本文關(guān)鍵詞：等幾何分析方法的本質(zhì)邊界條件處理研究　出處：《西北工業(yè)大學(xué)》2016年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 等幾何分析 本質(zhì)邊界條件 Dirichlet邊界條件 Nitsche法 水平集方程 Laplace-Bettrami方程 NURBS樣條

【摘要】：等幾何分析是近年來備受關(guān)注的一種新型數(shù)值分析方法。隨著問題規(guī)模越來越大、幾何模型越來越復(fù)雜,繁瑣的網(wǎng)格劃分成為制約有限元發(fā)展的瓶頸。等幾何分析采用CAD中精確描述幾何形狀的樣條函數(shù)進(jìn)行力學(xué)分析,設(shè)計(jì)模型和分析模型采用同一種幾何表示,解決CAD和CAE的模型異構(gòu)問題。論文選擇本質(zhì)邊界條件處理作為切入點(diǎn),以固體介質(zhì)傳熱、線彈性結(jié)構(gòu),不可壓縮流動(dòng)和曲面上的偏微分方程為應(yīng)用對(duì)象,研究等幾何分析在應(yīng)用中遇到的問題。由于樣條函數(shù)缺乏插值性,等幾何分析很難直接處理本質(zhì)邊界條件。如果像有限元一樣施加在單元結(jié)點(diǎn)上,會(huì)造成計(jì)算精度降低和收斂退化。針對(duì)這一問題,論文提出兩類本質(zhì)邊界條件處理方法。并在曲面偏微分方程幾何精確有限元求解的基礎(chǔ)上,提出參數(shù)曲面水平集方程及其數(shù)值計(jì)算。論文的主要研究?jī)?nèi)容包括:(1)提出基于樣條擬合的“強(qiáng)處理”方法“強(qiáng)處理”法是在有限元空間施加邊界約束的一類方法。基于樣條擬合的思想,論文提出三種“強(qiáng)處理”方法,包括:邊界配點(diǎn)法、最小二乘擴(kuò)展以及構(gòu)造等效插值基函數(shù)的轉(zhuǎn)換矩陣法。邊界配點(diǎn)法是在邊界上引入一組精心設(shè)計(jì)的插值點(diǎn),將本質(zhì)邊界條件處理轉(zhuǎn)化為樣條擬合問題。最小二乘擴(kuò)展是在配點(diǎn)法的基礎(chǔ)上,構(gòu)造邊界條件誤差殘余的最小二乘泛函,在求解精度、數(shù)值穩(wěn)定性和適用性上都要優(yōu)于配點(diǎn)法。論文第三章以固體介質(zhì)傳熱為例,與有限元的“直接施加法”對(duì)比,提出的方法在精度和收斂性上都有顯著提升。論文第五章從NURBS曲面提取對(duì)應(yīng)的有理Bézier單元,基于等幾何分析求解定義在參數(shù)曲面的Laplace-Beltrami方程。針對(duì)Bernstein多項(xiàng)式缺乏插值性的問題,提出一種等效插值多項(xiàng)式的矩陣轉(zhuǎn)化算法。轉(zhuǎn)換后的多項(xiàng)式滿足插值屬性,避免了本質(zhì)邊界條件處理的困難。由于有理Bézier單元可以精確描述球面、柱面等二次曲面表示的工業(yè)產(chǎn)品外形,在求解曲面偏微分方程時(shí)比有限元的精度更高。(2)提出基于Nitsche變分法的“弱處理”方法“弱處理”法是對(duì)等效積分形式進(jìn)行修正的一類方法。第三章將Nitsche法應(yīng)用到線彈性問題的位移邊界條件處理,通過拉格朗日乘子識(shí)別,構(gòu)造對(duì)應(yīng)的無約束泛函。并針對(duì)線彈性問題,證明了Nitsche變分法的條件穩(wěn)定性,給出穩(wěn)定系數(shù)的計(jì)算公式。與同類方法相比,Nitsche法具有自由度少、數(shù)值穩(wěn)定、懲罰項(xiàng)容易控制的優(yōu)勢(shì),第四章將其推廣到兩類重要的流體力學(xué)問題。對(duì)于速度場(chǎng)-壓強(qiáng)形式的二維Stokes流動(dòng),采用混合有限元求解需要構(gòu)造滿足inf-sup穩(wěn)定條件的單元格式。針對(duì)這一問題,引入流函數(shù)將其轉(zhuǎn)化四階偏微分方程,采用高階連續(xù)的NURBS基函數(shù)求解。最后混合采用Nitsche法和最小二乘擴(kuò)展法對(duì)該問題的兩組本質(zhì)邊界條件進(jìn)行處理。對(duì)于不可壓縮Navier-Stokes方程,論文推導(dǎo)出包含切向和法向全部速度Dirichlet邊界條件的新型等效積分公式。新公式更為一般化,可以“弱形式”處理無穿透壁面條件。對(duì)于生成的非線性方程組,給出切線剛度矩陣,采用Newton-Raphson法求解。(3)研究參數(shù)曲面水平集方程及其數(shù)值計(jì)算在傳統(tǒng)歐氏空間水平集的基礎(chǔ)上,論文提出參數(shù)曲面上的水平集方程,可應(yīng)用于曲面上的動(dòng)態(tài)界面追蹤問題。研究了參數(shù)曲面水平集方程的幾何性質(zhì),推導(dǎo)出約束在參數(shù)曲面上、隱式表示的空間曲線的切面法矢和曲率計(jì)算公式。最后采用樣條配點(diǎn)法數(shù)值求解參數(shù)曲面水平集方程。
[Abstract]:Geometric analysis is a new numerical analysis method has attracted much attention in recent years. With the increasingly large scale, the geometric model is more and more complex, cumbersome mesh has become a bottleneck restricting the development of finite element. The geometric analysis using a precise description of geometry in CAD splineto mechanics analysis, the design model and analysis model using the same geometry model solve the heterogeneous problem of CAD and CAE. The treatment of essential boundary conditions as a starting point, with solid medium heat transfer, linear elastic structure, incompressible flow and partial differential equations on the surface as the application object, on the geometric analysis of the problems in the application of the spline function. The lack of interpolation, geometric analysis is very difficult to directly deal with the essential boundary conditions. If a finite element like force on the element node, will cause the calculation accuracy and reduce the convergence deterioration. To solve this problem, this paper puts forward two kinds of essential boundary condition processing method. And based on the precise surface partial differential equation of finite element solution geometry, calculating equation and numerical parametric level set. The main research contents of this thesis include: (1) based on the spline fitting of the "strong" process method "processing method is applied in the finite element space boundary constraints for a class of spline fitting method. Based on the idea, the paper proposes three" strong processing methods, including the boundary collocation method, least squares expansion and conversion structure equivalent polynomial matrix method. The boundary collocation method is introducing a set of carefully designed the interpolation points on the boundary, the treatment of essential boundary conditions into spline fitting problem. The least square collocation method is extended based on the least squares functional structure boundary conditions of residual error, the accuracy of numerical solution. Stability and applicability is better than the collocation method. The third chapter in the solid medium heat transfer as an example, and directly applying the finite element method, proposed method has significantly improved the accuracy and convergence. The fifth chapter extracts the corresponding rational B zier unit from NURBS based on the surface. The geometric analysis solution is defined in the Laplace-Beltrami equation for Bernstein polynomial parametric surfaces. The lack of interpolation problems, put forward a matrix equivalent polynomial interpolation algorithm. The polynomial transformation after conversion satisfies the interpolation property, to avoid the difficulties in the treatment of essential boundary conditions. The rational B zier unit can accurately describe the sphere, cylinder two the surface shapes of industrial products, in solving surface partial differential equations than finite element accuracy. (2) based on Nitsche variational method of "weak" process method "weak processing method" Is a kind of modified method of equivalent integral form. The third chapter will deal with the displacement boundary condition Nitsche method is applied to the linear elastic problem, the Lagrange multiplier recognition, unconstrained functional structure. According to the corresponding linear problem, the stability conditions of Nitsche's variational method is proved, calculation formula of stability coefficient is given. Compared with other methods, the Nitsche method has less degree of freedom, numerical stability, the penalty is easy to control the advantage, the fourth chapter will be extended to the two important problems in fluid mechanics. For velocity field pressure in the form of two-dimensional Stokes flow, using mixed finite element solution need to be constructed to meet the stability conditions of inf-sup unit in format. This problem, introducing the stream function to convert the four order partial differential equation, by solving NURBS basis function continuous high order. Finally, by combining Nitsche method and least square method to expand the problem The two group of essential boundary conditions for processing. For the incompressible Navier-Stokes equation, this paper derives the new equivalent integral formula contains tangential and normal velocity Dirichlet all boundary conditions. The new formula is more general, can be "weak form" impenetrable wall conditions. For the nonlinear equations generated by the given tangent the stiffness matrix is calculated by Newton-Raphson method. (3) based on parametric level set equation and its numerical calculation in the traditional Euclidean space level set, the parameters on the surface level set equation, dynamic tracking interface can be applied to the surface of the problem. The study of geometrical parameters of surface level set equation, push derived constraints in parametric surface, calculation formula of plane normal vector and curvature of space curve is represented implicitly. Finally using the spline collocation method for numerical parametric level set equation.

【學(xué)位授予單位】：西北工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O241

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