,并對(duì)傅里葉級(jí)數(shù)的無(wú)限求和進(jìn)行有限截?cái)?將原本是求能帶結(jié)構(gòu)的問(wèn)題轉(zhuǎn)化為求矩陣特征值的問(wèn)題.最后,我們計(jì)算出了淺水波在按正方晶格和六角晶格兩種方式排列的無(wú)窮周期陣列上的能帶結(jié)構(gòu),并發(fā)現(xiàn)針對(duì)某些地形參數(shù),正立圓臺(tái)或倒立圓臺(tái)都可能形成相應(yīng)的完全頻隙,所謂完全頻隙是指在任何傳播方向,頻率落在此頻隙區(qū)間的波在相關(guān)周期地形上都是絕對(duì)禁止或無(wú)法存在.進(jìn)一步,我們分析討論了不同地形參數(shù)尤其是圓臺(tái)上下底面的填充率對(duì)頻隙寬度以及頻隙位置的影響.所得結(jié)果對(duì)近海工程中有限周期排列結(jié)構(gòu)物的建造和優(yōu)化具有理論性的參考價(jià)值.
[Abstract]:Since the 1990s, the energy band theory, which has been widely used in solid mechanics, has been extended to study the propagation of liquid surface waves on the periodic array structure of the seabed, and a lot of results have been obtained.Unfortunately, however, for more than two decades, almost all of the relevant studies have been confined to simple bottom structures with constant depths of segments or slices, such as two-dimensional rectangular step series and three-dimensional (effluent).Underwater or bottom drilling) vertical cylindrical arrays or vertical prism arrays, etc., but periodic arrays of infinite structures with variable water depth have almost never been studied. The reason is that in the case of varying water depth, the shallow water wave equation is used as the governing equation.It is necessary to solve the partial differential equation or ordinary differential equation with variable coefficients. If the Yehlet equation or the gentle slope equation is used, the wave number in the linear wave dispersion relation is an implicit function of the water depth function.Then it is necessary to solve partial differential equations or ordinary differential equations with implicit coefficients.In this dissertation, we study the band structure of linear shallow water waves across an infinite array of circular arrays arranged periodically on the seafloor.It is obvious that the water depth above the platform is continuously varying, which is rarely involved in the previous researches on the energy band structure of liquid surface waves.Because the linear shallow water wave is considered in this paper, the governing equation is linear shallow water wave equation (also called linear long wave equation).Based on the Fourier series expansion theory of binary periodic function, we expand the periodically varying water depth function into Fourier series, and at the same time, we expand the periodic part of the free liquid level elevation function into Fourier series, and the coefficients are to be determined.Then they are replaced into the governing equation and the infinite sum of Fourier series is truncated finitely. The problem of finding energy band structure is transformed into the problem of finding the eigenvalue of matrix.Finally, we calculate the energy band structure of shallow water waves on infinite periodic arrays arranged in square lattice and hexagonal lattice, and find that for some terrain parameters, both orthotropic and inverted stations may form a corresponding complete frequency gap.The so-called complete frequency gap means that the wave falling in the frequency interval in any direction of propagation is absolutely forbidden or unable to exist on the related periodic terrain.Furthermore, the influence of different topographic parameters, especially the filling ratio of the bottom surface of the platform, on the frequency gap width and the frequency slot position is analyzed and discussed.The obtained results are of theoretical reference value for the construction and optimization of finite periodic structures in offshore engineering.
【學(xué)位授予單位】：廣西民族大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：P751;O481.1

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本文編號(hào)：1711168