Numerical simulation of natural convection in a differentially heated tall enclosure using a spectral element method

Chebyshev spectral elements are applied to dissipation analysis of pore-pressure of roller compaction earth-rockfilled dams (ERD) during their construction. Nevertheless, the conventional finite element, for its excellent adaptability to complex geometrical configuration, is the most common way of spatial discretization for the pore-pressure solution of ERDs now [1]. The spectral element method, by means of the spectral isoparametric transformation, surmounts the disadvantages of disposing with complex geometry. According to the illustration of numerical examples, one can conclude that the spectral element methods have the following obvious advantages: 1) large spectral elements can be used in spectral element methods for the domains of homogeneous material; 2) in the application of large spectral elements to spatial discretization, only a few leading terms of Chebyshev interpolation polynomial are taken to arrive at the solutions of better accuracy; 3) spectral element methods have excellent convergence as well-known. Spectral method also is used to integrate the evolution equation in time to avoid the limitation of conditional stability of time-history integration

Numerical study of natural convection heat transfer inside a differentially heated square enclosure with adiabatic horizontal walls and vertical isothermal walls is investigated. Two insulated ribs are symmetrically located on horizontal walls. The governing non-linear equations are solved in a two-dimensional domain using a control volume method and the SIMPLER algorithm for the velocity–pressure coupling is employed. The results will be presented in forms of streamlines, isotherms and Nusselt number for Rayleigh number 106. It is shown that for small rib height the isotherms indicate the laminar boundary regime with high temperature gradient near the bottom of the hot surface and the top of cold one. However, as rib height increases this boundary layer is vanished. Also it is found that as the length and height of the ribs increase the mean Nusselt number decreases.

Transient wave propagations with the Noh-Bathe scheme and the spectral element method

The airborne electromagnetic (AEM) method has a high sampling rate and survey flexibility. However, traditional numerical modeling approaches must use high-resolution physical grids to guarantee modeling accuracy, especially for complex geological structures such as anisotropic earth. This can lead to huge computational costs. To solve this problem, we propose a spectral-element (SE) method for 3D AEM anisotropic modeling, which combines the advantages of spectral and finite-element methods. Thus, the SE method has accuracy as high as that of the spectral method and the ability to model complex geology inherited from the finite-element method. The SE method can improve the modeling accuracy within discrete grids and reduce the dependence of modeling results on the grids. This helps achieve high-accuracy anisotropic AEM modeling. We first introduced a rotating tensor of anisotropic conductivity to Maxwell’s equations and described the electrical field via SE basis functions based on GLL interpolation polynomials. We used the Galerkin weighted residual method to establish the linear equation system for the SE method, and we took a vertical magnetic dipole as the transmission source for our AEM modeling. We then applied fourth-order SE calculations with coarse physical grids to check the accuracy of our modeling results against a 1D semi-analytical solution for an anisotropic half-space model and verified the high accuracy of the SE. Moreover, we conducted AEM modeling for different anisotropic 3D abnormal bodies using two physical grid scales and three orders of SE to obtain the convergence conditions for different anisotropic abnormal bodies. Finally, we studied the identification of anisotropy for single anisotropic abnormal bodies, anisotropic surrounding rock, and single anisotropic abnormal body embedded in an anisotropic surrounding rock. This approach will play a key role in the inversion and interpretation of AEM data collected in regions with anisotropic geology.

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 nodal triangle-based spectral element method for the shallow water equations on the sphere

The main purpose of the current paper is to propose a new numerical scheme based on the spectral element procedure for simulating the neutral delay distributed‐order fractional damped diffusion‐wave equation. To this end, the temporal direction has been discretized by a finite difference formula with convergence order where 1<α<2. In the next, to obtain a full‐discrete scheme, we apply the spectral finite element method on the spatial direction. Furthermore, the unconditional stability of semidiscrete scheme and convergence order of full‐discrete scheme of new technique are discussed. Finally, 2 test problems have been considered to demonstrate the ability and efficiency of the proposed numerical technique.

The spectral element dynamical core has been demonstrated to be an accurate and scalable approach for solving the equations of motion in the atmosphere. However, it is also known that use of the spectral element method for tracer transport is costly and requires substantial parallel communication over a single time step. Consequently, recent efforts have turned to finding alternative transport schemes which maintain the scalability of the spectral element method without its significant cost. This article proposes a conservative semi‐Lagrangian approach for tracer transport which uses upstream trajectories to solve the transport equation on the native spectral element grid. This formulation, entitled the Flux‐Form Semi‐Lagrangian Spectral Element (FF‐SLSE) method, is highly accurate compared to many competing schemes, allows for large time steps, and requires fewer parallel communications over the same time interval than the spectral element method. In addition, the approach guarantees local conservation and is easily paired with a filter which can be used to ensure positivity. This article presents the dispersion relation for the 1D FF‐SLSE approach and demonstrates stability up to a Courant number of 2.44 with cubic basis. Several standard numerical tests are presented for the method in 2D to verify correctness, accuracy and robustness of the method, including a new test of a divergent flow in Carteisan geometry.

Development of a parallel spectral element code using SPMD constructs

In order for the increased use of fiber-reinforced composite structures to be financially feasible, employment of reliable and economical systems to detect damage and evaluate structural integrity is necessary. This task has traditionally been performed using off-line non-destructive testing (NDT) techniques. Safety enhancement programs and cost minimization schemes for repairs, however, have substantially increased the demand for real time integrity monitoring systems, i.e. structural health monitoring (SHM) systems, in the past few years. The real time feature imposes an additional constraint on SHM systems to be fast and computationally efficient. Among the existing approaches fulfilling these requirements, guided ultrasonic wave (GUW)-based methods are of particular interest, since they provide the possibility of finding small size defects, both at the surface and internal, and covering relatively large areas with reasonable hardware costs. Next to theses appealing features, there are certain complexities in utilizing GUWs for SHM of fiber-reinforced composites, that mainly arise from the multi-layer, anisotropic, and non-homogeneous nature of the material. In addition, the multi-mode character of GUWs further increases the complexity of the SHM problem in these materials. It is believed that computationally efficient methods for simulation of GUWs in composite structures can substantially contribute to the field of SHM. Such numerical tools do not only improve the understanding of the propagation of ultrasonic waves and their interaction with different damage types and boundary conditions, but can also make model-based damage identification techniques feasible in the context of on-line SHM. In this dissertation an improved framework for simulation of GUWs in composite structures is developed. The improvements are mainly brought about through the use of (i) physical constraints that reduces the dimensionality of the problem, (ii) improved approximation bases for spatial and temporal discretization of the governing equations, and (iii) efficient mathematical tools to enable the possibility of parallel computation. The formulated approach is a wavelet-based spectral finite element method (WSFEM), which offers the possibility of complete decoupling of the spatial and temporal discretization schemes, and results in parallel implementation of the temporal solution. Although the concept of the WSFEM was introduced a few years prior to this research, to the author's best knowledge, no general framework was proposed for dealing with 2D and 3D problems with inhomogeneity, anisotropy, geometrical complexity, and arbitrary boundary conditions. These issues are addressed in this dissertation in multiple steps as described below. 1- Improvement of the temporal discretization using compactly-supported wavelets, by computing the operators of the wavelet-Galerkin method over finite intervals, and demonstrating about 50% reduction in the number of sampling points, with the same accuracy, compared to the conventional wavelet-based approach. 2- Extension of the existing formulation of the 1D WSFEM based on an in-plane displacement field to 1D waveguides based on a 3D displacement field. In the 1D finite element formulation, spectral shape functions are employed which satisfy the governing equations, in which shear deformation and thickness contraction effects are also incorporated. The minimum number of elements for modeling 1D waveguides is used in this approach. 3- Formulation of a novel 2D WSFEM in which frequency-dependent basis functions are suggested for spatial discretization. Contrary to the conventional WSFEM, the presented scheme discretizes the spatial domain with 2D elements and does not require extra treatments for non-periodic boundary conditions. Superior properties of the formulation are shown in comparison with some time domain FEM schemes. 4- Generalization of the WSFEM and extension to 3D geometries. It is demonstrated that the standard spatial discretization schemes can be combined with the wavelet-Galerkin approach, to fully parallelize the temporal solution. A higher-order pseudo-spectral finite element method, i.e. spectral element method (SEM), is further adopted to attain spectral convergence properties over space and time. The developed WSFEM is subsequently employed in the passive time reversal (TR) method, which is a model-based approach for detection of load and damage location, and operates based on the time invariance of linear elastodynamic equations. It is shown that using the passive TR scheme, the problem of load and damage detection, which is essentially an inverse problem, can be solved in the form of a forward problem, thereby alleviating uniqueness and stability issues. A number of case studies and examples, numerical and experimental, are presented throughout this dissertation to better demonstrate the applicability of the proposed framework.

Earthquake disaster in the past has motivated studies on earthquake-resistant buildings to save lives and resources (Newmark and Hall 1982). With this objective, physical models and numerical methods are used to predict the dynamic behavior of structures. The numerical methods provide quantitative analyses of physical phenomena. They can be used to design structures with geometry and materials for an adequate dynamic behavior under seismic excitation. The most frequently used in structural dynamics are the finite element method (FEM) and the boundary element method (BEM). Based on wave propagation, the spectral finite element or spectral element method (SEM) was introduced by Beskos in 1978, organized and seemed by Doyle (1997) in the 1990s. It allows calculating relatively complex structures with different boundary conditions and discontinuities using simple theories. It combines important characteristics of FEM, dynamic stiffness method (DSM), and spectral analysis (Lee 2009). The dynamic stiffness matrices and shape functions used in SEM are exact within the scope of the underlying physical theory, and the method allows a reduced number of degrees of freedom. The matrices are depended on frequency, but using spectral analysis, the dynamic response can be easily composed by wave superposition. Harmonic, random, or damped transient excitations can be decomposed using the discrete Fourier transform (DFT). The discrete frequencies are used to calculate the spectral matrix and discrete responses. Then, the complete dynamic response is computed by the sum of frequency components (inverse DFT). As FEM, SEM uses the assembly of a global matrix using elementary matrices and spatial discretization. However, differently from FEM, only discontinuities and locations where loads are applied need to be meshed (Ahmida and Arruda 2001). It can be a useful tool because it joins DSMwith spectral analysis to produce accurate solutions in both frequency and time domains at a low computational cost due to a dramatic system reduction, making it more adequate for uncertainty and optimization analyses. Furthermore, it can easily be associated with other numerical methods such as FEM and BEM. The limitations of SEM are general nonlinear analysis, or when exact wave solutions are not available, as it occurs in geometrically complex structures. When using SEM it is important to be careful with the time–frequency transformations to avoid leakage and aliasing errors (Lee 2009). This chapter presents spectral element for straight and curved frame elements. First, the equations of motion are developed and physical and kinematic assumptions are presented. Then the spectrum relation is established and the spectral element is derived. Two simple examples – a twoand a three-dimensional

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

We introduce a new efficient spectral element approach to solve the two-dimensional magnetotelluric forward problem based on Gauss–Lobatto–Legendre polynomials. It combines the high accuracy of the spectral technique and the perfect flexibility of the finite element approach, which can significantly improve the calculation accuracy. This method mainly includes two steps: 1) transforming the boundary value problem in the partial differential form into the variational problem in the integral form and 2) solving large symmetric sparse systems based on the combination of incomplete LU factorization and the double conjugate gradient stability algorithm through the spectral element with quadrilateral meshes. We imply the spectral element method on a resistivity half-space model to obtain a simple analytical solution and find that the magnetic field solutions simulated by the spectral element approach matched closely to the exact solutions. The experiment result shows that the spectral element solution has high accuracy with coarse meshes. We further compare the numerical results of the spectral element, finite difference, and finite element approaches on the COMMEMI 2D-1 and smooth models, respectively. The numerical results of the spectral element procedure are highly consistent with the other two techniques. All these comparison results suggest that the spectral element technique can not only give high accuracy for modeling results but also provide more detailed information. In particular, a few nodes are required in this method relative to the finite difference and finite element methods, which can decrease the relative errors. We then deduce that the spectral element method might be an alternative approach to simulate the magnetotelluric responses in two- or three-dimensional structures.

In this work, a combination of the spectral stochastic finite element method (SSFEM) and the time-domain spectral element method (TSEM), referred to as the stochastic time domain spectral element method (STSEM), is presented to compute the stochasticity of strain and stress of a higher-order sandwich composite beam with spatial variability in the material properties. The method proposed in this work employs the efficiencies of both SSFEM and TSEM for the uncertainty analysis of a sandwich beam. The material properties of face sheets and core are considered as Gaussian random fields, which are discretized using the Karhunen-Loéve expansion, and polynomial chaos expansion is used to represent the response quantity. A numerical example is considered for which, first, a sensitivity analysis is performed to identify the most sensitive material properties. Then, the proposed STSEM is used to demonstrate the computational efficiency and numerical accuracy in comparison with Monte-Carlo simulation. Moreover, the effect of core depth on strain and stress variability is also examined.