Topology Optimization for Harmonic Excitation Structures with Minimum Length Scale Control Using the Discrete Variable Method

Continuumtopology optimization considering the vibration response is of great value in the engineering structure design.The aimof this studyis toaddress the topological designoptimizationof harmonic excitationstructureswith minimumlength scale control to facilitate structuralmanufacturing.Astructural topology design based on discrete variables is proposed to avoid localized vibration modes,gray regions and fuzzy boundaries in harmonic excitation topology optimization.The topological design model and sensitivity formulation are derived.The requirement of minimum size control is transformed into a geometric constraint using the discrete variables.Consequently,thin bars,small holes,and sharp corners,which are not conducive to the manufacturing process,can be eliminated from the design results.The present optimization design can efficiently achieve a 0–1 topology configuration with a significantly improved resonance frequency in a wide range of excitation frequencies.Additionally,the optimal solution for harmonic excitation topology optimization is not necessarily symmetric when the load and support are symmetric,which is a distinct difference fromthe static optimization design.Hence,one-half of the design domain cannot be selected according to the load and support symmetry.Numerical examples are presented to demonstrate the effectiveness of the discrete variable design for excitation frequency topology optimization,and to improve the design manufacturability.

Delay Restoration of Single-Track Railway Rescheduling

On the basis of analysis of the principle of delay restoration in a disturbed schedule, a heuristic algorithm for rescheduling trains is developed by restoring the total delay of the disturbed schedule. A discrete event topologic model is derived from the original undisturbed train diagram and a back propagation analysis method is used to label the maximum buffer time of each point in the model. In order to analyze the principle of delay restoration, the concept of critical delay is developed from the labeled maximum buffer time. The critical delay is the critical point of successful delay restoration. All the disturbed trains are classified into the strong-delayed trains and the weak-delayed trains by the criterion of the critical delay. Only the latter, in which actual delay is less than its critical delay, can be adjusted to a normal running state during time horizon considered. The heuristic algorithm is used to restore all the disturbed trains according to their critical details. The cores of the algorithm are the iterative repair technique and two repair methods for the two kinds of trains. The algorithm searches iteratively the space of possible conflicts caused by disturbed trains using an earfiest-delay-first heuristics and always attempts to repair the earliest constraint violation. The algorithm adjusts the weak-delayed trains directly back to the normal running state using the buffer time of the original train diagram. For the strong-delayed trains,the algorithm uses an utility function with some weighted attributes to determine the dynamic priority of the trains, and resolves the conflict according to the calculated dynamic priority. In the end, the experimental results show that the algorithm produces ＂good enough＂ schedules effectively and efficiently in disturbed situations.

Normal Systems over ANR's, Rigid Embeddings and Nonseparable Absorbing Sets

Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di- mensional absolute retracts. Michigan Math. J., 33, 291-313 （1986）] on strong Z-sets in ANR＇s and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w（U） ---- w（X） （where ＂w＂is the topological weight） for each open nonempty subset U of X, then X itself i,~ homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W（X, ＊） ---- {（xn）=l C X ： x~ = ＊ for almost all n} is homeomorphic to a pre-Hilbert space E with E EE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.