Exact integration in the boundary element method for two-dimensional elastostatic problems

This paper derives the exact integrations for the integrals in the boundary element analysis of two-dimensional elastostatics. For facilitation, the derivation is based on the simple forms of the fundamental functions by taking constant, discontinuous linear and discontinuous quadratic elements as examples. The efficiency and accuracy of the derived exact integrations are verified against five benchmark problems; the results indicate that the derived exact integrations significantly reduces the CPU time for forming the matrices of the boundary element analysis and solving the internal displacements.     

The labeled maximum matching problem

Given a graph G where a label is associated with each edge, we address the problem of looking for a maximum matching of G using the minimum number of different labels, namely the labeled maximum matching problem. It is a relatively new problem whose application is related to the timetabling problem. We prove it is NP-complete and present four different mathematical formulations. Moreover, we propose an exact algorithm based on a branch-and-bound approach to solve it. We evaluate the performance of our algorithm on a wide set of instances and compare our computational times with the ones required by CPLEX to solve the proposed mathematical formulations. Test results show the effectiveness of our procedure, that hugely outperforms the solver.     

单摆周期T0的理论分析与应用研究

理想单摆模型的周期与摆角无关,本文首先给出任意摆角条件下周期公式的精确解,同时用不同方法得出周期公式的近似解。通过T0-Tπ2lg与不同周期公式的比较,结果表明在误差允许的范围内,仍可用T0来代替T计算周期。     

3-D Elasticity Solutions of Simply Supported Laminated Rectangular Plates in Uniform Temperature Field

Based on the exact three dimensional (3-D) thermoelasticity theory, the elasticity solution of the simply supported laminated rectangular plates subjected to uniform temperature load is studied. An analytical method is presented to obtain the stress and displacement fields in the plate. First, the general solutions of a single-layer simply supported rectangular plate, which satisfies the governing differential equations and the boundary conditions at the lateral edges of the plate, is derived. Then, the displacement and stress relationships between the upper surface and the lower surface in the single-layer plate are obtained. According to the continuity of displacements, normal stress and shear stresses on the interface of two adjacent layers, the recursive formulae of displacements and stresses between the lowest layer and the top layer of the laminated plate are derived out by using the transfer matrix method. Finally, the unknown coefficients in the solutions are determined by the use of the upper surface and lower surface conditions of the plate. The distributions of displacements and stresses in the plate are obtained by substituting the unknown coefficients back to the recursive formulae and the solutions. The solutions obtained show excellent convergence. Comparing the numerical results with those obtained from the finite element method, the accuracy and correctness of the present method are demonstrated. Finally, the effects of temperature, thickness, layer number, and materials on the displacement and stress distributions of the plate are discussed in detail.     

Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σ_x=−N₀[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N₀/N_cr=0,0.5,0.8,0.95,1, where N_cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.     

Exclusive or Sum of Complex Terms expressions minimization

This paper demonstrates both theoretically, through theorem proofs, and experimentally, that existing minimization algorithms for ESOP (Exclusive or Sum Of Products) expressions can be successfully used for minimizing ESCT (Exclusive or Sum of Complex Terms) expressions as well. ESCT expressions are more general than ESOP ones and being reversible are very useful for logic design and for quantum circuits. The field of ESOP minimization has received considerable attention along the years and a great number of research works can be found in the literature, in contrast to the ESCT minimization problem with much fewer works addressing it. The open problems of finding exact ESCT expressions for completely specified functions of more than six variables as well as the problem of finding exact ESCT expressions for incompletely specified functions are solved here based on ESOP principles, for the first time.     

Error motion of a kinematic spindle

A prototype spindle for low-force, low-speed applications that is kinematically constrained in accordance with the principles of exact constraint was designed, fabricated, and tested. In the prototype spindle, the position and orientation of the shaft (rotor) are constrained at five contact points; four constraints are arranged radially around the rotor, and one constraint is located at the end of the rotor. This paper describes the design of the prototype spindle and presents a study of its error motion for the hardened steel rotor sliding on brass and UHMW-PE bonded to ceramic. The error motion is observed to be highly repeatable with less than 100 nm asynchronous errors, and a simple analytical model enables the prediction of the synchronous error motion using measurements of the rotor’s roundness profiles.     

A strongly polynomial algorithm for criticality of branching processes and consistency of stochastic context-free grammars

We provide a strongly polynomial algorithm for determining whether a given multi-type branching process is subcritical, critical, or supercritical. The same algorithm also decides consistency of stochastic context-free grammars.