PERIODIC SOLUTIONS OF THIRD-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

The sufficient condition for the existence of2 π- periodic solutions of the following third- order functional differential equations with variable coefficients a(t) x (t) +bx″2 k- 1(t) +cx′2 k- 1(t) + 2 k- 1 i=1 cixi(t) +g(x(t-τ) ) =p(t) =p(t+2π) is obtained.The approach is based on the abstract continuation theorem from Mawhin and the a- priori estimate of periodic solutions

Holder continuous weak solutions of the 2D Boussinesq equation with thermal diffusion

We show the existence of Holder continuous periodic weak solutions of the 2D Boussinesq equation with thermal diffusion which satisfy the prescribed kinetic energy.More precisely,for any smooth e(t):[0,1]→R+andε∈(0,110),there exist v∈C 110−ε([0,1]×T2)andθ∈C 1,120−εt 2 C 2,1 x 10−ε([0,1]×T2),which satisfy(1.1)in the sense of distribution and e(t)=ˆT2|v(t,x)|2 dx,∀t∈[0,1].

Pushing the limits of microwave absorption capability of carbon fiber in fabric design based on genetic algorithm

The field of electromagnetic wave absorption(EWA)requires the adaptability,tenability,and multifunction of high-performance materials in the future.The design and preparation of EWA materials aiming at performance requirements is the latest research hotspot.Here,a performancedriven strategy for simultaneously coordinating different target performances was proposed to optimize the structure of the periodical long continuous carbon/glass fiber fabric(PCGF)materials through algorithm and simulation.The optimized structure of the PCGF not only improves the impedance matching,but also introduces the induced orientation effect for a high cooperative loss of conductivity,resonance,and periodic structure.The flexible PCGF shows a broad effective absorption bandwidth(EAB)of 32.7 GHz covering a part of the C-band and the whole X-,Ku-,K-,and Ka-bands with a thickness(d)of only 0.92 mm and a density of 5.6×10^(−4) kg·cm^(−3).This highly designable fabric is promising for the EWA practical application owing to integrating the characteristics of good flexibility,acid and alkali resistance,bending resistance,excellent mechanical properties,and easy large-scale preparation.

Hybrid Method of combined difference and spectrum for time-domain maxwell’s equations

For the finite-difference time domain(FDTD)method,the electromagnetic scattering problem,which requires the characteristic structure size to be much smaller than the wavelength of the exciting source,is still a challenge.To circumvent this difficulty,this paper presents a novel hybrid numerical technique of combined difference and spectrum for time-domain Maxwell’s equations.With periodical continuation of each time-dependent quantity in Maxwell’s equations,the solutions before and after the continuation remain consistent in the first period,which results in the conversion of the continuous spectrum problem to a discrete one.The discrete spectrum of the field after continuation is obtained from difference methods for Maxwell’s curl equations in frequency-domain,and the time domain solution of the original problem is derived from their inverse Fourier transform.Due to its unconditional stability,the proposed scheme excels FDTD in resolving the aforementioned problems.In addition,this method can simulate dispersive media whose electric susceptibility cannot be expressed with Debye or Lorentz types of models.In dealing with boundary conditions,it can utilize the perfectly matched layer(PML)without extra codes.Numerical experiments demonstrate its effectiveness,easy implementation and high precision.

Theoretical aspects of selecting repeated unit cell model in micromechanical analysis using displacement-based finite element method

Repeated Unit Cell（RUC）is a useful tool in micromechanical analysis of composites using Displacement-based Finite Element（DFE）method,and merely applying Periodic Displacement Boundary Conditions（PDBCs）to RUC is almost a standard practice to conduct such analysis.Two basic questions arising from this practice are whether Periodic Traction Boundary Conditions（PTBCs,also known as traction continuity conditions）are guaranteed and whether the solution is independent of selection of RUCs.This paper presents the theoretical aspects to tackle these questions,which unify the strong form,weak form and DFE method of the micromechanical problem together.Specifically,the solution’s independence of selection of RUCs is dealt with on the strong form side,PTBCs are derived from the weak form as natural boundary conditions,and the validity of merely applying PDBCs in micromechanical Finite Element（FE）analysis is proved by referring to its intrinsic connection to the strong form and weak form.Key points in the theoretical aspects are demonstrated by illustrative examples,and the merits of setting micromechanical FE analysis under the background of a clear theoretical framework are highlighted in the efficient selection of RUCs for Uni Directional（UD）fiber-reinforced composites.