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This work presents a comprehensive methodology for designing meta-materials with desired non-linear elastic behaviors. The approach employs a modified asymptotic expansion based homogenization method for topology optimization with finite deformation. Design and optimization of meta-materials for targeted non-linear elastic response under various loading conditions is explored. 

Starting from a network of discrete beams, topology optimized structures are produced by simultaneously optimizing each beam’s width and the locations of each node within the network. Due to the sparse nature of a beam network and by utilizing gradient descent results in a drastic reduction in computational cost compared to existing methods. Two different optimization objectives are investigated: minimization of the strain energy occurring from loading, often referred to compliance minimization; and the design of structures with prescribed mechanical responses to an applied load. 

Abstract Turbulent rotations of the magnetic field vector are observed in the Alfvénic streams of the solar wind where the magnetic field strength remains close to a constant. They can lead to reversals of the radial magnetic field component or switchbacks. It is not ruled out from the data that the rotations are divisible into the sum of small random angular deflections. In this work, we develop tools aimed at the analysis of the one-point statistical properties of the directional fluctuations of the magnetic field vector in the solar wind. The angular fluctuations are modeled by a drift-diffusion process which admits the exponential distribution as steady-state solution. Realizations of the stochastic process are obtained by solving the corresponding Langevin equation. It is shown that the cumulative effects of consecutive small-angle deflections can yield frequent reversals of the magnetic field vector even when the concentration parameter of the directional data is large. The majority of the rotations are associated with nearly transverse magnetic field fluctuations in this case. 

This paper presents an ultimate motion methodology of a face-milling spiral bevel gear pair to synthesize the mating tooth surfaces with a predesigned fourth-order motion curve. The methodology is to control some contact points along the contact path in the process of tooth contact analysis via application of an extended local synthesis which permits some transmission errors rather than zero at the concerned contact point. The modified offset motion correction is selected to demonstrate the proposed methodology. Applied torque corresponding to an elastic approach of 0.00635 mm at the mean contact point is calculated and the loaded tooth contact analysis is performed. Numerical results show that the extended local synthesis can effectively control the transmission errors on the predesigned fourth-order motion curve at arbitrarily predesigned contact points along the contact path of the spiral bevel gear pair. The tooth contact pattern for the actual tooth pair is scattered into three segments since the rotational motion of the driven gear at any instant angular position is dependent on the tooth pair with the least transmission error among the three adjacent tooth pairs. The actual tooth contact patterns of the spiral bevel gear pair become continuous when meshing tooth surfaces are elastically deformed.