A VU-decomposition method for a second-order cone programming problem

A VU-decomposition method for solving a second-order cone problem is presented in this paper. It is first transformed into a nonlinear programming problem. Then, the structure of the Clarke subdifferential corresponding to the penalty function and some results of its VU-decomposition are given. Under a certain condition, a twice continuously differentiable trajectory is computed to produce a second-order expansion of the objective function. A conceptual algorithm for solving this problem with a superlinear convergence rate is given.     

基于二阶摄动法求解区间参数结构动力响应

在处理区间参数结构动力响应问题时,现有的分析方法大多局限于一阶区间分析方法. 如果参数的不确定量稍大,采用一阶区间分析方法对结构动力响应范围进行估计可能会失效,所以需要考虑二阶区间分析方法.但是采用基于区间运算的二阶区间分析方法得到的结果将会对动力响应范围过分高估. 为了克服以上缺点,首先基于二阶摄动法得到结构动力响应广义函数. 然后通过求解此动力响应函数的最大和最小值,将结构动力响应区间的问题转化为序列低维箱型约束下的二次规划问题. 最后采用DC 算法(di erence of convex functionsalgorithm) 对这些箱型约束下的二次规划问题进行求解. 这样可以在不引入过多计算量的情况下,避免了对动力响应范围的过分估计. 通过数值算例,将该方法和其他区间分析方法进行比较,验证了该方法的有效性与精确性.      

The application of finite element analysis to the solution of Stokes wave diffraction problems

A new finite-element based method of calculating non-linear wave loads on offshore structures in extreme seas is presented in this paper. The diffraction wave field is modelled using Stokes wave theory developed to second order. Wave loads and free surface elevations are obtained for fixed surface-piercing structures by solving a boundary value problem for the second-order velocity potential. Special attention has been given to the radiation condition for the second-order diffraction field. Results are presented for three test examples, the vertical cylinders of Kim and Yue and of Chakrabarti, and an elliptic cylinder. These results demonstrate that early problems with the application of second-order theory arising from inadequate radiation conditions have been overcome.     

Block elements in the theory of plates of irregular shape

The problem of studying the stress-strain state is considered for plates of irregular three-dimensional nonclassical shape made of a linear-elastic material, for example, for pyramidal or prismoidal plates. The block element method is used together with the vector eigenfunction method. This approach allows one to reduce the problem to solving a system of second-order integral equations with a second-order completely continuous operator and a separate equation in stresses and displacements.     

Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

In this paper, based on the second-order Taylor series expansion and the difference of convex functions algorithm for quadratic problems with box constraints(the DCA for QB), a new method is proposed to solve the static response problem of structures with fairly large uncertainties in interval parameters. Although current methods are effective for solving the static response problem of structures with interval parameters with small uncertainties, these methods may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, first, the general expression of the static response of structures in terms of structural parameters is derived based on the second-order Taylor series expansion. Then the problem of determining the bounds of the static response of uncertain structures is transformed into a series of quadratic problems with box constraints. These quadratic problems with box constraints can be solved using the DCA approach effectively. The numerical examples are given to illustrate the accuracy and the efficiency of the proposed method when comparing with other existing methods.     

A Second-Order Solution of Saint-Venant's Problem for an Elastic Pretwisted Bar Using Signorini's Perturbation Method

We use Signorini's expansion to analyse deformations of a straight, prismatic, isotropic, stress free, homogeneous body made of a second-order elastic material and loaded as follows. It is first twisted by an infinitesimal amount and then loaded by applying surface tractions, with nonzero resultant forces and/or moments, only at its end faces. The centroid of one end face is taken to be rigidly clamped. By using a semi-inverse method, the problem is reduced to that of solving two plane elliptic problems involving six arbitrary constants that characterize flexure, bending, extension, and torsion superimposed upon the infinitesimal twist. It is shown that the Clebsch hypothesis is not valid for this problem. A second-order Poisson's effect, not of the Saint-Venant type, and generalized Poynting effects may also occur in these problems. This revised version was published online in August 2006 with corrections to the Cover Date.     

A new approach to the solution of the Navier-Stokes equations

In the present paper a numerical algorithm is given for solving a standard problem in fluid dynamics, that of inviscid, irrotational, incompressible flow over an arbitrary symmetric profile. The purpose of the paper is to propose an alternative approach to solve certain fluid dynamic flows. This paper may be thought of as the first of a possible series of papers solving new and fundamental problems. In a sense, this new approach asks the question: what is the simplest and most efficient method of solving the problem considered by finite difference methods. It is believed that the following algorithm answers this question. Standard second-order finite difference techniques, such as SLOR and ADI, are used to solve numerically a mixed boundary value problem comprised of a pair of elliptic partial differential equations with constant coefficients.     

面内变刚度薄板弯曲问题的挠度?弯矩耦合神经网络方法

发展了一种求解面内变刚度功能梯度薄板弯曲问题的神经网络方法. 面内变刚度薄板弯曲问题的偏微分控制方程为一复杂的4阶偏微分方程, 传统的基于强形式的神经网络解法在求解该偏微分方程时可能会遇到难以收敛、边界条件难以处理的情况. 本文基于Kirchhoff薄板弯曲理论, 提出了一种直角坐标系下任意面内变刚度薄板弯曲问题的神经网络解法. 神经网络模型包含挠度网络与弯矩网络, 分别用于预测薄板的挠度与弯矩, 从而将求解4阶偏微分方程转换为求解一系列二阶偏微分方程组, 通过对挠度、弯矩试函数的构造可使得神经网络计算结果严格满足边界条件. 在误差的反向传播中, 根据本文提出的误差函数公式计算训练误差并结合Adam优化算法更新模型的内部参数. 求解了不同边界条件、形状的面内变刚度薄板弯曲问题, 并将所得计算结果与理论解、有限元解进行对比. 研究表明, 本文模型对于求解面内变刚度薄板弯曲问题具备适应性, 虽然模型中的弯矩网络收敛较挠度网络要慢, 但本文方法在试函数的构造上更为简单、适应性更强.      

Nonlinear Multigrid Methods for Numerical Solution of the Unsaturated Flow Equation in Two Space Dimensions

Picard and Newton iterations are widely used to solve numerically the nonlinear Richards’ equation (RE) governing water flow in unsaturated porous media. When solving RE in two space dimensions, direct methods applied to the linearized problem in the Newton/Picard iterations are inefficient. The numerical solving of RE in 2D with a nonlinear multigrid (MG) method that avoids Picard/Newton iterations is the focus of this work. The numerical approach is based on an implicit, second-order accurate time discretization combined with a second-order accurate finite difference spatial discretization. The test problems simulate infiltration of water in 2D unsaturated soils with hydraulic properties described by Broadbridge–White and van Genuchten–Mualem models. The numerical results show that nonlinear MG deserves to be taken into consideration for numerical solving of RE.     

A $$ \mathcal{U}\mathcal{V} $$-decomposed method for solving an MPEC problem

uv-decomposition method for solving a mathematical program with equilibrium constraints （MPEC） problem with linear complementarity constraints is presented. The problem is first converted into a nonlinear programming one. The structure of subdifferential a corresponding penalty function and results of its uv-decomposition are given. A conceptual algorithm for solving this problem with a superUnear convergence rate is then constructed in terms of the obtained results.     

Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re = 800 stable?

A detailed case study is made of one particular solution of the 2D incompressible Navier–Stokes equations. Careful mesh refinement studies were made using four different methods (and computer codes): (1) a high-order finite-element method solving the unsteady equations by time-marching; (2) a high-order finite-element method solving both the steady equations and the associated linear-stability problem; (3) a second-order finite difference method solving the unsteady equations in streamfunction form by time-marching; and (4) a spectral-element method solving the unsteady equations by time-marching. The unanimous conclusion is that the correct solution for flow over the backward-facing step at Re = 800 is steady—and it is stable, to both small and large perturbations.     

An efficient three-level explicit time-split scheme for solving two-dimensional unsteady nonlinear coupled Burgers' equations

This work deals with the numerical solutions of two-dimensional viscous coupled Burgers' equations with appropriate initial and boundary conditions using a three-level explicit time-split MacCormack approach. In this technique, the differential operators split the two-dimensional problem into two pieces so that the two-step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second-order accurate in time and fourth-order convergent in space, whereas it is second-order convergent in both time and space for high Reynolds numbers problems. This observation shows the utility and efficiency of the considered method compared with a broad range of numerical schemes widely studied in the literature for solving the two-dimensional time-dependent nonlinear coupled Burgers' equations. A large set of numerical examples that confirm the theoretical results are presented and critically discussed.     

A method for solving the factorized vorticity-stream function equations by finite elements

A new finite element method for solving the time-dependent incompressible Navier-Stokes equations with general boundary conditions is presented. The two second-order partial differential equations for the vorticity and the stream function are factorized, apart from the non-linear advection term, by eliminating the coupling due to the double specification on the stream function at (a part of) the boundary. This is achieved by reducing the no-slip boundary conditions to projection integral conditions for the vorticity field and by evaluating the relevant quantities involved according to an extension of the method of Glowinski and Pironneau for the biharmonic problem. Time integration schemes and iterative algorithms are introduced which require the solution only of banded linear systems of symmetric type. The proposed finite element formulation is compared with its finite difference equivalent by means of a few numerical examples. The results obtained using 4-noded bilinear elements provide an illustration of the superiority of the finite element based spatial discretization.     

Analytic Solution of an Inhomogeneous Kinetic Equation in the Problem of Second-Order Thermal Creep

An analytic solution of the problem of second-order thermal creep is obtained. A method for solving the half-space boundary value problem for an inhomogeneous linearized kinetic BGK equation forms the basis of the solution. The general solution of the input equation is constructed in the form of an expansion of the corresponding characteristic equation in terms of the eigenfunctions. Substitution of the solution in the boundary conditions leads to a Riemann boundary value problem. The unknown thermal creep velocity is found from the condition of solvability of the boundary value problem. The numerical analysis performed confirms the existence of negative thermophoresis (in the direction of the temperature gradient) for high-conductivity aerosol particles at low Knudsen numbers.     

基于微分包含的绳系卫星时间最优释放控制

考虑系绳弹性的影响,建立了绳系卫星系统三维动力学模型,研究了在状态和控制约束下的绳系卫星非线性时间最优控制问题. 为缩减系统变量,控制律设计没有采用通常的状态空间模型,而是基于二阶微分包含,将连续时间最优控制问题离散为大规模动态规化问题,最后通过数值模拟验证了该方法的有效性.      

Effective properties of a nonuniform beam under torsion

Effective characteristics are considered in the pure torsion problem for a nonuniform beam. The Saint-Venant semi-inverse method is used. A torsion stress function is introduced; this function can be found by solving a cross-sectional boundary value problem for a partial differential equation with variable coefficients. Two special boundary value problems are formulated for such an equation; after solving these problems, some effective characteristics are calculated in the case of torsion. It is shown that these effective characteristics satisfy the conditions of symmetry and positive definiteness. The case of an infinite in-plane layer of nonuniform thickness is discussed.     

ELASTIC DYNAMIC ANALYSIS OF MODERATELY THICK PLATE USING MESHLESS LRPIM

A meshless local radial point interpolation method （LRPIM） for solving elastic dy-namic problems of moderately thick plates is presented in this paper. The discretized system equation of the plate is obtained using a locally weighted residual method. It uses a radial basis function （RBF） coupled with a polynomial basis function as a trial function,and uses the quartic spline function as a test function of the weighted residual method. The shape function has the properties of the Kronecker delta function,and no additional treatment is done to impose essen-tial boundary conditions. The Newmark method for solving the dynamic problem is adopted in computation. Effects of sizes of the quadrature sub-domain and influence domain on the dynamic properties are investigated. The numerical results show that the presented method can give quite accurate results for the elastic dynamic problem of the moderately thick plate.     

连续体结构屈曲约束的ICM方法拓扑优化

基于ICM(独立、连续、映射)方法解决具有屈曲约束的连续体拓扑优化问题。建立以结构重量为目标,以屈曲临界力为约束的拓扑优化模型;采用独立的连续拓扑变量,借助泰勒展式将目标函数作二阶近似展开;借助瑞利商、泰勒展式、过滤函数将约束化为近似显函数,避免了灵敏度的计算;将优化模型转化为对偶规划,并利用序列二次规划求解,减少了设计变量的数目,缩小了模型的求解规模。给出三个算例,结果表明:该方法可有效地解决屈曲约束的连续体拓扑优化问题,能够得到合理的拓扑结构,并有较高的计算效率。     

Contact interaction between prestressed laminated shells of revolution and a flat foundation

A nonclassical model of shells that accounts for transverse shears and reduction is used to develop a method for solving the contact problem for inhomogeneous anisotropic shells of revolution subject to a field of mechanical and thermal loads. The prestresses are described by parametric terms in the linearized geometrically nonlinear equations of the second-order theory of flexible shells. The influence of the prestressed state of shells interacting with a flat surface on the contact area and the distribution of contact pressure is analyzed. Some computational features of the technique are discussed __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 67–77, October 2006.     

二阶流形元法与结构变形分析

在原有一阶流形元法的基础上开发了二阶流形元法数值仿真模型和相应的计算程序，并给出了计算实例。结果表明，二阶流形元法可以以较高的精度分析一般结构的变形和接触应力问题，对大变形问题独有优势。并能很好地模拟不连续介质的破坏过程及块体破坏后的运动。