The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations

The Laplace transform is applied to remove the time-dependent variable in the diffusion equation. For non-harmonic initial conditions this gives rise to a non-homogeneous modified Helmholtz equation which we solve by the method of fundamental solutions. To do this a particular solution must be obtained which we find through a method suggested by Atkinson. To avoid costly Gaussian quadratures, we approximate the particular solution using quasi-Monte-Carlo integration which has the advantage of ignoring the singularity in the integrand. The approximate transformed solution is then inverted numerically using Stehfest's algorithm. Two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2-D and 3-D. © 1998 John Wiley & Sons, Ltd.     

A Galerkin-reproducing kernel method: Application to the 2D nonlinear coupled Burgers' equations

The paper introduces a Galerkin method in the reproducing kernel Hilbert space. It is implemented as a meshless method based on spatial trial spaces spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. For the time-dependent PDEs it leads to a system of ordinary differential equations. The method is used for solving the 2D nonlinear coupled Burgers' equations having Dirichlet and mixed boundary conditions. The numerical solutions for different values of Reynolds number (Re) are compared with analytical solutions as well as other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Re in the case of Dirichlet boundary conditions.     

Modeling of dispersion and nonlinear characteristics of tapered photonic crystal fibers for applications in nonlinear optics

In this article, we investigate for the first time the dispersion and the nonlinear characteristics of the tapered photonic crystal fibers (PCFs) as a function of length z, via solving the eigenvalue equation of the guided mode using the finite-difference frequency-domain method. Since the structural parameters such as the air-hole diameter and the pitch of the microstructured cladding change along the tapered PCFs, dispersion and nonlinear properties change with the length as well. Therefore, it is important to know the exact behavior of such fiber parameters along z which is necessary for nonlinear optics applications. We simulate the z dependency of the zero-dispersion wavelength, dispersion slope, effective mode area, nonlinear parameter, and the confinement loss along the tapered PCFs and propose useful relations for describing dispersion and nonlinear parameters. The results of this article, which are in a very good agreement with the available experimental data, are important for simulating pulse propagation as well as investigating nonlinear effects such as supercontinuum generation and parametric amplification in tapered PCFs.     

纳米颗粒的短脉冲激光烧蚀制备及其非线性光学应用

本文主要介绍了纳米颗粒的短脉冲激光制备及其在非线性光学领域的应用。短脉冲激光制备纳米颗粒具有纯度高、操作简单和适用性广等优点,所制备的非线性纳米颗粒尺寸分布均匀,粒度小且可调控,在非线性光学材料中有着独特的地位。为了更深入地对此进行研究,本文介绍了纳米颗粒的光学非线性和激光的特点和原理。在此基础上,通过阐述短脉冲激光与物质相互作用的机理,说明了激光制备纳米颗粒所具有的优点,详细分析了制备条件对合成纳米颗粒的影响,并结合激光制备不同的纳米颗粒,介绍当前激光制备各类纳米颗粒的研究现状。激光制备纳米颗粒是一种操作简便、适用性广,且对环境友好的方法。

     

New mechanism for THz oscillation of the nonlinear refractive index in air: particle-like solutions

We describe a generalized nonlinear envelope equation governing the propagation of optical pulses, with superbroad spectrum, in air. The corresponding modified nonlinear term in the equation oscillates with terahertz frequency. The fluctuation is due to the group and phase velocity difference. In the partial case of femtosecond pulses that have a power slightly above that which is critical for self-focusing, an exact 3D + 1 particle-like solution is found.     

The time-marching method of fundamental solutions for wave equations

In this paper, a meshless numerical algorithm is developed for the solution of multi-dimensional wave equations with complicated domains. The proposed numerical method, which is truly meshless and quadrature-free, is based on the Houbolt finite difference (FD) scheme, the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). The wave equation is transformed into a Poisson-type equation with a time-dependent loading after the time domain is discretized by the Houbolt FD scheme. The Houbolt method is used to avoid the difficult problem of dealing with time evolution and the initial conditions to form the linear algebraic system. The MPS and MFS are then coupled to analyze the governing Poisson equation at each time step. In this paper we consider six numerical examples, namely, the problem of two-dimensional membrane vibrations, the wave propagation in a two-dimensional irregular domain, the wave propagation in an L-shaped geometry and wave vibration problems in the three-dimensional irregular domain, etc. Numerical validations of the robustness and the accuracy of the proposed method have proven that the meshless numerical model is a highly accurate and efficient tool for solving multi-dimensional wave equations with irregular geometries and even with non-smooth boundaries.     

Analysis of the second Piola-Kirchhoff stress in nonlinear thick and thin structures using exact geometry SaS solid-shell elements

This paper presents the finite rotation exact geometry four-node solid-shell element using the sampling surfaces (SaS) method. The SaS formulation is based on choosing inside the shell N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains and stresses leads to a robust higher-order shell formulation. The SaS are located at only Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. The proposed hybrid-mixed four-node solid-shell element is based on the Hu-Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated through efficient 3D analytical integration and its explicit form is given. As a result, the proposed exact geometry solid-shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement-based solid-shell elements.     

The Eulerian–Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations

The Eulerian–Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian–Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in the meantime the convective term in the Burgers’ equations is retrieved by the back-tracking scheme along the characteristics. The proposed numerical scheme is free from mesh generation and numerical integration and is a truly meshless method. Two-dimensional Burgers’ equations of one and two unknown variables with and without considering the disturbance of noisy data are analyzed. The numerical results are compared very well with the analytical solutions as well as the results by other numerical schemes. By observing these comparisons, the proposed meshless numerical scheme is convinced to be an accurate, stable and simple method for the solutions of the Burgers’ equations with irregular domain even using very coarse collocating points.     

A modified Newton‐type Koiter‐Newton method for tracing the geometrically nonlinear response of structures

The Koiter‐Newton (KN) method is a combination of local multimode polynomial approximations inspired by Koiter's initial postbuckling theory and global corrections using the standard Newton method. In the original formulation, the local polynomial approximation, called a reduced‐order model, is used to make significantly more accurate predictions compared to the standard linear prediction used in conjunction with Newton method. The correction to the exact equilibrium path relied exclusively on Newton‐Raphson method using the full model. In this paper, we proposed a modified Newton‐type KN method to trace the geometrically nonlinear response of structures. The developed predictor‐corrector strategy is applied to each predicted solution of the reduced‐order model. The reduced‐order model can be used also in the correction phase, and the exact full nonlinear model is applied only to calculate force residuals. Remainder terms in both the displacement expansion and the reduced‐order model are well considered and constantly updated during correction. The same augmented finite element model system is used for both the construction of the reduced‐order model and the iterations for correction. Hence, the developed method can be seen as a particular modified Newton method with a constant iteration matrix over the single KN step. This significantly reduces the computational cost of the method. As a side product, the method has better error control, leading to more robust step size adaptation strategies. Numerical results demonstrate the effectiveness of the method in treating nonlinear buckling problems.     

The method of approximate fundamental solutions (MAFS) for elliptic equations of general type with variable coefficients

The paper presents a meshless method for solving elliptic equations of general type with variable coefficients. It is based on the use of the delta-shaped functions and the method of approximate fundamental solutions first suggested for solving equations with constant coefficients. The method assumes that the solution domain is embedded in a square and the initial equation is extended onto the square with the help of the CICE −(Chebyshev interpolation + C-expansion) approximation scheme. As a result the coefficients of the equation are approximated by the truncated Fourier series over some orthogonal system in the square. The approximate fundamental solutions (AFSs) satisfy L[u]=I(x), where I(x) is the delta shaped function in the form of the truncated Fourier series. Thus, the AFSs due to the special form of the operator can be obtained in the similar form of truncated series. The next part of the MAFS follows the general scheme of the MFS. The numerical examples are presented and the results are compared with the analytical solutions. The comparison shows that the method presented provides a very high precision in solution of two-dimensional elliptic equations of general type with different boundary conditions (Dirichlet, Neumann, mixed) in arbitrary domains.     

Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays

In this article, we study the existence of multiple solutions of the integral boundary value problems for high-order nonlinear fractional differential equations with impulses and distributed delays. Some sufficient criteria will be established by the fixed point index theorem in cones. As application, one example is given to demonstrate the validity of our main results.     

衍射光学元件分析和设计中标量理论的局限性

以衍射光栅为例,用标量理论和严格耦合波理论的对比,分析衍射光学元件各参数对标量理论适用范围的影响。结果表明在光栅周期减小,刻蚀深度增加,光栅折射率增加,光束入射角度增加以及填充因子偏离 50%等情况下,标量理论的误差将逐渐增大,其中光栅周期和刻蚀深度对标量理论的影响较大,光栅周期小到 5 倍波长或者刻蚀深度大到 5 倍波长时,标量理论将不再适用。     

Stress intensity factor computation using the method of fundamental solutions: mixed‐mode problems

The method of fundamental solutions is applied to the computation of stress intensity factors in linear elastic fracture mechanics. The displacements are approximated by linear combinations of the fundamental solutions of the Cauchy–Navier equations of elasticity and the leading terms for the displacement near the crack tip. Two algorithms are developed, one using a single domain and one using domain decomposition. Numerical results are given. Copyright © 2006 John Wiley & Sons, Ltd.     

Potential field based geometric modelling using the method of fundamental solutions

We propose a new geometric modelling method based on the so‐called potential field (PF) modelling technique. The harmonic problem associated with this technique is solved numerically using the method of fundamental solutions (MFS). We investigate the applicability of the proposed approach to parametrically defined curves of varying complexity. Based on the MFS, we also provide definitions of the Boolean operations associated with the geometric modelling. Finally, we give practical applications of the method to computer‐aided design and manufacturing problems. Copyright © 2006 John Wiley & Sons, Ltd.     

Maximizing the bandwidth of coherent,mid-IR supercontinuum using highly nonlinear aperiodic nanofibers

We describe in detail a new procedure of maximizing the bandwidth of mid-infrared (mid-IR) supercontinuum (SC) in highly nonlinear microstructured As₂Se₃ and tellurite aperiodic nanofibers. By introducing aperiodic rings of first and secondary air holes into the cross-sections of our microstructured fiber designs, we achieve flattened and all-normal dispersion profiles over much broader bandwidths than would be possible with simple periodic designs. These fiber designs are optimized for efficient, broadband, and coherent SC generation in the mid-IR spectral region. Numerical simulations show that these designs enable the generation of a SC spanning over 2290?nm extending from 1140 to 3430?nm in 8?cm length of tellurite nanofiber with input energy of E?=?200?pJ and a SC bandwidth of over 4700?nm extending from 1795 to 6525?nm generated in only 8?mm-length of As₂Se₃-based nanofiber with input energy as low as E?=?100?pJ. This work provides a new type of broadband mid-IR SC source with flat spectral shape as well as excellent coherence and temporal properties by using aperiodic nanofibers with all-normal dispersion suitable for applications in ultrafast science, metrology, coherent control, non-destructive testing, spectroscopy, and optical coherence tomography in the mid-IR region.     

Iterative solution of a hybrid method for Maxwell's equations in the frequency domain

A hybrid method for solution of Maxwell's equations of electromagnetics in the frequency domain is developed as a combination between the method of moments and the approximation in physical optics. The equations are discretized by a Galerkin method and solved by an iterative block Gauss–Seidel method. The convergence of the iterations is studied theoretically and in numerical experiments. The accuracy of the hybrid method is compared to the method of moments for a cylinder with an incident field for different wavenumbers. Copyright © 2003 John Wiley & Sons, Ltd.     

Integral equation technique using normal mode method for nonlinear random vibration of clamped rectangular plates

In the present paper the authors have illustrated the application of the integral equation technique combined with modal superposition method for the nonlinear random analysis using equivalent linearization procedure. For this an all edges clamped isotropic rectangular plate is considered. It is subjected to stationary Gaussian white noise (zero mean) excitation which is random in time and deterministic (uniformly distributed) in space. Chu-Herrmann equations for large deflection of plates in terms of the displacements are solved. The results obtained have been compared with the ones in existing literature. A parametric study has also been conducted for different aspect ratios of the plate.     

A generalized Ritz method for partial differential equations in domains of arbitrary geometry using global shape functions

A boundary element method (BEM)-based variational method is presented for the solution of elliptic PDEs describing the mechanical response of general inhomogeneous anisotropic bodies of arbitrary geometry. The equations, which in general have variable coefficients, may be linear or nonlinear. Using the concept of the analog equation of Katsikadelis the original equation is converted into a linear membrane (Poisson) or a linear plate (biharmonic) equation, depending on the order of the PDE under a fictitious load, which is approximated with radial basis function series of multiquadric (MQ) type. The integral representation of the solution of the substitute equation yields shape functions, which are global and satisfy both essential and natural boundary conditions, hence the name generalized Ritz method. The minimization of the functional that produces the PDE as the associated Euler–Lagrange equation yields not only the Ritz coefficients but also permits the evaluation of optimal values for the shape parameters of the MQs as well as optimal position of their centers, minimizing thus the error. If a functional does not exists or cannot be constructed as it is the usual case of nonlinear PDEs, the Galerkin principle can be applied. Since the arising domain integrals are converted into boundary line integrals, the method is boundary-only and, therefore, it maintains all the advantages of the pure BEM. Example problems are studied, which illustrate the method and demonstrate its efficiency and great accuracy.