本文選題：等幾何分析 + 比例邊界有限元��； 參考：《大連理工大學(xué)》2013年博士論文

【摘要】：本文基于等幾何分析和比例邊界有限元方法,研究開(kāi)發(fā)了高精度、高效率的工程數(shù)值計(jì)算方法,將其應(yīng)用于大壩-庫(kù)水-地基系統(tǒng)的動(dòng)力分析,并拓展至結(jié)構(gòu)分析、電磁場(chǎng)分析和薄板彎曲和振動(dòng)分析等工程問(wèn)題的求解。 等幾何分析是近年來(lái)發(fā)展的一種新型的數(shù)值方法,旨在實(shí)現(xiàn)CAD和CAE的無(wú)縫統(tǒng)一。該方法將CAD中對(duì)幾何形狀進(jìn)行精確描述的樣條函數(shù)作為結(jié)構(gòu)分析的形函數(shù),可大大提高函數(shù)梯度場(chǎng)的計(jì)算精度,并在保持幾何形狀不變的前提下,方便地實(shí)現(xiàn)自適應(yīng)的非通訊性細(xì)分操作。比例邊界有限元方法是一種求解偏微分方程的半數(shù)值半解析解法,只需對(duì)求解區(qū)域的邊界進(jìn)行有限元離散,降低求解規(guī)模,但又無(wú)需基本解,特別適合于求解含有無(wú)限域、線彈性裂尖奇異應(yīng)力場(chǎng)等工程問(wèn)題。 本文將等幾何分析方法和比例邊界有限元方法相結(jié)合,提出了新型的比例邊界等幾何分析方法,相比于等幾何分析或者比例邊界有限元,比例邊界等幾何分析方法的計(jì)算精度和計(jì)算效率進(jìn)一步提高。因此將這種方法推廣應(yīng)用于實(shí)際工程中,將大大提高求解精度和效率。另外,此方法能和等幾何分析方法進(jìn)行無(wú)縫連接,利用等幾何分析和比例邊界等幾何分析的耦合方法,建立了大壩-庫(kù)水-無(wú)限地基時(shí)域動(dòng)力分析的計(jì)算模型,全面考慮了大壩-庫(kù)水、大壩-無(wú)限地基動(dòng)力相互作用對(duì)大壩地震響應(yīng)的影響。本文還將等幾何分析、比例邊界有限元方法、比例邊界等幾何分析的應(yīng)用領(lǐng)域進(jìn)行拓展,并對(duì)這些方法在實(shí)際應(yīng)用中所遇到的若干理論和數(shù)值問(wèn)題進(jìn)行了深入研究,并提出了有效的解決方法。主要內(nèi)容如下： (1)基于等幾何分析和比例邊界有限方法,提出了比例邊界等幾何分析方法。對(duì)彈性靜、動(dòng)力學(xué)以及電磁場(chǎng)問(wèn)題分別推導(dǎo)了其離散方程和求解格式,并進(jìn)行總結(jié)和比較。h-細(xì)分和p-細(xì)分過(guò)程僅需針對(duì)結(jié)構(gòu)環(huán)向表面邊界,在該方向上變量在相鄰單元交界面處可達(dá)到較高的連續(xù)階,而在徑向解具有解析特性。相比于其他的數(shù)值方法,可達(dá)到更高的收斂速度。針對(duì)該方法,還研究了各類邊界條件的施加策略。 (2)提出了大壩-庫(kù)水-地基系統(tǒng)動(dòng)力分析的等幾何分析方法-比例邊界等幾何分析方法時(shí)域耦合計(jì)算模型,全面考慮了大壩-庫(kù)水、大壩-無(wú)限地基動(dòng)力相互作用的影響,并將其應(yīng)用到高拱壩的地震動(dòng)力分析中,為大壩的抗震安全評(píng)價(jià)提供重要參考依據(jù)。其中在大壩-庫(kù)水動(dòng)力相互作用分析中,考慮了庫(kù)水可壓縮性、庫(kù)底淤沙對(duì)壓力波的吸收、無(wú)限水域的輻射阻尼效應(yīng),并構(gòu)造了壓力-力轉(zhuǎn)換矩陣,提高了流固耦合計(jì)算效率。在大壩-無(wú)限地基動(dòng)力相互作用分析中,采用求解加速度脈沖響應(yīng)函數(shù)的高效穩(wěn)定算法,并構(gòu)造了高效的加速度脈沖響應(yīng)函數(shù)的離散策略和自適應(yīng)策略,大大提高了相互作用力的計(jì)算精度和效率。利用該模型,分析了大壩-庫(kù)水-地基動(dòng)力相互作用對(duì)重力壩、拱壩系統(tǒng)地震響應(yīng)的影響。 (3)針對(duì)比例邊界有限元、等幾何分析、比例邊界等幾何分析在新的應(yīng)用領(lǐng)域拓展過(guò)程中遇到的若干理論和數(shù)值問(wèn)題,提出了有效的處理方法。構(gòu)造了新型的NURBS曲面的裁剪交點(diǎn)搜索算法及單元局部重構(gòu)方式。相對(duì)于傳統(tǒng)的以全域單元為搜索對(duì)象的搜索策略,新型的搜索策略則沿著裁剪曲線的切線方向逐一搜索,提高了搜索效率。提出了Lagrange乘子法解決重控制點(diǎn)問(wèn)題和非齊次邊值問(wèn)題。發(fā)展了等幾何分析在電磁場(chǎng)分析中的應(yīng)用,對(duì)靜電場(chǎng)問(wèn)題和波導(dǎo)本征問(wèn)題進(jìn)行求解,顯著地提高了計(jì)算效率和計(jì)算精度,對(duì)電容和波導(dǎo)等電子元件的設(shè)計(jì)具有重要參考意義。發(fā)展了等幾何分析在薄板彎曲與振動(dòng)分析中的應(yīng)用,無(wú)需引入轉(zhuǎn)角自由度便可滿足C’連續(xù)的要求,顯著地減少了計(jì)算自由度,并對(duì)穩(wěn)定參數(shù)的取值進(jìn)行了討論。提出了相似中心由固定型擴(kuò)展為移動(dòng)型的處理方法,為比例邊界有限元的進(jìn)一步拓展,提供了有利條件。 通過(guò)本論文的研究可看出,等幾何分析方法、比例邊界等幾何分析方法在實(shí)際工程中具有廣闊的應(yīng)用前景,相關(guān)問(wèn)題還需做更為廣泛和深入的研究。
[Abstract]:Based on the equal geometric analysis and the proportional boundary finite element method, this paper develops a high precision and high efficiency numerical calculation method, and applies it to the dynamic analysis of the dam reservoir water foundation system, and extends to the structural analysis, the electromagnetic field analysis and the solution of the thin plate bending and vibration analysis.
Equal geometry analysis is a new numerical method developed in recent years, which aims to realize the seamless integration of CAD and CAE. This method uses the spline function which describes the geometric shape accurately as the shape function of the structural analysis, which can greatly improve the calculation precision of the function gradient field, and it is convenient to keep the geometric shape unchanged. The present self-adaptive non communication subdivision operation. The proportional boundary finite element method is a semi numerical semi analytic solution for solving partial differential equations. Only the finite element method is needed to solve the boundary of the solving region, and the solution size is reduced, but it does not need the basic solution. It is especially suitable for solving the engineering questions including the infinite domain, the singular stress field of the line elastic crack tip and so on. Question.
In this paper, a new geometric analysis method is proposed by combining the geometric analysis method with the proportional boundary finite element method. Compared with the geometric analysis or the proportional boundary finite element, the calculation precision and the calculation efficiency of the geometric analysis method are further higher than that of the equal geometric analysis or the proportional boundary finite element, so this method is popularized and applied to the actual work. In the process, the solution accuracy and efficiency will be greatly improved. In addition, the method can be connected seamlessly with the geometric analysis method, and the calculation model of the time domain dynamic analysis of the dam reservoir water infinite foundation is established by using the coupling method of geometric analysis and proportional boundary, such as the geometric analysis and the proportional boundary, and the dam reservoir water and the dynamic phase of the dam infinite foundation are considered in the whole. The influence of interaction on the seismic response of the dam. This paper also extends the application fields such as geometric analysis, proportional boundary finite element method, proportional boundary and other geometric analysis, and studies some theoretical and numerical problems encountered in practical applications, and puts forward effective solutions. Below:
(1) based on the equal geometric analysis and proportional boundary finite method, the geometric analysis method of proportional boundary is proposed. The discrete equation and the solution form are derived for the elastic static, dynamic and electromagnetic problems. The.H- subdivision and the p- subdivision process only need a needle to the surface boundary of the structure ring, and the variable in this direction is in the phase. A higher continuous order can be achieved at the interface of the adjacent unit, and the radial solution has an analytical characteristic. Compared with the other numerical methods, a higher convergence rate can be achieved. In this method, the application strategies of various boundary conditions are also studied.
(2) a time-domain coupling calculation model is proposed for the geometric analysis method of the dynamic analysis of the dam reservoir water foundation system dynamic analysis, such as the proportional boundary and the geometric analysis method, which comprehensively considers the influence of the dam reservoir water, the dynamic interaction of the dam and the infinite foundation, and applies it to the seismic dynamic analysis of the high arch dam, which provides the seismic safety evaluation for the dam. In the dam reservoir hydrodynamic interaction analysis, the compressibility of the reservoir water, the absorption of the sediment to the pressure wave and the radiation damping effect of the infinite water are considered in the analysis of dam reservoir hydrodynamic interaction, and the pressure force conversion matrix is constructed to improve the calculation efficiency of the fluid solid coupling. The efficient stability algorithm of the velocity impulse response function and the discrete strategy and adaptive strategy of the efficient acceleration impulse response function are constructed, and the calculation precision and efficiency of the interaction force are greatly improved. The effect of the dynamic interaction of dam, water and foundation on the seismic response of the gravity dam and the arch dam system is analyzed.
(3) aiming at some theoretical and numerical problems encountered in the development of new application fields such as proportional boundary finite element, equal geometric analysis, proportional boundary and other geometric analysis in the new application field, an effective processing method is proposed. A new NURBS surface cutting intersection search algorithm and unit local reconstruction method are constructed. The search strategy of the search object and the new search strategy are searched one by one along the tangent direction of the clipping curve to improve the search efficiency. The Lagrange multiplier method is proposed to solve the heavy control point problem and the non homogeneous boundary value problem. The application of the equal geometric analysis in the electromagnetic field analysis is developed, and the electrostatic field and the waveguide eigenproblem are solved. The solution has greatly improved the calculation efficiency and calculation accuracy. It has important reference significance for the design of electronic components such as capacitance and waveguide. The application of equal geometry analysis to the analysis of bending and vibration of thin plates has been developed. Without the introduction of angle freedom, it can satisfy the requirement of C 'continuity and significantly reduce the degree of freedom of calculation and to stabilize the parameters. The value is discussed. The method of processing similar center from fixed extension to movable type is proposed, which provides favorable conditions for further expansion of proportional boundary finite element.
Through the study of this paper, we can see that geometric analysis methods such as geometric analysis, proportional boundary and other geometric analysis methods have broad application prospects in practical engineering, and the related problems need to be more extensive and in-depth research.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2013
【分類號(hào)】：TU311.4

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