Spectral Element Method Research Articles

On the Origin of Görtler Vortices in Flow over a Multi-Element Airfoil

The flow characteristics of a 30P30N three-element high-lift airfoil at low Reynolds numbers O104 are examined through three-dimensional simulations using a high-order spectral element method. This study primarily investigates the flow structures of the slat cove and Görtler vortices formed on the upper surface of the main airfoil. Spanwise instability grows exponentially in the slat cove with a constant wavelength, corresponding to that of the subsequently formed Görtler vortices. Görtler number calculations show that curvature-induced centrifugal instability at the slat cusp leads to the subsequent formation of Görtler vortices. Proper orthogonal decomposition (POD) is used to analyze the development of flow structures in the slat cove in different time ranges. At early time, the flow in the slat cove is dominated by shear layers that evolve into spanwise perturbations. These perturbations further evolve into distinct bell-shaped structures close to the slat cusp and are advected to the upper surface of the main airfoil, leading to the formation of Görtler vortices.

Seafloor Topographic Data Processing in Near-Seafloor Acoustic Field Simulation

Near-seafloor acoustic field characteristics are essential prior knowledge for near-seafloor underwater acoustic engineering. Scholte waves are a crucial component of the near-seafloor acoustic fields. These fields, when considering Scholte waves, are susceptible to seafloor relief. However, open-source bathymetric datasets generally lack sufficient precision. Therefore, acoustic field simulations using open-source data can contain significant errors or even introduce erroneous propagation characteristics. The spectral element method (SEM) is an example of exploring the influence of an inadequate spatial-sampling rate and sea-depth precision on acoustic field simulations. In this article, appropriate methods for topographic processing are presented. The results indicate that the terrain can be corrected using cubic spline interpolation in cases of an inadequate spatial-sampling rate. Where there is insufficient sea-depth precision, this study proposes a terrain processing method. The first step involves sequentially determining the interpolation points for the rising and falling edge, depressions, bulges, and horizontal segments. Then, it adopts cubic spline interpolation. The SEM examples effectively verify this effect. Given the limited research on terrain correction in acoustic field simulations, this study introduces a low-complexity method that can effectively support exploring acoustic fields affected by seafloor terrain.

Simulations of Ground Motions for Mw 7.9 Nepal Earthquake

ABSTRACT This paper focuses on developing a model for the Indian region to simulate ground motion using the spectral element method. The topography, bathymetry, and the 3-dimensional material properties are available in the global database are considered for the region. The simulated model is validated by generating ground motions for the Nepal earthquake, April 25, 2015, which is one of the most devastating earthquakes in recent years, with a magnitude of 7.9 Mw. These waveforms of the earthquake’s main shock and aftershocks are well recorded by the instruments available in the region. The uncertainty of the source is addressed by considering three slip models from global database. The simulated results are shown to be satisfactory when compared with recorded ground motions. The goodness-of-fit score is calculated for the synthetic ground motions, followed by the comparison with global ground motion models. The results achieve a scored of more than four, indicating a fair fit for ground motions. Furthermore, the developed in this study can be used for the generate time histories, which is useful for hazard analysis.

Spectral element method for random vibration analysis of the vehicle-track coupling system

In this paper, we propose a new random vibration analysis model for the vertical vehicle-track coupling system, which combines the symplectic method and the spectral element method (SEM) to enhance calculation efficiency and accuracy. The model assumed that the vehicle is composed of a multi-rigid body with 10 DOFs, and its motion equation is derived using the traditional frequency analysis method. Tracks are considered periodic structures, and the part between sleeper support points is considered a substructure. The motion equation of the periodic track-substructure is established by spectral element method (SEM), and the propagation of vibration wave between the substructures is calculated by the symplectic method. The frequency responses of the vehicle-track coupling system are calculated by using the German low-interference irregularity spectrum and short-wave irregularity spectrum as excitation. Then, the rationality of this method is verified by comparing it with the results from the traditional symplectic method and simulations using Universal Mechanism software. The comparison shows that this method surpasses the traditional symplectic method in high-frequency accuracy while maintaining high computational efficiency. Although the finite element method also has the capability for high-frequency analysis, this method can calculate the high-frequency vibration of an infinitely long track using just one element, making it more suitable for the efficient analysis of vehicle-track coupling systems. Finally, the model is applied to analyze the distribution characteristics of the vibration response of the coupled system in the frequency domain.

Energy bounds for discontinuous Galerkin spectral element approximations of well-posed overset grid problems for hyperbolic systems

We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an appropriate norm, the energy of the approximation of the latter is bounded by data only for fixed polynomial order, mesh, and time. In the absence of dissipation, coupling of the overlapping domains is destabilizing by allowing positive eigenvalues in the system to be integrated in time. This coupling can be stabilized in one space dimension by using the upwind numerical flux. To help provide additional dissipation, we introduce a novel penalty method that applies dissipation at arbitrary points within the overlap region and depends only on the difference between the solutions. We present numerical experiments in one space dimension to illustrate the implementation of the well-posed penalty formulation, and show spectral convergence of the approximations when sufficient dissipation is applied.

MultiShape: a spectral element method, with applications to Dynamic Density Functional Theory and PDE-constrained optimization

Abstract A new numerical framework is developed to solve general nonlinear and nonlocal PDEs on complicated two-dimensional domains. This enables the solution of a wide range of both steady and time-dependent problems on nonstandard geometries, as well as providing the ability to impose nonlinear and nonlocal boundary conditions (typical of those arising in the modelling of physical phenomena) in a flexible and automated way. This spectral element methodology, which we called MultiShape, is compatible with other state-of-the-art numerical methods, such as differential–algebraic equation solvers and optimization algorithms. MultiShape is an open-source Matlab library, in which the numerical implementation is designed to be user-friendly: the problem set-up and computations are done automatically through intuitive operator definitions and notation. Validation tests are presented, before we showcase the power and versatility of MultiShape with three motivating examples in Dynamic Density Functional Theory and PDE-constrained optimization.

Kinematic rupture modeling of broadband ground motion from the 2022 MS6.9 Menyuan earthquake

AbstractWe propose a novel kinematic rupture modeling procedure for synthesizing broadband ground motions derived from the frequency-wavenumber integration algorithm. This procedure addresses two key issues in characterizing the rupture processes relevant to broadband seismic radiation: an accurate Green's function and a well-constrained kinematic source model. For the first issue, we derive the theoretical Green's function based on an improved dynamic stiffness matrix approach that effectively handles wave propagation in a 1D crustal velocity structure across a broad frequency band. For the second issue, we generate the hybrid source model that combines asperity slip and random slip over the fault plane to effectively implement constraints on the radiated energy during the whole rupture process. The accuracy and effectiveness of the proposed methodology are verified by comparing with the surface acceleration traces and Fourier spectra calculated by spectral element method. With the hybrid source model and crustal velocity structure applicable to the target area, the broadband (0–10 Hz) ground motion of the 2022 MS6.9 Menyuan earthquake is synthesized. The amplitude, duration, and frequency content of the synthetic motions are systematically compared with those of the available observed records and ground motion attenuation relationships, as well as the spatial distribution characteristics of the near-field ground motions from the earthquake scenarios are presented. In conclusion, the case study of the Menyuan MS6.9 earthquake demonstrates that the presented modeling procedure can estimate broadband ground motions rapidly and reliably from a physics-based kinematic rupture perspective.

Coupling model between building post structure and floors with the spectral element method

With the densification of urban area and public transport, the structure-borne noise disturbance due to railway traffic is a major issue. In order to mitigate it, ground-borne vibration propagation in building has to be investigated. The case of post-beam constructions necessitates 3D modeling with choices between complex, precise and computationally time-consuming approaches (3D finite element method for example), and fast but less accurate approaches like the statistical energy analysis (SEA) which need a strong modal density. For such post-beam constructions, the spectral element method (SEM) appears to be a good compromise. This method allows reducing the number of elements to represent the truss structure compared to the finite element method. However, the building walls and floor are difficult to consider with the SEM. In this paper, the challenge of coupling post-beam structures and floors is examined, a method is proposed and results are discussed. The aim is to develop a method applicable to different buildings while keeping advantage of SEM low discretization.

The modeling of acoustic black hole beam by spectral element method

A technique developed in recent decades for the passive control of vibrations and noise is the Acoustic Black Hole (ABH). The structural ABH effect is achieved by incorporating a local modification in the thickness of a thin structure, typically a beam or a plate, with a variation in thickness according to a spatial power law. The combination of the variation in the thickness power law and a local increase in damping, provided by the concomitant application of layers of viscoelastic material, produces a significant reduction in the wave propagation speed and an increase of attenuation properties. As an elastic wave propagates within an ABH, its propagation speed decreases smoothly and continuously, in theoretical case, when the structure's thickness disappears at the end of the ABH, the wave speed drops to zero. For practical applications, ABH is typically combined with dissipative materials to obtain significantly higher structural loss factors. This work aims to establish a numerical approach for modeling and designing a beam-type structural one-dimensional ABH device for vibration control device using the Spectral Element Method (SEM) and verified by the Finite Element Method (FEM). Examples are performed, and the results are compared with those found in the literature.

Design of vibrational filters based on 3D lattice material elongated structure

This study examines the potential of elongated periodic three-dimensional lattice structures for the design of vibration filters using so-called absolute band gaps. These filters are generally obtained by modifying the total stiffness of the structure, which decreases the propagation velocity of all wave polarisations. With a view to optimisation, we are implementing an original structure consisting of interconnected 3D beams with a sinusoidal accordion shape to control the speed of longitudinal waves. In addition, a central junction in the shape of a deformed cross controls the speed of transverse waves. To calculate the propagation of elastic waves in such structures, we use spectral element methods that take into account the longitudinal, torsional and bending motions of three-dimensional beams. The dispersion curves are obtained by applying the Floquet-Bloch periodicity conditions. They are analysed by a detailed study of the wave motion and polarisation. Finally, the geometrical parameters of the lattice cell are optimised using a genetic algorithm to generate the widest absolute band gap at the lowest frequency.

Inverse design of acoustic metamaterial-based vibration control for offshore wind turbines

This paper presents an inverse design approach for mechanical metamaterial turbines utilising a deep learning algorithm. We first establish a theoretical model for metamaterial turbines using the spectral element method to analyse their vibration characteristics. Subsequently, we predict the vibration transmission properties of these turbines and achieve on-demand inverse design of metastructures through a fully connected deep-learning neural network model. The efficacy of the forward prediction and reverse design network models is validated using an inverse neural network. Our results demonstrate a strong agreement between the predicted and target values, affirming the feasibility of employing deep learning for on-demand meta-turbine design. This intelligent design approach holds promise for expediting and enhancing the design of metamaterials tailored for low-frequency vibration isolation in engineering structures.

Nonlinear Seismic Response of an Alluvial Basin Modelled by Spectral Element Method: Implementation of a Davidenkov Constitutive Model

ABSTRACT A subroutine incorporating soil nonlinearity is presented using SPECFEM3D Cartesian, an open-access Spectral Element Method (SEM)-based code for full waveform simulation. Specifically, the adaptive time-stepping scheme is developed in the time marching algorithm, enabling the computation to be effective. The verification of the method is first demonstrated via comparison with results calculated by DEEPSOIL. Then, applications for a trapezoid basin are implemented to study the effects of impedance ratio (IR), magnitude, and basin shape on ground motion. Finally, we focus on the nonlinear ground motion simulation of the Shidian basin in southwestern China and compare it with the linear results.

Passive Source Reverse Time Migration Based on the Spectral Element Method

AbstractIncreasing deployment of dense arrays has facilitated detailed structure imaging for tectonic investigation, hazard assessment and resource exploration. Strong velocity heterogeneity and topographic changes have to be considered during passive source imaging. However, it is quite challenging for ray‐based methods, such as Kirchhoff migration or the widely used teleseismic receiver function, to handle these problems. In this study, we propose a 3‐D passive source reverse time migration strategy based on the spectral element method. It is realized by decomposing the time reversal full elastic wavefield into amplitude‐preserved vector P and S wavefields by solving the corresponding weak‐form solutions, followed by a dot‐product imaging condition to get images for the subsurface structures. It enables us to use regional 3‐D migration velocity models and take topographic variations into account, helping us to locate reflectors at more accurate positions than traditional 1‐D model‐based methods, like teleseismic receiver functions. Two synthetic tests are used to demonstrate the advantages of the proposed method to handle topographic variations and complex velocity heterogeneities. Furthermore, applications to the Laramie array data using both teleseismic P and S waves enable us to identify several south‐dipping structures beneath the Laramie basin in southeast Wyoming, which are interpreted as the Cheyenne Belt suture zone and agree with, and improve upon previous geological interpretations.

Low-Reynolds-number droplet motion in shear flow micro-confined by a rough substrate

A three-dimensional spectral boundary element method has been employed to compute for the dynamics of the droplet motion driven by shear flow near a single solid substrate with a rough surface. The droplet size is comparable with the surface features of the substrate. This is a problem that has barely been explored but with applications in biomedical research and heat management. This work numerically investigated the influences of surface roughness features, such as the roughness amplitude and wavelength, on the droplet deformation and velocities. We observe that a greater amplitude or wavelength leads to larger variations in droplet velocity perpendicular to the substrate. The droplet velocity along the substrate increases when the amplitude is reduced or when the wavelength increases. The effects of capillary number and viscosity ratios have also been studied. The droplet deformation and its velocity increases as we increase the capillary number, while the viscosity ratio shows a non-monotonic influence on the droplet behavior. The predicted droplet behaviors, including deformation, velocities, and trajectories, can provide physical insight, help to understand the droplet behavior in microfluidic devices without a perfectly smooth surface, and contribute in the design and operation of those devices.

Efficient spectral element method for the Euler equations on unbounded domains

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

A spectral element approach for response spectrum estimation of frame structures with uncertain parameters subjected to stationary stochastic excitations

Determination of the response spectra of structures under excitation is crucial for vibration control, reliability analysis, fatigue life prediction, and optimization design of structures. However, available methods can be time-consuming and even erroneous when dealing with structural uncertainty, making them difficult to apply to actual complex structures. This paper proposes a spectral element approach for response spectrum estimation of structures with uncertain parameters subjected to stationary stochastic excitations. First, the spectral element method and Karhunen-Loève expansion technology are utilized to represent the response spectra of structures with uncertain parameters to reduce computation cost. Then, a dimension-reduction technology of random variables is utilized to simplify the numerical computations while retaining accuracy. Based on the above, the method for the envelope response spectrum with practical applicability is proposed in this study. The method can accurately and efficiently estimate the response spectrum of structures with uncertain parameters. The proposed approach is validated by comparison with the Monte Carlo method using three numerical examples.

Deep neural helmholtz operators for 3D elastic wave propagation and inversion

Summary Numerical simulations of seismic wave propagation in heterogeneous 3D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is particularly problematic for full waveform inversion, which typically involves numerous runs of the forward process. In machine learning there has been considerable recent work in the area of operator learning, with a new class of models called neural operators allowing for data-driven solutions to partial differential equations. Recent works in seismology have shown that when neural operators are adequately trained, they can significantly shorten the compute time for wave propagation. However, the memory required for the 3D time domain equations may be prohibitive. In this study, we show that these limitations can be overcome by solving the wave equations in the frequency domain, also known as the Helmholtz equations, since the solutions for a set of frequencies can be determined in parallel. The 3D Helmholtz neural operator is 40 times more memory-efficient than an equivalent time-domain version. We employ a Helmholtz neural operator for 2D and 3D elastic wave modeling, achieving two orders of magnitude acceleration compared to a baseline spectral element method. The neural operator accurately generalizes to variable velocity structures and can be evaluated on denser input meshes than used in the training simulations. We also show that when solving for wavefields strictly on the surface, the accuracy can be significantly improved via a graph neural operator layer. In leveraging automatic differentiation, the proposed method can serve as an alternative to the adjoint-state approach for 3D full-waveform inversion, reducing the computation time by a factor of 350.

A quasi-3D SinZZ model-driven multi-field Chebyshev FEM for nonlinear vibration control in multilayer multiferroic composite plates

This paper presents an advanced numerical method for modeling and mitigating nonlinear vibrations in thin multilayered fiber-reinforced multiferroic composite plates, addressing complex multi-physical interactions. The proposed methodology leverages a multi-physical coupling Chebyshev finite element formulation, utilizing high-order shape functions derived from Chebyshev polynomials. By integrating the strengths of spectral element methods and Legendre spectral finite element methods, this approach effectively overcomes challenges such as shear locking and spurious zero energy modes while ensuring high convergence rates in multi-physical problems. A quasi-3D refined model, incorporating the Murakami zig-zag model and sinusoidal shear deformation theory, is employed to accurately capture the nonlinear Von Kármán strain–displacement relationship and the magneto-electro-elastic coupling in multilayer structures with thickness-dependent material properties. To suppress nonlinear vibrations, the study utilizes a closed-loop multiphysical Chebyshev finite element model for time-domain analysis of viscoelastically damped systems, employing the Golla–Hughes–McTavish model. The results underscore the significant influence of multiferroic properties and the strategic distribution of ferroelectric fibers within the substrate on the dynamic behavior of the plate. The numerical validation, supported by rigorous verification, demonstrates the robustness of the proposed method in effectively simulating and controlling multilayer fiber-reinforced multiferroic composite plates. Additionally, this research highlights the potential for significant vibration reduction through semi-active damping mechanisms, offering valuable insights for practical applications in industries where precision and stability are critical.

Revisiting the 1934 Mw 8.2 Bihar–Nepal earthquake—Simulation of broadband ground motions

SUMMARY The 1934 Mw 8.2 Bihar–Nepal earthquake was one of the devastating earthquakes, which made seismologists realize the importance of proper seismic hazard analysis and design aspects in India. The event occurred way before proper seismic networks were implemented and hence there are no recorded ground motions available for this event. This study, thus aims to generate possible ground motions for the 1934 Mw 8.2 Bihar–Nepal event. The complex geographical features, ambiguous source information and lack of ground motion data make the simulation and validation of ground motions very difficult. In this regard, the broad-band (BB) ground motions are simulated and validated for the most recent well-documented Himalayan event, that is, the 2015 Mw 7.9 Nepal earthquake in order to calibrate the model and simulation methodology. For this purpose, the computational model is presented for a region of 1000 km × 670 km (longitude 80–89 °E and latitude 23–30 °N) in the Indo-Gangetic Basin to simulate the low-frequency (LF) ground motions using spectral element method. These deterministically simulated LF ground motions are combined with stochastically simulated high-frequency (HF) ground motions based on an improved seismological model . The seismic moment and dimensions of the rupture plane are used to generate ten samples for the finite fault source model having different slip distribution along the rupture plane as a random field. The BB ground motions (0.01–25 Hz) are obtained by merging LF and HF ground motions in the time domain by matching them at a frequency of ∼0.3 Hz. Such BB results are simulated at a grid of stations and at locations where modified Mercalli intensity (MMI) values are available. The estimated MMI values and the observed MMI values are compared to emphasize the efficacy of the model. The maximum PGA estimated from the simulated ground motions in horizontal and vertical directions are observed to be 0.48 g and 0.4 g. Further, 5 per cent damped response spectra and spectral amplification are analysed concerning the sediment depth of the Indo-Gangetic Basin. The results from the study can serve as inputs for dynamic analysis and the design of earthquake-resistant structures across different locations in the Indo-Gangetic Basin.

A fourth order Runge-Kutta type of exponential time differencing and triangular spectral element method for two dimensional nonlinear Maxwell's equations

In this paper, we study a numerical scheme to solve the nonlinear Maxwell's equations. The discrete scheme is based on the triangular spectral element method (TSEM) in space and the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method in time. TSEM has the advantages of spectral accuracy and geometric flexibility. The ETD method involves exact integration of the linear part of the governing equation followed by an approximation of an integral involving the nonlinear terms. The RK4 scheme is introduced for the time integration of the nonlinear terms. The stability region of the ETDRK4 method is depicted. Moreover, the contour integral in the complex plan is utilized and improved to compute the matrix function required by the implementation of ETDRK4. The numerical results demonstrate that our proposed method is of exponential convergence with the order of basis function in space and fourth order accuracy in time.