【作者】 刘欢；

【导师】 程爱杰； 王宏；

【作者基本信息】 山东大学 ， 计算数学， 2019， 博士

【摘要】 经典的整数阶扩散方程最早由德国生理学家Fick在研究营养物质在生物细胞膜内的传播过程时提出,并随后由Einstein与Pearson分别运用第一性原理和随机游走的方法导出。他们的工作都有两个共同假设,即扩散过程中的粒子运动存在有限的平均自由程以及平均等待时间。在这两个假设下,粒子的运动服从经典的Gauss分布,对应的Fokker-Planck方程即为经典的扩散方程。但是很多试验表明这两个假设对非均匀介质是不成立的,粒子的运动呈现出了重尾等待时间、长程空间相互作用或强非对称羽流等反常扩散现象。由于整数阶扩散方程的解具有指数衰减的特性,使得其不能很好的刻画这些现象。而分数阶偏微分方程的解是幂律衰减的,所以比经典的整数阶扩散方程更能精确地模拟反常扩散过程。实际上,反常扩散现象广泛的存在于自然界和各种工程领域当中,如粘弹性流体、河流与地下水中的生物堵塞过程、河口泥沙输运过程中出现的长尾羽流现象等。所以关于分数阶偏微分方程的数学和数值研究具有很强的理论和实际意义,并且已经成为了近些年的一个热点研究方向。在固体力学当中,经典的偏微分方程模型假设其所有的内力都是局部作用的。然而,由于其在位移场上的可微性假设,偏微分方程模型不能很精确地模拟位移场不连续的问题,例如材料的损伤演化或断裂破坏等。美国Sandia国家实验室的Silling研究员在2000年提出了近场动力学理论[98]用于解决偏微分方程模型中的连续性假设与实际问题不连续的矛盾,其基本思想是用非局部的积分模型代替原来的偏微分方程模型。在近场动力学理论中,本构模型依赖于有限的变形向量,而不是依赖于经典本构模型中的变形梯度,因此近场动力学理论更适合用于模拟固体力学中位移场不连续的情形。但是,与经典的整数阶偏微分方程不同的是,分数阶偏微分方程和近场动力学模型都具有非局部的特性,其相应的数值离散会产生稠密甚至是满的系数矩阵或者在时间上具有历史依赖性。我们以Riesz空间分数阶扩散方程为例来说明这些问题,对其进行数值离散后导出的矩阵为满阵。我们需要(?)(N2)的内存单元来存储系数矩阵,其中N为空间未知量的个数。同时,如果我们使用直接解法如高斯消去法对矩阵求逆,则相应的计算量为(?)(N3)。若我们使用Krylov子空间迭代算法求解,则在每一迭代步里需要(?)(N2)的计算量计算矩阵和向量的乘积,但是其内存需求依然为O(N2)。这些困难都是在处理整数阶偏微分方程时不曾遇到的,尤其在面对大规模或者高维问题时更为突出。本文主要研究了分数阶偏微分方程及近场动力学模型的高效数值算法及其在相场模拟、断裂力学中的应用。其主要结构如下:第一章,我们简单介绍了分数阶微积分及近场动力学模型的基本定义与研究背景,同时还回顾了两种具有特殊结构的矩阵:Toeplitz矩阵和循环矩阵的相关知识。第二章,对一维稳态的键基近场动力学模型,我们提出了一系列快速的间断Galerkin有限元(DG)方法。更精确地说,为了处理位移在网格节点处发生间断的情形,我们在局部加密网格上提出了一种预处理的快速分片常数DG方法。我们还在一致剖分网格上发展了一种快速的预处理分片线性DG方法,这种方法在处理位移场在计算节点处发生跳跃间断的情形时具有二阶精度,否则只有半阶精度。基于这两个结果,我们又给出了一种快速的杂交DG格式去处理位移场在网格单元内部发生跳跃间断的情况。在不含跳跃间断的网格单元,我们使用网格步长为h的分片常数DG格式,而在含有跳跃间断的网格单元我们使用网格步长为h2的分片线性DG格式,所以杂交DG格式在整体上具有一阶精度。数值实验显示这些方法可以有效的提高数值模拟的精度和效率。第三章,我们研究了时间分数阶Allen-Cahn和Cahn-Hilliard相场模型来解释非均匀多孔介质中的反常次扩散输运行为或某些材料的记忆效应。由于时间分数阶导数的非局部性,在每一时间步我们都需要前面所有时间层的信息,因此其内存需求为O(MN),而总共的计算量为O(M2N),其中M,N分别是时间方向和空间方向的剖分数。利用文献[65]中的sum-of-exponentials(SOE)近似Caputo分数阶导数中的奇异核,我们分别对时间分数阶Allen-Cahn和Cahn-Hilliard相场模型给出了高效的有限差分格式和Fourier谱格式来减少数值格式的内存需求和计算复杂度。目前,因为时间分数阶相场模型的能量耗散定律依然没有得到证明,所以对它们的能量耗散行为进行数值研究是很有必要的。对于时间分数阶Cahn-Hilliard方程,我们发现分数阶α越大,其能量衰减越快。因为时间分数阶偏微分方程描述的是次扩散过程,所以这个观察结果在物理上是合理的。然而,对于时间分数阶Allen-Cahn方程,我们却得到了相反的结论,这似乎有点不合理,具体原因还有待进一步的调查。另外,我们还研究了时间分数阶Cahn-Hilliard方程的粗化动力学行为,数值实验显示其能量衰减具有O(t-α/3)的标度律。当时间分数阶Cahn-Hilliard方程退化成整数阶Cahn-Hilliard方程的时候,其结果与我们熟知的(?)(t-1/3)标度律是吻合的。第四章,我们研究了时空分数阶Allen-Cahn方程的快速有限元方法,其中空间分数阶导数为超奇异积分定义的分数阶Laplacian算子。同样由于分数阶导数的非局部性,直接算法如高斯消去法需要(?)(N2+MN)的内存存储系数矩阵以及(?)(M2N+MN3)的计算量,其中M,N分别是时间方向和空间方向的剖分数。通过分析系数矩阵的特殊结构以及使用分治算法处理时间分数阶导数,我们得到了一种高效且无损耗的分治有限元算法。此算法只需要(?)(MN)的存储量以及每一迭代步(?)(MN(log N+log2 M))的计算量。我们使用了能量二次化方法处理模型中的非线性项,从而弱化了显格式或者全隐格式中可能出现的时间步长限制[111]。最后我们研究了时空分数阶Allen-Cahn方程的模拟能力以及能量耗散性质。数值实验验证了所给格式的高效性和时空分数阶Allen-Cahn方程的可调节的界面宽度和灵活的能量耗散行为。第五章,我们对两维的Riesz空间分数阶扩散方程提出了一种快速的Crank-Nicolson交替方向有限体积方法,并在离散范数意义下给出了其稳定性和收敛性证明。结果表明这种格式是无条件稳定的,且在时间和空间上都具有二阶精度。在每个时间步里,相比于经典的有限体积方法,交替方向有限体积方法可以将计算量从(?)(N3)=(?)(N3Ny3)降低到(?)(Nx3Ny+NxNy3),其中Nx,Ny分别为空间x方向和y方向的剖分数。我们通过使用置换矩阵,发展了一种无压缩损耗的高效Crank-Nicolson交替方向有限体积方法。这种方法可以进一步将数值格式的存储量降低到(?)(N),而在每一迭代步里,可以将计算量进一步降低到(?)(NlogN)。最后我们通过几个数值实验验证了所提出的格式的高效性以及实用性。更多还原

【Abstract】 Classical integer-order diffusion equation was first proposed by Fick during his study of the propagation of nutrients within the membranes of living organ-isms,and then derived by Einstein and Pearson based on the first principle and the random walk,respectively.There are two common assumptions in their orig-inal derivations,i.e.,(i)the existence of a mean free path and(ii)the existence of a mean waiting time in the underlying particle movements.Based on these assumptions,the movement of the particles obeys the classic Gauss distribution,and the Fokker-Planck equation is the classical diffusion equation.But many experiments show that these two assumptions are not satisfied for the inhomoge-neous medium,the movement of the particles exhibits heavy-tailed waiting time,long-range spatial interaction or strong asymmetric plume.Since the fundamental solutions to classical integer-order diffusion equation are exponentially decaying,thus it cannot accurately describe these anomalous diffusion phenomena.The solutions to fractional diffusion equations are power law decaying,thus they pro-vide more accurate descriptions on these challenging phenomena.Anomalous diffusion phenomena exist widely in nature and various engineering fields,such as viscoelastic fluid,biological blockage in rivers and groundwater and the long tail plume during the sediment transport in estuary.Therefore,the mathematics and numerical studies on fractional partial differential equations(FPDEs)have strong theoretical and practical significances,and have become a popular research field.In solid mechanics,the classical PDEs model assume that all of the internal forces are local.However,because of its differentiability assumptions on the displacement field,the PDEs model can not accurately describe the discontinuous problems,such as the damage evolution of materials and fracture damage.Silling[98]developed the peridynamic(PD)theory to overcome this difficulty,whose basic idea is to replace the original PDEs model by nonlocal integral model.In the PD theory,the constitutive model relies on the finite deformation vector rather than the deformation gradient,thus it is particular suitable to describe the problems with discontinuous displacement field in the solid mechanics.Compared to classical integer-order PDEs,both of the FPDEs and the PD model exhibit nonlocal features,numerical discretizations of these models give rise to dense or full coefficient matrices or historical dependence.We use the Riesz space-fractional diffusion equation to expose the idea,the coefficient matrix arising from its numerical discretizaiton is full,which can be stored in(?)(N2)memory with N being the number of spatial unknowns.At each time step,direct solvers such as the Gauss elimination can be used to inverse the coefficient matrix with the computational cost of(?)(N3).The Krylov subspace iteration methods can be used to solve the linear system with the computational complexity of(?)(N2)per iteration,but the memory requirements are still(?)(N2).These difficulties are never encountered when dealing with integer-order PDEs,and especially for large scale or high dimensional problems.This thesis mainly study the efficient numerical methods for FPDEs and related nonlocal models with applications to phase field simulations and fracture mechanics.The structure of this thesis are organized as follows:In chapter 1,we firstly outline the fractional operators and PD theory,and then we recall two special matrices,including the Toeplitz matrix and the circulant matrix.In chapter 2,we develop a series of fast discontinuous Galerkin(DG)finite element methods for a bond-based linear PD model in one space dimension.More precisely,we develop a preconditioned fast piecewise-constant DG scheme on a geometrically graded locally refined composite mesh to handle the scenario that the jump discontinuity of the displacement field occurs at one of the nodes in the original uniform partition.We also develop a preconditioned fast piecewise-linear DG scheme on a uniform mesh that has a second-order convergence rate when the jump discontinuity occurs at one of the computational nodes or has a convergence rate of one-half order otherwise.Motivated by these results,we develop a preconditioned fast hybrid DG scheme that is discretized on a locally uniformly refined composite mesh to numerically simulate the PD model where the jump discontinuity of the displacement field occurs inside a computational cell.We use a piecewise-constant DG scheme on the uniform mesh of mesh size h and a piecewise-linear DG scheme on the locally uniformly refined mesh of mesh size h2,so that the hybraid DG scheme has an overall convergence rate of first-order.Numerical experiments show the accuracy and efficiency of the proposed schemes.In chapter 3,we study time-fractional Allen-Cahn and Cahn-Hilliard phase-field models to account for the anomalously subdiffusive transport behavior in heterogeneous porous materials or memory effect of certain materials.Due to the nonlocal behavior of the time-fractional operators,at each time step,the informations in all previous time steps are needed which leads to an overall com-putational complexity of(?)(M2N)and memory requirements of(?)(MN),where M,N are the number of time steps and spatial unknowns.We utilize the sum-of-exponentials(SOE)[65]approximation to develop efficient finite difference scheme and Fourier spectral scheme to effectively treat the significantly increased memory requirement and computational complexity.Since the proof of the energy dissipation law is still an open problem,thus it is worthy to numerically study the energy decay behavior of the time-fractional phase-field models.For the time-fractional Cahn-Hilliard equation,we observe that the bigger the fractional order a,the faster the energy decays.This conclu-sion is physically reasonable,since the time-fractional operator is corresponding to the sub-diffusion process.However,we obtained an opposite conclusion for the time-fracitonal Allen-Cahn equation.Moreover,we also study the coarsening dynamics of the time-fractional Cahn-Hilliard equation,numerical results reveal that the scaling law for the energy decays as(?)(t-α/3),which is consistent with the well-known result(?)(t-1/3)of the classical Cahn-Hilliard equation.In chapter 4,we investigate the finite element approximation of a space-time fractional Allen-Cahn equation,where the classical Laplacian operator is replaced by the integral fractional Laplacian.Due to the nonlocal features of frac-tional derivatives,conventional solvers require(?)(N2+MN)memory units and(?)(M2N+MN3)computational complexity.By exploring the special structure of the stiffness matrix and utilizing the divide-and-conquer strategy,we developed an efficient linearized divide-and-conquer Galerkin finite element method with-out resorting to any lossy compression.This new method reduces the memory requirement to(?)(MN)and the operation count to(?)(MN(log2 M+log N)).We employ the "Invariant Energy Quadratization" approach to alleviate the time step constraint.Numerical experiments are presented to demonstrate the efficiency of the new methods and the flexibility of turnable sharpness and decay behavior of the space-time fractional Allen-Cahn equation.In chapter 5,we develop a Crank-Nicolson alternating direction implicit finite volume(CN-ADI-FV)method for time-dependent Riesz space-fractional diffusion equation in two space dimensions.Norm-based stability and convergence analy-sis are given to show that the developed method is unconditional stable and of second-order accuracy both in space and time.Compared with the conventional finite volume scheme,the developed CN-ADI-FV scheme reduces the computa-tional complexity of original finite volume scheme from(?)(N3)=(?)(Nx3Ny3)to(?)(Nx3Ny+NxNy3),where Nx and Ny are the number of partitions in x and y directions,respectively.By using the permutation matrix,we develop an ef-ficient lossless CN-ADI-FV scheme which only has(?)(N)memory requirement and(?)(N log N)computational complexity per iteration.Several numerical ex-periments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large-scale modeling and simulations.更多还原