Optimal Control for Multiscale Elliptic Equations with Rough Coefficients

This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough $L^\infty$ coefficients, which has not only complex coupling between nonseparable scales and nonlinearity, but also important applications in composite materials and geophysics. We use one of the recently developed numerical homogenization techniques, the so-called Rough Polyharmonic Splines (RPS) and its generalization (GRPS) for the efficient resolution of the elliptic operator on the coarse scale. Those methods have optimal convergence rate which do not rely on the regularity of the coefficients nor the concepts of scale-separation or periodicity. As the iterative solution of the nonlinearly coupled OCP-OPT formulation for the optimal control problem requires solving the corresponding (state and co-state) multiscale elliptic equations many times with different right hand sides, numerical homogenization approach only requires one-time pre-computation on the fine scale and the following iterations can be done with computational cost proportional to coarse degrees of freedom. Numerical experiments are presented to validate the theoretical analysis.     

On a Class of Quasilinear Elliptic Equations

We consider a class of quasilinear elliptic boundary problems, including the following Modified Nonlinear Schrödinger Equation as a special case: $$\begin{cases} ∆u+ \frac{1}{2} u∆(u^2)−V(x)u+|u|^{q−2}u=0 \ \ \ in \ Ω, \\u=0 \ \ \ \ \ \ \ ~ ~ ~ on \ ∂Ω, \end{cases}$$ where $Ω$ is the entire space $\mathbb{R}^N$ or $Ω ⊂ \mathbb{R}^N$ is a bounded domain with smooth boundary, $q∈(2,22^∗]$ with $2^∗=2N/(N−2)$ being the critical Sobolev exponent and $22^∗= 4N/(N−2).$ We review the general methods developed in the last twenty years or so for the studies of existence, multiplicity, nodal property of the solutions within this range of nonlinearity up to the new critical exponent $4N/(N−2),$ which is a unique feature for this class of problems. We also discuss some related and more general problems.     

Surface Acoustic Wave Characterization of Equivalent Young's Moduli for Patterned Films北大核心CSCD

Based on the equivalent elasticity theory for layered materials, the micro-mechanics equivalent models for single and dual damascene structures were established. The equivalent elastic constant of the patterned structure was introduced, to establish the propagation model for the surface acoustic waves propagating in the layered structure of the patterned film/ substrate, and the theoretical dispersion curves of the surface acoustic waves were calculated with Green’s function and the matrix method. The finite element method was used to calculate 24 numerical examples of damascene structures with different volume ratios, and the results were compared with those of the strain energy method. The results show that, the average relative errors of the equivalent Young’s moduli of the 300 nm-thick dual damascene film and the 100 nm-thick single damascene film are 2.06% and 2.27%, respectively. The research verifies the correctness of the equivalent patterned structure model and the feasibility of the surface acoustic wave method to characterize the mechanical properties of patterned films, and provides a reference for the development of suitable chemico-mechanical polishing technologies for patterned films under low pressure. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Sharp Condition for Global Existence of Supercritical Nonlinear Schr?dinger Equation with a Partial Confinement

We study the L²-supercritical nonlinear Schr?dinger equation(NLS) with a partial confinement,which is the limit case of the cigar-shaped model in Bose-Einstein condensate(BEC). By constructing a cross constrained variational problem and establishing the invariant manifolds of the evolution fow, we show a sharp condition for global existence.     

交叉增益调制型波长转换器噪声特性的研究

对基于半导体光放大器的交叉增益调制型波长转换器相向和同向两种工作方式进行了研究。实验上分别实现了两种工作方式在1550nm波段的波长转换,比较了相同外部条件下两种工作方式的噪声特性,在全面考虑放大自发辐射噪声及其消耗载流子,增益谱的色散性和有源区内部损耗等因素的基础上,运用放大器的分段模型,计算了不同条件下两种方式的输出噪声功率,结果表明：相向型工作方式受放大自发辐射噪声的影响要远大于同向工作方式,输出放大自发辐射噪声功率的大小主要取决于载流子浓度的高低,而载流子空间分布的不均匀程度则决定两种工作方式输出噪声功率的差值大小。     

A W-Band Frequency Tracing Transceiver System

In millimeter-wave, phase-locked oscillators are often used because the frequency drift of oscillators is very large. However, they are very complex. In this paper, a W-band frequency tracing transceiver system is presented and tested, which is used for digital communication. Although the communication distance is less than that of phase-locked systems, the scheme is simple and the cost is low.     

Micromachined Uncooled IR Bolometer Linear Array Using VO2 Thin Films

Mixed vanadium oxide thin films, as VO₂ for the main composition are materials for uncooled microbolometer due to their high temperature coefficient of resistance (TCR) at room temperature. This paper describes the design and fabrication of 8-element linear array IR uncooled microbolometers using the films and micromachining technology. The characteristics of the array is investigated in the spectral region of 8–12 m. The fabricated detectors exhibit responsivity of up to 10 KV/W, typical detectivity of 1.89×10⁸ cmHz^1/2/W, and thermal time constant of 11 ms, at 296 K and at a frequency of 30 Hz. Furthermore, The uncorrected response uniformity of the linear array bolometers is less than 20%.     

Fabrication of MgO nanobelts using a halide source and their structural characterization

MgO nanobelts have been fabricated by chemical vapor deposition using MgCl₃ as starting material. The products consist of a large quantity of belt-like nanostructures with typical lengths in the range of several tens to several hundreds of micrometers; some of them even have lengths on the order of a millimeter. The typical thickness and width-to-thickness ratio of the MgO nanobelts are in the range of 20 to 100 nm and about 5 to 10, respectively. The size and morphology of the MgO nanobelts were measured by transmission electron microscopy. Investigations of X-ray diffraction patterns and using high-resolution transmission electron microscopy indicate that the nanobelts have a cubic structure and are single-crystalline. Received: 23 August 2001 / Accepted: 27 August 2001 / Published online: 2 October 2001