Accuracy Of Spectral Element Method Research Articles

Reduced proper orthogonal decomposition spectral element method for the solution of 2D multi-term time fractional mixed diffusion and diffusion-wave equations in linear and nonlinear modes

At the present work, we propose a fast and high order numerical method for the solution of 2D multi-term time fractional mixed diffusion and diffusion-wave equation on regular and irregular convex and non-convex domains. We use a finite difference scheme based on a Crank-Nicolson method to discretize this equation in time variable and prove that the presented semi-discrete scheme is unconditionally stable. Due to the features of geometric flexibility and high order accuracy of spectral element method (SEM), this method is used to discretize the equation in space components. Afterwards, an error estimate of fully discrete scheme is given. In order to save the elapsed CPU time of simulation, a reduced-order model based on the proper orthogonal decomposition (POD) method is employed. Also, we provide the error estimate between the reduced POD-SEM solutions and usual SEM solutions. Moreover, we use the non-uniform temporal discretization to overcome the initial singularity. To investigate the accuracy of method, the numerical simulations are provided on regular and irregular convex and non-convex domains. Also, to illustrate the efficiency of proposed method, we compare the numerical solutions with the results of other methods developed in literature on regular and irregular domains from both accuracy and CPU time point of views.

Accuracy of Spectral Element Method for Wave, Parabolic, and Schrödinger Equations

The spectral element method constructed by the $Q^k$ ($k\geq 2$) continuous finite element method with $(k+1)$-point Gauss-Lobatto quadrature on rectangular meshes is a popular high order scheme for solving wave equations in various applications. It can also be regarded as a finite difference scheme on all Gauss-Lobatto points. We prove that this finite difference scheme is $(k+2)$-order accurate in discrete 2-norm for smooth solutions. The same proof can be extended to the spectral element method solving linear parabolic and Schr\"odinger equations. The main result also applies to the spectral element method on curvilinear meshes that can be smoothly mapped to rectangular meshes on the unit square.

Density ratio effects on the topology of coherent turbulent structures in two-way coupled particle-laden channel flows

This investigation considers the effect of density ratio on the modification of coherent turbulent structures in particle-laden channel flows at a shear Reynolds number of Reτ = 180. Direct numerical simulation and Lagrangian particle tracking are used to accurately predict the motion of particles dispersed within the flow at three Stokes numbers, St+ = 0.1, 50, and 92. Particle–fluid coupling is achieved through a local element-based force feedback field in the Navier–Stokes equations, which are solved using a seventh-order accurate spectral element method. After an initial transitory period wherein the effects of particle–fluid interaction are emphasized, the low density ratio particles are found to enhance the turbulence field, increasing the frequency of Q-criterion satisfying regions, while the inertial particles suppress the turbulence, reducing the number of quasistreamwise vortices. Results indicate that the topology of the quasistreamwise vortices is altered by the presence of the particles in the viscous sublayer, the buffer layer, and the log-law region such that the distribution of the third invariant of the deviatoric tensor, R, is widened by the presence of tracer-like particles and made thinner by the inertial particles. This effect reduces the amount of unstable focus/compressing regions and stable focus/stretching regions, which account for the streamwise vortical structures observed in these types of flow. Investigating the instantaneous coupling force field surrounding turbulent structures and low speed streaks shows that particles exert their greatest influence on the fluid in the regions noted, and mechanisms by which the particulate phase interacts with turbulent vortices are analyzed.

An accurate triangular spectral element method-based numerical simulation for acoustic problems in complex geometries

An accurate triangular spectral element method (TSEM) is developed to simulate acoustic problems in complex computational domains. With Fekete points and Koornwinder-Dubiner polynomials introduced, triangular elements are used in the present method to substitute quadrilateral elements in traditional spectral element method (SEM). The efficiency of discretizing complex geometry is enhanced while high accuracy of SEM is remained. The weak form of the second-order governing equations derived from the linearized Euler equations (LEEs) are solved, and perfectly matched layer (PML) boundary condition is implemented. Three benchmark problems with analytical solutions are employed to testify the exponential convergence rate, convenient implementation of solid wall boundary condition and capable discretization in complex geometries of the present method respectively. An application on Helmholtz resonator (HR) is presented as well to demonstrate the possibility of using the present method in practical engineering. The numerical resonance frequency of HR reaches an excellent agreement with the theoretical result.

3D frequency-domain elastic wave modeling with the spectral element method using a massively parallel direct solver

ABSTRACTComplex topography, the free-surface boundary condition, and anelastic properties of media should be accounted for in the frame of onshore geophysical prospecting imaging, such as full-waveform inversion (FWI). In this context, an accurate and efficient forward-modeling engine is mandatory. We have performed 3D frequency-domain anisotropic elastic wave modeling by using the highly accurate spectral element method and a sparse multifrontal direct solver. An efficient approach similar to computing the matrix-vector product in the time domain is used to build the matrix. We validate the numerical results by comparing with analytical solutions. A parallel direct solver, the sparse direct multifrontal massively parallel solver (MUMPS), is used to solve the linear system. We find that a hybrid implementation of message passing interface and open multiprocessing is more efficient in flops and memory cost. The influence of the deformed mesh, free-surface boundary condition, and heterogeneity of media on MUMPS performance is negligible. Complexity analysis suggests that the memory complexity of MUMPS agrees with the theoretical order [Formula: see text] (or [Formula: see text] with an efficient matrix reordering method) for an [Formula: see text] grid when nontrivial topography is considered. With the available resources, we conduct a moderate scale modeling with a subset of the SEAM Phase II Foothills model, where 60 wavelengths in the [Formula: see text]-axis are propagated. Computing one gradient of FWI based on this model using the frequency-domain modeling is shown to require similar or fewer computational resources than what would be required for a time-domain solver, depending on the number of sources, while larger memory is necessary. An estimation of the increasing trend indicates that approximately 20 Tb of memory would be required for a [Formula: see text] wavelength modeling. The limit of MUMPS scalability hinders the application to larger scale applications.

Spectral element method for numerical simulation of ETHD enhanced heat transfer in an enclosure with uniform and sinusoidal temperature boundary conditions

This paper presents a novel numerical framework based on the accurate spectral element method (SEM) for electro-thermo-hydrodynamics (ETHD), in which the coupled electric, flow, and heat fields are solved simultaneously. The electric field potential is estimated by the classic Galerkin variation with the electric field vectors evaluated by the spectral approximation, and the streamline upwind Petrov-Galerkin variation and consistent approximate upwind technique are applied to stabilize the numerical simulation of charge density transport. As for the flow and heat transfer, the improved time-splitting method developed in our previous work is applied to deal with the coupled pressure and velocity in the momentum equations. Three carefully selected cases of analytical solution, pure thermal and pure electric convection, and electro-thermo-convection are performed to validate the algorithm. Then, the electro-thermo-convection in an enclosure with uniform temperature distribution is simulated and the average Nusselt numbers at different electric Rayleigh numbers are presented accurately. Additionally, the sinusoidal temperature boundary conditions are considered that the contours for stream function, temperature, and charge density are analyzed in detail to depict the heat transfer enhancement with ETHD effects.

SEM-based numerical framework for non-Darcy mixed convection in a porous cavity with three adiabatic thin baffles on the hot wall

PurposeThe purpose of this paper is to develop a numerical framework based on the accurate spectral element method (SEM) to simulate the mixed convective heat transfer within a porous enclosure with three adiabatic thin baffles of different lengths in nine cases and analyze the effects of several parameters.Design/methodology/approachThe authors develop an improved time-splitting method to solve the Darcy–Brinkman–Forchheimer model. No extra assumptions are introduced for the intermediate velocity, and the final velocity field satisfies the incompressible constraint strictly compared with the classical method. The governing equations are split into a pure convection problem, a Stokes problem and a thermal diffusion problem. The least-squares variation is adopted for the Stokes problem, and the Galerkin variation is used for the other two problems, such that the pressure and velocity can be discretized with the same interpolation order, which benefits the numerical accuracy and program design.FindingsRegarding the method, the excellent spectral accuracy, the capability of discretizing complex computational regions and the improved time-splitting methods make SEM an effective tool to accurately predict the non-Darcy convective heat transfer; as for the numerical tests, it is observed that weakened convection and heat transfer are induced by the increasing length of the baffles. The flow and heat transfer in channel 1 is only related to the length of baffle 1 because of the downward-driven right sidewall, and it is more difficult for baffle 3 to form the secondary flow on its tip.Originality/valueA novel numerical framework for Darcy–Brinkman–Forchheimer model is developed, expanding the application of SEM for simulating non-Darcy convective heat transfer to improve the numerical accuracy. Numerical results and analysis for flow and heat fields could help designers understand the control of heat transfer using adiabatic baffles better.

A probabilistic approach for the detection of bolt loosening in periodically supported structures endowed with bolted flange joints

Abstract From the literature, only very limited research activities have been carried out for fault diagnose of periodic structural system following model-based approaches. This paper focuses on the development of a practical methodology for modeling and detecting bolt loosening on periodically supported beam-type structure endowed with bolted flange joints, representing typical supported pipeline system in industry, through using measured modal parameters. Within the framework of periodic system, an efficient analytical model of the complete periodic system is first developed for dynamic analysis in frequency domain. The highly accurate spectral element method is employed to formulate the supercell-based dynamic stiffness matrix (DSM) of periodic cell containing bolted flange connection in the midspan, and the transfer matrix-based method is also developed for assembling the system DSM of entire periodic structural system through the obtained DSM of each individual cell, where the computational effort required in dynamic analysis of the complete periodic system with a large amount of repeated cells is almost comparable to a single cell. Then, in the proposed methodology, the statistical detection of bolt loosening is accomplished through two phases. The most plausible model class with appropriate parameterization complexity is first recognized by following the Bayesian model class selection strategy in the first phase. In the subsequent phase, the posterior probability density function of the stiffness scaling parameters is identified following the particle filter-based approach. To demonstrate and validate the proposed methodology, this paper reports not only the theoretical development but also a comprehensive series of numerical and experimental case studies, and corresponding results achieved are very encouraging.

Regularization modelling for large‐eddy simulation in wall‐bounded turbulence: An explicit filtering‐based approach

SummaryA comprehensive analysis of a filtering‐based regularization model for large‐eddy simulation has been carried out for fully developed turbulent channel flow, and its potency as a large‐eddy simulation candidate has been tested by comparison with eddy‐viscosity subgrid‐scale models. The exponentially accurate spectral element method, particularly open‐source code Nek5000, has been used for such computations. The simulations in the channel flow have been performed at moderately high bulk Reynolds number on the basis of half channel width, 20 000, and compared against the direct numerical simulation database for the mean flow profile, second‐order statistics, and their streamwise energy spectra. The results show that the filtering‐based regularization model is in reasonably good agreement with the subgrid‐scale models, and they also elucidate valuable insights on the filter‐based regularization.

Effect of artificial length scales in large eddy simulation of a neutral atmospheric boundary layer flow: A simple solution to log-layer mismatch

A large eddy simulation (LES) methodology coupled with near-wall modeling has been implemented in the current study for high Re neutral atmospheric boundary layer flows using an exponentially accurate spectral element method in an open-source research code Nek5000. The effect of artificial length scales due to subgrid scale (SGS) and near wall modeling (NWM) on the scaling laws and structure of the inner and outer layer eddies is studied using varying SGS and NWM parameters in the spectral element framework. The study provides an understanding of the various length scales and dynamics of the eddies affected by the LES model and also the fundamental physics behind the inner and outer layer eddies which are responsible for the correct behavior of the mean statistics in accordance with the definition of equilibrium layers by Townsend. An economical and accurate LES model based on capturing the near wall coherent eddies has been designed, which is successful in eliminating the artificial length scale effects like the log-layer mismatch or the secondary peak generation in the streamwise variance.

The Implementation of Spectral Element Method in a CAE System for the Solution of Elasticity Problems on Hybrid Curvilinear Meshes

Modern high-performance computing systems allow us to explore and implement new technologies and mathematical modeling algorithms into industrial software systems of engineering analysis. For a long time the finite element method (FEM) was considered as the basic approach to mathematical simulation of elasticity theory problems; it provided the problems solution within an engineering error. However, modern high-tech equipment allows us to implement design solutions with a high enough accuracy, which requires more sophisticated approaches within the mathematical simulation of elasticity problems in industrial packages of engineering analysis. One of such approaches is the spectral element method (SEM). The implementation of SEM in a CAE system for the solution of elasticity problems is considered. An important feature of the proposed variant of SEM implementation is a support of hybrid curvilinear meshes. The main advantages of SEM over the FEM are discussed. The shape functions for different classes of spectral elements are written. Some results of computations are given for model problems that have analytical solutions. The results show the better accuracy of SEM in comparison with FEM for the same meshes.

Exponentially Accurate Spectral Element Method for Fourth Order Elliptic Problems

In this paper, a fully non-conforming least-squares spectral element method for fourth order elliptic problems on smooth domains is presented. The proposed method works for a general fourth order elliptic operator with non-homogeneous data in two dimensions and gives exponentially accurate solutions. We derive differentiability estimates and prove our main stability estimate theorem using a non-conforming spectral element method. We then formulate a numerical scheme using a block diagonal preconditioner. Error estimates are also proven for the proposed method. We provide the computational complexity of our method and present results of numerical simulations that have been performed to validate the theory.

Empirical assessment of the validity limits of the surface wave full ray theory using realistic 3-D Earth models

The surface wave full ray theory (FRT) is an efficient tool to calculate synthetic waveforms of surface waves. It combines the concept of local modes with exact ray tracing as a function of frequency, providing a more complete description of surface wave propagation than the widely used great circle approximation (GCA). The purpose of this study is to evaluate the ability of the FRT approach to model teleseismic long-period surface waveforms (T ∼ 45–150 s) in the context of current 3-D Earth models to empirically assess its validity domain and its scope for future studies in seismic tomography. To achieve this goal, we compute vertical and horizontal component fundamental mode synthetic Rayleigh waveforms using the FRT, which are compared with calculations using the highly accurate spectral element method. We use 13 global earth models including 3-D crustal and mantle structure, which are derived by successively varying the strength and lengthscale of heterogeneity in current tomographic models. For completeness, GCA waveforms are also compared with the spectral element method. We find that the FRT accurately predicts the phase and amplitude of long-period Rayleigh waves (T ∼ 45–150 s) for almost all the models considered, with errors in the modelling of the phase (amplitude) of Rayleigh waves being smaller than 5 per cent (10 per cent) in most cases. The largest errors in phase and amplitude are observed for T ∼ 45 s and for the three roughest earth models considered that exhibit shear wave anomalies of up to ∼20 per cent, which is much larger than in current global tomographic models. In addition, we find that overall the GCA does not predict Rayleigh wave amplitudes well, except for the longest wave periods (T ∼ 150 s) and the smoothest models considered. Although the GCA accurately predicts Rayleigh wave phase for current earth models such as S20RTS and S40RTS, FRT's phase errors are smaller, notably for the shortest wave periods considered (T ∼ 45 s and T ∼ 60 s). This suggests that the FRT approach is a useful means to build the next generation of elastic and anelastic surface wave tomography models. Finally, we observe a clear correlation between the FRT amplitude and phase errors and the roughness of the models. This allows us to quantify the limits of validity of the FRT in terms of model roughness thresholds, which can serve as useful guides in future seismic tomographic studies.

On the Accuracy of Spectral Element Method in Electromagnetic Scattering Problems

Spectral element method (SEM), which is known of its high accuracy, has been recently applied in solving electromagnetic problems governed by Maxwell's equations. This paper investigates the accuracy of SEM in two- dimensional, frequency-domain electromagnetic scattering problems where Helmholtz equation acts as the governing partial differential equation (PDE). As experience in meshing a problem in finite element method is important to obtain accurate results, the choice of elements in SEM, on the other hand, is important too. The aspect ratio in this paper is taken into account while studying the accuracy in a single element by utilizing the Green's function. In addition, the scalar field scattered by a circular cylinder placed in front of an incident plane wave is solved after truncating the domain by perfectly matched layer. Numerical results show that one should carefully discretize the problem and keeping the aspect ratio close to unity as much as possible to guarantee accurate results.

Computation of linear transmittance of thermal bridges in precast concrete sandwich panels

Precast concrete lightened sandwich panels are widely used building elements. They are made by two concrete wythes separated by a layer of lightweight material: the central layer is inhomogeneous due to the presence of concrete ribs which tie the external wythe and act as thermal bridges.Computation of thermal transmittance of sandwich panels is clearly described in European Standards, but in many cases it requires numerical simulations to determine the linear transmittance ψ associated with lightweight material-concrete interfaces in the inhomogeneous layer. Although simple, these simulations represent a critical issue for many panel manufacturers and they would much rather prefer correlations to compute ψ.In this work we present a correlation based on an artificial neural network (ANN) to estimate linear trasmittauce values for current Italian sandwich panel production. Five input parameters are considered: rib width, lightweight material conductivity, and thickness of the three panel layers. To obtain the data which are necessary to train and test the ANN, a fast and accurate Spectral Element Method is used to solve Laplace equation in the neighborhood of a rib. 5460 ψ values are collected which ensure an accurate network response.

Single-mode bubble evolution simulations of Rayleigh Taylor instability with spectral element method and a viscous model

Based on the single-mode potential model of Jacobs and Rikanati, Firstly, a viscous single-mode bubble evolution model of Rayleigh Taylor instability (RTI) is developed in this study. Viscous effects of RTI’s early stage growth for low Atwood number have been explained. In addition, direct numerical simulations of single mode RTI are studied with Navier–Stokes equations and a transport-diffusive equation for miscible fluids, in which these equations are discritized with discontinuous Galerkin (DG) spectral element method. The turbulent mixing of RTI has kinetic energy dissipation, and the dissipation rate is determined by the inertial and viscous effects. Therefore, the numerical techniques must include a dissipation mechanism for kinetic energy. For this reason, the high accurate spectral element method is employed in this study. Agreement between the theoretical model and the numerical results shows that simulations of RTI is feasible using the mathematical miscible fluid model. The results also suggest that a high order numerical method may provide the capability of simulating small scale fluctuations in turbulent flows with RTI.

A VARIABLE-ORDER SPECTRAL ELEMENT METHOD FOR INCOMPRESSIBLE VISCOUS FLOW SIMULATION

A new variable-order spectral element scheme is proposed in this work for the numerical solution of the steady incompressible Navier–Stokes equations in primitive variables. The spectral orders of polynomial expansion in each spatial direction for each element are specified by the user in advance. Next, an interface-nodes-replacement procedure is initialized to recover the conformity of the mesh. The interpolation functions are modified to accommodate the new sets of nodes. The present formulation does not require any interface iteration scheme; in addition, any complicated data structure is also avoided. Implementation of this strategy toward the existing spectral element framework is direct and simple. Additionally, the performance of the method is examined by many numerical experiments. Improved flexibility and efficiency of the accurate spectral element method make it a highly promising candidate for solving practical fluid dynamics problems.