辐射扩散方程基于虚拟元方法的一个保正守恒格式

作为近年来广受关注的一种数值方法，虚拟元方法具有很多优势。但在求解实际问题导出的一些辐射扩散方程时，该方法可能无法保证数值解的非负性及一般多边形网格上的局部守恒性。针对辐射扩散方程，利用非线性两点流逼近方法作为后处理措施，提出了一种基于虚拟元方法的保正守恒格式。该格式通过最低阶虚拟元方法得到数值解的单元顶点值，再利用非线性两点流逼近方法得到数值解的非负单元中心值，同时使格式满足局部守恒性。任意多边形网格上的数值结果表明，该格式具有保正性和解的近似二阶收敛速度，对于处理含强间断或非线性扩散系数的辐射扩散问题均有较强的适应性。     

拆除爆破数值模拟研究进展

评述了数值模拟技术在拆除爆破中应用的重要性,回顾了拆除爆破数值模拟的主要方法,论述了各种方法的优缺点。结合工程应用实例介绍了笔者及团队研发的离散元框架内的网格实体模型。分析了当前拆除爆破数值模拟技术存在的主要问题,对拆除爆破数值模拟技术的进一步发展进行了展望。     

四元数量子力学中一类特征值反问题及其应用

本文研究了四元数量子力学中一类要求其解是正规或可对角化四元数矩阵的特征值反问题。并给出了其有解的充要条件和通解的表达式。讨论了四元数量子力学中带有谱约束的最小二乘解反问题,得到了此问题有解的充分条件。最后给出了数值算法和数值例子。     

超椭圆中厚板的自由振动

本文用一种新型的数值方法棗微分容积法求解任意边界条件下超椭圆形中厚板的自由振动问题。通过微分容积法将中厚板自由振动的控制微分方程和边界条件离散成为一组齐次的线性代数方程,这是一典型的特征值问题,用子空间迭代法可求出其特征值和特征向量。文中通过一些数值算例研究了该方法的收敛性和数值精度,展示了该方法的可行性和有效性。     

低频域结构噪声问题数值解法

对形体不规则的物体，数值计算其结构噪声问题是获得解答的直接办法，文中综述了结构噪声问题数值解法的基本思路，首先介绍了数值求解的两基本方法；有限元和边界元法的基本数学原理，操作方法及各自的局限性；其后描述了结构噪声耦合问题的数值计算公式的构造运用不同数值方法求解公式的特点；针对边界元法构造的公式特征值问题的不足，给出了几种边界元法结构噪声特征值分析的方法，文章结尾，提出了目前商业有限元软件分析噪声问题的不足及改进方向。     

组合杂交四边形元的多重网格预处理共轭梯度方法

组合杂交元方法是一种求解弹性力学问题的稳定化有限元方法．为了快速求解组合杂交元离散得到的大型、稀疏、对称正定系统,本文研究了多重网格预处理共轭梯度方法．首先,通过选用合适的网格转移算子和光滑策略,得到了有效的多重网格预处理器．其次,通过分析数值试验结果证明所得到的多重网格预处理共轭梯度方法是有效可行的,利用该预处理方法大大降低了系数矩阵的条件数,提高了计算效率．此外,对于一类高性能的组合杂交元,多重网格预处理共轭梯度方法在网格畸变时依然收敛.     

考虑频变阻尼的黏弹性阻尼层板结构模态分析方法

结构动力中的模态分析可归结为数学上矩阵特征值问题的求解。研究了一种求解有限元非线性特征值问题的数值方法,即RSRR,该方法通过对系统矩阵逆矩阵的采样,构造可靠的特征空间用于非线性特征值问题的求解,比现有基于围道积分的非线性特征值解法稳定性更好、精度更高。采用基于Layerwise离散层理论的Layerwise板单元建立黏弹性阻尼结构有限元模型比混合单元建模方法简单方便,结合Layerwise板单元建模方法,将RSRR拓展应用于黏弹性阻尼结构的模态分析,算例结果表明RSRR求解精度高、稳定性好,是黏弹性阻尼结构模态分析的有效数值方法。     

特征值问题迭代伽略金法与Rayleigh商加速

该文讨论特征值问题非协调有限元和混合有限元的加速计算方法。基于迭代伽略金法和Rayleigh商加速技巧,我们建立了特征值问题Wilson非协调有限元和Ciarlet-Raviart混合有限元的加速计算方案。这些新方案把在细网格上解一个特征值问题简化为在粗网格上解一个特征值问题和在细网格上解一个线性方程。文中证明了新方案的计算结果仍然保持了渐近最优精度阶,并用数值实验验证了理论结果。     

圆板轴对称自由振动的微分容积解法

本文用一种新型的数值方法——微分容积法求解任意边界条件下圆形中厚板的轴对称自由振动问题。通过微分容积法将中厚板轴对称自由振动的控制微分方程和边界条件离散成为一组关于各配点位移的齐次的线性代数方程，这是一典型的特征值问题，求解该特征值问题便可求得板的自振频率和振型。文中用一些数值算例研究了该方法的收敛性和数值精度，展示了该方法的可行性和有效性。     

基于NOWs的并行多重网格计算

分析了工和站网络环境（ＮＯＷｓ）下并行多重网格计算中阻碍并行效率发挥的关键因素，分析了Ｓｃｈｗａｒｚ并行对并行效率和数值效率的影响，总结了将其应用到求解ＳＰＤ椭圆方程，间断问题、ｓｔｏｋｅｓ方程，高Ｒｅｙｎｏｌｄｓ不可压Ｎａｖｉｅｒ０Ｓｔｏｋｅｓ方程和三维跨声速Ｅｕｌｅｒ方程中所取得的重要结论。     

Numerical recipes for elastodynamic virtual element methods with explicit time integration

We present a general framework to solve elastodynamic problems by means of the virtual element method (VEM) with explicit time integration. In particular, the VEM is extended to analyze nearly incompressible solids using the B-bar method. We show that, to establish a B-bar formulation in the VEM setting, one simply needs to modify the stability term to stabilize only the deviatoric part of the stiffness matrix, which requires no additional computational effort. Convergence of the numerical solution is addressed in relation to stability, mass lumping scheme, element size, and distortion of arbitrary elements, either convex or nonconvex. For the estimation of the critical time step, two approaches are presented, ie, the maximum eigenvalue of a system of mass and stiffness matrices and an effective element length. Computational results demonstrate that small edges on convex polygonal elements do not significantly affect the critical time step, whereas convergence of the VEM solution is observed regardless of the stability term and the element shape in both two and three dimensions. This extensive investigation provides numerical recipes for elastodynamic VEMs with explicit time integration and related problems.     

Least squares element method for boundary eigenvalue problems

Linear and non-linear boundary eigenvalue problems are discretized by a new finite element like method. The reason for the new construction principle is the non-linear dependence of the dynamic stiffness element matrix on an eigenparameter. The dynamic stiffness element matrix is evaluated for a fixed number of parameters and is then elementwise replaced by a polynomial in the eigenparameter by solving least squares problems. A fast solver is introduced for the resulting non-linear matrix eigenvalue problem. It consists of a combination of bisection method and inverse iteration. The superiority of the newconstructionprinciple in comparison with the finite or dynamic element method is demonstrated finally for some numerical examples.     

几类典型网格下三维弹性问题的代数多层网格法

有限元方法是数值求解三维弹性问题的一类重要的离散化方法.在有限元分析中,网格的几何形状及网格质量会对有限元离散代数系统的求解产生很大影响.该文系统研究了几类典型网格对几种常用AMG法计算效率的影响,并进行了详细的性能测试与比较.利用容易获知的部分几何与分析信息(如方程类型,节点自由度信息),再结合经典AMG法中的网格粗...     

二维新型快速多极虚边界元配点法

以二维弹性问题为研究背景, 提出了一种二维新型快速多极虚边界元配点法的求解思想, 即采用新型的快速多极展开和运用广义极小残值法来求解传统的虚边界元配点法方程。相对常规快速多极展开技术, 该文针对二维弹性问题在原有的快速多极虚边界元法展开格式的基础上, 通过引入对角化的概念, 以更新展开传递格式, 欲达到进一步提高计算效率的目的。数值算例说明了该方法的可行性, 计算效率和计算精度。此外, 该文方法的思想具有一般性, 应用上具有扩展性。     

Virtual boundary element-integral collocation method for the plane magnetoelectroelastic solids

This paper presents a virtual boundary element—integral collocation method (VBEM) for the plane magnetoelectroelastic solids, which is based on the basic idea of the virtual boundary element method for elasticity and the fundamental solutions of the plane magnetoelectroelastic solids. Besides sharing all the advantages of the conventional boundary element method (BEM) over domain discretization methods, it avoids the computation of singular integral on the boundary by introducing the virtual boundary. In the end, several numerical examples are performed to demonstrate the performance of this method, and the results show that they agree well with the exact solutions. The method is one of the efficient numerical methods used to analyze megnatoelectroelastic solids.     

基于X-SBFEM的裂纹体非网格重剖分耦合模型研究

扩展有限元法利用了非网格重剖分技术,但需要基于裂尖解析解构造复杂的插值基函数,计算精度受网格疏密和插值基函数等因素影响。比例边界有限元法则在求解无限域和裂尖奇异性问题优势明显,两者衔接于有限元法理论内,可建立一种结合二者优势的断裂耦合数值模型。该文从虚功原理出发,利用位移协调与力平衡机制,提出了一种断裂计算的新方法X-SBFEM,达到了扩展有限元模拟裂纹主体、比例边界有限元模拟裂尖的目的。在数值算例中,通过边裂纹和混合型裂纹的应力强度因子计算,并与理论解对比,验证了该方法的准确性和有效性。     

用不完全模态测量数据修正有限元分析模型

提出一种基于不完全模态测量数据同时修正有限元质量矩阵与刚度矩阵的有效数值方法。运用代数特征值反问题的理论与方法，得到了满足正交关系及特征方程的最逼近有限元质量矩阵及刚度矩阵的唯一的修正质量矩阵与刚度矩阵（最优修正矩阵）。该方法有一个简洁的表达式，修正过程简单而且容易实现。数值算例表明，修正模型与模态试验数据具有非常好的一致性。     

Radial integration boundary element method for acoustic eigenvalue problems

In this paper, the radial integration boundary element method is developed to solve acoustic eigenvalue problems for the sake of eliminating the frequency dependency of the coefficient matrices in traditional boundary element method. The radial integration method is presented to transform domain integrals to boundary integrals. In this case, the unknown acoustic variable contained in domain integrals is approximated with the use of compactly supported radial basis functions and the combination of radial basis functions and global functions. As a domain integrals transformation method, the radial integration method is based on pure mathematical treatments and eliminates the dependence on particular solutions of the dual reciprocity method and the particular integral method. Eventually, the acoustic eigenvalue analysis procedure based on the radial integration method resorts to a generalized eigenvalue problem rather than an enhanced determinant search method or a standard eigenvalue analysis with matrices of large size, just like the multiple reciprocity method. Several numerical examples are presented to demonstrate the validity and accuracy of the proposed approach.     

Solution of the Helmholtz eigenvalue problem via the boundary element method

The numerical solution of the Helmholtz eigenvalue problem is considered. The application of the boundary element method reduces it to that of a non-linear eigenvalue problem. Through a polynomial approximation with respect to the wavenumber, the non-linear eigenvalue problem is reduced to a standard generalized eigenvalue problem. The method is applied to the test problems of a three-dimensional sphere with an axisymmetric boundary condition and a two-dimensional square.