一类非局部广义弹性模型混合形式的有限元数值方法
发布时间：2018-12-17 03:22
【摘要】：本文考虑如下一类非局部广义弹性模型:其中,Ω =(0,1),d~+0和d~-0分别表示左右扩散系数,表示源汇项,0D_y~β和yD_1~β分别表示β阶左、右Riemann-Liouville分数阶导数算子.在此模型中,同时出现了积分运算和求导运算,于是,引入一个中间变量来求解此类积分微分方程的想法是自然的.通过令被积函数中分数阶导数的部分作为中间变量,我们将原问题分解为一个1-α阶积分方程和一个β阶微分方程,由此可以定义其混合形式的变分格式.因为得到的两个等价方程不需要满足一定的耦合关系,可独立求解,我们只需证明双线性形式在空间H-(1-α)/2(Ω)×H0β/2(Ω)中具有强制性和连续性,根据Lax-Milgram引理即可得到混合问题的变分解的适定性.在分数阶积分方程解的适定性的讨论中,我们也得到了一种关于一类第一型Fredholm积分方程在空间H-(1-α)/2(Ω)中可解性的判定准则.基于混合形式的变分原理,进一步定义了混合形式的有限元离散格式,并证明了此格式数值解的存在唯一性.针对这一数值模拟,我们利用插值算子和L2投影算子的误差估计性质分别给出了关于中间变量和最终变量的能量模估计.数值试验的结果验证了此格式的准确性.由于分数阶算子具有非局部性质,在由此得到的离散格式中,线性方程组的系数矩阵多为稠密矩阵.对于一个N阶问题而言,矩阵的存储量为O(N2),直接求解(如Gauss消元法)的计算量为O(N~3),伴随着N的增大,问题的复杂度将使得计算时间过长而丧失了算法的高效性.于是,我们要为此类问题的求解寻找一种实现加速的计算方法.当我们选择分片常数多项式函数和分片线性多项式函数分别近似中间变量和最终变量时,经过计算发现,与离散格式相对应的系数矩阵具有或部分具有Toeplitz结构.我们知道,Toeplitz矩阵的存储量可降低为O(N),且Toeplitz矩阵-向量积的计算量为O(N log N),因此,我们可以在共轭梯度法的基础上设计一种求解此类线性方程组的快速算法,使得矩阵的存储量为O(N),每步迭代的计算量为O(N log N).对一些条件数不好的矩阵而言,加入合适的预处理子可以进一步减少迭代次数从而提高计算效率.数值试验的结果验证了此快速算法的有效性.
[Abstract]:In this paper, we consider a class of nonlocal generalized elastic models, where 惟 = (0 ~ 1), d ~ 0 and d ~ 0 denote the left and right diffusivity coefficient, denote the source term, and denote the left and right Riemann-Liouville fractional derivative operators of 尾 order respectively. In this model, there are integral operations and derivation operations at the same time, so it is natural to introduce an intermediate variable to solve this kind of integrodifferential equation. By taking the fractional derivative part of the integrable function as the intermediate variable, we decompose the original problem into an integral equation of order 1- 伪 and a differential equation of order 尾, and then define its mixed form variational scheme. Because the two equivalent equations do not need to satisfy a certain coupling relation and can be solved independently, we only need to prove that the bilinear form is mandatory and continuous in the space H- (1- 伪) / 2 (惟) 脳 H0 尾 / 2 (惟). According to the Lax-Milgram 's Lemma, we can obtain the proper definiteness of the variational decomposition of the mixed problem. In the discussion of the fitness of solutions for fractional integral equations, we also obtain a criterion for the solvability of a class of first type Fredholm integral equations in space H- (1- 伪) / 2 (惟). Based on the variational principle of mixed form, the finite element discrete scheme of mixed form is further defined, and the existence and uniqueness of the numerical solution of the scheme are proved. For this numerical simulation, we give the energy modulus estimates for intermediate variables and final variables by using the error estimation properties of interpolation operator and L2 projection operator. The results of numerical experiments verify the accuracy of the scheme. Because of the nonlocal property of fractional order operators, the coefficient matrices of linear equations are dense matrices in the discrete schemes. For a problem of order N, the storage of matrix is O (N2), and the computation of direct solution (such as Gauss elimination method) is O (N3), which is accompanied by the increase of N. The complexity of the problem will make the computation time too long and lose the efficiency of the algorithm. Therefore, we need to find an accelerated computing method for solving this kind of problem. When we select piecewise constant polynomial function and piecewise linear polynomial function to approximate intermediate variable and final variable respectively, we find that the coefficient matrix corresponding to discrete scheme has or partly has Toeplitz structure. We know that the storage of Toeplitz matrix can be reduced to O (N), and the computation of Toeplitz matrix-vector product is O (N log N),. Therefore, we can design a fast algorithm for solving this kind of linear equations on the basis of conjugate gradient method. So that the memory of the matrix is O (N), the computation of each iteration is O (N log N). For some matrices with poor condition number, adding a suitable preprocessor can further reduce the number of iterations and improve the computational efficiency. The effectiveness of this fast algorithm is verified by numerical experiments.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
本文编号：2383567
[Abstract]:In this paper, we consider a class of nonlocal generalized elastic models, where 惟 = (0 ~ 1), d ~ 0 and d ~ 0 denote the left and right diffusivity coefficient, denote the source term, and denote the left and right Riemann-Liouville fractional derivative operators of 尾 order respectively. In this model, there are integral operations and derivation operations at the same time, so it is natural to introduce an intermediate variable to solve this kind of integrodifferential equation. By taking the fractional derivative part of the integrable function as the intermediate variable, we decompose the original problem into an integral equation of order 1- 伪 and a differential equation of order 尾, and then define its mixed form variational scheme. Because the two equivalent equations do not need to satisfy a certain coupling relation and can be solved independently, we only need to prove that the bilinear form is mandatory and continuous in the space H- (1- 伪) / 2 (惟) 脳 H0 尾 / 2 (惟). According to the Lax-Milgram 's Lemma, we can obtain the proper definiteness of the variational decomposition of the mixed problem. In the discussion of the fitness of solutions for fractional integral equations, we also obtain a criterion for the solvability of a class of first type Fredholm integral equations in space H- (1- 伪) / 2 (惟). Based on the variational principle of mixed form, the finite element discrete scheme of mixed form is further defined, and the existence and uniqueness of the numerical solution of the scheme are proved. For this numerical simulation, we give the energy modulus estimates for intermediate variables and final variables by using the error estimation properties of interpolation operator and L2 projection operator. The results of numerical experiments verify the accuracy of the scheme. Because of the nonlocal property of fractional order operators, the coefficient matrices of linear equations are dense matrices in the discrete schemes. For a problem of order N, the storage of matrix is O (N2), and the computation of direct solution (such as Gauss elimination method) is O (N3), which is accompanied by the increase of N. The complexity of the problem will make the computation time too long and lose the efficiency of the algorithm. Therefore, we need to find an accelerated computing method for solving this kind of problem. When we select piecewise constant polynomial function and piecewise linear polynomial function to approximate intermediate variable and final variable respectively, we find that the coefficient matrix corresponding to discrete scheme has or partly has Toeplitz structure. We know that the storage of Toeplitz matrix can be reduced to O (N), and the computation of Toeplitz matrix-vector product is O (N log N),. Therefore, we can design a fast algorithm for solving this kind of linear equations on the basis of conjugate gradient method. So that the memory of the matrix is O (N), the computation of each iteration is O (N log N). For some matrices with poor condition number, adding a suitable preprocessor can further reduce the number of iterations and improve the computational efficiency. The effectiveness of this fast algorithm is verified by numerical experiments.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
【相似文献】
相关期刊论文 前1条
1 徐俊明;关于图论中混合形式的Menger定理的证明[J];华中工学院学报;1984年02期
相关硕士学位论文 前1条
1 贾略羚;一类非局部广义弹性模型混合形式的有限元数值方法[D];山东师范大学;2017年
，本文编号：2383567
本文链接：https://www.wllwen.com/kejilunwen/yysx/2383567.html
