【摘要】：分數(shù)階微積分概念歷史悠久,最早源于1695年9月L' Hospital寫給萊布尼茨的信件中。在分數(shù)階微積分被提出至今300多年時間中,由于在物理和力學等學科并未獲得廣泛的關(guān)注與應(yīng)用,而僅僅作為數(shù)學領(lǐng)域中的純理論問題被諸多數(shù)學學者研究,這其中包括Euler, Lacroix, Abel, Liouville, Riemann等。隨著對復(fù)雜物理現(xiàn)象認識程度的加深和計算機模擬能力的提高,力學與工程問題的分數(shù)階導(dǎo)數(shù)建模越來越引起人們的重視。尤其對于擴散現(xiàn)象,研究工作者發(fā)現(xiàn)越來越多的擴散現(xiàn)象不滿足Fick定律,這樣的擴散過程稱為反常擴散過程。在描述這些復(fù)雜系統(tǒng)時,由于反常擴散所具有的歷史依賴與全域相關(guān)的特性恰好可以由分數(shù)階導(dǎo)數(shù)來表示,因此較于整數(shù)階動力學方程,分數(shù)階動力學方程更能有效的描述([18,50,57,5])。無論在理論分析還是數(shù)值計算方面,新的分數(shù)階動力學方程都為數(shù)學工作者提出了新的挑戰(zhàn)。在數(shù)值計算方面,現(xiàn)在已經(jīng)有很多數(shù)值求解方法,例如有限差分法([14,42]),有限體積法([91]),有限元法([60,25]),譜方法([35,94])等。這些計算方法已經(jīng)被廣泛的應(yīng)用于反常擴散的數(shù)值模擬中。但是這些方法的誤差分析均有很強的正則性假設(shè)。通過本文第二章中的反例,我們可以看到即使分數(shù)階方程的擴散系數(shù)和右端均充分光滑,我們依然不能保證解的正則性,這是分數(shù)階動力學方程區(qū)分于整數(shù)階動力學方程的一個很重要的特性(在整數(shù)階動力學方程情況下,根據(jù)方程正則性理論,方程系數(shù)和右端的光滑性可以保證方程解的正則性)。通過這一點,我們知道上述數(shù)值方法誤差分析的假設(shè)條件缺少理論支撐,另外在方程解的正則性不強的情況下,即使方程的系數(shù)和右端光滑,我們采用高階差分,高次有限元以及標準譜方法均不能達到很好的誤差收斂情況。在計算效率方面,分數(shù)階動力學方程離散得出的系數(shù)矩陣通常為滿陣,如果我們假設(shè)矩陣的階數(shù)或問題的規(guī)模為N,則系數(shù)矩陣的存儲量為O(N2),如果用常用的直接方法求解分數(shù)階動力學方程離散得出的線性系統(tǒng),則計算復(fù)雜度為O(N3)。因此,一直以來,尤其是求解大規(guī)�；蛘叨嗑S問題時,分數(shù)階動力學模擬是很費時間的。為解決這一問題,王宏等著名學者通過分析系數(shù)矩陣的代數(shù)結(jié)構(gòu),運用快速傅立葉變換,成功的將系數(shù)矩陣存儲量降為O(N),將求解線性方程組的計算量降為每一步Krylov子空間迭代O(N log N)的計算量([79,80])。固體材料和結(jié)構(gòu)的破壞問題一直是力學研究的經(jīng)典問題,也是機械、航空航天、土木、水利和化工等領(lǐng)域關(guān)注的重點。在近場動力學理論提出之前,隨著斷裂力學、損傷力學等學科的發(fā)展和計算機軟件硬件水平的提高,研究者提出了各種不同的力學模型和數(shù)值方法來模擬固體材料和漸進破壞的全過程。這些模型均是建立在連續(xù)介質(zhì)假設(shè)之上的,他們假設(shè)介質(zhì)所有的內(nèi)部力均為接觸力,最終控制方程絕大多數(shù)由偏微分方程所描述。傳統(tǒng)的有限元法和有限差分法同樣建立在連續(xù)介質(zhì)假設(shè)的思想上,在模擬時必須明確知道斷裂的位置與尺寸,這在很多現(xiàn)實應(yīng)用中很難實現(xiàn),另外隨著斷裂的發(fā)展,傳統(tǒng)的有限元法或者有限差分法必須重新劃分網(wǎng)格,有很強的網(wǎng)格依賴性([33])。隨著不連續(xù)有限元方法的發(fā)展,在模擬固體材料斷裂及發(fā)展問題上取得了一些進步,但是在模擬高維復(fù)雜斷裂系統(tǒng)時仍有很強的局限性。為了克服連續(xù)介質(zhì)力學假定與固體材料不連續(xù)這一基本矛盾,2000年,Silling基于非局部作用建模,提出了近場動力學模型,它是用積分思想表述的積分方程([68])。這一模型不在基于連續(xù)介質(zhì)假設(shè)和求解微分方程來模擬破壞問題,而是將固體看成由由一些包含所有物質(zhì)信息的帶質(zhì)量的物質(zhì)點組成,點與點之間存在著相互作用,隨著點與點之間距離的增加,這個作用力在減弱,因此通常人們選取一個點的δ鄰域為其作用力的影響域。在該理論框架下,不連續(xù)現(xiàn)象自然產(chǎn)生,同時這一理論突破了分子動力學在計算尺度上的局限,在宏、細、微觀尺度均可表現(xiàn)出較高的求解精度。在近場動力學提出之后,很多數(shù)值方法例如無網(wǎng)格方法([64，63,70])、有限元方法([15]),基于積分的有限差分方法([77])等被提出求解近場動力學模型。在有限元情況下,已經(jīng)被證明數(shù)值解滿足最優(yōu)誤差估計。然而這些方法都有一個共同的特點,特別在求解多維問題時,由于離散所得到的系數(shù)矩陣為稠密矩陣或者滿陣(這取決于影響域δ的大小)。因此,類似分數(shù)階動力學方程,系數(shù)矩陣的存儲量為O(N2),求解最后線性系統(tǒng)的計算復(fù)雜度為O(N3)。另外如果用有限元法求解近場動力學模型,每一個系數(shù)矩陣的元素均需要計算2d次重積分,其中d是維數(shù),但由于積分核含有奇性,則計算這一積分是很耗時的。王宏等學者同樣根據(jù)矩陣的代數(shù)結(jié)構(gòu)與快速傅立葉變換,成功的將矩陣存儲量降為O(N)，將求解最后線性系統(tǒng)的計算量降為每Krylov子空間迭代O(N log N)的計算量([81，，82])�；谝陨峡紤],我們分別研究了分數(shù)階擴散方程的保高精度譜Galerkin方法和近場動力學模型的快速配置算法.本文的安排如下：在第一章中,我們給出了在本文剩余部分用到的一些基本概念,包括分數(shù)階導(dǎo)數(shù)的Riemann-Liouville導(dǎo)數(shù)定義和Caputo定義及其一些基本性質(zhì),另外我們給出了一些特殊矩陣的定義及其一些性質(zhì)。在第二章中我們給出了一種分數(shù)階擴散方程的保高精度譜Galerkin方法,這種方法可以保證在方程系數(shù)和右端都充分光滑的條件下,即使真解沒有足夠的光滑性,我們也可以保證解的高精度。并且數(shù)值解比標準譜Galerkin方法得到的數(shù)值解精度要好,因為在真解沒有足夠光滑性的條件下.標準譜Galerkin方法并不能達到高精度。我們同時證明了該方法的誤差估計。這一章中給出的算例說明了這一保高精度譜方法的有效性。在第三章中我們提出了求解二維近場動力學模型的快速配置方法。在這一章中,我們仔細分析了由配置法離散得出的系數(shù)矩陣,經(jīng)過分析我們得出系數(shù)矩陣與任何向量的乘積可以由三個block-Toeplitz-Toeplitz-block (BT-TB)矩陣與向量的乘積得到,所以系數(shù)矩陣與向量的乘法的計算復(fù)雜度為O(N log N),如果用Krylov子空間迭代法求解該線性系統(tǒng),則每一步迭代的計算量可以由O(N2)降為O(NlogN).同時從本文也可以得到該矩陣的計算機存儲量可以由O(N2)降為O(N).這一章中給出的算例說明了快速配置方法的有效性。在第四章中針對在近場動力學模型中積分核函數(shù)奇性大,運用Krylov子空間迭代求解由于配置法離散得出的線性系統(tǒng)迭代次數(shù)比較多的情況,我們提出了兩種預(yù)條件矩陣。第一種預(yù)條件是block-Cireulant-Toeplitz-block (BCTB)型的,第二種預(yù)條件是block-Circulant-Circulant-block (BCCB)型的。這兩種預(yù)條件對降低求解線性系統(tǒng)迭代次數(shù)是有效的,并且通過這一章給出的例子我們可以看到第二種預(yù)條件因為計算預(yù)條件矩陣的逆比較快速,所以計算時間會更快。在第五章中我們運用加罰的思想提出了求解一般凸區(qū)域非局部擴散模型的快速配置方法。這種方法是通過加罰將原來凸區(qū)域上的問題擴展為包含該凸區(qū)域的矩形的問題,經(jīng)過配置法離散,我們可以看到系數(shù)矩陣是一個BTTB矩陣與一個對角矩陣的和,通過這種矩陣結(jié)構(gòu),我們將系數(shù)矩陣的存儲由O(N2)降為O(N),將每一步Krylov子空間的計算量由O(N2)降為O(N log N).數(shù)值算例說明了這種方法的有效性。
[Abstract]:The concept of fractional calculus has a long history and originated from L'Hospital's letter to Leibniz in September 1695. Since it was put forward more than 300 years ago, it has been studied by many mathematicians only as a pure theoretical problem in the field of mathematics because it has not been widely concerned and applied in physics and mechanics. These include Euler, Lacroix, Abel, Liouville, Riemann and so on. With the deepening of understanding of complex physical phenomena and the improvement of computer simulation ability, the fractional derivative modeling of mechanical and engineering problems has attracted more and more attention. In describing these complex systems, the fractional-order dynamic equation is more effective than the integer-order dynamic equation ([18,50,57,5]) because the historical dependence and global dependence of the anomalous diffusion can be expressed by fractional derivatives. New fractional dynamic equations pose new challenges to mathematicians both in theoretical analysis and numerical calculation. In numerical calculation, there are many numerical methods, such as finite difference method ([14,42]), finite volume method ([91]), finite element method ([60,25]), spectral method ([35,94]) and so on. It is widely used in numerical simulation of anomalous diffusion. But the error analysis of these methods has strong regularity assumptions. Through the counter examples in Chapter 2 of this paper, we can see that even though the diffusion coefficients and the right end of the fractional order equation are smooth enough, we still can not guarantee the regularity of the solution, which is the fractional order dynamics square. A very important property of the equation that distinguishes it from the integer-order dynamic equation (in the case of the integer-order dynamic equation, according to the regularity theory of the equation, the coefficients of the equation and the smoothness of the right-hand end can guarantee the regularity of the solution of the equation). From this point, we know that the assumptions for error analysis of the above numerical methods lack theoretical support, and in addition, in the case of the integer-order dynamic equation, When the regularity of the solution of the equation is not strong, even if the coefficients of the equation and the right end of the equation are smooth, we can not achieve good error convergence by using high-order difference, high-order finite element and standard spectral method. The storage capacity of the coefficient matrix is O(N2) if the order or the scale of the problem is N, and the computational complexity is O(N3) if the linear system is discretized by solving the fractional-order dynamic equation with the usual direct method. Wang Hong and other famous scholars analyzed the algebraic structure of coefficient matrix and used fast Fourier transform to reduce the storage of coefficient matrix to O (N) and the computation of solving linear equations to O (N log N) iteration in every Krylov subspace ([79,80]). Classical problems are also the focus of attention in mechanical, aerospace, civil, hydraulic and chemical fields. Before the theory of near-field dynamics was put forward, with the development of fracture mechanics, damage mechanics and the improvement of computer software and hardware, researchers proposed various mechanical models and numerical methods to simulate solids. These models are based on the continuum assumption that all the internal forces in the medium are contact forces, and the ultimate governing equations are mostly described by partial differential equations. With the development of fracture, the traditional finite element method or finite difference method must be re-meshed, which has strong grid dependence ([33]). With the development of discontinuous finite element method, it is difficult to simulate the fracture and development of solid materials. Some progress has been made, but there are still strong limitations in simulating high-dimensional complex fracture systems. In order to overcome the basic contradiction between continuum mechanics assumption and solid material discontinuity, in 2000, Silling proposed a near field dynamic model based on nonlocal action modeling, which is an integral equation ([68]) expressed by integral theory. It is not based on the continuum hypothesis and the solution of differential equations to simulate the failure problem. Instead, the solid is considered to consist of some mass-bearing points containing all the material information. There is an interaction between points. As the distance between points increases, the interaction force is weakening, so people usually choose a point. In the framework of this theory, discontinuities occur naturally, and this theory breaks through the limitations of molecular dynamics on computational scales. It can show high accuracy in macro, fine and micro scales. Finite element method ([15]), integral-based finite difference method ([77]) and so on are proposed to solve the near-field dynamic model. In the case of finite element method, it has been proved that the numerical solution satisfies the optimal error estimate. Matrix or full matrix (depending on the size of the influence domain delta). Therefore, similar to the fractional-order dynamic equation, the storage of the coefficient matrix is O (N2), and the computational complexity of solving the final linear system is O (N3). In addition, if the finite element method is used to solve the near-field dynamic model, the elements of each coefficient matrix need to compute the second-order multiple integral, where D is According to the algebraic structure of the matrix and the fast Fourier transform, Wang Hong and other scholars succeeded in reducing the storage of the matrix to O (N), and the computation of solving the final linear system to O (N log N) iteration per Krylov subspace ([81,82]). In this paper, we study the high-precision spectral Galerkin method for fractional-order diffusion equations and the fast collocation algorithm for near-field dynamic models. In the first chapter, we give some basic concepts used in the rest of this paper, including the Riemann-Liouville derivative definition and the Caputo definition of fractional-order derivatives. In the second chapter, we give a high-precision spectral Galerkin method for fractional diffusion equations, which guarantees sufficient smoothness of the coefficients and the right end of the equation, even if the true solution does not have enough smoothness. They can also guarantee the high accuracy of the solution, and the numerical solution is more accurate than the standard spectral Galerkin method, because the true solution is not smooth enough. The standard spectral Galerkin method can not achieve high accuracy. We also prove the error estimate of the method. In Chapter 3, we propose a fast collocation method for solving two-dimensional near-field dynamic models. In this chapter, we analyze the coefficient matrices discretized by collocation method carefully. After analysis, we conclude that the product of the coefficient matrix and any vector can be obtained by three block-Toeplitz-Toeplitz-block (BT-BT-BT). The product of TB matrix and vector is obtained, so the computational complexity of the multiplication of coefficient matrix and vector is O (N log N). If the linear system is solved by Krylov subspace iteration method, the computational complexity of each iteration can be reduced from O (N 2) to O (N log N). At the same time, the computer storage of the matrix can be reduced from O (N 2) to O (N). In chapter 4, two preconditioned matrices are proposed to solve the linear system with large number of iterations due to the discretization of the collocation method. The first preconditioned matrix is given. The condition is block-Cireulant-Toeplitz-block (BCTB) type, and the second precondition is block-Circulant-Circulant-block (BCCB) type. These two preconditions are effective in reducing the number of iterations for solving linear systems, and we can see from the example given in this chapter that the second precondition is because the inverse ratio of the preconditioned matrix is calculated. In Chapter 5, we propose a fast collocation method for solving nonlocal diffusion models in convex domains. This method extends the problem on convex domains to a rectangular problem containing the convex domains by adding penalties. After the collocation method is discretized, we can see the system. Number matrix is the sum of a BTTB matrix and a diagonal matrix. By using this matrix structure, we reduce the storage of coefficient matrix from O (N2) to O (N), and the computational complexity of each Krylov subspace from O (N2) to O (N log N). Numerical examples show the effectiveness of this method.
【學位授予單位】：山東大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O241.82

【參考文獻】

相關(guān)期刊論文 前1條

1 HUANG JianFei;NIE NingMing;TANG YiFa;;A second order finite difference-spectral method for space fractional diffusion equations[J];Science China(Mathematics);2014年06期

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