DFT modal analysis of spectral element methods for the 2D elastic wave equation

The DFT modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete Fourier transform domain sampled in the mesh nodes. This technique provides a natural matching of exact and approximate modes of propagation. We extend this technique to spectral element methods for the 2D isotropic elastic wave equation, by using a Rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation, taking full advantage of the tensor product representation of the spectral element matrices. Numerical experiments illustrate the dependence of dispersion errors on the grid resolution, polynomial degree, and discretization in time. We consider spectral element methods with Chebyshev and Legendre collocation points.     

An improved isogeometrical analysis approach to functionally graded plane elasticity problems

The isogeometric analysis method is extended for addressing the plane elasticity problems with functionally graded materials. The proposed method which employs an improved form of the isogeometric analysis approach allows gradation of material properties through the patches and is given the name Generalized Iso-Geometrical Analysis (GIGA). The gradations of materials, which are considered as imaginary surfaces over the computational domain, are defined in a fully isoparametric formulation by using the same NURBS basis functions employed for the construction of the geometry and the approximation of the solution. The basic concept of the developed approach is concisely explained and its relation to the standard isogeometric analysis method is pointed out. It is shown that the difficulties encountered in the finite element analysis of the functionally graded materials are alleviated to a large degree by employing the mentioned method. Different numerical examples are presented and compared with available analytical solutions as well as the conventional and graded finite element methods to demonstrate the performance and accuracy of the proposed approach. The presented procedure can also be employed for solving other partial differential equations with non-constant coefficients.     

On error estimates of an exponential wave integrator sine pseudospectral method for the Klein–Gordon–Zakharov system

In this article, we propose an exponential wave integrator sine pseudospectral (EWI‐SP) method for solving the Klein–Gordon–Zakharov (KGZ) system. The numerical method is based on a Deuflhard‐type exponential wave integrator for temporal integrations and the sine pseudospectral method for spatial discretizations. The scheme is fully explicit, time reversible and very efficient due to the fast algorithm. Rigorous finite time error estimates are established for the EWI‐SP method in energy space with no CFL‐type conditions which show that the method has second order accuracy in time and spectral accuracy in space. Extensive numerical experiments and comparisons are done to confirm the theoretical studies. Numerical results suggest the EWI‐SP allows large time steps and mesh size in practical computing. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 266–291, 2016     

流体饱和多孔隙介质波动方程小波有限差分法

研究流体饱和多孔隙介质中波动方程的数值模拟．针对求解二维弹性波方程问题,提出小波有限差分法．该方法综合了小波多分辨分析计算灵活、计算效率高特性和有限差分易于实现的优点．数值模拟的结果显示,此方法对于求解流体饱和多孔隙介质方程的数值模拟是有效稳定的．     

Estudio de la estabilidad y dispersión del problema de propagación de ondas sísmicas en 2-D utilizando el método de diferencias finitas generalizadas

This paper shows the solution to the problem of seismic wave propagation in 2-D using generalized finite difference (GFD) explicit schemes. Regular and irregular meshes can be used with this method.As we are using an explicit method, it is necessary to obtain the stability condition by using the von Neumann analysis. We also obtained the star dispersion formulas for the phase velocities for the P and S waves, as well as the ones for the group velocities.As the control over the irregularity in the mesh is very important in the application of this method, we have defined an index of irregularity for the star (IIS) and another for the cloud (IIC), analyzing its relationship with the dispersion and time step used in the calculations.     

Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach

This paper deals with the linear free vibration analysis of Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. The geometry of the beam as well as the displacement field are defined using the NURBS basis functions which present the basic concept of the isogeometric analysis. A novel approach based on the fundamental relations of the differential geometry and Cauchy continuum beam model is presented and applied to derive the stiffness and consistent mass matrices of the corresponding spatial curved beam element. In the Bernoulli–Euler beam element only translational and torsional inertia are taken into account, while the Rayleigh beam element takes all inertial terms into consideration. Due to their formulation, isogeometric beam elements can be used for the dynamic analysis of spatial curved beams. Several illustrative examples have been chosen in order to check the convergence and accuracy of the proposed method. The results have been compared with the available data from the literature as well as with the finite element solutions.     

Static and dynamic analyses of isogeometric curvilinearly stiffened plates

The isogeometric analysis (IGA) is a new approach which builds a seamless connection between Computer Aided Design (CAD) and Computer Aided Engineering (CAE). This approach which uses the B-Splines or the Non-Uniform Rational B-Splines (NURBS) as a geometric representation of the object is a discretization technology for numerical analysis. The IGA has advantages of capturing exact geometry and making the flexibility of refinement, which results in higher calculation accuracy. To study the static and dynamic characteristics of curvilinearly stiffened plates, the NURBS based isogeometric analysis approach is developed in this paper. We use this approach to analyze the static deformation, the free vibration and the vibration behavior in the presence of in-plane loads of curvilinearly stiffened plates. Furthermore, the large deformation and the large amplitude vibration of the curvilinearly stiffened plates are also studied based on the von Karman's large deformation theory. One of the superiorities of the present method in the analysis of the stiffened plates is that the element number is much less than commercial finite element software, whereas another advantage is that the mesh refinement process is much more convenient compared with traditional finite element method (FEM). Some numerical examples are shown to validate the correctness and superiority of the present method by comparing with the results from commercial software and finite element analysis.     

一维气液两相漂移模型全隐式AUSMV算法研究

气液两相漂移模型显式AUSMV（advection upstream splitting method combined with flux vector splitting method）算法的时间步长受限于CFL（Courant-Friedrichs-Lewy）条件，为了提高计算效率，建立了一种全隐式AUSMV算法求解气液两相漂移模型.采用AUSM格式结合FVS（flux vector splitting）格式构造连续方程和运动方程的对流项数值通量，AUSM格式构造压力项数值通量.离散控制方程是非线性方程组，采用六阶Newton(牛顿)法结合数值Jacobi矩阵求解.计算经典算例Zuber-Findlay激波管问题和复杂漂移关系变质量流动问题，结果分析表明:全隐式AUSMV算法，色散效应小，无数值震荡，计算精度高.在压力波波速高的条件下，可以显著提高计算效率，耗散效应小.     

Relaxing the CFL condition for the wave equation on adaptive meshes

The Courant-Friedrichs-Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. A simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of wave propagation in functionally graded piezoelectric composite plates reinforced with graphene platelets

This paper presents a semi-analytical approach to investigate wave propagation characteristics in functionally graded graphene reinforced piezoelectric composite plates. Three patterns of graphene platelets (GPLs) describe the layer-wise variation of material properties in the thickness direction. Based on the Reissner-Mindlin plate theory and the isogeometric analysis, elastodynamic wave equation for the piezoelectric composite plate is derived by Hamilton’s principle and parameterized with the non-uniform rational B-splines (NURBS). The equation is transformed into a second-order polynomial eigenvalue problem with regard to wave dispersion. Then, the semi-analytical approach is validated by comparing with the existing results and the convergence on computing dispersion behaviors is also demonstrated. The effects of various distributions, volume fraction, size parameters and piezoelectricity of GPLs as well as different geometry parameters of the composite plate on dispersion characteristics are discussed in detail. The results show great potential of graphene reinforcements in design of smart composite structures and application for structural health monitoring.     

Conservation of energy,momentum and actions in numerical discretizations of non-linear wave equations

For classes of symplectic and symmetric time-stepping methods— trigonometric integrators and the Störmer–Verlet or leapfrog method—applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.     

Dynamic responses of Euler–Bernoulli beam subjected to moving vehicles using isogeometric approach

Dynamic analysis of beam structures subjected to moving vehicles using an isogeometric Euler–Bernoulli formulation is presented in this paper. The method utilizes B-Splines or Non-Uniform Rational–Splines (NURBS) as the basis functions for both geometric and analysis implementation. The rotation-free technique has been incorporated into the formulation by using only one deflection variable with excluding the rotational degrees of freedom adopted for each control point. Then, it enables to use a few degrees of freedom (Dofs) to achieve a highly accurate solution. The validations of the proposed method included a complicated moving vehicle and rough pavement effects are compared to the precisely analytical results. Compared with most existing methods of finite element method (FEM) and readily analytical solutions, the present technique indicated the effectiveness of present isogeometric method and its well accurate prediction for suitable simulating the interaction model of the bridge structures and complicated vehicles.     

Spectral element method for acoustic wave simulation in heterogeneous media

In this paper, we present a spectral element method for studying acoustic wave propagation in complex geological structures. Due to complexity (both lithological and stratigraphical), the use of numerical methods of higher accuracy and flexibility is needed to achieve the correct results. The spectral element method shows more accurate results compared to the low-order finite element, the conventional finite difference and the pseudospectral methods. High accuracy is reached even for rather long wave propagation times and dispersion errors are essentially eliminated; pirregular interfaces between different media can be well described so that numerical artifacts or noises are not at all introduced. The method is tested against analytical solutions both in the two-dimensional homogenous and heterogeneous media. The results of different simulations are presented.     

一维Burgers方程和KdV方程的广义有限谱方法

给出了高精度的广义有限谱方法．为使方法在时间离散方面保持高精度,采用了Adams-Bashforth 预报格式和Adams-Moulton校正格式,为了避免由Korteweg-de Vries(KdV)方程的弥散项引起的数值振荡, 给出了两种数值稳定器．以Legendre多项式、Chebyshev多项式和Hermite多项式为基函数作为例子,给出的方法与具有分析解的Burgers方程的非线性对流扩散问题和KdV方程的单孤独波和双孤独波传播问题进行了比较,结果非常吻合．     

Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.     

Patch coupling with the isogeometric dual mortar approach

The use of a common set of basis functions for design and analysis is the main paradigm of isogeometric analysis. The characteristics of the commonly used non-uniform rational B-splines (NURBS) surfaces require methods to handle non-conforming meshes to attain an efficient computational framework. The isogeometric mortar method uses constrained approximation spaces to enforce a coupling of deformations at the interface between patches in a weak manner. This method neither requires additional degrees of freedom nor the choice of empirical parameters. The main drawback of the standard isogeometric mortar approach is the non-local support of the mortar basis functions along the interface. This yields a large number of nodes per element for elements adjacent to the interface. Thus, the computational costs increase significantly for mesh refinement. This issue is remedied by the use of dual basis functions for the mortar method, which is referred to as dual mortar method. In this contribution several choices for the dual basis functions for B-splines are proposed and compared. A special focus is set on the support of the dual basis functions and on the support of the resulting mortar basis functions. Numerical examples show the influence of the choice for the dual basis functions on the accuracy of the global stress distribution, on the fulfillment of the interface conditions and on numerical efficiency. The use of approximate dual basis functions is shown to be competitive to computations of conforming meshes in terms of accuracy and efficiency. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lamb wave propagation using Wave Finite Element Method

Some of the available techniques for Lamb wave propagation simulation are the Finite Element Method (FEM), the Boundary Element Method and the Finite Difference Method. The FEM is the best method when complex damage, geometry or boundary is involved. However, high Lamb wave frequency requires very small element size thus high computational cost in FEM analysis. By using the existence of periodicity in plates, an attempt to reduce this computational cost is done using Wave FEM. The applicability of this method to model Lamb wave propagation in plate is first assessed in this paper for the 1-D wave propagation and compared with FEM explicit method. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

B-spline based finite element method in one-dimensional discontinuous elastic wave propagation

The B-spline variant of the finite element method (FEM) is tested in one-dimensional discontinuous elastic wave propagation. The B-spline based FEM (called Isogeometric analysis IGA) uses spline functions as testing and shape functions in the Galerkin continuous content. Here, the accuracy of stress distribution and spurious oscillations of the B-spline based FEM are studied in numerical modeling of one-dimensional propagation of stress discontinuities in a bar, where the analytical solution is known. For time integration, the Newmark method, implicit form of the generalized-α method, the central difference method and the predictor/multi-corrector method are tested and compared. The use of lumped and consistent mass matrices in explicit time integration is discussed. Due to accuracy, the consistent mass matrix is preferred in explicit time integration in IGA.     

Isogeometric analysis of 3D straight beam-type structures by Carrera Unified Formulation

This paper applies the isogeometric analysis (IGA) based on unified one-dimensional (1D) models to study static, free vibration and dynamic responses of metallic and laminated composite straight beam structures. By employing the Carrera Unified Formulation (CUF), 3D displacement fields are expanded as 1D generalized displacement unknowns over the cross-section domain. 2D hierarchical Legendre expansions (HLE) are adopted in the local area for the refinement of cross-section kinematics. In contrast, B-spline functions are used to approximate 1D generalized displacement unknowns, satisfying the requirement of interelement high-order continuity. Consequently, IGA-based weak-form governing equations can be derived using the principle of virtual work and written in terms of fundamental nuclei, which are independent of the class and order of beam theory. Several geometrically linear analyses are conducted to address the enhanced capability of the proposed approach, which is prominent in the detection of shear stresses, higher-order modes and stress wave propagation problems. Besides, 3D-like behaviors can be captured by the present IGA-based CUF-HLE method with reduced computational costs compared with 3D finite element method (FEM) and FEM-based CUF-HLE method.     

Algorithms for numerical solution of the modified equal width wave equation using collocation method

Quintic B-spline collocation algorithms for numerical solution of the modified equal width wave (MEW) equation have been proposed. The algorithms are based on Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. Quintic B-spline collocation method over the finite intervals is also applied to the time split MEW equation and space split MEW equation. Results for the three algorithms are compared by studying the propagation of the solitary wave, interaction of the solitary waves, wave generation and birth of solitons.