Spectral Element Method Research Articles

Far-field blast responses of sandwich arbitrary polygonal reinforced plate system

This paper presents a spectral element model for investigating the dynamic response characteristics of the arbitrary polygonal built-up reinforced plate system subjected to far-field blast load. In the researched system, the plate elements are made of sandwich materials with functionally graded carbon nanotube reinforced (FG-CNTRC) panels and three-dimensional graphene foam (3D-GrF) core. The stiffness matrix assembly method is incorporated into the spectral element method (SEM) for establishing the overall dynamic formulas of arbitrary assembled plates. Thereinto, the governing equations of motion of plate elements are derived by the Hamilton's principle based on the simple first-order shear deformation theory (S-FSDT) retaining only four displacement variables, which provides an advantage to maintain interelement continuity conditions completely. After that, the natural frequencies and transient responses of the systems with different built-up forms are solved by introducing a golden section search algorithm and Fourier series expansion of the blast load. The admirable predictive accuracy of the present model is verified by comparing the obtained results and the reference solution from finite element method (FEM). Moreover, the effects of the CNT distribution form, CNT volume fraction, 3D-GrF pore distribution form, 3D-GrF porosity, the thickness ratio of the sandwich panel, TNT mass and explosion radius on the transient vibration properties of the system are revealed through a detailed parametric study.

Computational framework for efficient high-fidelity optimization of bio-inspired propulsion and its application to accelerating swimmers

A new computational framework for high-fidelity optimization of kinematic gaits during self-propelled undulatory swimming is developed. A computational framework utilizes a spectral-element method on moving body-fitted grids for a simulation of self-propelled swimming, and a surrogate-based optimization (SBO) procedure. A new volume-conservation method for reconstruction of a swimmer's geometry during the undulatory motion is proposed to ensure numerical stability of the fluid-structure interaction solver in an incompressible flow framework. A surrogate-based optimization algorithm that utilizes a Kriging response surface method is adopted and further developed in this work to manage the optimization process in the presence of physiological constraints on the fish body motion. A grid convergence of the optimization results is established, and the influence of the polynomial refinement on the results of optimization procedure is assessed. The increase in polynomial order does not change the optimum gaits of locomotion or relative efficiency rankings between the modes, but it results in slightly lower predicted efficiency for all the modes. The optimum solution is characterized by a kinematic gait that generates the reverse Karman vortex street associated with high propulsive efficiency. Efficiency of sub-optimum modes is found to increase with both the tail amplitude and the effective flapping length of the swimmer, and a new scaling law is proposed to capture these trends. Lastly, the SBO algorithm converged to an optimized gate with significantly less function evaluations than typically observed for evolutionary algorithms. This suggests that the SBO framework is a well suited alternative for high-fidelity optimization of fluid and structure problems.

Energy spectral element method for longitudinal acoustic propagation in ducts

Sound field behavior in an acoustic enclosure is an important part of the design of transportation vehicle passenger cabin, concert halls, conference rooms, and etc. Different analysis methods are available and have strengths and weaknesses. Low frequency band envelopes and negligible absorption in the walls can be modeled by Modal Analysis or Finite Element Method. However, as the frequency band increases, both methods become computationally intensive and Statistical Energy Analysis or the Sabine model can be an efficient approach. However, these methods do not take into account any spatial variation within the enclosure. The Energy Flow Analysis (EFA) solution for acoustic enclosures can be done analytically. In this paper, the Energy Spectral Element Method (ESEM) is formulated and applied to predict the spatial distribution of energy flux and density of acoustic ducts at high frequencies. ESEM is a matrix methodology based on EFA to solve acoustic and structural vibration problems. In this work, numerical models involving simple and coupled one-dimensional acoustic ducts are generated by ESEM, and the results are compared with energy densities calculated from the pressure fields predicted by the Spectral Element Method (SEM).

Modal spectral element method with modified Legendre polynomials to analyze binary crossed gratings.

In a previous paper, a modal spectral element method (SEM), the originality of which comes from the use of a hierarchical basis built with modified Legendre polynomials, was shown to be very powerful for the analysis of lamellar gratings. In this work, keeping the same ingredients, the method has been extended to the general case of binary crossed gratings. The geometric versatility of the SEM is illustrated with gratings whose patterns are not aligned with the boundaries of the elementary cell. The method is validated by a comparison to the Fourier modal method (FMM) in the case of anisotropic crossed gratings and with the FMM with adaptive spatial resolution in the case of a square-hole array in a silver film.

Spectral-Element Spectral-Integral (SESI) Method for Doubly Periodic Problems With 3-D Scatterers Embedded in Multiple Regions of Layered Media

In this work, a hybrid 3-D spectral-element spectral-integral (SESI) method is developed by combining the spectral element method (SEM) and spectral integral method (SIM) for the fast evaluation of electromagnetic (EM) scattering from double Bloch (Floquet) periodic problems with 3-D scatterers embedded in multiple regions of layered media. This 3-D SESI method is a further extension of the previously developed 2-D SESI method. In the proposed method, the computational domain is divided into several SEM and SIM subdomains, where the subdomains are coupled by Riemann transmission conditions (RTCs) to allow the use of nonconforming meshes for these subdomains. Meanwhile, the 3-D periodic layered medium Green’s function (PLMGF) is derived and employed to simulate 3-D structures in layered media without their volumetric discretization. Four numerical experiments show that the developed SIM method is highly efficient and accurate; the SESI method is more efficient than the traditional finite element method (FEM), thus is a promising solver for designing and optimizing frequency selective surfaces (FSSs), metamaterials, and metasurfaces.

A Full-Vectorial Spectral Element Method With Generalized Sheet Transition Conditions for High-Efficiency Metasurface/Metafilm Simulation

Recently, metasurfaces as 2-D metamaterials have been widely applied due to their advantages in low loss, ease of fabrication, and tunability in the phase, magnitude, and polarization of incident electromagnetic waves. However, due to the subwavelength-thickness properties of metasurfaces, their direct modeling and simulation are challenging because of the heavy computational burden. In this work, a high-efficiency spectral element method with generalized sheet transition conditions (SEM-GSTCs) is proposed by replacing finite thickness metasurfaces with zero-thickness equivalent boundary conditions to avoid dense elements and the associated enormous number of unknowns. Moreover, both the scalar and the full-vectorial SEM-GSTC are implemented. Some illustrative numerical examples are shown to verify that the proposed SEM-GSTC not only can obtain highly accurate numerical results but also is effective in addressing complex functions ranging from isotropic to fully bianisotropic metasurfaces.

Boundary conditions determined by experimentally measured surface properties for wave-based time-domain simulations

Boundary conditions are essential and necessary when simulating sound wave propagation inside an enclosed space. In this work, complex-valued surface acoustical properties for the actual materials are obtained from impedance tube measurements. This work applies model-based Bayesian inference to estimate the exact parameters for arbitrary boundary conditions. The Bayesian approach presented in this work demonstrates the capability to predict the complex-valued surface boundary function of frequency in broadband ranges. This approach is equivalently applicable in different boundary models in time domain depending on wave-based numerical methods used for acoustic modelings, such as the spectral element method or finite-difference time-domain method. With the accurately estimated boundary models, the efficacy of the boundary conditions is explored within the time-domain approaches.

A two-grid spectral method to study of dynamics of dense discrete systems governed by Rosenau-Burgers' equation

A numerical formulation with second-order accuracy in the time direction and spectral accuracy in the space variable is proposed for solving a nonlinear high-dimensional Rosenau-Burgers equation. We show how the spectral element method and the two-grid idea can be used together. The two-grid idea is combined with the spectral element method for simulating the nonlinear high-dimensional Rosenau-Burgers' equation. The proposed technique is based on the three-level algorithm. The existence and uniqueness of the solutions to Steps 1, 2, and 3 are investigated and also the error analysis is discussed.

Large-Scale direct numerical simulations of turbulence using GPUs and modern Fortran

We present our approach to making direct numerical simulations of turbulence with applications in sustainable shipping. We use modern Fortran and the spectral element method to leverage and scale on supercomputers powered by the Nvidia A100 and the recent AMD Instinct MI250X GPUs, while still providing support for user software developed in Fortran. We demonstrate the efficiency of our approach by performing the world’s first direct numerical simulation of the flow around a Flettner rotor at Re = 30,000 and its interaction with a turbulent boundary layer. We present a performance comparison between the AMD Instinct MI250X and Nvidia A100 GPUs for scalable computational fluid dynamics. Our results show that one MI250X offers performance on par with two A100 GPUs and has a similar power efficiency based on readings from on-chip energy sensors.

Analysis of Acoustic–Magnetic Fields Induced by Underwater Pressure Wave in a Finite-Depth Ocean

As underwater disturbances (natural or artificial) occur in the ocean, moving seawater crossing the geomagnetic fields will produce weak circular currents. These currents can induce measurable magnetic fields, which might be useful for monitoring ocean internal waves using aeromagnetic survey. In this research, a spectral-element method (SEM) based on Gauss–Lobatto–Legendre (GLL) polynomials is presented to characterize the magnetic field induced by the underwater pressure waves. A concise mathematical model is established through combining the acoustic wave equations and Maxwell’s equations. Specifically, the acoustic–magnetic coupling simulation adopts the nodal-based SEM for acoustic analysis and edge-based SEM for electromagnetic analysis. The proposed SEM has spectral accuracy, as the error exponentially decreases with the order of the basis functions. Additionally, by adopting an independent modeling and mesh scheme in two solvers, respectively, the waste of computing resources is avoided. The experimental analysis demonstrates that the induced magnetic fields mechanically propagate with the acoustic wave, producing the pseudo-radiation phenomenon. The signals of these magnetic fields may extend for tens of kilometers and exist for hours under certain circumstances, which provide a theoretical basis for underwater target identification via high-sensitivity atomic magnetometer.

Fast, low-memory numerical methods for radiative transfer via hp-adaptive mesh refinement

Numerical solutions to the radiative transfer equation are typically computationally expensive. The large expense arises because the solution has a high dimensionality with NM degrees of freedom, where the N and M arise from spatial and angular degrees of freedom, respectively. Here, a numerical method is presented that aims for fast and low-memory calculations, in the sense of computational cost and memory requirements of only O(N). The method uses a discontinuous Galerkin (DG) spectral element method and hp-adaptive mesh refinement to reduce the number of spatial degrees of freedom from N to n, thereby reducing the total cost and memory to nM, with the aim of achieving nM approximately equal to N. After this reduction in memory to O(N), in order to ensure a computational cost of O(N), a suitable preconditioner is identified and utilized. Numerical examples are presented in two spatial dimensions to allow calculation of high-resolution reference solutions for comparison, while the methodology is general and applies in either two or three spatial dimensions. The numerical examples show large memory reduction ratios N/n and fast O(N) computational cost. A variety of examples is shown, including smooth spatial variations or steep gradients, and Rayleigh (isotropic) or Mie (anisotropic) scattering. The methods could enable more tractable computations for many applications, such as medical imaging and weather and climate prediction.

Numerical and experimental investigations on flexural vibration characteristics of metamaterial beam with carbon fiber-reinforced polymers

To effectively reduce vibration, a metamaterial beam with carbon fiber-reinforced polymer (CFRP) is proposed. The dispersion curves are calculated by the transfer matrix method (TM). The frequency response is investigated by the spectral element method (SEM). Displacement fields at different representative frequencies are displayed by the finite element method (FEM). The effects of the geometry, material parameters, and cross-ply angle of CFRP on the band gap are considered numerically, three types of beams are discussed and compared. From the results, it was found that the cross-ply angle and length of CFRP have significant influence on the band gaps. The frequency and width of the band gaps are different when the CFRP layers are imparity. Finally, a CFRP beam with ten periods was fabricated and the experiment was carried out to test transmission characteristics. Although the number of periods is only ten, it was found that the experimental results agree well with the numerical results in the low-frequency range. The results obtained in the paper can provide some useful reference for understanding and optimization of the design for metamaterial beam with wide band gaps, lightweight and high strength.

Wave propagation in thin pretwisted composite strips with an embedded delamination

In this work, a thin pretwisted and delaminated composite strip is modeled using the variational asymptotic method (VAM). The original 3D problem is reduced to a 2D cross-sectional analysis and a 1D problem along the length of the strip. Damage, in the form of an embedded delamination, is accounted in the strip by following a sublaminate approach. The 1D governing elastodynamic equations of the strip model is solved using the spectral finite element (SFE) method, wherein the structure is modeled as a wave guide. In SFE method, the governing elastodynamic equations are solved in transformed frequency domain to obtain the dynamic stiffness matrix. The guided wave response of the delaminated composite structure for different loads and boundary conditions are subsequently computed. Modal and wave response from SFE using the reduced 1D model are validated by comparing results with those reported in the literature and in some cases with the results obtained from a 3D model using a direct finite element (FE) package. Modal and wave propagation analysis is performed for various dimensions of delamination and composite layup. The analysis provides insight into the dynamic characteristic of partially delaminated strip. The usage of SFE is expected to yield a faster result and lesser computation time than the conventional direct FE method for analyzing and monitoring the delaminated strip exposed to high frequency excitations. Additionally, the capability of the model to investigate the inverse problem of damage detection is demonstrated.

A matrix–free high–order solver for the numerical solution of cardiac electrophysiology

We propose a matrix–free solver for the numerical solution of the cardiac electrophysiology model consisting of the monodomain nonlinear reaction–diffusion equation coupled with a system of ordinary differential equations for the ionic species. Our numerical approximation is based on the high–order Spectral Element Method (SEM) to achieve accurate numerical discretization while employing a much smaller number of Degrees of Freedom than first–order Finite Elements. We combine vectorization with sum–factorization, thus allowing for a very efficient use of high–order polynomials in a high performance computing framework. We validate the effectiveness of our matrix–free solver in a variety of applications and perform different electrophysiological simulations ranging from a simple slab of cardiac tissue to a realistic four–chamber heart geometry. We compare SEM to SEM with Numerical Integration (SEM–NI), showing that they provide comparable results in terms of accuracy and efficiency. In both cases, increasing the local polynomial degree p leads to better numerical results and smaller computational times than reducing the mesh size h. We also implement a matrix–free Geometric Multigrid preconditioner that results in a comparable number of linear solver iterations with respect to a state–of–the–art matrix–based Algebraic Multigrid preconditioner. As a matter of fact, the matrix–free solver proposed here yields up to 45× speed–up with respect to a conventional matrix–based solver.

Antipodal waveform observations of seismic waves diffracting and refracting at the base of Earth's outer core

We approach this analysis of the inner-outer core boundary (IOCB) with antipodal waveform data in the distance range 179.0°–180° to test the hypothesis whether propagation at the base of the outer core is commensurate with diffraction and/or refraction. The propagation paths observed cover about two-thirds of the IOCB surface. Seismic data from seven diameters are examined—Tonga to Algeria (station code TAM), Sulawesi to Amazon (PTGA), northern Chile to Hainan Island (QIZ), two between central Chile and the mainland China (XAN, ENH), and two diameters from New Zealand to both Portugal (PTO) and Spain (ECAL). The adequacy of global Earth models—both 1-D and 1-D core +3D mantle—in fitting the antipodal observations is found to be deficient. Lateral heterogeneity among the propagation paths are mapped and projected to the Earth's surface for context. We stack data to increase signal-to-noise, and model waveform data via the 3D spectral element method on the EarthSimulator4. With the exception of one path to ENH China, none of the global Earth models match amplitudes of diffracted waves or the stacked amplitudes relative to PKIKP, which travels along the antipodal diameter. Energetic arrivals observed in the waveform data set are modeled as a combination of refraction within a low velocity zone (LVZ) at the base of the outer core, and diffraction around this structure. The observations at the base of the outer core may be further subdivided between paths with a thin low velocity zone (QIZ, XAN, PTGA), and those with a thicker, slower LVZ (TAM). The former paths are characterized by a thin LVZ (20–50 km thick, layer–gradient, respectively) with a velocity of approximately 10.0 km/s. The second group of paths shows large amplitude, complicated waveforms for TAM, corroborated by data from PTO and ECAL. TAM waveforms are modeled with a negative gradient from 100 to 50–75 km thickness above the IOCB, to a basal velocity of 8.8 km/s at IOCB. We also modeled the prospect of non-zero rigidity at the bottom of the outer core, but found no evidence of sensitivity in the core data. The structures fitting the data correlate across the inner–outer core boundary, with a velocity anomaly in PKPbc data juxtaposed immediately above the TAM path beneath SE Asia. More generally, the TAM IOCB path when projected up to the core-mantle boundary (CMB) apparently aligns with regions between the large low shear velocity provinces (LLSVP) at the CMB. Other paths with thinner or negligible LVZ at the IOCB partially underlie the LLSVP when projected to the CMB.

Accelerated sound propagation using an error-free Fourier method coupled with the spectral-element method

Simulating acoustics using numerical methods efficiently and accurately has been an active research area for the last decades and has applications in computer games, VR/AR, and architectural design. However, their extensive computation time makes these methods challenging for large scenes and broad frequency ranges. This work attempts to accelerate the simulations using rectangular decomposition, enabling error-free propagation in the bulk of the domain consisting of air. We exploit the analytical solution to the wave equation calculated using the Fast Fourier Transform with near-optimal spatial and temporal discretizations satisfying the Nyquist criterium. Coupling with the spectral-element method near the boundaries results in a method capable of handling complex geometries with realistic boundaries, though with the caveat that additional errors and computational overhead may result from the interface. This talk will investigate the accuracy and efficiency of the proposed domain decomposition method compared to a spectral-element method running in the entire domain.

Toward a robust detection of viscous and turbulent flow regions using unsupervised machine learning

We propose an invariant feature space for the detection of viscous-dominated and turbulent regions (i.e., boundary layers and wakes). The developed methodology uses the principal invariants of the strain and rotational rate tensors as input to an unsupervised Machine Learning Gaussian mixture model. The selected feature space is independent of the coordinate frame used to generate the processed data, as it relies on the principal invariants of the strain and rotational rate, which are Galilean invariants. This methodology allows us to identify two distinct flow regions: a viscous-dominated, rotational region (a boundary layer and a wake region) and an inviscid, irrotational region (an outer flow region). We have tested the methodology on a laminar and a turbulent (using Large Eddy Simulation) case for flows past a circular cylinder at Re = 40 and Re = 3900 and a laminar flow around an airfoil at Re=1×105. The simulations have been conducted using a high-order nodal Discontinuous Galerkin Spectral Element Method. The results obtained are analyzed to show that Gaussian mixture clustering provides an effective identification method of viscous-dominated and rotational regions in the flow. We also include comparisons with traditional sensors to show that the proposed clustering does not depend on the selection of an arbitrary threshold, as required when using traditional sensors.

Effect of Prandtl number on the flow instabilities in thermocapillary liquid bridges between two coaxial disks with different radii

The effect of Prandtl number on the instability of thermocapillary liquid bridges has been extensively studied, but the vast majority of efforts were put into typical cylindrical liquid bridge between two equal ends. The crystal growth practices have the propensity to form liquid bridges between unequal disks, thus the dependence of instability patterns and underlying mechanisms on the radius ratio is crucial but remains unclear. By fixing the radius ratio at Γr=0.5, we systematically explore the effects of Prandtl number and heating strategy, namely the upper heating and the bottom heating settings, on flow instabilities in thermocapillary liquid bridges between coaxial disks with different radii under microgravity through linear stability analysis based on spectral element method. In contrast to typical cylindrical liquid bridge (Γr=1), the present analysis indicates that the instability is always an oscillatory bifurcation even for small Prandtl number Pr=0.001 in the bottom heating scenario. What's more, the instability remains a stationary bifurcation and the critical Marangoni number experiences significant increase with the rise of Pr in the region of 0.001≤Pr≤1.2 for the upper heating situation. For all liquid bridges, the energy analysis results show that the instability under different Prandtl numbers can be roughly divided into three types.

A Finite/Spectral Element Hybrid Method for Modeling and Band-Gap Characterization of Metamaterial Sandwich Plates.

In this study, elastic metamaterial sandwich plates with axially deformed Timoshenko beam cores, considering both the out-of-plane and in-plane deformations of the face plates, are designed and the vibration band-gap properties are explored. The beam cores act as local resonators that can bear axial force, bending moment and shearing force. The finite element method (FEM) and the spectral element method (SEM) are combined to create the finite/spectral element hybrid method (FE-SEHM) for establishing the dynamic model and calculating the frequency response functions (FRFs) of the elastic metamaterial sandwich plate with axially deformed beam cores. It is observed that the metamaterial sandwich plate possesses both the axial and transverse vibration band-gaps of the beams, and the two kinds of band-gaps are independent. Compared with the metamaterial sandwich plates with rod cores, those with axially deformed beam cores have more extensive application ranges for vibration reduction.

Analysis of the sound insulation performance of periodic wall structures by a virtual acoustic laboratory

The propagation of acoustic waves in periodic structures, also known as phononic crystals or PCs, is prohibited in certain frequency ranges, which are referred to as the frequency band-gaps. The existence, the location and the width of the frequency band-gaps are mainly determined by the geometrical parameters and the material properties of the PCs. In this work, a virtual acoustic laboratory based on a fully coupled fluid-structure interaction (FSI) model is developed to determine the sound insulation capacity of one-dimensional (1D) and two-dimensional (2D) periodic walls. The FSI model is discretized using the frequency-domain spectral element method (FDSEM), which is an advanced finite element method (FEM) using special high-order shape functions. Following the guidelines of the ISO10140, the setup of the developed FSI model allows us to take into account the essential physical phenomena, especially the interaction of the wall structure with the fluid domains (air). The FSI model based on the FDSEM increases the computational efficiency and accuracy in comparison with the standard FEM. Several numerical examples will be presented and discussed to show that the designed periodic walls in certain frequency ranges within the band-gaps may exhibit much better sound insulation capabilities than monolithic walls.