本文選題：邊界元法　切入點(diǎn)：Burton-Miller邊界積分方程　出處：《清華大學(xué)》2015年博士論文　論文類(lèi)型：學(xué)位論文

【摘要】：航天器在發(fā)射階段除經(jīng)受運(yùn)載火箭向上傳遞的機(jī)械振動(dòng)之外,還有排氣噪聲和氣動(dòng)噪聲經(jīng)整流罩傳遞到航天器表面。因此,薄壁結(jié)構(gòu)的聲固耦合問(wèn)題是航天器力學(xué)環(huán)境預(yù)示的重要組成部分,對(duì)于指導(dǎo)航天器設(shè)計(jì)有重要作用。噪聲激勵(lì)的頻率范圍可達(dá)10~10000Hz,中高頻段呈現(xiàn)明顯的隨機(jī)特性,只能采用統(tǒng)計(jì)能量分析等方法,而低頻段主要呈現(xiàn)確定性的耦合振動(dòng),邊界元法是一種可供選擇的分析方法,相關(guān)研究也有重要的理論意義。本文建立了一套求解薄壁結(jié)構(gòu)聲固耦合問(wèn)題的高精度邊界元法的框架,在以下四個(gè)方面取得了創(chuàng)新成果。第一,提出了聲場(chǎng)問(wèn)題的一種新的高精度邊界元法。這種新方法基于聲場(chǎng)的Burton-Miller邊界積分方程,采用保持邊界原始幾何形狀的聲壓連續(xù)單元,在初始設(shè)定比較合理網(wǎng)格的基礎(chǔ)上,充分保證核函數(shù)與形函數(shù)乘積在單元上積分的精度,求解得到初始解,同時(shí)用相鄰單元間聲壓梯度的相對(duì)間斷值作為離散誤差指示來(lái)顯示解的精度,并指導(dǎo)網(wǎng)格細(xì)分重新計(jì)算,再通過(guò)比較兩次計(jì)算結(jié)果來(lái)判斷收斂情況,決定是否還要進(jìn)一步細(xì)分網(wǎng)格,直至得到滿意的收斂解。文中以球形邊界為例,構(gòu)造了四類(lèi)球面參數(shù)單元,用剛球散射聲場(chǎng)問(wèn)題對(duì)誤差指示進(jìn)行了驗(yàn)證,并用它求解了較復(fù)雜的多球散射問(wèn)題。第二,發(fā)展了球面單元上弱奇異積分和超奇異積分的計(jì)算方法。推導(dǎo)了球面單元上各種奇異積分的最終格式。用Guiggiani方法求解了球面單元的超奇異積分,高精度的計(jì)算結(jié)果表明了超奇異積分計(jì)算的準(zhǔn)確性。并將高效偉提出的徑向積分方法用于求解聲學(xué)問(wèn)題中的超奇異積分計(jì)算,與Guiggiani方法進(jìn)行了對(duì)比。第三,為建立三維薄壁結(jié)構(gòu)彈性動(dòng)力學(xué)頻域分析的高精度邊界元法,發(fā)展了保證核函數(shù)與形函數(shù)乘積在單元上積分精度的高效方法,其中包括:推導(dǎo)了自由項(xiàng)的最終格式,實(shí)現(xiàn)了自由項(xiàng)的直接計(jì)算;實(shí)現(xiàn)了球面單元、8節(jié)點(diǎn)等參單元上Cauchy主值積分的直接計(jì)算。第四,提出了將聲場(chǎng)頻域分析與三維薄壁結(jié)構(gòu)彈性動(dòng)力學(xué)頻域分析直接耦合的聲固耦合問(wèn)題的高精度邊界元法計(jì)算方案,為求解聲固耦合問(wèn)題提供了新思路。耦合后的方程是全邊界元方程,因此邊界元方法中的快速算法將可方便地引入,為高性能邊界元法(即引入快速算法的高精度邊界元法)的建立提供了基礎(chǔ)。
[Abstract]:In addition to the mechanical vibration transmitted upward by the launch vehicle during the launch phase, the spacecraft also has exhaust noise and aerodynamic noise transmitted to the surface of the spacecraft through the fairing. The acousto-solid coupling problem of thin-walled structures is an important part of spacecraft mechanical environment prediction, and plays an important role in guiding spacecraft design. The frequency range of noise excitation can reach 10 ~ 10000Hz. The method of statistical energy analysis can only be used, while the low frequency band mainly presents deterministic coupled vibration. The boundary element method is an alternative analysis method. In this paper, a set of high-precision boundary element method for solving acousto-solid coupling problem of thin-walled structures is established, and some innovative results are obtained in the following four aspects. A new high-precision boundary element method for acoustic field problems is proposed, which is based on the Burton-Miller boundary integral equation of sound field and adopts the sound pressure continuous element with preserving the original geometry shape of the sound field. The accuracy of the product of kernel function and shape function is fully guaranteed on the element, and the initial solution is obtained. The relative discontinuous value of sound pressure gradient between adjacent elements is used as the indication of discrete error to show the accuracy of the solution, and the mesh subdivision recalculation is guided. The convergence is judged by comparing the results of two calculations, and it is decided whether to subdivide the mesh further until a satisfactory convergence solution is obtained. In this paper, four kinds of spherical parameter elements are constructed, taking the spherical boundary as an example. The error indication is verified by the sound field problem of rigid sphere scattering, and the more complex multi-sphere scattering problem is solved. In this paper, the calculation methods of weak singular integrals and hypersingular integrals on spherical elements are developed. The final forms of various singular integrals on spherical elements are derived. The hypersingular integrals of spherical elements are solved by Guiggiani method. The high precision calculation results show the accuracy of the hypersingular integral calculation. The radial integral method proposed by Hexiwei is compared with the Guiggiani method in solving the acoustic problem. In order to establish a high-precision boundary element method for the frequency-domain analysis of three-dimensional thin-walled structures in elastic dynamics, an efficient method to ensure the integration accuracy of the product of kernel function and shape function on the element is developed, which includes: the final format of the free term is derived. Direct calculation of free term and direct calculation of Cauchy principal integral on 8 node isoparametric element of spherical element are realized. 4th, A high-precision boundary element method is proposed to calculate the acousto-solid coupling problem in which the acoustic field frequency domain analysis is directly coupled with the three-dimensional thin-walled structure elastic dynamics frequency domain analysis. The coupled equation is the whole boundary element equation, so the fast algorithm in the boundary element method can be introduced conveniently. It provides the foundation for the establishment of high performance boundary element method (i.e. the high precision boundary element method with fast algorithm).
【學(xué)位授予單位】：清華大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：V414.4

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