Method of lines for temperature field of functionally graded materials

The finite element method (FEM) and the boundary element method (BEM) are often adopted. Howev er, they are not convenient to spatially vary thermal properties of functionally graded material (FGM). Therefore, the method of lines (MOL) is introduced to solve the temperature field of FGM. The basic idea of the method is to semi-discretize the governing equation into a system of ordinary differential equations (ODEs) defined on discrete lines by means of the finite difference method. The temperature field of FGM can be obtained by solving the ODEs. The functions of thermal properties are directly embodied in these equations and these properties are not discretized in the domain. Thus, difficulty of FEM and BEM is overcome by the method. As a numerical example, the temperature field of a plane problem is analyzed for FGMs through varying thermal conductivity coefficient by the MOL.     

Method of lines for temperature field of functionally graded materials

The finite element method (FEM) and the boundary element method (BEM) are often adopted. However, they are not convenient to spatially vary thermal properties of functionally graded material (FGM). Therefore, the method of lines (MOL) is introduced to solve the temperature field of FGM. The basic idea of the method is to semi-discretize the governing equation into a system of ordinary differential equations (ODEs) defined on discrete lines by means of the finite difference method. The temperature field of FGM can be obtained by solving the ODEs. The functions of thermal properties are directly embodied in these equations and these properties are not discretized in the domain. Thus, difficulty of FEM and BEM is overcome by the method. As a numerical example, the temperature field of a plane problem is analyzed for FGMs through varying thermal conductivity coefficient by the MOL. Foundation item: Project (90305023 and 59731020) supported by the National Natural Science Foundation of China     

无限长条功能梯度材料的反平面裂纹问题

研究了无限长条正交各向异性功能梯度材料在平面剪切作用下的Yoffe裂纹的动力学问题，材料的两上方向的剪切模量假定为指数模型，通过采用积分变换-积分方程方法，求得了裂纹尖端的动态应力场和动态应力强度因子，并研究了裂纹运动速度、几何尺寸、梯度参数和不均匀系数对动态应力强度因子的影响，结果表明，裂纹尖端应力具有的奇异性，裂纹的运动速度越大，应力强度因子越大；材料的模量梯度越大，应力强度因子越低；不均匀系数越大，应力强度因子越小。     

样条函数线法在壳体非线性分析中的应用

采用样条函数线法分析了圆柱壳及具有封闭截面壳体的几何非线性问题.用样条函数插值将二维非线性偏微分问题化为一组用径向结线位移增量表示的非线性常微分方程,然后用常微分方程求解器迭代求解;还导出了用于非线性分析的样条函数线法增量方程;最后给出了算例.     

样条函数线法分析壳体非线性问题

采用样条函数线法分析了圆柱壳及具有封闭截面壳体的几何非线性问题．用样条函数插值将二维非线性偏微分问题化为一组用径向结线位移增量表示的非线性常微分方程,然后用常微分方程求解器迭代求解;还导出了用于非线性分析的样条函数线法增量方程;最后给出了算例．     

功能梯度材料的数字散斑相关方法实验研究

利用数字散斑相关方法对功能梯度材料I型静态断裂特性进行实验研究,并利用有限元软件进行数值模拟.首先通过在环氧树脂中添加玻璃微珠,利用重力沉降法制备功能梯度材料试件.为对比材料梯度对静态断裂行为的影响,准备了三种梯度变化规律的试件.然后对功能梯度材料试件进行3点弯曲实验,通过CCD采集试件散斑场,利用数字散斑相关方法计算应力强度因子.通过与有限元模拟结果进行对比,验证了实验结果的正确性.最后分析不同梯度变化规律对材料断裂性能的影响,证明材料弹性梯度对位于弹性模量较小边的静态裂纹具有保护作用.     

Computational uncertainty principle in nonlinear ordinary differential equations (I)——Numerical results

In a majority of cases of long-time numerical integration for initial-value problems, roundoff error has received little attention. Using twenty-nine numerical methods, the influence of round-off error on numerical solutions is generally studied through a large number of numerical experiments. Here we find that there exists a strong dependence on machine precision (which is a new kind of dependence different from the sensitive dependence on initial conditions), maximally effective computation time (MECT) and optimal stepsize (OS) in solving nonlinear ordinary differential equations (ODEs) in finite machine precision. And an optimal searching method for evaluating MECT and OS under finite machine precision is presented. The relationships between MECT, OS, the order of numerical method and machine precision are found. Numerical results show that round-off error plays a significant role in the above phenomena. Moreover, we find two universal relations which are independent of the types of ODEs, initial val     

Computational uncertainty principle in nonlinear ordinary differential equations--Numerical results

In a majority of cases of long-time numerical integration for initial-value problems, round-off error has received little attention. Using twenty-nine numerical methods, the influence of round-off error on numerical solutions is generally studied through a large number of numerical experiments. Here we find that there exists a strong dependence on machine precision (which is a new kind of dependence different from the sensitive dependence on initial conditions), maximally effective computation time (MECT) and optimal stepsize (OS) in solving nonlinear ordinary differential equations (ODEs) in finite machine precision. And an optimal searching method for evaluating MECT and OS under finite machine precision is presented. The relationships between MECT, OS, the order of numerical method and machine precision are found. Numerical results show that round-off error plays a significant role in the above phenomena. Moreover, we find two universal relations which are independent of the types of ODEs, initial values and numerical schemes. Based on the results of numerical experiments, we present a computational uncertainty principle, which is a great challenge to the reliability of long-time numerical integration for nonlinear ODEs.     

The dynamic behavior of two collinear cracks in magneto-electro-elastic composites under harmonic anti-plane shear waves

The dynamic behavior of two collinear cracks in magneto-electro-elastic composites under harmonic anti-plane shear waves is studied using the Schmidt method for the permeable crack surface conditions. By using the Fourier transform, the problem can be solved with a set of triple integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. In solving the triple integral equations, the jump of the displacements across the crack surface is expanded in a series of Jacobi polynomials. It can be obtained that the stress field is independent of the electric field and the magnetic flux.     

抛物型方程反问题的遗传算法

针对二维抛物型方程参数反演问题,利用遗传算法求解此反演问题,把参数反演问题转化为优化问题,通过演化计算方法求解.该方法的研究可为偏微分方程的参数识别提供一个有效的工具.     

二维问题快速多极虚边界元法

将快速多极展开算法和广义极小残值法应用于虚边界元法的方程求解中.以二维弹性力学问题为研究背景,提出了二维问题快速多极虚边界元法的思想.该方法利用二维复平面上的基本解,并将其展开为适合于快速多极算法的格式,即变革计算结构(或模式),使解方程的计算量和储存量与所求问题的自由度数成线性比例.此点充分体现出该方法数值模拟大规模自由度问题的能力.数值算例说明了该方法的可行性,计算效率和计算精度,同时,该方法的思想具有一般性,应用上具有扩展性.     

辅助激励源区域分解算法在电磁散射问题中的应用

针对电大问题，提出了一种基于辅助激励源的区域分解算法．首先把原求解区域划分成若干个子区域，显著地降低了原问题的规模和复杂度．在子区域分界面上引入虚拟的辅助激励源以交换信息，建立了相应的矩阵方程及其快速求解算法．一旦获得了辅助激励源，即可并行地计算各子区域内部的场．与传统算法相比，该算法有效地提高了计算效率和计算精度，降低了存储量；此外，它特别适合于求解具有几何重复性特征的结构，如栅格、光子带隙／电磁带隙、频率选择表面等．     

重心插值配点法求解初值问题

将计算区间采用第二类Chebyshev点离散,利用数值稳定性好、计算精度高的重心Lagrange插值近似未知函数,建立未知函数各阶导数在计算节点上的微分矩阵,提出数值求解微分方程初值问题的重心插值配点法。采用重心插值配点法将微分方程及其初始条件离散为线性代数方程。将初始条件离散代数方程直接附加到微分方程离散代数方程组,得到n个变量n 2个方程的代数方程组,采用最小二乘法法求解线性代数方程,得到节点的函数值。进而利用微分矩阵直接计算得到未知函数在节点的一阶导数和二阶导数值。数值算例表明本文方法具有计算公式简单、程序实施方便和计算精度高的优点。     

光滑点插值法应用于流固耦合的比较研究

针对传统有限元法（FEM）固体模型刚度过硬导致低阶单元求解精度较低的问题,采用光滑点插值方法（S-PIM）. S-PIM得益于梯度光滑技术能软化固体模型刚度,基于容易剖分的线性背景网格能改善固体求解精度. 采用不同的光滑域构建方式可以得到不同的固体求解器,从而在不同程度上提高计算精度. 本研究以浸没光滑点插值法（IS-PIM）为基础,在流固耦合（FSI）模型中采用较成熟的半隐式特征分离法（CBS）作为流体求解器,分别采用有限元法、边基光滑点插值方法（ES-PIM）以及点基局部光滑点插值方法（NPS-PIM）作为固体求解器,比较不同固体求解器条件下的计算精度和效率. 结果表明,与边基光滑点插值方法和有限元法相比,在流固耦合模型中采用点基局部光滑点插值法可以得到更准确的固体模型刚度,也更有利于计算精度和计算效率的提高.     

基于热力耦合的钢筋混凝土锈胀开裂分析

为对锈蚀钢筋混凝土构件开裂问题进行分析,鉴于氯离子扩散控制方程与瞬态热传导控制方程的相似性,根据传统的Fick定律和已有的氯盐环境下钢筋非均匀锈胀模型以及热弹性力学理论,提出钢筋混凝土锈胀开裂模拟的热力耦合方法.对氯离子的扩散采用瞬态热传导方法进行分析模拟,提取各时间步钢筋单元表面的氯离子质量浓度即温度作为混凝土的温度荷载,引入钢筋锈胀的时变等效温度膨胀系数,采用升温膨胀对钢筋的锈胀以及由此引起的混凝土开裂过程进行有限元分析.数值算例表明,该方法可以有效地模拟和分析钢筋的非均匀锈胀和混凝土的裂缝扩展行为.     

常微分方程中的隐式混合单步连续块方法（英文）

推广了相关文献中用一类k-维隐式混合块方法求解微分方程初值,得到离散的数值近似解.对这些离散的数值解做连续化延伸,发现2k＋2阶隐式混合单步连续块方法存在且惟一.同时还讨论了这类连续块方法在常微分方程中的A-稳定性及一些相关性质.     

Flexural strength of zirconia/stainless steel functionally graded materials

Zirconia/stainless steel (ZrO₂/SUS316L) functionally graded materials (FGMs) were fabricated by tape casting and laminating. Microstructures of FGMs were observed by optical microscope. Fracture behavior of FGMs in different loading modes and influences of different gradient changes on flexural strength were investigated. The results show that ZrO₂/ SUS316L FGMs with graded components at interlayers are obtained after they are sintered in vacuum and pressureless condition at 1 350 °C. The I–II mixed mode crack creates in composite layer and grows to both sides zigzag while loading on ZrO₂ layer. Flexural strengths are 496.4, 421.7 and 387.5 MPa when gradient changes are 10%, 15% and 20%, but flexural strengths of the corresponding fracture layers are 387.1, 334.6 and 282.3 MPa since cracks of FGMs are affected by three-dimensional stress, respectively. The cracks are generated in ZrO₂ layer and extend to SUS316L layer while loading is added on SUS316L layer, flexural strength does not change with the graded components and keeps consistent basically.     

Hydro-mechanical modeling of impermeable discontinuity in rock by extended finite element method

The extended finite element method(XFEM) is a numerical method for modeling discontinuities within the classical finite element framework. The computation mesh in XFEM is independent of the discontinuities, such that remeshing for moving discontinuities can be overcome. The extended finite element method is presented for hydro-mechanical modeling of impermeable discontinuities in rock. The governing equation of XFEM for hydraulic fracture modeling is derived by the virtual work principle of the fracture problem considering the water pressure on crack surface. The coupling relationship between water pressure gradient on crack surface and fracture opening width is obtained by semi-analytical and semi-numerical method. This method simplifies coupling analysis iteration and improves computational precision. Finally, the efficiency of the proposed method for modeling hydraulic fracture problems is verified by two examples and the advantages of the XFEM for hydraulic fracturing analysis are displayed.     

A Differential Continuation－Regularization Method for Solving Inverse Problems of Acoustic Wave Equations

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电大散射问题的复合场积分方程快速傅里叶变换算法

针对电大散射问题,提出了一种基于快速傅里叶变换的新算法．整个求解区域被分成近区和远区两部分：近区部分采用传统矩量法计算;远区部分将格林函数在背景笛卡尔网格上进行拉格朗日插值,使其阻抗元素可用Toeplitz矩阵描述,并采用快速傅里叶变换加快矩阵向量积．复合场积分方程消除了电场积分方程和磁场积分方程的内谐振问题,保持了矩量法的准确性,且求解矩阵方程的迭代次数减少为电场积分方程的10％．