The Lagrange–Galerkin Spectral Element Method on Unstructured Quadrilateral Grids

This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral and spectral element methods. (2) We compare between spectral methods, spectral element methods, finite element methods and their derivedp-version,h-version, andhp-version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods.

In this article, to solve the frequency-domain airborne electromagnetic (EM) modeling problem, we express the electric and magnetic field by the vector magnetic potential and the scalar electric potential and derive the governing equation using the Galerkin weighted residual method under the Coulomb gauge. To avoid the strong singularity of the solution near the transmitting source, we directly solve the relatively slow-changing secondary potential. By introducing the spectral element method (SEM) based on unstructured tetrahedral grids, in which the EM field in each element is characterized by high-order Proriol–Koornwinder–Dubiner (PKD) orthogonal polynomials, we can obtain stable and accurate numerical solutions. By establishing the mapping relationship between the physical domain, the right-angled tetrahedral reference domain, and the orthogonal hexahedral domain, we can easily accomplish the element matrix analysis and calculation. The numerical examples show that, when the SEM is used to simulate the airborne EM response, a high accuracy can be obtained even if a very coarse grid is used. Furthermore, since the tetrahedral grids can easily model the complex boundaries, the SEM based on unstructured grids has significant advantages for simulating complex underground structures.

This paper introduces a Galerkin spectral element method (GSEM) for hybrid beam. The considered beam model depends upon the first order shear deformation theory (FSDT) is taken into account. The Galerkin strategy is used to formulate the spectral stiffness matrix based on the frequency. The spectral element method (SEM) is assessed by contrasting its solution with the exact analytical results accessible from literature as well as with the results by finite element method (FEM). This technique shows accuracy and computational efficiency with minimum number of element.

Time/wave domain analysis for axially moving pre-stressed nanobeam by wavelet-based spectral element method

Exponentially accurate spectral and spectral element methods for fractional ODEs

In this article we consider the spectral Galerkin method with the implicit/explicit Euler scheme for the two‐dimensional Navier–Stokes equations with the L2 initial data. Due to the poor smoothness of the solution on [0,1), we use the the spectral Galerkin method based on high‐dimensional spectral space HM and small time step Δt2 on this interval. While on [1,∞), we use the spectral Galerkin method based on low‐dimensional spectral space Hm(m = O(M1/2)) and large time step Δt. For the spectral Galerkin method, we provide the standard H2‐stability and the L2‐error analysis. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007

In recent years, the spectral element method (SEM) has been extensively applied to computational electromagnetics for large scale problems. As a special high-order finite element method, however, the SEM also includes DC spurious modes as in the traditional finite element method. Since 2015, the mixed spectral element method has been developed to remove such spurious modes by incorporating Gauss's law into the traditional SEM (N. Liu, L. Tobon, Y. Tang, Q. H. Liu, “Mixed Spectral Element Method for 2D Maxwell Eigenvalue Problem,” Communications in Computational Physics, vol. 17, no. 2, pp. 458-486, 2015; N. Liu, L. Tobon, Y. Zhao, Y. Tang, Q. H. Liu, “Mixed Spectral Element Method for 3-D Maxwell Eigenvalue Problem,” IEEE Trans. Microwave Theory Tech., vol. 64, no. 2, pp. 317-325, 2015).

Spectral element method in time for rapidly actuated systems

High-order discontinuous element-based schemes for the inviscid shallow water equations: Spectral multidomain penalty and discontinuous Galerkin methods

The use of mesh-based methods for modeling elongated composite structures with inhomogeneities leads to an increase in computational costs when discretizing the waveguide part of greater linear dimensions, while the semi-analytical numerical methods do not allow one to directly describe the structures with local inhomogeneities of an arbitrary shape. To compensate for the shortcomings of these two classes of numerical methods, we propose a hybrid scheme based on the spectral element method (SEM) and the semi-analytical finite element method (SAFEM) for studying the anti-plane vibrations of a composite structure in the frequency domain. Thus, for a waveguide, this scheme makes it possible to represent the solution via the SAFEM as a sum of modes or guided waves, and the adjacent regions are discretized using the SEM. The displacement and stress continuity conditions are imposed on the common boundary of two domains. To couple the solutions, we introduce an auxiliary function of displacement, which is approximated by applying the same basis functions as those used in SEM and SAFEM (Lagrange interpolation polynomials on the Gauss–Legendre–Lobatto nodal points). The unknown expansion coefficients of this function are determined by the Galerkin and collocation methods. It has been established that both methods provide the same accuracy. The results obtained by the hybrid approach employing the Galerkin and collocation methods are compared with the results calculated in the standard finite element software. It is shown that they are in good agreement. The presented hybrid approach can be straightforwardly extended to the case of in-plane motion, but it requires significant refinement for a three-dimensional case.

In the previous chapters, with an exception of GPG, the finite element formulations are based on the Galerkin methods in which test functions are chosen to be the same as the trial functions. This is not required in the weighted residual methods. Weighted residual methods other than the Galerkin methods include spectral element methods (SEM), least square methods (LSM), moment methods, or collocation methods, in which the test functions or weighting functions are not necessarily the same as the trial functions. In spectral element methods (SEM), polynomials in terms of nodal values of the variables are combined with special functions such as Chebyshev or Legendre polynomials. For least square methods, the test functions are constructed by the derivative of the residual with respect to the nodal values of the variables. Some arbitrary functions are chosen as test functions for the moment and collocation methods. Recently, the weighted residual concept has been used in meshless configurations, known as the finite point method (FPM), partition of unity method, meshless cloud method, or element-free method.

Although the spectral element method (SEM) has been well recognized as an exact continuum element method, its application has been limited mostly to one‐dimensional (1D) structures, or plates that can be transformed into 1D‐like problems by assuming the displacements in one direction of the plate in terms of known functions. We propose a spectral element model for the transverse vibration of a finite membrane subjected to arbitrary boundary conditions. The proposed model is developed by using the boundary splitting method and the waveguide FEM‐based spectral super element method in combination. The performance of the proposed spectral element model is numerically validated by comparison with exact solutions and solutions using the standard finite element method (FEM).

Abstract. The non-hydrostatic (NH) compressible Euler equations for dry atmosphere were solved in a simplified two-dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. By using horizontal SEM, which decomposes the physical domain into smaller pieces with a small communication stencil, a high level of scalability can be achieved. By using vertical FDM, an easy method for coupling the dynamics and existing physics packages can be provided. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL) quadrature points. The FDM employs a third-order upwind-biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative and integral terms. For temporal integration, a time-split, third-order Runge–Kutta (RK3) integration technique was applied. The Euler equations that were used here are in flux form based on the hydrostatic pressure vertical coordinate. The equations are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma–pressure vertical coordinate was implemented in this model. We validated the model by conducting the widely used standard tests: linear hydrostatic mountain wave, tracer advection, and gravity wave over the Schär-type mountain, as well as density current, inertia–gravity wave, and rising thermal bubble. The results from these tests demonstrated that the model using the horizontal SEM and the vertical FDM is accurate and robust provided sufficient diffusion is applied. The results with various horizontal resolutions also showed convergence of second-order accuracy due to the accuracy of the time integration scheme and that of the vertical direction, although high-order basis functions were used in the horizontal. By using the 2-D slice model, we effectively showed that the combined spatial discretization method of the spectral element and finite difference methods in the horizontal and vertical directions, respectively, offers a viable method for development of an NH dynamical core.

The objective of this paper is to present the Spectral Element Method (SEM) as an accurate and efficient design tool for static and dynamic simulations of cantilever based MEMS devices. The microcantilever under consideration is modeled as a Timoshenko beam and discretized using the spectral element formulation that accounts for fringing field and the nonlinearity arising from the electrostatic driving force. The static analysis has been carried out using Picard’s iteration method and the static pull-in displacement and voltage have been calculated. An eigenvalue analysis of this beam is also carried out to determine its natural frequencies. In addition, the dynamics of this cantilever is studied using the explicit Newmark predictor-corrector method to generate the time history. In all cases, the results have been compared to the one-dimensional Finite Element Method and three-dimensional finite element method (implemented through the commercial package COMSOL Multiphysics) to examine the accuracy and computational speed of the proposed SEM. The results of the simulations were also compared to those obtained by experiments in the existing scientific literature. These comparisons lead to the inference that the SEM is able to reproduce the static and dynamic response of the beam to a high degree of accuracy. It was also found that several numerical features inherent in the SEM lead to a significantly faster computation than the corresponding finite element method for equivalent degrees of freedom. This advantage was verified by using the SEM to carry out static and dynamic simulations of variable width microcantilevers. We therefore propose that the SEM is a viable tool for the MEMS community to accurately and quickly determine the static and dynamic pull-in parameters, frequency eigenvalues, and static and dynamic behavior at the design stage.

On the comparison of semi-analytical methods for the stability analysis of delay differential equations