Spectral Element Research Articles

Immediate traceability of ash based on laser-induced breakdown spectroscopy and machine learning

This article reports on an advanced level physical laboratory experiment designed for college level undergraduate education and for scholars who need specialized training on using and interpreting Laser-Induced Breakdown Spectroscopy. The technical principle, experimental operation and sample preparation of Laser-Induced Breakdown Spectroscopy are introduced in detail. A presentation and discussion of the use of Laser-Induced Breakdown Spectroscopy in the traceability of four common samples is emphasized. Combining Laser-Induced Breakdown Spectroscopy with Machine Learning, two distinct datasets are constructed through the extraction of spectral features. Dimensionality reduction of spectral data is performed using Linear Discriminant Analysis, while the Random Forest model is employed for provenance classification. Finally, the interpretability of the Random Forest model is leveraged to explore the contributions of different spectral elements to provenance tracing. Results demonstrated the system’s effectiveness in not only accurately identifying ash types but also in elucidating the influential chemical components, offering significant implications for material analysis and environmental monitoring. On an educational standpoint, this paper will allow any reader, in particular, undergraduate and graduate students, to gain a better understanding of the theory and practice of Laser-Induced Breakdown Spectroscopy and machine learning.

Mini-Workshop: Multivariate Orthogonal Polynomials: New synergies with Numerical Analysis

Multivariate polynomials and, in particular, multivariate orthogonal polynomials (MOPs) are research areas within the fields of special functions, Lie groups, quantum groups, computer algebra to name only some of them. However, there are many important areas in the field of numerical analysis where multivariate polynomials (of high order) play a crucial role: approximation by spectral methods and finite elements, discrete calculus, polynomial trace liftings, exact sequence properties, sparsity, efficient and stable recursions, analysis of the geometry of the zeros. The miniworkshop brought together experts from the fields of MOPs and numerical analysis of partial differential equations.

Error Propagation and Control in 2D and 3D Hybrid Seismic Wave Simulations for Box Tomography

ABSTRACT To enhance the local resolution of global waveform tomography models, particularly in areas of interest within the Earth’s deep structures, a higher resolution localized tomography approach (referred to as “box tomography”) is crucial for a more detailed understanding of the Earth’s internal structure and geodynamics. Because the small-scale features targeted by box tomography are finer than those in global reference models, distinct spatial meshes are necessary for global and local (hybrid) forward simulations. Within the spectral element method (SEM) framework, we employ the intrinsic Lagrangian spatial interpolation to compute and store hybrid inputs (displacement/potential) in the global numerical simulation. These hybrid inputs are subsequently imposed into the localized domain during the iterative box tomography. However, inaccurate spatial Lagrange interpolation can lead to imprecise hybrid inputs, and this error can propagate from the global simulation to the hybrid simulation. It is essential to quantitatively analyze this error propagation and control it to ensure the credibility of box tomography. We introduce a unique spatial window function into the conventional “direct discrete differentiation” hybrid method. When the local mesh and structure align with those in the global simulation, the synthetic hybrid waveforms match the global ones, serving as a reference for quantitatively assessing error propagation stemming from changes in the local spatial mesh during hybrid simulation. Significantly, the relative waveform error arising due to spatial Lagrange interpolation is around 5% when employing the traditional SEM with five Gauss–Lobatto–Legendre points per minimum wavelength in the 3D global simulation through SPECFEM3D_GLOBE. Ultimately, we achieve hybrid waveforms with an accuracy of about 1.5% by increasing the spectral elements by about 1.5 times in the standard global simulation.

A time-domain spectral finite element method for acoustoelasticity: Modeling the effect of mechanical loading on guided wave propagation

Ultrasonic testing techniques such as guided wave-based structural health monitoring aim to evaluate the integrity of a material with sensors and actuators that operate in situ, i.e. while the material is in use. Since ultrasonic wave propagation is sensitive to environmental conditions such as pre-deformation of the structure, the design and performance evaluation of monitoring systems in this context is a complicated task that requires quantitative data and the associated modeling effort. In our work, we propose a set of numerical tools to solve the problem of mechanical wave propagation in materials subjected to pre-deformation. This type of configuration is usually treated in the domain of acoustoelasticity. A relevant modeling approach is to consider two different problems: a quasi-static nonlinear problem for the large displacement field of the structure and a linearized time-domain wave propagation problem. After carefully reviewing the modeling ingredients to represent the configurations of interest, we propose an original combination of numerical tools that leads to a computationally efficient algorithm. More specifically, we use 3D shell elements for the quasi-static nonlinear problem and the time-domain spectral finite element method to numerically solve the wave propagation problem. Our approach can represent any type of material constitutive law, geometry or mechanical solicitation. We present realistic numerical results on 3D cases related to the monitoring of both isotropic and anisotropic materials, illustrating the genericity and efficiency of our method. We also validate our approach by comparing it to experimental data from the literature.

Correction: A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

Correction: A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

Vibration analysis of cylindrical shell discontinuously coupled with annular plate with arbitrary boundary conditions

Discontinuously coupled cylindrical shell-plates or plate-like structures are common components of underwater vehicles, and their vibration characteristics change significantly due to modal coupling. The vibration of discontinuously coupled annular plate-cylindrical shell structures is analyzed based on the wave propagation method(WPM) and virtual spring(VS). Taking uniform annular plates and cylindrical shells as spectral elements, the displacement solutions are described by the trigonometric function and Bessel function respectively, and the vibration responses of the annular plate-cylindrical shell with arbitrary boundaries are obtained. Based on VS and weighted least square norm least square method(WLSNLSM), the discontinuous coupling between the annular plate and cylindrical shell is described as the non-uniform distribution of stiffness of virtual spring by continuity condition, and the dynamic stiffness matrices of continuously and discontinuously coupled structure can be obtained simultaneously. The reason for vibration characteristic change under discontinuous coupling is explained from the perspective of modal superposition. Finally, a finite element model is established to verify the accuracy of the proposed method. The influence of general discontinuous coupling on the dynamic behavior of shell structure is analyzed by numerical examples, which provides useful guidance for structural vibration analysis with discontinuous coupling.

A Timoshenko-Ehrenfest beam model for simulating Langevin transducer dynamics

The Langevin transducer is a widely used power ultrasonic device across both medical and industrial applications, from orthopaedic surgery to drilling and welding. It is a sandwich-type device which typically consists of piezoelectric ceramic rings between two metallic end-masses. The transducer is commonly operated at a particular resonant mode to deliver ultrasonic vibrations to a target structure of interest, and is generally modelled using approaches including the transfer matrix method and electromechanical equivalent circuit methods. Here, we propose a variational framework to study the dynamics of the Langevin transducer based on the Timoshenko-Ehrenfest beam theory and Hamilton's principle. The variational equation derived from this model is then discretized by the standard finite element method with spectral elements. To verify the proposed one-dimensional electromechanical model, the computed resonance frequencies, or natural frequencies, of the one-dimensional model are compared to those of a three-dimensional finite element model with respect to varying geometry parameters characterizing the transducer. The results of the reduced one-dimensional and full three-dimensional models are then compared to those measured through an experimental investigation consisting of laser Doppler vibrometry. This is undertaken for the first ten eigenfrequencies, including longitudinal and bending modes, where normalized amplitudes and vibration mode shapes are reported. The close correlation between modelling and experiment demonstrates that the proposed one-dimensional electromechanical model can deliver results consistent with the full three-dimensional model and from experiment, thus verifying a rapid and reliable method for studying the free vibrations and dynamics of the Langevin transducer which accounts for the axial vibrations of the piezoelectric ceramic stack in all cases.

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.

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.

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 domain decomposition solver for spectral stochastic finite element: an approximate sparse expansion approach

Spectral stochastic finite element (SSFE) has been widely used in the uncertainty quantification of real-life problems. However, the prohibitive computational burden prevents the application of the method in practical engineering systems because an enormous augmented system has to be solved. Although the domain decomposition method has been introduced to SSFE to improve the efficiency for the solution of the augmented system, there still exist significant challenges in solving the extended Schur complement (e-SC) system from domain decomposition method. In this paper, we develop an approximate sparse expansion-based domain decomposition solver to generalize the application of SSFE. An approximate sparse expansion is first presented for the subdomain-level augmented matrix so that the computational cost in each iteration of the preconditioned conjugate gradient is greatly alleviated. Based on the developed sparse expansion, we further establish an approximate sparse preconditioner to accelerate the convergence of the preconditioned conjugate gradient. The developed approximate sparse expansion-based domain decomposition solver is then incorporated in the context of SSFE. Since the difficulties of solving the e-SC system have been overcome, the developed approximate sparse expansion-based solver greatly improves the computational efficiency of the solution of the e-SC system, and thereby, the SSFE is capable of dealing with large-scale engineering systems. Two numerical examples demonstrate that the developed method can significantly enhance the efficiency for the stochastic response analysis of practical engineering systems.

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.

Neko: A modern, portable, and scalable framework for high-fidelity computational fluid dynamics

Computational fluid dynamics (CFD), in particular applied to turbulent flows, is a research area with great engineering and fundamental physical interest. However, already at moderately high Reynolds numbers the computational cost becomes prohibitive as the range of active spatial and temporal scales is quickly widening. Specifically scale-resolving simulations, including large-eddy simulation (LES) and direct numerical simulations (DNS), thus need to rely on modern efficient numerical methods and corresponding software implementations. Recent trends and advancements, including more diverse and heterogeneous hardware in High-Performance Computing (HPC), are challenging software developers in their pursuit for good performance and numerical stability. The well-known maxim “software outlives hardware” may no longer necessarily hold true, and developers are today forced to re-factor their codebases to leverage these powerful new systems. In this paper, we present Neko, a new portable framework for high-order spectral element discretization, targeting turbulent flows in moderately complex geometries. Neko is fully available as open software. Unlike prior works, Neko adopts a modern object-oriented approach in Fortran 2008, allowing multi-tier abstractions of the solver stack and facilitating hardware backends ranging from general-purpose processors (CPUs) down to exotic vector processors and FPGAs. We show that Neko’s performance and accuracy are comparable to NekRS, and thus on-par with Nek5000’s successor on modern CPU machines. Furthermore, we develop a performance model, which we use to discuss challenges and opportunities for high-order solvers on emerging hardware.

Analysis of Planetary Scale Waves Using Idealized Sudden Stratospheric Warming Simulations in Different Dynamical Cores

AbstractPlanetary waves from the troposphere are known to play an important role in sudden stratospheric warming (SSW). To evaluate the representation of large‐scale waves in idealized SSW simulations, three dynamical cores of the National Center for Atmospheric Research Community Atmosphere Model were tested in this study: the Eulerian (EUL) spectral‐transform, finite‐volume (FV), and spectral element (SE) models. Notable differences were observed among the dynamical cores, with FV being unable to generate major SSWs and simulating minor SSWs less effectively than the other models. Wave decomposition analysis revealed distinct planetary wave propagation patterns during the development of each event. Zonal Wave Number 2 wave activities primarily determined SSW types, and ZWN1 led SSWs to recovery during the ensuing periods. However, FV failed to adequately propagate large‐scale waves in minor SSW events, preventing the reversal of stratospheric zonal‐mean zonal winds. Interestingly, FV showed decreased upward Eliassen–Palm flux compared to that of other dynamical cores, even during climatology. Additional tests were performed to examine the reasons for FV's atypical results, but the dynamic impacts on planetary waves remain poorly understood.

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.