Spectral Element Method Research Articles

Size effect in polymeric lattice materials with size-dependent Poisson's ratio caused by Cosserat elasticity

Polymeric lattice materials with micro/nano-structures are attractive for applications in a wide range of bioengineering systems. Resent experimental results show that elastic constitutive law of polymer materials is in line with the Cosserat elasticity. In this work, a Cosserat continuum spectral element method is employed to explore the size-dependent mechanical performance of polymer polymeric lattice with horseshoe microstructures, efficiently. The mechanical performance predicted by the proposed method agrees very well with the experiment data. Our results demonstrate that size effects are significant in polymeric lattice materials. The size-dependent negative Poisson's ratio is found in the polymeric lattice materials with the same topological structure due to the size effect caused by the Cosserat elasticity of the polymer materials. It could be implied that it is possible to continuously adjust the negative Poisson's ratio of the polymeric lattice material over a wide range by only changing its microstructural size.

Introduction to the Distributional Finite Difference Method for 3D Seismic Wave Propagation and Comparison With the Spectral Element Method

AbstractWe have extended the distributional finite difference method (DFDM) to simulate the seismic‐wave propagation in 3D regional earth models. DFDM shares similarities to the discontinuous finite element method on a global scale and to the finite difference method locally. Instead of using linear staggered finite‐difference operators, we employ DFDM operators based on B‐splines and a definition of derivatives in the sense of distributions, to obtain accurate spatial derivatives. The weighted residuals method used in DFDM's locally weak formalism of spatial derivatives accurately and naturally accounts for the free surface, curvilinear meshing, and solid‐fluid coupling, for which it only requires setting the shear modulus and the continuity condition to zero. The computational efficiency of DFDM is comparable to the spectral element method (SEM) due to the more accurate mass matrix and a new band‐diagonal mass factorization. Numerical examples show that the superior accuracy of the band‐diagonal mass and stiffness matrices in DFDM enables fewer points per wavelength, approaching the spectral limit, and compensates for the increased computational burden due to four Lebedev staggered grids. Specifically, DFDM needs 2.5 points per wavelength, compared to the five points per wavelength required in SEM for 0.5% waveform error in a homogeneous model. Notably, while maintaining the same accuracy in the solid domain, DFDM demonstrates superior accuracy in the fluid domain compared to SEM. To validate its accuracy and flexibility, we present various 3D benchmarks involving homogeneous and heterogeneous regional elastic models and solid‐fluid coupling in both Cartesian and spherical settings.

A Verification Suite of Test Cases for the Barotropic Solver of Ocean Models

AbstractThe development of any atmosphere or ocean model warrants a suite of test cases (TCs) to verify its spatial and temporal discretizations, order of accuracy, stability, reproducibility, portability, scalability, etc. In this paper, we present a suite of shallow water TCs designed to verify the barotropic solver of atmosphere and ocean models. These include the non‐dispersive coastal Kelvin wave; the dispersive inertia‐gravity wave; the dispersive planetary and topographic Rossby waves; the barotropic tide; and a non‐linear manufactured solution. These TCs check the implementation of the linear pressure gradient term; the linear constant or variable‐coefficient Coriolis and bathymetry terms; and the non‐linear advection terms. Simulation results are presented for a variety of time‐stepping methods as well as two spatial discretizations: a mimetic finite volume method based on the TRiSK scheme, and a high‐order discontinuous Galerkin spectral element method. The experimental procedure for conducting these numerical experiments is detailed. It underscores several key considerations that vary depending on the chosen spatial discretization method. Finally, convergence studies of every TC are conducted with refinement in both space and time, only in space, and only in time. The convergence slopes match the expected theoretical predictions.

Pure TE and TM modes in the metallic waveguide filled homogeneously with fully anisotropic and lossless medium

AbstractThis letter gives a sufficient condition for the existence of the pure transverse electric (TE) and transverse magnetic (TM) modes in the metallic waveguide filled with a homogeneous, fully anisotropic and lossless medium. The condition is a relation between the permittivity and permeability tensors of the fully anisotropic medium. The theory given in the letter is an extension of the classic electromagnetic waveguide theory. Furthermore, a mixed spectral element method is applied to simulate this type of anisotropic waveguide problem. At last, a numerical experiment shows that the sufficient condition supported in the letter does confirm the existence of the pure TE and TM modes in the anisotropic waveguide.

Numerical investigation of mixing performance in supersonic cold flow over dual-cavity in scramjet

Mixing performance in the combustion chamber is important in achieving the efficiency of scramjet for its compact structure and the short residence time caused by high velocity. This study focuses on predicting the mixing performance in a viscous supersonic flow past the cavity flame holder in the scramjet, using the discontinuous spectral element method with direct numerical simulation. The arrangement and distribution of a series of cavities are calculated at various inlet velocities. By comparing the contours of Mach number and static temperature, as well as analyzing the airflow residence time with a new calculating formula and the drag based on the numerical results, it is concluded that higher inlet velocities result in faster stabilization. It also leads to longer airflow residence time when the supersonic flow passes through the dual-cavity with a tandem connection rather than the parallel one or the single-cavity structures. As for the shortened rear wall of cavities, these structures can decrease the drag quickly, but they also decrease the airflow residence time seriously, which destroys the mixing performance. In addition, these conclusions are applied to a practical case of the cavity flame holder, verifying the effectiveness of tandem dual-cavity structures in enhancing the mixing performance by increasing the airflow residence time and maintaining or reducing the drag. This study can provide valuable suggestions in further design of cavity flame holders for different flow conditions.

Development of a prediction model of dynamic deflections for asphalt pavements under FWD impulse loading

Pavement deflections are critical indicators in pavement design and acceptance programs. Falling weight deflectometer (FWD) technique has been widely adopted to measure the dynamic deflections in pavement engineering. Although the dynamic effect of FWD impulse loading on deflections is significant, FWD deflections are still commonly interpreted and examined using the static analysis programs in acceptance projects because of their ease of implementation and high computational efficiency. This may lead to an inability to effectively control the construction quality of pavements. Accordingly, this study aims to propose a simple and accurate approach for converting static deflection to dynamic deflection. Layered elastic theory (LET) and spectral element method (SEM) were adopted to generate the theoretical static and dynamic deflection databases, respectively. Based on the databases, a simple approach is proposed to predict the dynamic deflection only using four static deflections. The proposed approach shows lower computational workload and higher accuracy than other traditional formula forms which include equivalent layer moduli and thicknesses. In the field engineering cases, the calculated static deflections can be 2 ∼ 3 times higher than the measured ones, while the predicted dynamic deflections from the proposed approach match the theoretical dynamic deflections well and are comparable to the measured deflections for both asphalt pavements with asphalt binder or inorganic binder bases and granular bases. This indicates that dynamic deflections can more effectively control the construction quality of asphalt pavements in the FWD deflection-based design program. Meanwhile, the drawbacks of traditional prediction methods are investigated and evaluated. The proposed approach exhibits simpler form and higher accuracy, while avoiding the drawbacks of traditional methods. Combining with the common static LET-based programs, the proposed approach provides a more reasonable and practical way for the construction quality acceptance of asphalt pavements using FWD data.

Implicit-explicit time integration for the immersed wave equation

Efficient solution strategies for wave propagation problems with complex geometries are essential in many engineering fields. Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. We consider the finite cell method that extends the weak form from the physical into the extended domain, multiplied by a small value, to stabilize badly cut cells. Combined with explicit time integration methods, the presence of finite cells with very little support in the physical domain results in tiny critical time step sizes. While the finite cell stabilization limits how small the critical time step size can become, this limit is still restrictive for many applications. Explicit transient analyses commonly use the spectral element method due to its natural way of obtaining diagonal mass matrices through nodal lumping. The resulting method is very efficient since nodal lumping renders the solution of the equation systems trivial while not reducing the accuracy or the critical time step size. The combination of the spectral element and finite cell methods is called the spectral cell method. Unfortunately, a direct application of nodal lumping in the spectral cell method is impossible due to the special quadrature necessary to treat the discontinuous integrand inside the cut cells. The existing approaches to lump the mass matrices of cut cells significantly reduce the accuracy of the approximation.We analyze an implicit-explicit (IMEX) time integration method to exploit the advantages of the nodal lumping scheme for uncut cells on one side and the unconditional stability of implicit time integration schemes for cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use explicit second-order central differences to integrate the uncut degrees of freedom that lead to a diagonal block in the mass matrix and an implicit trapezoidal Newmark method to integrate the remaining degrees of freedom (those supported by at least one cut cell). The immersed Newmark IMEX approach preserves the high-order convergence rates and the geometric flexibility of the finite cell method while retaining the efficiency of the nodal lumping scheme for uncut cells without compromising the critical time step size. We analyze a simple system of spring-coupled masses to highlight some of the essential characteristics of Newmark IMEX time integration. We then solve the scalar wave equation with immersed boundaries on two- and three-dimensional examples with significant geometric complexity to show that our approach is more efficient than state-of-the-art time integration schemes when comparing accuracy and runtime. While we focus on the finite and spectral cell methods, we expect other immersed methods to benefit equally from Newmark IMEX time integration.

Adjoint inversion of antipodal PKPab waveforms for P wave velocity anomaly at the base of the lower mantle

We perform an adjoint inversion by using antipodal PKPab phases to estimate the heterogeneous Vp structure at the base of the lowermost mantle. We have carefully examined antipodal stations with high S/N ratios during the past 30 years and selected 20 source-receiver pairs with the epicentral distances >179.0 degree and the Mw <7.0. We have used the spectral-element method on the Earth Simulator of JAMSTEC to calculate the synthetic seismograms for a heterogeneous mantle Vp model with the accuracy of period about 8 s (110 km wavelength at the CMB). We have set up the time window to retrieve the PKPab phase from the vertical component of both observed and computed seismograms to calculate the adjoint source to obtain the sensitivity kernels of Vp in the mantle. The computed Vp sensitivity kernel for each event shows the characteristic annulus pattern in the lowermost mantle, which covers a large area of the CMB. The twenty PKPab earthquake-station pairs in this antipodal study contribute the equivalent of about 3140 measurements at the CMB—compared with 1871 previously studied—and provide new data. Therefore, although the number of individual source-receiver pairs is not large, the summed sensitivity kernels of the PKPab phase for Vp structure at the base of the mantle may be sufficient to model heterogeneity of the Vp structure at the CMB. We summed each event kernel to set up a sensitivity kernel of Vp in the lowermost mantle and iterated the inversion to estimate a heterogeneous structure in D″. Although we have iterations that dominantly affect both South America and the South Pacific, the summary final model shows features within South of Africa, South Pacific, SE Australia, and Central & South America. Keeping the Vs model fixed, we map the CMB Vp heterogeneity measured by the parameter Rs,p = dlnVs/dlnVp and find qualitative, proximal correspondence with the degree-2 pattern of Large Low Velocity Shear Provinces observed in shear tomographic models: in the Pacific and Africa Rs,p > 2.0, whereas the surrounding edges of the edges of the Pacific show Rs,p < 2.0.

Accuracy assessment of discontinuous Galerkin spectral element method in simulating supersonic free jets

Accuracy assessment of discontinuous Galerkin spectral element method in simulating supersonic free jets

Efficient spectral and spectral element methods for Sobolev equation with diagonalization technique

In this paper, we first introduce a new series of Legendre basis functions by using the matrix decomposition technique, which are simultaneously orthogonal in both L2- and H1-inner products. Then, we construct efficient space-time spectral method for linear Sobolev equation using spectral approximation in space and multi-domain collocation approximation in time, which can be implemented in a synchronous parallel fashion. Next, we propose a novel Legendre spectral element method for solving nonlinear Sobolev equation with Burgers' type term, which reduce the non-zero entries of linear systems and computational cost. Some rigorous error estimates are carried out for one-dimensional Sobolev equation. Numerical experiments illustrate the effectiveness and accuracy of the suggested approaches.

Mechanical Analysis of Semi-Rigid Base Asphalt Pavement under the Influence of Groundwater with the Spectral Element Method

Over prolonged exposure to groundwater conditions, semi-rigid base asphalt pavements can undergo significant changes in their internal moisture field, resulting in substantial variations in the pavement’s stiffness and, consequently, affecting the overall load-bearing capacity and stability of the road structure. This paper employs FWD non-destructive testing equipment to assess its mechanical performance and conduct data analysis and conducts a mechanical response study of asphalt road surfaces, considering the influence of roadbed moisture levels. Using the dynamic analytical theory, the fundamental equations and stiffness matrices for a linear elastic half-space model were established, leading to the development of a computational model for the mechanical response of semi-rigid base asphalt pavements under FWD dynamic loading, with an examination of the surface deflection in relation to changes in groundwater levels. Numerical examples and engineering applications were employed to validate the proposed model. The research findings indicate: With the passage of time, surface deflection values initially increase and then decrease, exhibiting a sinusoidal variation pattern similar to that of the applied load. As the distance from the loading center increases, the moment of peak deflection continually lags behind. The average absolute relative error between the results obtained using the method proposed in this study and the traditional ABAQUS finite element method was only 0.70%. The correlation coefficient between the theoretically computed deflection curve and the measured deflection curve using the spectral element method was greater than 0.9, with an average absolute relative error of 4.92% between the theoretical peak deflection and the measured peak deflection. As the groundwater level rises, surface deflection noticeably increases, with an approximately 40% increase in deflection values at the loading center. These research findings can be utilized to analyze the dynamic deflection of semi-rigid base asphalt pavements under various groundwater conditions, providing significant practical value for assessing road structural performance and serviceability.

Controllable flexural wave in laminated metabeam with embedded multiple resonators

Acoustic metamaterials and bandgap engineering have been research areas that have drawn intensive and extensive research attention in recent years. The exploration of low-frequency bandgaps as well as the corresponding bandgap modulation has been a subject of much concern by numerous research scholars and engineers. In this paper, a new laminated beam metastructure which uses piezoelectric springs to connect local resonators to the laminated beam is proposed. Considering the intrinsic relationship of materials and loads due to resonator system, the Euler-Bernoulli beam theory is employed to derive the governing equation of the system, and the band structure is solved by the spectral element method. Subsequently, we solve and analyze the effects of external excitation sources including electric and magnetic fields on the bandgap, and also investigate the influence resonator number and mass. Finally, the modulation effect of piezoelectric spring stiffness on the bandgap is analyzed. The result of this paper provides a sound theoretical guide for the application and control of multi-frequency domain bandgaps. It also serves new ideas for vibration isolation and suppression of metabeam structures in real-life engineering applications.

Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods

We extend the monolithic convex limiting (MCL) methodology to nodal discontinuous Galerkin spectral-element methods (DGSEMS). The use of Legendre-Gauss-Lobatto (LGL) quadrature endows collocated DGSEM space discretizations of nonlinear hyperbolic problems with properties that greatly simplify the design of invariant domain-preserving high-resolution schemes. Compared to many other continuous and discontinuous Galerkin method variants, a particular advantage of the LGL spectral operator is the availability of a natural decomposition into a compatible subcell flux discretization. Representing a high-order spatial semi-discretization in terms of intermediate states, we perform flux limiting in a manner that keeps these states and the results of Runge-Kutta stages in convex invariant domains. In addition, local bounds may be imposed on scalar quantities of interest. In contrast to limiting approaches based on predictor-corrector algorithms, our MCL procedure for LGL-DGSEM yields nonlinear flux approximations that are independent of the time-step size and can be further modified to enforce entropy stability. To demonstrate the robustness of MCL/DGSEM schemes for the compressible Euler equations, we run simulations for challenging setups featuring strong shocks, steep density gradients, and vortex dominated flows.

Fractional Spectral and Fractional Finite Element Methods: A Comprehensive Review and Future Prospects

Fractional Spectral and Fractional Finite Element Methods: A Comprehensive Review and Future Prospects

Random medium modeling and underwater acoustic field simulation

The von Karman random medium model is established based on the small scale random medium theory in this paper. Combined with the finite element theory, the numerical simulation of the sound field under this model is realized by using the spectral-element method, and the characteristics of the sound field are compared with those under the uniform medium underwater environment. The results show that the correlation length reflects the variation of the heterogeneous size, and the scattering wave generated by random disturbance may provide a more reasonable explanation for the amplitude and energy anomalies in some trial data.

Band folding induced broadband vibration suppression of star-shaped metamaterials: Theory and experiment

Metamaterials with bandgap features can block wave propagation in specific frequency ranges and thus have wide applications for vibration mitigation and noise reduction. In this study, we renovated the design of star-shaped metamaterial (SSM) to make it capable of producing lower and wider vibration bandgaps. Unlike traditional periodic truss structures in the literature, the designed SSM structure breaks the spatial symmetry, opening the degenerate points formed by the band folding effect, thereby producing extra bandgaps for vibration suppression. First of all, a finite element (FE) model of the SSM is established, and its band structures and vibration transmittance are calculated. The preliminary result validated our hypothesis, demonstrating that the proposed SSM structure created additional bandgaps for suppressing low-frequency and broadband vibrations. Subsequently, the frequency response analyses are conducted using the spectral element method (SEM) and experimental test. The effects of the structural parameters on the bandgaps of the proposed SSM are studied. In addition, we further extended and generalized the band folding design concept. We showed that one can produce more bandgaps by designing high-order SSMs. In general, this study presents an approach for designing truss structures with broadband vibration suppression performance.

Flexural wave bandgaps of corrugated-core sandwich metabeam with inertial amplified resonators using the spectral element method

In this paper, we present a theoretical investigation of flexural wave propagation and vibration bandgap properties in corrugated-core sandwich metabeam with inertial amplified resonators. The complete analytical solutions for the flexural wave propagation with free boundary conditions are obtained through the spectral element method. Finite element calculation of transmission characteristics of sandwich metabeam with inertial amplified resonators are carried out to validate the accuracy of the proposed complete analytical solutions. In addition, the vibration reduction effect of sandwich metabeam with inertial amplified resonators is compared with that of the corrugated-core metabeam with local resonance. The influences of system parameters on the flexural wave attenuation are analyzed. Results show that the complete analytical solution for the proposed sandwich metabeam structure can be effectively used to predict the vibrational properties. In addition, significant enlargement of low-frequency bandgap can be obtained by using an inertial amplification sandwich metabeam structure. The proposed sandwich metabeam structure has good bandgap adjustment ability.

Efficiently solving fractional delay differential equations of variable order via an adjusted spectral element approach

This paper presents a new approach for solving fractional delay differential equations of variable order using the spectral element method. The proposed method overcomes the limitations of traditional spectral methods, such as poor approximation in long intervals and inefficiency in high degrees. By introducing a variable order differentiation matrix and using basic Lagrangian functions to approximate the solution in each element, the method achieves high accuracy and efficiency. A penalty method is also applied to minimize the jump of fluxes at interface points, and the effectiveness of this approach is analyzed. Finally, three benchmark problems are solved, and the convergence analysis demonstrates the effectiveness and efficiency of the proposed method. In essence, this paper offers a significant contribution to the literature on fractional differential equations and their numerical solution methodologies.

Energy Exascale Computational Fluid Dynamics Simulations With the Spectral Element Method

Abstract Development and application of the open-source GPU-based fluid-thermal simulation code, NekRS, are described. Time advancement is based on an efficient kth-order accurate timesplit formulation coupled with scalable iterative solvers. Spatial discretization is based on the high-order spectral element method (SEM), which affords the use of fast, low-memory, matrix-free operator evaluation. Recent developments include support for nonconforming meshes using overset grids and for GPU-based Lagrangian particle tracking. Results of large-eddy simulations of atmospheric boundary layers for wind-energy applications as well as extensive nuclear energy applications are presented.

Vibration isolation by relative resonance perceived in combination of rigid bodies and elastic beams

This paper develops a novel low-frequency vibration isolation device based on rigid body dynamics called a Rigid Elastic Vibration Isolator (REVI). The REVI model is realized by coupling four elastic beams with two rigid bodies monolithically made using 3D printing. The system operates based on the vibration of the intermediate rigid bodies, which leads to the anti-resonance phenomenon at low frequencies. The dynamic analysis of the proposed REVI system has been meticulously investigated through analytical methods and real-world experimentation. The analytical method uses the Spectral element method to obtain the dynamic response, which is also validated by the experimental findings. Furthermore, the REVI transmittance sensitivity analysis was conducted by adjusting the rigid mass and system load and exploring methods for generating wide low-frequency bandgaps. The inclination angle of the REVI system is also varied, revealing the bandgap characteristics regarding negative transmittance level. The parametric study varying the geometric properties of the REVI system enhances our understanding of the bandgap and attenuation characteristics within the attenuation band. The REVI mechanism is practical and easily implemented, allowing for accurate and repeatable modeling. Moreover, the analytical observations assist in refining the shape of the REVI mechanism to achieve the necessary bandgap for the desired transmittance.