Static and Dynamic Analysis of Electrostatically Actuated Microcantilevers Using the Spectral Element 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).

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.

A time-domain spectral element method with C1 continuity for static and dynamic analysis of frame structures

Finite element method(FEM) is one of the most popularly used method analyzing the dynamic behaviors of structures. But unless number of finite elements is large enough, the results from FEM some what different from exact analytical solutions, especially at high frequency range. On the other hand, as the spectral analysis method(SAM) deals directly with the governing equations of a structure, the results from this melthod cannot but be exact regardless of any frequency range. However, the SAM can be applied only to the case where a structure is subjected to the concentrated loads, despite a structure could be unddergone distributed loads more generally. In this paper, therefore, new spectral analysis algorithm is introduced through the spectral element method(SEM), so that it can be applied to anlystructures whether they are subjected to the concentrated loads or to the distributed loads. The results from this new SEM are compared with both the results from FEM and the exact analytical solutions. As expected, the results from new SEM algorithm are found to be almost identical to the exact analytical solutions while those from FEM are not agreed well with the exact analytical solutions as the mode number increases.

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

Solution of radiative heat transfer in graded index media by least square spectral element method

Spectral element method in time for rapidly actuated systems

The dynamic responses and vibration suppression of frame structures considering the shear deformation are investigated using the spectral element method (SEM). Vibration suppression of the frame structures is studied through the active control technique using the piezoelectric rods. The local dynamic stiffness matrices of the spectral elements for the Timoshenko beams and longitudinal vibration of the rods are obtained in the Fourier phase space and that for the axial-bending-shear coupled vibration of the piezoelectric rod is also gotten. From the transformation between the local and global coordinates, the global dynamic stiffness matrix for the whole frame structures containing the piezoelectric rods is obtained. From the numerical simulation, the dynamic responses of the frame structures can be effectively and exactly calculated by the SEM comparing with the finite element method. It can be found that the dynamic responses of the frame structures can be effectively suppressed by the active control technique based on the piezoelectric rods. The influences of the structural parameters on the dynamic responses and active control effect are also analyzed.

In this work, the forced response of rotating rings exhibiting nonperiodic and periodic variations in material properties and geometry is assessed by means of the spectral element method (SEM). Based on the Euler–Bernoulli beam theory, a spectral element for a planar rotating ring is derived. This spectral element allows the investigation of the effects of structural damping, internal pressure and elastic foundations in the harmonic response of rotating rings. The dynamic response of rotating rings including periodic imperfections that lead to band gap effects is addressed. The spectral element formulation provides exact solutions within the range of validity of the applied theory using a reduced number of degrees-of-freedom. Thus, it contributes to reducing the computational time. It also provides a straightforward way to solve structural dynamics problems including arbitrary boundary conditions and discontinuities. The proposed formulation is validated by comparison with analytical solutions, which are available only for uniform homogenous rings.

SUMMARY Simulating poroelastic waves in large-scale 3-D problems having porous media coupled with elastic solids and fluids is computationally challenging for traditional methods. It is well established that the spectral element method (SEM) is more effective than the traditional methods like the finite element method (FEM) when dealing with complex geophysical problems, for its high-order accuracy with exponential convergence. However, at present, little research has been done for SEM in the frequency domain, which will be more efficient than the time-domain SEM for narrowband simulations with multiple sources, material dispersion and attenuation. Herein, we systematically develop a SEM in the frequency domain to simulate coupled poroelastic, elastic and acoustic waves in anisotropic (i.e. porosity, permeability and elastic coefficients with anisotropy), heterogeneous, and lossy media. Furthermore, we completely remove the dimension inconsistency between the displacement field and the pressure in porous media to reduce the condition number of the system matrix by around 16 orders of magnitude while maintaining the symmetry of the system matrix. To solve the multiphysics coupling problems, we apply different coupling conditions to different interface types, and use basis functions to discretize the corresponding governing equations. Numerical examples show that the proposed SEM can obtain higher accuracy with much fewer unknowns compared with the FEM and has the capacity to solve the large-scale real coupling problems.

ABSTRACTThis article aims at numerically simulating the 3D seismic wave propagation from rupture to structures. A two-step method coupling the spectral element method (SEM) and the finite-element method (FEM) is proposed based on the domain reduction method to simultaneously simulate the seismic wave propagation in large-scale regions and analyze the dynamic behavior of structures in local sites. First, the concept of the proposed two-step method is illustrated. In the first step, the seismic wave propagation of the entire area, involving the source, propagation media, and local region of interest, is simulated using the SEM. In the second step, the dynamic analysis of structure-foundation system with local geological and topographical conditions is implemented using the FEM in a fine mesh based on the results in the first step. Subsequently, the FEM grid size is evaluated to match the SEM results, and the proposed SEM–FEM procedure is verified using both point-source and finite-fault model in a layered flat model. Finally, two analysis examples are presented using the proposed procedure. The analysis results show that the proposed SEM–FEM procedure can well consider the effects of local geological and topographical conditions on synthesized ground motions and can be applied to the rupture-to-structure simulations in earthquake engineering.

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 spectral element method (SEM) is an advanced implementation of the finite element method (FEM), in which the internal nodes of each element are arranged at positions corresponding to the zeros of certain families of orthogonal polynomials. High accuracy can be obtained with SEM using a relatively small number of elements if compared with a FEM corresponding implementation. In this work, a new SEM in cylindrical coordinates is explored for the analysis of field propagation in inhomogeneous and anisotropic-filled waveguides. Numerical results show that our technique is efficient and accurate for modeling such waveguide devices.

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).

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