Spectral Element Method Research Articles

Zero-thermal-expansion metamaterial with broadband vibration suppression

Although conventional metamaterials possess extraordinary properties, they cannot meet the practical requirements of engineering structures subjected to both temperature fluctuations and vibration excitations. To bridge this research gap, this paper proposes a novel dual-functional metamaterial with zero thermal expansion and broadband vibration suppression. Inspired by the thermal mismatch and band gap effects, we present an innovative design strategy that incorporates star-shaped re-entrant lattices and locally rudder-shaped struts with two-phase materials. To effectively guide the coupled design, a theoretical model accounting for stretching-bending deformations is established to predict the thermoelastic behavior of the metamaterial. Moreover, a parameterized dynamic model using the spectral element method is developed to study the vibration characteristics. Particularly, the spectral formulation of curved Timoshenko beams is derived, including in-plane and out-of-plane vibrations. The comparisons between theoretical predictions and finite element simulations validate the accuracy of analytical models in characterizing thermal deformations and vibration responses. Finally, the case studies shed light on the underlying formation mechanisms of zero thermal expansion and multiple band gaps, while also unveiling the effects of geometric parameters on the performance of such dual-functional metamaterials. Our innovative design strategy and analytical methodology not only offer appealing alternatives for engineering applications but also push the boundaries of metamaterial properties by enabling the transition from single- to dual-functionality.

Spatial discontinuous Galerkin spectral element method for a family of chromatography models in CADET

Packed bed liquid chromatography is widely applied in academia and industry. Model-based methods are increasingly utilized for process development and optimization, demanding multitudes of complex simulations. We derive spatial arbitrary order discontinuous Galerkin (DG) discretizations for three commonly used chromatography models, including the general rate model (GRM). The methods are integrated in the open source CADET software, making efficient implementations publicly available for the first time. The DG CADET code is validated and benchmarked against the original finite volume CADET code. We observe great performance advantages for DG, depending on the discrete problem size. For a four-component steric mass action GRM, we achieve a speed-up of an order of magnitude for an error range typical for engineering applications. We explore the performance of a collocation Legendre–Gauß–Lobatto (LGL) quadrature DG method in comparison to an exact integration DG method. Our performance benchmarks indicate a slight advantage for collocation DG.

Time parallelism and Newton-adaptivity of the two-derivative deferred correction discontinuous Galerkin method

In this work, we consider a high-order discretization of compressible viscous flows allowing parallelization both in space and time.The discontinuous Galerkin spectral element method, which is well-suited for massively parallel simulations, is used for spatial discretization. The main novelty in this work is the additional demonstration of time-parallel capabilities within an implicit two-derivative timestepping procedure to further increase the parallel speedup. Temporal parallelism is made possible by a predictor-corrector-type time discretization that allows to split the associated workload onto multiple processors.We identify a homogeneous load balance with respect to the linear (GMRES) iterations on each processor as a key for parallel efficiency. To homogenize the load and to enable practical simulations, an adaptive strategy for Newton’s method is introduced. It is shown that the time-parallel method provides a parallel efficiency of approx. 60−70% on 4−7 computational partitions. Moreover, the capabilities of the novel method for the simulation of large-scale problems are illustrated with a mixed temporal and spatial parallelization on more than 1000 processors.

High-order methods for hypersonic flows with strong shocks and real chemistry

We compare high-order methods including spectral difference (SD), flux reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element method (ES-DGSEM), modal discontinuous Galerkin methods, and WENO to select the best candidate to simulate strong shock waves characteristic of hypersonic flows. We consider several benchmarks, including the Leblanc and modified shock-density wave interaction problems that require robust stabilization and positivity-preserving properties for a successful flow realization. We also perform simulations of the three-species Sod problem with simplified chemistry with the chemical reaction source terms introduced in the Euler equations. The ES-DGSEM scheme exhibits the highest stability, negligible numerical oscillations, and requires the least computational effort in resolving reactive flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to hypersonic Euler equations by deriving a new set of two-point entropy conservative fluxes for a five-species gas model. In this paper, hypersonic Euler equations refer to the multi-species Euler equations for which the internal energy and thermodynamic properties are computed using the Rigid-Rotor Harmonic-Oscillator model. Stabilization for capturing strong shock waves occurs by blending high-order entropy conservative fluxes with low-order finite volume fluxes constructed using the HLLC Riemann solver. The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod problem. To this end, we adopt the Mutation++ library to compute the reaction source terms, thermodynamic properties, and transport coefficients. We also investigate the effect of real chemistry versus ideal chemistry, and the results demonstrate that the ideal chemistry assumption fails at high temperatures, hence real chemistry must be employed for accurate predictions. Finally, we consider a viscous hypersonic flow problem to verify the transport coefficients and reaction source terms determined by the Mutation++ library.

Advanced Numerical Methods for Graphene Simulation with Equivalent Boundary Conditions: A Review

Since the discovery of graphene, due to its excellent optical, thermal, mechanical and electrical properties, it has a broad application prospect in energy, materials, biomedicine, electromagnetism and other fields. A great quantity of researches on the physical mechanism of graphene has been applied to engineering in electromagnetism and optics. To study the properties of graphene, different kinds of numerical methods such as the mixed finite element method (Mixed FEM), the mixed spectral element method (Mixed SEM), Method of Auxiliary Sources (MAS), discontinuous Galerkin time-domain method (DGTD) and interior penalty discontinuous Galerkin time domain (IPDG) have been developed for simulating the electromagnetic field effects of graphene and equivalent boundary conditions such as impedance transmission boundary condition (ITBC), surface current boundary condition (SCBC), impedance matrix boundary condition (IMBC) and surface impedance boundary condition (SIBC) have been employed to replace graphene in the computational domain. In this work, the numerical methods with equivalent boundary conditions are reviewed, and some examples are provided to illustrate their applicability.

An efficient spectral element method for two-dimensional magnetotelluric modeling

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.

Guided wave and genetic algorithm-based inversion for characterization of pile foundations

The characterization of geometric and mechanical properties is of great importance for the reuse of unknown foundations. In this paper, we propose a pile characterization method to non-invasively estimate the length and mechanical properties of a pile simultaneously and automatically. The forward model is established based on the guided wave theory for a cylindrical pile and a genetic algorithm-based method is proposed for the inversion process. The guided wave model is built using the spectral element method, which can generate the dispersion relation for the given physical properties. The resonance data (including the resonance frequency and the resonance number) and pile length can be related to the phase velocity, which can be derived by assuming either the displacement control or stress control boundary conditions. The loss function in the inversion method is defined by linking the dispersion relation of the forward model and the resonance analysis for those two boundary conditions. The proposed method is validated against the experimental data. By using the proposed method, pile characterization can be achieved automatically. The average accuracy of the proposed method for characterizing the pile properties (such as pile length, shear wave velocity, and longitudinal wave velocity) and the pile length is within 5% error for each variable.

Ultrasonic guided waves through Coalesence of matrix cracks and delaminations in laminated composites

Composite fabric-matrix interface mismatch during manufacturing and applied loads cause matrix micro-cracks and their evolution toward interface delamination and stiffness reduction. Fatigue and impact loads can grow these matrix cracks further as early damage precursors, which are not easily detectable. We simulate ultrasonic guided wave propagation in laminated composite with matrix crack mixed delamination site to understand better detection possibilities for these damage precursors using the ultrasonic inspection method. Since extremely small wavelength ultrasonic waves attenuate rapidly in composites, whereas long wavelength guided waves may not produce high sensitivity to small-scale defects, therefore we attempt to analyze super-resolution quantitative scheme with suitable frequencies to correlate defect parameters. We employ a time-domain spectral finite element method for fast computation of high-frequency wavefield while accurately resolving the matrix crack-delamination geometric features. Its high-order field interpolation enables us to model the nearfield interactions of the waves with the transverse micro-cracks and further with an evolved macro-scale delamination. The computational scheme is aimed to scale to complex defect local field simulations and cover larger size composite structures. This is achieved due to a mathematically consistent diagonalized mass matrix property which makes the implicit time marching-based computation highly efficient for highly resolved and large-size finite element systems. The transient dynamic stress field around the cracks modulates the guided wave mode packets. We analyze the characteristics of the signal sensitivities due to the defect features, most notably, an interesting correlation between matrix crack density and frequency modulation or sub-band contribution. Wave interaction with the defect region creates weak but distinct reflection patterns while transmitting through the region. The study provides useful and new understanding towards defect size estimation and indicative measure of thickness wise layers of matrix cracks and its coalescence into delamination in terms of energy spectral density of the sub-bands.

Temperature-controlled tunable zigzag metamaterial beams to manipulate flexural waves

The study of artificially structured materials to control waves is ongoing, owing to their unprecedented properties. However, their narrow working frequency ranges significantly hinder their applications. In this study, a kind of tunable zigzag metamaterial beams with wider working frequency ranges are proposed by incorporating springs and damping elements into our previously proposed pure zigzag beams. The complex band structures of these tunable zigzag metamaterial beams are characterized by the spectral element method (SEM), and a parametric analysis of spring rigidity and damping is conducted. The results show that the full stopbands of our tunable zigzag metamaterial beams are highly sensitive to spring rigidity, which confirms their good tunability. Actuators serving as spring and damping elements are then designed and 3D printed using a thermally sensitive material, VeroPureWhite. The actuators are glued to pure zigzag aluminum beams to generate integrated tunable zigzag metamaterial beams. The transmission spectra are measured for the integrated tunable zigzag metamaterial beams under several thermal loading cycles. Both the numerical and experimental results demonstrate the wide tunability of our proposed tunable metamaterial beams.

Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting.

The discontinuous Galerkin spectral element method (DGSEM) is a compact and high-order method applicable to complex meshes. However, the aliasing errors in simulating under-resolved vortex flows and non-physical oscillations in simulating shock waves may lead to instability of the DGSEM. In this paper, an entropy-stable DGSEM (ESDGSEM) based on subcell limiting is proposed to improve the non-linear stability of the method. First, we discuss the stability and resolution of the entropy-stable DGSEM based on different solution points. Second, a provably entropy-stable DGSEM based on subcell limiting is established on Legendre-Gauss (LG) solution points. Numerical experiments demonstrate that the ESDGSEM-LG scheme is superior in non-linear stability and resolution, and ESDGSEM-LG with subcell limiting is robust in shock-capturing.

Polarized radiative transfer in seawater-in-oil emulsions floated on seawater considering the impact of oil absorption on seawater droplet scattering.

Among various remote sensing approaches, optical polarization remote sensing shows great advantages in identifying oil-water emulsions in seawater and has become one of the most promising detection technologies. Herein, we focus on exploring the sensitivity of polarized radiative transfer properties for oil emulsion polarization detection to the influence factors of viewing angle, droplet volume fraction and radius, incident wavelength, and emulsion thickness. The radiative properties of seawater droplets dispersed in crude oil are calculated using the improved Lorenz-Mie theory considering the absorption of crude oil as the host medium, after which the reflected Stokes vector and the degree of linear polarization (DOLP) of seawater-in-oil emulsions floating on seawater are obtained using the spectral element method. By analyzing the calculation results of a 0° viewing azimuth angle, the detection wavelength and viewing zenith angles corresponding to the highest sensitivity of the DOLP to the above factors are significantly different; thus, quantitative remote sensing detection of the droplet volume fraction, droplet diameter, and emulsion thickness is possible. Exploring the sensitivity of polarized remote sensing signals for oil emulsion polarization detection to the above factors is a prerequisite for quantitative polarization detection of oil emulsions.

Curvature effect on the instabilities of lid-driven flow in a toroidal square cavity

The effect of curvature on the three-dimensional instabilities of lid-driven flow in a toroidal square cavity is numerically studied using the linear stability analysis based on the spectral element method. The critical stability curves are determined, and four types of instability modes are predicted. For small curvatures (δ ≤ 0.2), the most dangerous mode is a steady short-wavelength mode (λc ≈ 0.4), which is consistent with that in the straight square cavity. In a narrow range of 0.22 ≤ δ ≤ 0.28, the critical mode is oscillatory and of medium wavelength (0.84 < λc < 1.07). For δ ≥ 0.29, two types of steady long-wavelength modes (λc > 1.26) become the critical mode. The disturbance energy analyses show that all the instabilities are caused by the centrifugal instability mechanism.

Dispersion Analysis of Three-Dimensional Elastic Wave Propagation in Transversely Isotropic Media Using Optimally Blended Spectral-Element Method

The accuracy of the optimally blended spectral-element method for wave propagation in a homogeneous transversely isotropic elastic medium was investigated. A nonstandard quadrature rule obtained by combining Gauss quadrature rule and Gauss–Lobatto–Legendre quadrature rule was used to compute elementary matrixes. The mass and stiffness matrixes were represented as the triple tensor-product of elementary matrixes as second-order tensors. The solution of the eigenvalue problem representing the semidiscretized version of elastic wave equation for plane harmonic wave propagation was obtained using the Rayleigh quotient approximation technique. The resulting eigenvalues subsequently were used to estimate the phase and group velocity of three bulk waves. The variations of the errors in these velocities of the bulk waves with the number of grid points per wavelength were depicted graphically. These variations were shown for different polynomial orders and various angles of deviation from the symmetry axis. The effect of a change in the order of time discretization on error variation also was shown. The solution obtained by the optimally blended spectral-element method was found to be more accurate than that from the classical spectral-element method for low-order polynomials. The improvement in solution accuracy was demonstrated through dispersion analysis.

3D Airborne EM Forward Modeling Based on Finite-Element Method with Goal-Oriented Adaptive Octree Mesh

The finite-element (FE) method for three-dimensional (3D) airborne electromagnetic (AEM) modeling can flexibly simulate complex geological structures at high accuracy. However, it has low efficiency and high computational requirements. To solve these problems, one needs to generate meshes more reasonably. In view of this, we develop an adaptive octree meshing scheme for frequency-domain AEM modeling. The octree meshes have the characteristics of regularity and flexibility, while the adaptive algorithm can effectively refine the mesh locally. In our adaptive mesh generation, the posterior errors and weighted coefficients are used to construct the final weighted posterior errors. We verify the accuracy of our method by comparing its results with semi-analytical solutions for a half-space model. Furthermore, we use the spectral-element (SE) method and our method to calculate EM responses for an abnormal block model and compare their computational costs. The results show that our adaptive scheme has obviously technical advantages over SE method for AEM modeling with multiple frequencies and multiple survey stations. Finally, we calculate a model with complex geological structures to verify the feasibility of our algorithm in complex geological circumstances.

Flexural–torsional vibration reduction of an eccentric phononic crystal pipe conveying fluid

Flexural vibration and torsional vibration pervasively exist in slender structures, which often lead to noise emission and fatigue failure of structures. In order to effectively suppress coupled flexural–torsional vibrations of defective devices transporting fluid, this paper introduces an eccentric fluid-conveying pipe axially made of periodically varying materials, and the flexural–torsional bandgap (BG) characteristics of such phononic crystal pipe system are investigated. With consideration of warping effect, coupled flexural–torsional motion equations are established. The complex band structure of sub-structure and frequency response function and attenuation configuration of topological structure are attained by applying the spectral element method. In comparison with literatures only yielding total merging BG of flexural and torsional vibrations, this paper presents a novel result of completely separate flexural BG and torsional BG, which enables individual manipulation of the two elastic waves. Meanwhile, broadband frequency BGs are harvested in the proposed pipe structure, especially for torsional vibration, demonstrating an excellent vibration isolation property. Complicated regulation mechanism of eccentric, warping effects and some critical parameters is explored. This study is expected to provide a new approach for flexural–torsional vibration reduction of engineering fluid-transporting devices.

An efficient 2.5‐D forward algorithm based on the spectral element method for airborne transient electromagnetics data

AbstractThe airborne transient electromagnetics method has been widely used in geophysical exploration in recent years, but it still faces challenges in balancing the accuracy and efficiency of electromagnetic data interpretation. As far as forward modelling is concerned, the higher the dimension of the geophysical model, the higher the accuracy of data interpretation, but, correspondingly, the more computing resources need to be consumed, which will greatly reduce investigation efficiency and practicability. In this paper, the spectral element method is first introduced for solving the 2.5‐dimensional forward modelling of the airborne transient electromagnetic system, which has a smaller computing scale than three‐dimensional modelling and is closer to the actual geological structure than either one‐ or two‐dimensional modelling. In the forward algorithm, a non‐uniform quadrilateral structured mesh is adopted to simplify the computing scale, and the Talbot algorithm rather than Gaver–Stefest algorithm is applied to the inverse Laplace transform to improve the numerical precision of this conversion. Moreover, we use parallel computing technology to improve the algorithm efficiency while keeping satisfactory accuracy. The study shows that, whether a low‐resistivity or high‐resistivity layered geophysical model, the numerical solutions of the proposed spectral element method forward algorithm agree well with the analytical solutions of the corresponding models; furthermore, the key factors affecting the accuracy of the numerical solution are analysed by experiments. Finally, we successfully applied it to the 2.5‐dimensional geoelectric model simulation.

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge–Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.

Investigation of a magnetorheological elastomer metamaterial sandwich beam with tunable graded stiffness for broadband vibration attenuation

Metamaterials with local resonance show promising application prospects in low-frequency vibration attenuation. However, with the drawback of narrow band gap, such potential is greatly limited. In order to broaden the local resonant band gap, a semi-active graded magnetorheological elastomer (MRE) metamaterial sandwich beam (GMREMSB) with real-time tunable graded stiffness was proposed and investigated in this study. For theoretical calculation, a mass-spring model was established for the GMREMSB. Then the calculated band gap and transmissibility using Timoshenko beam theory and spectral element method were compared. An experimental test was also conducted for verification. The results show that the bandwidth of the proposed GMREMSB can be widened by the graded stiffness arranged in ascending order. The experimental band gap of the GMREMSB under the graded current of 0.0–0.5–1.0 A is 6.4 Hz wider than the band gap of the periodic structure with the single current of 0.0 A and is 5.0 Hz wider than that of 1.0 A. The growth rate reaches 15.06% and 11.39%, respectively.

Adaptive hp refinement for spectral elements in numerical relativity

When a numerical simulation has to handle a physics problem with a wide range of time-dependent length scales, dynamically adaptive discretizations can be the method of choice. We present a major upgrade to the numerical relativity code bamps in the form of fully adaptive, physics-agnostic hp refinement. We describe the foundations of mesh refinement in the context of spectral element methods, the precise algorithm used to perform refinement in bamps, as well as several indicator functions used to drive it. Finally, we test the performance, scaling, and the accuracy of the code in treating several 1D and 2D example problems, showing clear improvements over static mesh configurations. In particular, we consider a simple nonlinear wave equation, the evolution of a real scalar field minimally coupled to gravity, as well as nonlinear gravitational waves.

Self-adaptive physics-driven deep learning for seismic wave modeling in complex topography

Solving for the scattered wavefield is a key scientific problem in the field of seismology and earthquake engineering. Physics-informed neural networks (PINNs) developed in recent years have great potential in possibly increasing the flexibility and efficacy of seismic modeling and inversion. Inspired by self-adaptive physics-informed neural networks (SA-PINNs), we introduce a framework for modeling seismic waves in complex topography The relevant theoretical model construction was performed using the one-dimensional (1D) wave equation as an example. Using SA-PINNs and combining them with sparse initial wavefield data formed by the spectral element method (SEM), we carry out a numerical simulation of two-dimensional (2D) SH wave propagation to realize typical cases such as infinite/semi-infinite domain and arc-shaped canyon/hill topography. For complex scattered wavefields, a sequential learning method with time-domain decomposition was introduced in SA-PINNs to improve the scalability and solution accuracy of the network. The accuracy and reliability of the proposed method to simulate wave propagation in complex topography were verified by comparing the displacement seismograms calculated by the SA-PINNs method with those calculated by the SEM. The results show that the SA-PINNs have the advantage of gridless and fine-grained simulation and can realize numerical simulation conditions, such as free surface and side-boundary wavefield transmission.