本文選題：等幾何分析　切入點(diǎn)：Fortran　出處：《山東大學(xué)》2016年博士論文　論文類(lèi)型：學(xué)位論文

【摘要】：等幾何分析(IGA,Isogeometric Analysis)于2005年被提出,旨在將設(shè)計(jì)、分析和優(yōu)化集成在一起。等幾何分析利用準(zhǔn)確描述幾何形狀的基函數(shù)(如NURBS, Non Uniform Rational B-spline)作為計(jì)算解空間的函數(shù),統(tǒng)一的表達(dá)形式使得分析模型和幾何模型可以無(wú)障礙地交互。隨著工程實(shí)際中模型的日益復(fù)雜化,有限元分析中模型的網(wǎng)格生成耗費(fèi)了大量的時(shí)間,而等幾何分析將計(jì)算機(jī)輔助幾何設(shè)計(jì)(CAD,Computer Aided Design)與計(jì)算機(jī)輔助工程分析(CAE,Computer Aided Engineering)進(jìn)行了無(wú)縫結(jié)合,很好地解決了這一問(wèn)題。等幾何分析已成為當(dāng)前工程分析發(fā)展的趨勢(shì),將會(huì)對(duì)CAD和CAE產(chǎn)生重大影響�，F(xiàn)有的有限元環(huán)境很多基于Fortran語(yǔ)言,利用Fortran對(duì)等幾何分析進(jìn)行編程有希望實(shí)現(xiàn)等幾何分析與現(xiàn)有有限元分析的集成。作者利用Fortran語(yǔ)言對(duì)NURBS等幾何分析的理論以及實(shí)現(xiàn)方法進(jìn)行了研究,并將基于Fortran的等幾何分析用于不同的力學(xué)問(wèn)題以提高等幾何分析的計(jì)算效率和準(zhǔn)確性。為了提高程序的運(yùn)行效率,同時(shí)也為了使程序能夠方便地被不同的算例調(diào)用,文中采用Fortran語(yǔ)言中廣泛使用的模塊化編程。為了使Fortran編程能對(duì)不同的算例兼容,作者最大化的將共用子程序,如高斯積分,NURBS基函數(shù)及其導(dǎo)數(shù)的計(jì)算,連接矩陣等,編入同一模塊,對(duì)Fortran用于等幾何分析的研究做了大量的基礎(chǔ)性工作。另外,現(xiàn)有的等幾何分析軟件在解決大型稀疏矩陣問(wèn)題時(shí),計(jì)算效率低且不穩(wěn)定,作者將一種大型稀疏矩陣的求解器GSS(Grus Sparse Solver)植入Fortran編程中,使得等幾何分析的效率大大提高。文中提供可下載的相關(guān)算例的Fortran源程序。NURBS基函數(shù)通常不滿(mǎn)足克羅內(nèi)克函數(shù)的性質(zhì),即不具有插值性�？刂泣c(diǎn)不在邊界上時(shí),難以直接施加位移邊界條件。作者采用罰函數(shù)法處理邊界條件,但罰因子取值的大小會(huì)影響計(jì)算結(jié)果。作者通過(guò)不同的算例探索合適的罰因子取值大小,結(jié)果表明,罰因子的取值在高于整體剛度矩陣中絕對(duì)值最大值2-3個(gè)數(shù)量級(jí)時(shí),計(jì)算結(jié)果較為準(zhǔn)確。此外,提高模型的網(wǎng)格質(zhì)量可以減輕罰因子的影響,但高質(zhì)量的網(wǎng)格勢(shì)必會(huì)導(dǎo)致計(jì)算工作量的增加。作者完善了基于罰函數(shù)法處理邊界條件的Kirchhoff-Love板殼理論。NURBS函數(shù)不僅用來(lái)描述單元形狀和位移場(chǎng),而且還提供了Kirchhoff-Love理論所需要的高階連續(xù)函數(shù)。將基于Fortran的等幾何分析用于不同的板殼算例模擬,證明了其準(zhǔn)確性和快速收斂性,同時(shí)說(shuō)明了等幾何分析即使在粗糙的網(wǎng)格水平上,也能得到準(zhǔn)確的計(jì)算結(jié)果。作者還結(jié)合擴(kuò)展有限元與等幾何分析對(duì)斷裂力學(xué)問(wèn)題進(jìn)行了研究,證明了基于Fortran的擴(kuò)展等幾何分析模擬不連續(xù)問(wèn)題的有效性,給出了選擇強(qiáng)化控制點(diǎn)的方法并對(duì)裂紋不連續(xù)域和尖端位移場(chǎng)分別利用Heavisde方程和裂紋尖端方程進(jìn)行了強(qiáng)化。通過(guò)與擴(kuò)展有限元用于模擬相同的裂紋模型作比較,說(shuō)明了擴(kuò)展等幾何分析僅需更少的單元便可獲得準(zhǔn)確的結(jié)果。在對(duì)不同算例位移和應(yīng)力場(chǎng)的表達(dá)上,高階函數(shù)的應(yīng)用使得其光滑、連續(xù)。為了實(shí)現(xiàn)局部細(xì)化,作者利用多面片技術(shù)對(duì)帶孔平板問(wèn)題模型進(jìn)行了分片處理,并在每一片模型上進(jìn)行了位移和應(yīng)力計(jì)算,得到了準(zhǔn)確的結(jié)果。在模擬斷裂力學(xué)問(wèn)題時(shí),利用線(xiàn)性節(jié)點(diǎn)值的插入對(duì)裂紋區(qū)域進(jìn)行了局部細(xì)化,減少了計(jì)算誤差。根據(jù)不同的算例應(yīng)用不同程度的細(xì)化方法所獲得的結(jié)果來(lái)看,提高網(wǎng)格的細(xì)化質(zhì)量,可以使計(jì)算結(jié)果更加準(zhǔn)確,位移和應(yīng)力場(chǎng)表達(dá)的更加連續(xù)。綜上,作者開(kāi)發(fā)了基于Fortran的新的等幾何分析工具,并將其用于不同的力學(xué)問(wèn)題以驗(yàn)證其準(zhǔn)確性和高效性。同時(shí),完善了等幾何分析中的相關(guān)理論,對(duì)等幾何分析中出現(xiàn)的問(wèn)題進(jìn)行了分析,并提出合理的改進(jìn)方法。然而由于時(shí)間和條件有限,還需對(duì)程序進(jìn)行優(yōu)化,也需要將Fortran編程應(yīng)用于更多的等幾何分析算法。
[Abstract]:The geometric analysis (IGA, Isogeometric Analysis) was put forward in 2005, aims to design, analysis and optimization are integrated together. The geometric analysis using the accurate description of the geometry of the base function (such as NURBS, Non Uniform Rational B-spline) as a function of solution space, a uniform expression makes the analysis model and the geometric model can be barrier free interaction with the model. In practical engineering becomes more and more complex, grid generation in the finite element analysis model of the spent a lot of time, and the geometric analysis of computer aided geometric design (CAD Computer, Aided Design) analysis and Computer Aided Engineering (CAE Computer, Aided Engineering) for seamless integration, a good solution this problem. The geometric analysis has become the development trend of current engineering analysis, will have a significant impact on CAD and CAE. Many of the existing finite element environment based on F Ortran language, using Fortran programming equivalence geometric analysis to achieve the integration of geometric analysis with existing finite element analysis. The author analysis by Fortran language on the NURBS geometry theory and realization method is studied, and the analysis of computational efficiency and accuracy of the geometric analysis on different mechanical problems in order to improve the Fortran. Based on the geometry. In order to improve the efficiency of the program, but also to make the program can easily be different examples called modular programming Fortran language widely used in this paper. In order to make the Fortran programming can be compatible with the example of different authors, the maximum common subroutines, such as the Gauss integral calculation, NURBS basis function and derivative, connection matrix, into one module of Fortran for the research of geometric analysis foundation has done a lot of work. In addition, the existing etc. Geometric analysis software in solving large sparse matrix problems, the computational efficiency is low and unstable, the author of a large sparse matrix solver GSS (Grus Sparse Solver) with Fortran programming, the efficiency is greatly improved. The geometric analysis provides downloadable relevant examples of Fortran source program based on.NURBS function normally do not meet the properties of Kronecker function in this paper, which has no interpolation. Control points are not on the boundary, it is difficult to directly impose displacement boundary conditions. The author adopts the penalty function method to deal with the boundary conditions, but the penalty factor value will affect the results. The author through different examples to explore the value of penalty factor, right the results show that the value of penalty factor in higher than the maximum absolute value of 2-3 orders of magnitude in the global stiffness matrix, the calculation result is accurate. In addition, to improve the quality of the mesh model can reduce the penalty for The influence of the grid, but high quality will inevitably lead to the increase of computational effort. The author improves the penalty function method to deal with the boundary conditions of the Kirchhoff-Love shell theory based on.NURBS function is not only used to describe the element shape and the displacement field, but also provides a high order continuous function required by the Kirchhoff-Love theory. The Fortran analysis for different geometry etc. the numerical simulation based on the shell, to prove its accuracy and fast convergence, and discusses the geometric analysis even in rough grid level, also can get accurate results. The author also analyzes the fracture mechanics problems of finite element and geometric expansion, proved the validity analysis of simulation of discontinuous the problem of Fortran extension based on geometry, given the choice of strengthening method of control points and the crack tip displacement field and discontinuous domain respectively by Heavisd The e equation and the equation of crack tip has been strengthened. With the extended finite element method is used to simulate the crack model of the same comparison, illustrates the extension of geometric analysis only less unit can obtain accurate results. In different examples of displacement and the expression of stress field on the application of higher order functions makes it smooth in order to realize the continuous, local refinement, the problem with the plate model were divided by multi slice technique, and the displacement and stress were calculated in each model, get accurate results. In the simulation of fracture mechanics problems, using the linear insert node value by local refinement the crack area, reduce the calculation error. According to the different cases considered refinement methods different application results show that the improved grid refinement quality, can make the results more accurate, the displacement and stress field The more continuous. In summary, the author developed based on new geometric analysis tool Fortran, and used different mechanical problems to verify its accuracy and efficiency. At the same time, improve the relevant theory of geometry analysis, the problems of equivalence in geometric analysis are analyzed, and put forward the improvement method is reasonable however, due to the limited time and conditions, but also need to optimize the process, also need to be used in Fortran programming more geometric analysis algorithm.

【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O342

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