Numerical computation of stress induced microstructure

The mesh transformation method is applied on a two-dimensional elastic crystal model to study the formation of laminated microstructure in austenite-martensite phase transition when certain external loads are applied. Numerical experiments show that simple laminated microstructures with various volume fractions and twin width can be obtained by varying the loads. Numerical experiments also show that second order laminated microstructure with branched needle-like laminates can also be obtained by certain loads.     

Stability of microstructures for some martensitic transformations

We analyze the stability of laminated microstructure for martensitic crystals that undergo cubic to trigonal, orthorhombic to triclinic, and trigonal to monoclinic transformations. We show that the microstructure is unique and stable for all laminates except when the lattice parameters satisfy certain identities.     

Chebyshev center based column generation

The classical column generation approach often shows a very slow convergence. Many different acceleration techniques have been proposed recently to improve the convergence. Here, we briefly survey these methods and propose a novel algorithm based on the Chebyshev center of the dual polyhedron. The Chebyshev center can be obtained by solving a linear program; consequently, the proposed method can be applied with small modifications on the classical column generation procedure. We also show that the performance of our algorithm can be enhanced by introducing proximity parameters which enable the position of the Chebyshev center to be adjusted. Numerical experiments are conducted on the binpacking, vehicle routing problem with time windows, and the generalized assignment problem. The computational results of these experiments demonstrate the effectiveness of our proposed method.     

Hamilton体系下层合板弱粘接模型的应用

层合板的层间粘接模型对整个层合板的结构有重要影响．运用Hamilton正则方程对层合板层间的不同类型的粘接模型进行了分析．结合弹性材料修正后的Hellinger Reissner变分原理和插值函数,构建了直角坐标系下8节点层合板的每一层的线性方程;考虑到脱层板的连接界面处应力和位移的关系,改进了现有的常用弱粘接模型,建立不同粘接模型的控制方程;最后通过求解整个板的控制方程,得到层合板的层间应力和位移．数值算例验证了该模型的正确性,并研究了层间界面为线性和非线性时的问题．结果表明:应用改进后的弱粘接模型,能够更好地模拟层合板的弱界面失效过程．     

Finite Order Rank-One Convex Envelopes and Computation of Microstructures with Laminates in Laminates

Finite order rank-one convex envelopes are introduced and it is shown that the i-th order laminated microstructures, or laminates in laminates, can be solved by any of the k-th order rank-one convex envelopes with k i. It is also shown that in finite element approximations of microstructures, replacing the non-quasiconvex potential energy density by its k-th order rank-one convex envelope, one can generally obtain sharper numerical results. Especially, for crystalline microstructures with laminates in laminates of order no greater than k + 1, numerical results with up to the computer precision can be obtained. Numerical examples on the first and second order rank-one convex envelopes for the Ericksen-James two-dimensional model for elastic crystals are given. A numerical example on finite element approximations of a crystalline microstructure by using the first order rank-one convex envelope and the periodic relaxation method is also presented. The methods turn out to be very successful for microstructures with laminates in laminates.     

Buckling and free vibration analyses of laminated composite plates by using two new hyperbolic shear-deformation theories

Two new hyperbolic displacement models, HPSDT1 and HPSDT2, are used for the buckling and free vibration analyses of simply supported orthotropic laminated composite plates. The models contain hyperbolic expressions to account for the parabolic distributions of transverse shear stresses and to satisfy the zero shear-stress conditions at the top and bottom surfaces of the plates. The equation of motion for thick laminated rectangular plates subjected to in-plane loads is deduced through the use of Hamilton’s principle. Closed-form solutions are obtained by using the Navier technique, and then the buckling loads and the fundamental frequencies are found by solving eigenvalue problems. The accuracy of the models presented is demonstrated by comparing the results obtained with solutions of other higher-order models given in the literature. It is found that the theories proposed can predict the fundamental frequencies and buckling loads of cross-ply laminated composite plates rather accurately. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 217–230, March–April, 2008.     

非均匀弹性地基圆板轴对称弯曲的一般解

本文在阶梯折算法的基础上,提出一个新的方法——精确解析法,得到了非均匀弹性地基圆板弯曲的一般解.文中导出了在任意轴对称载荷和边界条件下求解非均匀弹性地基圆板和中心带孔圆板弯曲的一般公式,并给出一致收敛于精确解的证明.文中得到的一般解可直接计算无弹性地基圆板的弯曲问题.问题最后归结为求解一个二元一次代数方程.文末给出算例,算例表明无论内力和位移均可得到满意的结果.     

Restarted Full Orthogonalization Method for Shifted Linear Systems

Restarted GMRES is known to be inefficient for solving shifted systems when the shifts are handled simultaneously. Variants have been proposed to enhance its performance. We show that another restarted method, restarted Full Orthogonalization Method (FOM), can effectively be employed. The total number of iterations of restarted FOM applied to all shifted systems simultaneously is the same as that obtained by applying restarted FOM to the shifted system with slowest convergence rate, while the computational cost grows only sub-linearly with the number of shifts. Numerical experiments are reported.     

Chaotic and bifurcation dynamics of a simply-supported thermo-elastic circular plate with variable thickness in large deflection

This paper presents the conditions that can possibly lead to chaotic motion and bifurcation behavior for a simply-supported large deflection thermo-elastic circular plate with variable thickness by utilizing the criteria of fractal dimensions, maximum Lyapunov exponents and bifurcation diagrams. The governing partial differential equation of the simply supported thermo-elastic circular plate with variable thickness is first derived by means of Galerkin method. Several different features including Fourier spectra, phase plot, Poincar’e map and bifurcation diagrams are numerically computed. These features are used to characterize the dynamic behavior of the plate subjected to various excitations of lateral loads and thermal loads. Numerical examples are presented to verify the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. Numerical modeling results indicate that large deflection motion of a thermo-elastic circular plate with variable thickness possesses chaotic motions and bifurcation motion under different lateral loads and thermal loads. The simulation results also indicate that the periodic motion of a circular plate can be obtained for the convex or the concave circular plate. The dynamic motion of the circular plate is periodic for the cases including (1) the lateral loading frequency is within a specific range, (2) thermal and lateral loadings are operated in a specific range and (3) the thickness parameter is less than a specific critical value for the convex circular plate or greater than a specific critical value for the concave circular plate. The modeling results show that the proposed method can be employed to predict the non-linear dynamics of any large deflection circular plate with variable thickness.     

A mesh transformation method for computing microstructures

Summary. A numerical method is established to solve the problem of minimizing a nonquasiconvex potential energy. Convergence of the method is proved both in the case on its own and in the case when it is combined with a weak boundary condition. Numerical examples are given to show that the method, especially when applied together with a continuation method and some other numerical techniques, is not only successful and efficient in solving problems with laminated microstructures but also capable of computing more complicated microstructures. Received March 17, 2000 / Published online April 5, 2001     

Incorporating the pricing decisions into the dynamic fleet management problem

This paper presents a model to coordinate the pricing and fleet management decisions of a freight carrier. We consider a setting where the loads faced by the carrier over a certain time horizon are deterministic functions of the prices. We want to find what prices the carrier should charge so that its pricing and fleet management decisions jointly maximize the profits. Our solution approach is an iterative one. At each iteration, we solve the fleet management problem with fixed prices, and then, adjust these prices by using the primal-dual solution to the fleet management problem so as to obtain ‘better’ prices. Computational experiments show that our approach yields high-quality solutions and can efficiently be applied on large problems.     

Boundary collocation method for a cracked rectangular plate with double external tension

In this article, the boundary collocation method is employed to investigate the problems of a central crack in a rectangular plate which applied double external tension on the outer boundary under the assumption that the dimensions of the plate are much larger than that of the crack. A set of stress functions has also been proposed based on the theoretical analysis which satisfies the condition that there is no external force on the crack surfaces. It is only necessary to consider the condition on the external boundary. Using boundary collocation method, the linear algebra equations at collocation points are obtained. The least squares method is used to obtain the solution of the equations, so that the unknown coefficients can be obtained. According to the expression of the stress intensity factor at crack tip, we can obtain the numerical results of stress intensity factor. Numerical experiments show that the results coincide with the exact solution of the infinite plate. In particular, this case of the double external tension applied on the outer boundary is seldom studied by boundary collocation method.     

Computation of worst geometric imperfection profiles of composite cylindrical shell panels by minimizing the non-linear buckling load

In this article the “most unfavorable” shape of initial geometric imperfection profile for laminated cylindrical shell panel is obtained analytically by minimizing the limit point load. The partial differential equations governing the shell stability problem are reduced to a set of non-linear algebraic equations using Galerkin's technique. The non-linear equilibrium path is traced by employing Newton–Raphson method in conjunction with the Riks approach. A double Fourier series is used to represent the initial geometric imperfection profile for the cylindrical shell panel. The optimum values of these Fourier coefficients are determined by minimizing the limit point load using genetic algorithm. The results are determined for simply supported composite cylindrical shell panel. Numerical results show that more number of terms is needed in Fourier series representation to obtain the “worst” geometric imperfection profile which gives lower limit load compared to single term representation of imperfection. We have incorporated constraints on the shape of imperfection to avoid unrealistic limit point loads (due to imperfection shape) as we have assumed that the imperfection is due to machining/manufactuting.     

多维区域上椭圆型方程Dirichlet问题的勒让德多小波多层扩充算法

用小波伽辽金方法求解多维区域上椭圆型方程齐次Dirichlet问题,构造了近似解空间的两个等价的勒让德多小波基,使得快速求解离散后的线性方程组的多层扩充算法得以实现.数值算例表明该算法是有效的.     

Lattice Boltzmann model for two-dimensional generalized sine-Gordon equation

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.     

A feasible decomposition method for constrained equations and its application to complementarity problems

In this paper, by analyzing the propositions of solution of the convex quadratic programming with nonnegative constraints, we propose a feasible decomposition method for constrained equations. Under mild conditions, the global convergence can be obtained. The method is applied to the complementary problems. Numerical results are also given to show the efficiency of the proposed method.     

A High-Order Piecewise Quartic Spline Rule for Hadamard Integral and Its Application of the Cavity Scattering

We develop a fourth-order piecewise quartic spline rule for Hadamard integral. The quadrature formula of Hadamard integral is obtained by replacing the integrand function with the piecewise quartic spline interpolation function. We establish corresponding error estimates and analyze the numerical stability. The rule can achieve fourth-order convergence at any point in the interval, even when the singular point coincides with the grid point. Since the derivative information of the integrand is not required, the rule can be easily applied to solve many practical problems. Finally, the quadrature formula is applied to solve the electromagnetic scattering from cavities with different wave numbers, which improves the whole accuracy of the solution. Numerical experiments are presented to show the efficiency and accuracy of the theoretical analysis.     

Finite element dynamic analysis of unsymmetric composite laminated beams with shear effect and rotary inertia under the action of moving loads

The finite element dynamic response of an unsymmetric composite laminated orthotropic beam, subjected to moving loads, has been studied. One-dimensional finite element based on classical lamination theory, first-order shear deformation theory, and higher-order shear deformation theory having 16, 20 and 24 degrees of freedom, respectively, are developed to study the effects of extension, bending, and transverse shear deformation. The theories also account for the Poisson effect, thus, the lateral strains and curvatures can be expressed in terms of the axial and transverse strains and curvatures and the characteristic couplings (bend–stretch, shear–stretch and bend–twist couplings) are not lost. The dynamic response of symmetric cross-ply and unsymmetric angle-ply laminated beams under the action of a moving load have been compared to the results of an isotropic simple beam. The formulation also has been applied to the static and free vibration analysis.     

Superconvergence of the composite Simpson’s rule for a certain finite-part integral and its applications

The superconvergence phenomenon of the composite Simpson’s rule for the finite-part integral with a third-order singularity is studied. The superconvergence points are located and the superconvergence estimate is obtained. Some applications of the superconvergence result, including the evaluation of the finite-part integrals and the solution of a certain finite-part integral equation, are also discussed and two algorithms are suggested. Numerical experiments are presented to confirm the superconvergence analysis and to show the efficiency of the algorithms.