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Because of the introduction of modal coupling, discontinuous boundaries have a significant impact on the vibration characteristics of annular plates. Based on the spectral element method (SEM) and parametric equivalent method (PEM), the vibration calculation and analysis of internally perforated annular plate with discontinuous boundary are explained. Firstly, a uniform annular plate is regarded as a spectral element, and the vibration response of an annular plate with general elastic boundaries is obtained based on the motion equation of the annular plate and the virtual spring method (VSM). The displacement solution is described by the Bessel function, and a fitting method of discontinuous spring stiffness coefficient based on the weighted least square norm least square method is proposed. Based on the force equilibrium condition, the vibration response of an annular plate with a discontinuous boundary is derived, and the reasons for the change of vibration characteristics with a discontinuous boundary are explained from modal superposition. The dynamic stiffness matrix method is used to establish the vibration equation of an annular plate with a discontinuous boundary, and the influence of internal openings on the vibration characteristics of an annular plate is effectively calculated by the parametric equivalent method. Finally, a finite element simulation model is established to test the accuracy and stability of the proposed method. The influence of general discontinuous boundary conditions on structural dynamic behavior is analyzed by numerical examples, which provides useful guidance for structural vibration analysis with discontinuous boundary conditions. [ABSTRACT FROM AUTHOR]

The characterization of the subsurface profile properties that are susceptible to liquefaction during earthquakes is of great importance in accident prevention. Field data can be used in a stochastic inverse problem to estimate the material properties using Markov chain Monte Carlo Methods (McMC) that incorporates prior knowledge about the unknown parameters as well as the data available. Here, we numerically study the Random Walk (RW) and Differential Evolution (DE) variations of the Metropolis algorithm in the context of seismic modeling considering the viscoelastic wave equation. The algorithms are tested with data from the 1987 Superstition Hills earthquake recorded at the Wildlife Site. The results show the effectiveness and accuracy of these algorithms, with emphasis to DE which reached convergence earlier compared to RW. [ABSTRACT FROM AUTHOR]

Fundamental mode surface wave data have often been used to construct global shear velocity models of the upper mantle under the so-called 'path average approximation', an efficient approach from the computational point of view. With the advent of full-waveform inversion and numerical wavefield computations, such as afforded by the spectral element method, accounting for the effects of the crust accurately becomes challenging. Here, we assess the merits of accounting for crustal and uppermost mantle effects on surface and body waveforms using fundamental mode dispersion data and a smooth representation of the shallow structure. For this we take as reference a model obtained by full-waveform inversion and wavefield computations using the spectral element method, model SEMUCB-WM1 and compare the waveform fits of synthetics to different parts of three component observed teleseismic records, in the period band 32–300 s for body waves and 40–300 s for surface waves and their overtones for three different models. The latter are: a dispersion-only based model (model Disp_20s_iter5), and two models modified from SEMUCB-WM1 by successively replacing the top 200 km (model Merged _200 km) and top 80 km (model Merged _80 km), respectively, by a model constrained solely by fundamental mode surface wave dispersion data between periods of 20 and 150 s. The crustal part of these three models (resp. SEMUCB-WM1) is constrained from dispersion data in the period range 20–60 s (resp. 25–60 s), using the concept of homogenization which is tailored to simplify complex geological features, enhancing the computational efficiency of our seismic modelling. We evaluate the fits to observed waveforms provided by these three models compared to those of SEMUCB-WM1 by computing three component synthetics using the spectral element method for five globally distributed events recorded at 200+stations, using several measures of misfit. While fits to waveforms for model 3 are similar to those for SEMUCB-WM1, the other two models provide increasingly poorer fits as the distance travelled by the corresponding seismic wave increases and/or as it samples deeper in the mantle. In particular, models 1 and 2 are biased towards fast shear velocities, on average. Our results suggest that, given a comparable frequency band, models constructed using fundamental mode surface wave data alone and the path average approximation, fail to provide acceptable fits to the corresponding waveforms. However, the shallow part of such a 3-D radially anisotropic model can be a good starting model for further full waveform inversion using numerical wavefield computations. Moreover, the shallow part of such a model, including its smooth crustal model, and down to a maximum depth that depends on the frequency band considered, can be fixed in full-waveform inversion iterations for deeper structure. This can save significant computational time when higher resolution is sought in the deeper mantle. In the future, additional constraints for the construction of the homogenized model of the crust can be implemented from independent short period studies, either globally or regionally. [ABSTRACT FROM AUTHOR]

This paper presents spectral element based approach for determining free planner wave propagation in two-dimensional periodic square-frame-grid-lattices (SqL). Implementing the newly developed nonlinear eigenvalue solver in conjunction with Bloch–Floquet periodic boundary condition, band structures for various configurations of SqL with attached mass or spring-mass resonator are analysed. Variation of the attenuation bandwidth is studied for different mass and frequencies of the attachments to the SqL. At the natural frequency of the resonator, an attenuation bandgap can be perceived; therefore, by tuning the natural frequency of the resonator, sub-wavelength band-gap can be obtained at the desired frequency for enhanced vibration or noise isolation. However, in few cases, mostly while the resonators are attached at the center point of the SqL, the attenuation bandgap near the natural frequency of the resonator may disappear as the center point lying in the node for that free wave frequency. Additionally, existence of dirac cone, eigenfrequency-loci veering are observed in the band-structures. The iso-frequency contours are also investigated to develop an insight comprehension about the underlying physics of energy transmission and mechanics of wave propagation. Various salient wave propagation features, namely self-collimation, lensing, and negative refraction of group velocity are identified and elucidated in this paper. [ABSTRACT FROM AUTHOR]

In this article, wave propagation in a one-dimensional (1-D) composite periodic frame structure is analyzed using spectral finite elements (SFE) method together with Bloch's theorem. Each element in the periodic frame structure is modeled as a first-order shear deformation theory (FSDT) beam element and wave equations for the beam elements are derived using FSDT considering axial, flexural, and shear deformations. Next, these equations are solved using the frequency domain SFE method to obtain an elemental dynamic stiffness matrix which on assembly gives the dynamic stiffness matrix for the unit cell of the periodic structure. A polynomial eigenvalue problem is formulated thereafter using Bloch's theorem to evaluate the dispersion coefficients and wave amplitude ratios. Finally, the entire periodic frame is modeled as homogenized two-noded elements capturing the wave response of the PS with a drastic reduction in computational cost. The results obtained using the present method are validated with finite element simulation using ANSYS mechanical Ansys Parametric Design Language (APDL). The frequency band gaps observed in the frequency domain responses are investigated and corroborated with those predicted by the dispersion relations. This is followed by a detailed parametric study on the influence of different parameters on the band gap characteristics. The present method provides a scheme of wave propagation analysis of composite periodic frames with substantial computational efficiency. [ABSTRACT FROM AUTHOR]

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In this paper, we propose the Legendre-Galerkin Network (LGNet), a novel machine learning-based numerical solver for parametric partial differential equations (PDEs) using spectral methods. Spectral methods leverage orthogonal function expansions, such as Fourier series and Legendre polynomials, to achieve highly accurate solutions with a reduced number of grid points. Our framework combines the advantages of spectral methods, including accuracy, efficiency, and generalization, with the capabilities of deep neural networks. By integrating deep neural networks into the spectral framework, our approach reduces computational costs that enable real-time predictions. The mathematical foundation of the LGNet solver is robust and reliable, incorporating a well-developed loss function derived from the weak formulation. This ensures precise approximation of solutions while maintaining consistency with boundary conditions. The proposed LGNet solver offers a compelling solution that harnesses the strengths of both spectral methods and deep neural networks, providing an effective tool for solving parametric PDEs. [ABSTRACT FROM AUTHOR]

Aerodynamic noise resulting from the flow around cylinders is a significant engineering challenge in aviation and wind engineering. The phenomenon of alternating vortex shedding in the flow leads to vibration and noise generation. However, accurately describing both the flow field and the sound field is challenging due to the significant difference in magnitude between them. To tackle this issue, this work introduces the application of the spectral element method (SEM) and flow-acoustic monolithic simulation for solving the two-dimensional compressible Navier–Stokes equations at low Reynolds numbers. This study is to investigate the reduction of flow-induced noise through the implementation of slotting technology on a circular cylinder. This study focuses on examining two different slit width ratios, s/d = 0.15 and 0.25, with a slit angle of attack of 0°. A comparative analysis is conducted between a complete circular cylinder and a slotted circular cylinder. The findings indicate that the slotted cylinder exhibits reduced intensity of vortex shedding and an extended region of downstream vortex generation compared to the complete cylinder. Notably, when s/d = 0.25, the slotted cylinder demonstrates minimal noise generation. Even at s/d = 0.15, a significant reduction in flow-induced noise is observed. These results highlight the potential of utilizing slotting technology on cylinders to effectively mitigate aerodynamic noise. The application of SEM and flow-acoustic monolithic simulation shows their relevance in analyzing and designing noise mitigation techniques in aerodynamics. This work can develop innovative solutions to reduce noise and improve the performance of various applications in aviation and wind engineering. [ABSTRACT FROM AUTHOR]

The interest in phononic crystals and acoustic metamaterials has been an intensive subject of research in recent years. Finding a robust way to significantly expand or actively control the bandgap has received extensive attention. In this study, we propose a prestressed metamaterial beam attached with multiply local resonators connected by actively tunable piezoelectric springs. The Euler–Bernoulli beam theory and Timoshenko beam theory are applied in the theoretical analysis of the system. Further, the spectral element method is utilized to analytically compute the dispersion relation and transmission ratio and excellent agreement with reference to the benchmark is reported. The influences of an external axial force on the bandgap range and attenuation behavior are further studied. Subsequently, the effect of resonator number and mass on the local resonance bandgap structure is investigated in two parametric studies. The active control of bandgap range and frequency is then verified. By analyzing frequency response function, the tunable transmission ratio of a supercell can be observed. To conclude, this paper not only provides a guideline for designs of wave attenuation with multiple frequency regimes in a one-dimensional system, but it can also be extended to sub-wavelength wave manipulation designs. [ABSTRACT FROM AUTHOR]

A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems. [ABSTRACT FROM AUTHOR]

Quantifying the sizes of the damages in structures using guided waves is an emerging technology. It needs a suitable mathematical model that represents the behavior of the guided waves as they travel through the structure. Though finite element based theoretical models can provide insight into the behavior, they are computationally very costly. Spectral element is a promising solution in this regard. Quantifying the damage using a modeconverted wave based spectral element has not been reported yet. In this work, a spectral element that represents the characteristics of wave propagation, scattering and mode conversion caused by asymmetric notch-type damage in an isotropic waveguide is presented. This frequency domain damage-spectral element is formulated through a combination of three structural waveguides by enforcing appropriate force equilibrium conditions. The spectral element is able to represent the wave scattering and mode conversion due to the presence of the damage. This spectral element is employed in analyzing the wave propagation in a beam that has damage and the characteristics obtained are in agreement with the results expected based on time-of-flight calculations and those obtained using the finite element method. The relation between the depth of the damage and the magnitude of the mode-converted wave for notch-type damage is established using this spectral element and compared with the experimental result. Usage of the same spectral element in quantifying the length of the damage based on the reflections of the primary propagating wave from the damage ends is also demonstrated. [ABSTRACT FROM AUTHOR]

Bioengineering techniques are being increasingly adopted as a sustainable solution to soil slope instability. Despite their recognized benefits, however, the mechanistic contribution of vegetation to slope stability remains inadequately understood due to the intricate nature of soil–root interactions and the complexity of root architectures. Most existing research predominantly offers qualitative assessments of vegetation effectiveness. This study aims to numerically substantiate the role of vegetation as a bioengineering technique for soil slope stabilization. Various plant species used commonly for soil stabilization were identified, and undisturbed soil samples were collected to quantify the shear strength parameters of the soils from both barren and vegetated slopes, along with root tensile strengths. A comprehensive topographic survey was conducted to capture the precise topography of the study area. Utilizing these primary data, in this work we develop 3D models for five representative plants within each species category and employ the Spectral Element Method (SEM), an advanced higher-order formulation of the finite element method (FEM), within the 3D domain to evaluate the factor of safety for the soil slopes. The SEM offers superior accuracy and stability in numerical computations. The results obtained through the SEM were corroborated through the FEM modeling and were found to be consistent with other established methodologies. This innovative approach of 3D SEM aims to quantitatively assess the impact of vegetation on soil slope stability and provide a more rigorous understanding of bioengineering applications in geotechnical engineering. [ABSTRACT FROM AUTHOR]

Acoustic waves are essential tools for guiding underwater activities. For many years, numerical modeling of ocean acoustic propagation has been a major research focus in underwater acoustics. Normal mode theory, one of the earliest and most extensively studied methods in this field, is renowned for its well-established theoretical framework. The core of normal mode theory involves the numerical solution of modal equations. In classical normal mode models, these equations are typically discretized using low-order finite difference methods, which, while broadly applicable, suffer from a limited convergence rate. The spectral element method, widely used in the seismic field, is recognized for its spectral precision and flexibility. In this article, we propose a normal mode model discretized using the spectral element method. The weak form of the modal equation directly satisfies boundary and interface conditions without requiring additional operations. The entire computational domain can be divided into segments of varying number and length, configured according to environmental conditions. The perfectly matched layer technique is employed to simulate acoustic half-space boundary conditions, effectively addressing the high computational costs and numerical instability associated with traditional artificial absorbing layers. Based on these algorithms, we have developed a numerical program (SEM). This research verifies the accuracy of the spectral element model through three different types of numerical experiments. [ABSTRACT FROM AUTHOR]

AbstractThe vibration of ship and marine structures directly impacts their structural performance and noise levels. Therefore, the vibration band gap characteristics of two-dimensional periodic grillage structures are investigated based on spectral element method in this article. Firstly, the governing motion equation of out-of-plane vibration of periodic grid structures is established in light of the Bloch theorem, and the shear spring oscillator system is introduced to simulate the first-order vertical shear vibration of bottom plate. As a result, the periodic oscillator coupling grid structure model can be proposed. Based on the periodic grid structures, the influence of material distribution on vibration band gap is obtained. Meanwhile, the dispersion relations and vibration transmission of periodic coupling model are calculated and the coupling effect of local resonance system on the band gap characteristics is analyzed. At last, a general approach for equating periodic grillage unit to numerical model is outlined and summarized on basis of the similarity of dynamic properties. The effectiveness and accuracy of the numerical method in out-of-plane vibration band gap analysis of periodic grillage structures are validated through the computation and model test. [ABSTRACT FROM AUTHOR]

The objective of this paper is to develop, verify, and experimentally validate a mesh-free spectral graph theory-based approach for rapid prediction of thermal history in metal parts made using the wire arc additive manufacturing (WAAM) process. Accurate and rapid prediction of the thermal history is a critical prerequisite for functional quality assurance of WAAM parts. In the spectral graph method, the WAAM part is represented as a set of discrete nodes encompassed by a network graph. The thermal history is obtained by solving the heat equation on the network graph. The spectral graph theory approach thus bypasses the cumbersome computational burden associated with mesh generation in the finite element method. To validate the spectral graph theory approach, experimental temperature data is acquired for multi-layer mild steel WAAM parts processed under six different combinations of shape and inter-layer dwell time conditions. Further, each experiment was replicated resulting in a total of 12 parts. The accuracy of the thermal trends predicted by the spectral graph theory approach was quantified in terms of the symmetric mean absolute percent error (SMAPE) and root mean squared error (RMSE, °C). The thermal history trends were predicted with SMAPE < 5% and RMSE < 11 °C relative to the experimental temperature measurements. The computation time to obtain the thermal history of each 21-layer part was approximately 3.5 to 4 h. The spectral graph theory predictions were further verified with a commercial finite element software package (Simufact-Welding). For a similar level of SMAPE and RMSE as the spectral graph theory approach, the FE-based Simufact predictions required over 7.5 h. The computational advantage of the spectral graph theory approach is particularly valuable for devising physics-based process parameter optimization and feedforward control strategies to mitigate flaw formation in WAAM parts. [ABSTRACT FROM AUTHOR]

The development of new methods for the synthesis of nanosystems with plasmonic properties and for their analysis is an important field of modern photonics. Here, low-temperature laser synthesis of quasi-two-dimensional gold nanonetworks (with individual nanowire diameters of 5 nm) is carried out in superfluid helium. The plasmonic properties of nanosystems obtained in this way are investigated by Raman scattering for the first time. Possible features of the near field in these systems are studied using three-dimensional numerical electrodynamic modeling by the finite element method in the visible to near-infrared spectral range (400–1000 nm). The complex internal structure of hot spots arising around two intersecting 5-nm nanowires (the elementary model unit) and the diversity of their localization types, showing tip-to-gap transition, are demonstrated. [ABSTRACT FROM AUTHOR]

Isospectral surfaces provide a rich family of benchmark problems. In this paper the efficacy of a hp -finite element method in such Laplace-Beltrami eigenvalue problems has been studied. In addition, for the p -version natural auxiliary space error estimators have been shown to be effective also in this context. As part of the numerical experiments further numerical evidence of the validity of the so-called Quarter Sphere Conjecture has been produced. For the isospectral surface pairs constructed via transplantation, the extension for perforated surfaces is derived. The lack of such constraints for surfaces with isothermal coordinates is also demonstrated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we consider parabolic partial differential equations in (2+1) dimensions, with a focus on optimal convergence for solutions that are smooth in time but have corner singularities in space owing to the non-smoothness of the spatial boundary. The method employs geometric mesh along with a nonconforming least squares spectral element approximation in space coupled with the implicit Crank-Nicolson finite difference scheme in time. The unconditional stability of the scheme is rigorously proved and the error estimate which is almost optimal second order accurate in time and exponentially accurate in space is established. At each time instant, the subdivided problem is solved in parallel via the preconditioned conjugate gradient method in different processors by using an almost optimal preconditioner, in the sense that the condition number of the resulting system is O ((l n N) 2) (N being the degree of spatial approximation), without having to store any stiffness (or mass) matrix or load vector and the inter-element communication is established through Message Passing Interface (MPI). Specific numerical examples are presented with regard to the performance of the scheme. [ABSTRACT FROM AUTHOR]

This paper presents the investigation of a metastructure, built with periodic arrays of inertial amplification (IA) cantilever-type resonators on a host beam, to improve its low-frequency flexural wave attenuation performance. The IA mechanism is composed of an additional mass and three rigid bars, which are connected by hinges and embedded on the host beam and cantilever-type resonators, respectively. The dynamical model of an IA cantilever-type resonant beam (IACRB) is established by the spectral element method, and its bandgap characteristics and vibration transmissibility are verified numerically and experimentally. The bandgaps of the lumped mass cantilever-type resonant beam are compared to those of the IACRB, showing that the latter has better low-frequency wave attenuation ability. To reveal the underlying physics, the effects of the amplification ratio, IA span, and length ratio of cantilever-type resonators on the band diagram of the IACRB are investigated. It is found that the variation of the amplification ratio results in the bandgap near-coupling phenomenon and the bandgap transposition phenomenon. The increase of IA span by changing position 1 weakens the coupling effect of local resonance and IA. However, the increase of IA span by changing position 2 creates the super-wide pseudo-bandgap as well, which is potential in engineering practices. The variation of the length ratio of cantilever-type resonators decreases its first bending modal frequency and shifts the IA bandgap to a lower frequency range slightly. The modal analysis results of the IACRB show that the proposed mechanism affects the modal distributions of the host structure. The modal frequencies existing in bandgaps are caused by the local vibration mode of the ends of the IACRB. [ABSTRACT FROM AUTHOR]

ABSTRACTTopographic influences of the ground motions in 3D geological configurations have been addressed using Spectral Element Method as implemented in SPECFEM3D code developed to simulate seismic wave propagation. Hexahedral mesh model was constructed to which the 1987 Jinka earthquake source characteristics were applied. The simulation results have shown that topography has significant effects on ground motion and changes PGA values by ± 55% compared to the flat topography. PGA amplification variation between valley and hill/ridge top is observed to be in the order of 1.6. The study indicates that there is PGA amplification on top of hills and de-amplifications occur at the base. Valleys and flat areas generally show low and negative amplification factors. It is observed that topographic features play significant roles in concentrating or diffusing seismic energy. This was confirmed by observing the behaviour of PGA values along selected profiles and by comparing PGA maps with and without topography. The study shows that topography has serious consequences and must be taken into account to minimise damage to infrastructures and safeguard human lives and suggests the importance of incorporating the effects of topographic features in any seismic hazard analysis schemes in densely populated mountainous regions.

Topographic influences of the ground motions in 3D geological configurations have been addressed using Spectral Element Method as implemented in SPECFEM3D code developed to simulate seismic wave propagation. Hexahedral mesh model was constructed to which the 1987 Jinka earthquake source characteristics were applied. The simulation results have shown that topography has significant effects on ground motion and changes PGA values by ± 55% compared to the flat topography. PGA amplification variation between valley and hill/ridge top is observed to be in the order of 1.6. The study indicates that there is PGA amplification on top of hills and de-amplifications occur at the base. Valleys and flat areas generally show low and negative amplification factors. It is observed that topographic features play significant roles in concentrating or diffusing seismic energy. This was confirmed by observing the behaviour of PGA values along selected profiles and by comparing PGA maps with and without topography. The study shows that topography has serious consequences and must be taken into account to minimise damage to infrastructures and safeguard human lives and suggests the importance of incorporating the effects of topographic features in any seismic hazard analysis schemes in densely populated mountainous regions. [ABSTRACT FROM AUTHOR]

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems. • Two energetic spectral-element schemes in time for parabolic systems. • Maintain mass conservation and energy dissipation. • Superconvergence of the implicit method proven without use of supercloseness. • Linear, energy stable and efficient semi-implicit method. • Semi-implicit scheme as iterative implicit solver excels existing methods. [ABSTRACT FROM AUTHOR]

The classical staggered ℙN‐ℙN−2$$ {\mathbb{P}}_N\hbox{-} {\mathbb{P}}_{N-2} $$ spectral element method (SEM) is revisited and extended to quasi‐static magnetohydrodynamic (MHD) flows. In this realm, which is valid in the limit of vanishing magnetic Reynolds number, the evaluation of the Lorentz force in the momentum equation requires the electric current density, governed by Ohm's law and a charge conservation condition derived from Ampère's law, to be determined. Once discretized with the SEM, this translates into solving one additional problem for the electric potential involving the so‐called consistent Poisson operator. The method is well suited for fully three‐dimensional flows in complex geometries. Changes in resolution requirements aside, consideration of the electromagnetic quantities is estimated to increase the computational cost associated with MHD by about 40% relative to hydrodynamics. The accuracy and the capabilities of the scheme is demonstrated on a set of common flows from the MHD literature. Exponential convergence with polynomial order is confirmed for the electric current density. [ABSTRACT FROM AUTHOR]

Abstract This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral element method for spatial discretization. We prove that optimal adaptivity can be achieved by combining adaptive time steps with adaptive spectral mesh. We introduce two sets of error indicators for time and space, respectively, and derive optimal estimates.

AbstractAnalytical solutions for the Boltzmann transport equation are typically limited to specific scenarios, prompting the research transport community to extensively explore numerical approaches with special attention to the discrete ordinates (SN) formulation. The spectral nodal methods are widely recognized for their capability to preserve the analytical solution of the SN mathematical model. However, these formulations exhibit notable complexity with high-order polynomial approximations for the transverse leakage terms, motivating researchers to pursue simpler hybrid approaches by partially preserving the analytical solution. Both spectral nodal and hybrid spectral nodal methods introduce nonstandard auxiliary equations not amenable to standard iterative schemes due to a highly coupling in angular directions. Following these ideas, we present a finite element – spectral nodal hybrid method referred to as Extended Diamond Difference-Constant Nodal (EDD-CN) method for two-dimensional fixed-source discrete ordinate transport problems with constant nodal approximation for the leakage terms. The proposed method seeks to preserve eigenfunctions associated with the dominant pair of eigenvalues in the transverse integrated SN equations. In addition, we propose a novel iterative scheme called Decoupled Cell Iteration (DCI) scheme which decouples the angular directions in the nonstandard auxiliary equations and follows conventional iteration transport sweeps for solving SN problems. [ABSTRACT FROM AUTHOR]

A novel metamaterial periodic multi-span beam with elastic supports and local resonators is designed and studied. The vibration characteristics of the designed metamaterial multi-span beam structure are investigated by the spectral element method (SEM) combined with the transfer matrix method (TMM). The accuracy and feasibility of the theoretical methods are validated by the finite element method (FEM) and vibration experiment. It is demonstrated that the metamaterial periodic multi-span beam with elastic supports can produce local resonance bandgaps in the low-frequency regions and Bragg bandgaps in the middle- and high-frequency regions. For some support stiffnesses, the amplitudes and widths of the bandgaps for the proposed structure are larger than those of the metamaterial multi-span beam with simply supported boundaries. Thus, the positions and ranges of the Bragg bandgaps can be adjusted by changing the stiffness of the elastic supports so as to improve the vibration reduction performance of the structure. [ABSTRACT FROM AUTHOR]

Introduction/purpose: The dynamic analysis of a seismic response of a concrete vault dam is a complex problem in which the representation of the behavior of the material requires some form of a nonlinear model, especially if the concrete is subjected to a significant stress load of the ground. In the case of large movements of the latter, large cracks may form in some areas of the dam, especially at the base of the dam and near sudden changes in geometry. Methods: This analysis was based on a numerical simulation of the dynamic effect. This work was carried out using the finite element method with the ANSYS 12.1 program. The dam was modelled in two dimensions. Four types of analysis were performed: static analysis, modal analysis, seismic analysis under excitation of two accelerograms (Asnam 1980 and Boumerdes 2003), and spectral analysis. Results: This analysis showed the vulnerability of the Brezina dam to the Boumerdes earthquake with high stresses at the base of the structure. Conclusions: Based on this study, it was concluded that if the Brezina dam suffers an earthquake of a greater intensity than that of Boumerdes, this will cause structural damage and cracks that will compromise the dam’s watertightness as well as its durability. [ABSTRACT FROM AUTHOR]

For Volterra integro-differential equations (VIDEs) with weakly singular kernels, their solutions are singular at the initial time. This property brings a great challenge to traditional numerical methods. Here, we investigate the numerical approximation for the solution of the nonlinear model with weakly singular kernels. Due to its characteristic, we split the interval and focus on the first one to save operation. We employ the corresponding singular functions as basis functions in the first interval to simulate its singular behavior, and take the Legendre polynomials as basis functions in the other one. Then the corresponding hp-version spectral method is proposed, the existence and uniqueness of solution to the numerical scheme are proved, the hp-version optimal convergence is derived. Numerical experiments verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We present an extension of the summation-by-parts (SBP) framework to tensorproduct spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e., satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol--Koornwinder--Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using nontensorial multidimensional SBP operators and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees. [ABSTRACT FROM AUTHOR]

We introduce a fast direct solver for variable-coefficient elliptic PDEs on surfaces based on the hierarchical Poincaré--Steklov method. The method takes as input an unstructured, high-order quadrilateral mesh of a surface and discretizes surface differential operators on each element using a high-order spectral collocation scheme. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in O (N log N) operations for a mesh with N degrees of freedom. The resulting fast direct solver may be used to accelerate high-order implicit time-stepping schemes, as the precomputed operators can be reused for fast elliptic solves on surfaces. On a standard laptop, precomputation for a 12th-order surface mesh with over 1 million degrees of freedom takes 10 seconds, while subsequent solves take only 0.25 seconds. We apply the method to a range of problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace--Beltrami problem, the Hodge decomposition of a tangential vector field, and some time-dependent nonlinear reaction-diffusion systems. [ABSTRACT FROM AUTHOR]

Based on the domain reduction method, this study employs an SEM-FEM hybrid workflow which integrates the advantages of the spectral element method (SEM) for flexible and highly efficient simulation of seismic wave propagation in a three-dimensional (3D) regional-scale geophysics model and the finite element method (FEM) for fine simulation of structural response including soil-structure interaction, and performs a physics-based simulation from initial fault rupture on an ancient wood structure. After verification of the hybrid workflow, a large-scale model of an ancient wood structure in the Beijing area, The Tower of Buddhist Incense, is established and its responses under the 1665 Tongxian earthquake and the 1730 Yiheyuan earthquake are simulated. The results from the simulated ground motion and seismic response of the wood structure under the two earthquakes demonstrate that this hybrid workflow can be employed to efficiently provide insight into the relationships between geophysical parameters and the structural response, and is of great significance toward accurate input for seismic simulation of structures under specific site and fault conditions. [ABSTRACT FROM AUTHOR]

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 L 2 - and H 1 -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. [ABSTRACT FROM AUTHOR]

In this paper, we study the two-parameter spectral element method based on weighted shifted orthogonal polynomials for solving singularly perturbed diffusion equation on an interval [0, 1] which are modeled with singular parameters. We continue our study to estimate the lower bound of the weighted orthogonal polynomial coefficient and the upper bound of a posteriori error estimates of the method through different weighted norms to minimize the computational cost. Numerical examples are implemented to study the applicability and efficiency of the technique. The obtained error bounds for the coefficient of orthogonal polynomials and the posteriori estimates fall within the bounds derived in the theoretical section. It is also observed that the two weighted norms decreases when the values of N 1 and N 2 increases for the three choices of and for different values of x and y. The quality and accuracy of the solution can be realized through figures and tables. [ABSTRACT FROM AUTHOR]

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. • Created a variational framework based on the Timoshenko-Ehrenfest beam theory. • Proposed one-dimensional electromechanical model. • The model was solved by the spectral element method. • Verified the robustness of 1D model by comparing it to 3D models and experiments. [ABSTRACT FROM AUTHOR]

Seismic exploration has caught widespread attention in metal ore exploration due to its higher resolution. However, the presence of topography and complex underground structures in metal ore exploration complicates seismic records. Therefore, it is essential to apply a numerical simulation method suitable for metal ore exploration to study the propagation law of seismic waves in shallow and ore-forming zones, providing reliable theoretical support for multi-component seismic techniques. In particular, the presence of topography generates strong-amplitude surface waves, scattered waves, and converted waves, which consistently distort seismic records and affect the imaging accuracy of the metallogenic belts. Additionally, the propagation of seismic waves is also affected by the anisotropy and viscoelasticity of the underground medium. This paper proposes an elastic wave finite-difference numerical simulation method suitable for irregularly topographical and complex medium conditions, named the comprehensive parameter correction method, which implements a free-surface boundary condition based on the concept of medium averaging. It is algorithmically simple and implies no additional computational costs. Meanwhile, the results obtained by this method are highly consistent with those of the spectral element method, demonstrating its accuracy. By presenting several numerical simulation cases and illustrating the impact of topography and medium conditions on seismic records, this paper demonstrates the necessity of considering irregularly topographical and complex medium conditions in metal ore exploration. In conclusion, the numerical simulation method we propose provides a solid theoretical foundation for the application of seismic exploration methods in metal ore exploration. [ABSTRACT FROM AUTHOR]

The local resonance phononic crystals have advantages in obtaining low frequency band gaps, while the local resonance band gap obtained by traditional structures is inconvenient for engineering applications due to its narrow width. In this manuscript, a theoretical framework for the design and analysis of the Outward Hierarchical Local Resonance Euler Beam (OHLREB) structure has been introduced, which considers the foundational constraints, non-rigid interfaces, and the damping characteristics of the coating layer, leading to a significant enhancement in the low-frequency band gap properties. To analyze the band gap properties of OHLREB, the improved transfer matrix method is derived to obtain the band structure, and the spectral element method is applied to calculate the frequency response function. Furthermore, the band gap mechanism is investigated based on modal analysis of primitive cell and the time domain analysis and energy period analysis of OHLREB structure. The results indicate that improved transfer matrix method can offer a more streamlined and transparent process for solving eigenvalues and determining attenuation intervals, which can be used for the theoretical analysis of band gap regulation. It overcomes the constraints of traditional approaches by reducing the computational burden associated with matrix inversions and multiplications, thereby enhancing the efficiency of band gap analysis. The band gap mechanism investigation demonstrates that the energy conversion period is considerably longer than the load period in band gap frequency, leading to multiple conversions of the kinetic energy of the scatterer and the elastic strain energy of the coating layer within a single energy cycle, which leads to the strong local resonance and results in the attenuation of elastic waves in the band gap range. The parameter analysis is carried out to reveal the regulation laws of band gap, the results reveal that band gap intervals and attenuation efficiency can be optimized to obtain low frequency and broadband by configurating properly the control parameters, such as the matrix reaction modulus, interface stiffness, the coating layer damping, and the scatterer combination. Based on the analysis results, the design and control directions for obtaining the wider and lower frequency band gaps are given. The conclusions obtained in this manuscript can provide theoretical support and practical design perspectives for multi-frequency vibration control field. [ABSTRACT FROM AUTHOR]

The design, simulation, and beam dynamics analysis of the integrated illumination system of the scanning electron microscope (SEM) have been reported. The dimensions and capacity of the SEM components, such as the electron gun, electron column, and lenses, were determined using the finite element method. The optimization of the illumination system was performed, and characteristics beam parameters were studied analytically. The gun consisted of a simple diode structure having; cathode, Wehnelt, and anode electrodes. A tungsten cathode of diameter 150 μm was used to emit the electrons thermionically. The electrons beam was generated at 2700 K with acceleration potential of 15 kV. Pair of electromagnetic lenses producing field strengths, 221 mT and 253 mT, respectively, focused the beam at 56.5 mm from cathode, with beam diameter of 9 μm in the post anode region. The calculated value of beam brightness was 2.97 × 106 A/(mm-rad)2. The final beam spot thus obtained was stable and perfectly symmetric along the two transverse directions. Gun assembly and lenses are thermally stable at the operational temperature. The illumination system geometry offers full-beam collimation control with reduced column length, making it suitable for electron diffraction studies, where focusing of the beam onto a sample is desired. [ABSTRACT FROM AUTHOR]

Background: Singular perturbation troubles in everyday differential equations (ODEs) involve a small parameter ϵ that causes answers to show off behaviors on multiple scales, main to massive challenges in both theoretical and numerical evaluation. Objective: This paper at targets to comprehensively look at the theoretical and numerical techniques for reading and solving singular perturbation troubles in ODEs. Focus is located on know-how the behaviors precipitated through the small parameter and developing strong numerical techniques to accurately seize these behaviors. Methods: Theoretical approaches employed include matched asymptotic expansions and multiple scale analysis. These methods decompose the solution domain into regions with different scales, constructing and matching approximate solutions to ensure smooth transitions. Numerical techniques such as finite difference methods, finite element methods, and spectral methods are utilized, with particular emphasis on adaptive mesh refinement to handle boundary layers effectively Results: Theoretical evaluation demonstrates the effectiveness of matched asymptotic expansions and multiple scale analysis in presenting correct approximations and easy transitions among one of a kind solution regions. Numerical techniques showed high accuracy and performance, particularly whilst blended with adaptive techniques. Finite distinction strategies with non-uniform grids and adaptive mesh refinement, finite detail techniques with adaptive mesh strategies, and spectral strategies with special handling of boundary layers all proved successful. Stability and convergence analyses showed the reliability of those techniques. Conclusions: This comprehensive evaluation highlights the strengths and applicability of each theoretical and numerical strategies in tackling singular perturbation issues in ODEs. The mixed use of these techniques lets in for correct and green answers, presenting treasured equipment for applications in diverse clinical and engineering fields. [ABSTRACT FROM AUTHOR]

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In this paper we investigate a model for quantum gravity on finite noncommutative spaces using the theory of blobbed topological recursion. The model is based on a particular class of random finite real spectral triples (A,H,D,γ,J), called random matrix geometries of type (1,0), with a fixed fermion space (A,H,γ,J), and a distribution of the form e-S(D)dD over the moduli space of Dirac operators. The action functional S(D) is considered to be a sum of terms of the form ∏si=1Tr(Dni) for arbitrary s⩾1. The Schwinger-Dyson equations satisfied by the connected correlators Wn of the corresponding multi-trace formal 1-Hermitian matrix model are derived by a differential geometric approach. It is shown that the coefficients Wg,n of the large N expansion of Wn's enumerate discrete surfaces, called stuffed maps, whose building blocks are of particular topologies. The spectral curve (Σ,ω0,1,ω0,2) of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential ω0,2 in terms of the formal parameters of the model. [ABSTRACT FROM AUTHOR]

An effective and accurate solver for the direct-current-resistivity forward-modeling problem has become a cutting-edge research topic. However, computational limitations arise due to the substantial amount of data involved, hindering the widespread use of three-dimensional forward modeling, which is otherwise considered the most effective approach for identifying geo-electrical anomalies. An efficient compromise, or potentially an alternative, is found in two-and-a-half-dimensional (2.5D) modeling, which employs a three-dimensional current source within a two-dimensional subsurface medium. Consequently, a Legendre spectral-element algorithm is developed specifically for 2.5D direct-current-resistivity forward modeling, taking into account the presence of topography. This numerical algorithm can combine the complex geometric flexibility of the finite-element method with the high precision of the spectral method. To solve the wavenumber-domain electrical potential variational problem, which is converted into the two-dimensional Helmholtz equation with mixed boundary conditions, the Gauss–Lobatto–Legendre (GLL) quadrature is employed in all discrete quadrilateral spectral elements, ensuring identical Legendre polynomial interpolation and quadrature points. The Legendre spectral-element method is applied to solve a two-dimensional Helmholtz equation and a resistivity half-space model. Numerical experiments demonstrate that the proposed approach yields highly accurate numerical results, even with a coarse mesh. Additionally, the Legendre spectral-element algorithm is employed to simulate the apparent resistivity distortions caused by surface topographical variations in the direct-current resistivity Wenner-alpha array. These numerical results affirm the substantial impact of topographical variations on the apparent resistivity data obtained in the field. Consequently, when interpreting field data, it is crucial to consider topographic effects to the extent they can be simulated. Moreover, our numerical method can be extended and implemented for a more accurate computation of three-dimensional direct-current-resistivity forward modeling. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This comprehensive overview presents our continued efforts in high-order finite difference method (FDM) development for adaptive numerical dissipation control in the long-time integration of direct numerical simulation (DNS), large eddy simulation (LES), and implicit LES (ILES) computations of compressible turbulence for gas dynamics and MHD. The focus is on turbulence with shock wave numerical simulations using the adaptive blending of high-order structure-preserving non-dissipative methods (classical central, Padé (compact), and dispersion relation-preserving (DRP)) with high-order shock-capturing methods in such a way that high-order shock-capturing methods are active only in the vicinity of shock/shear waves, and high-gradient and spurious high-frequency oscillation regions guided via flow sensors. Any efficient and high-resolution high-order shock-capturing methods are good candidates for the blending of methods procedure. Typically, the adaptive blending of more than one method falls under two camps: hybrid methods and nonlinear filter methods. They are applicable to unstructured finite volume, finite element, discontinuous Galerkin, and spectral element methods. This work represents the culmination of over 20 years of high-order FDM developments and hands-on experience by the authors and collaborators in adaptive numerical dissipation control using the "high order nonlinear filter approach". Extensions of these FDM versions to curvilinear nonuniform, freestream-preserving moving grids and time-varying deforming grids were also developed. By examining the construction of these two approaches using the high-order multistage type of temporal discretization, the nonlinear filter approach is made more efficient and less CPU-intensive while obtaining similar accuracy. A representative variety of test cases that compare the various blending of high-order methods with standalone standard methods is illustrated. Due to the fact that our nonlinear filter methods are not well known in compressible turbulence with shock waves, the intent of this comprehensive overview is for general audiences who are not familiar with our nonlinear filter methods. For readers interested in the implementation of our methods into their computer code, it is hoped that the long overview will be helpful. [ABSTRACT FROM AUTHOR]

This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral element method for spatial discretization. We prove that optimal adaptivity can be achieved by combining adaptive time steps with adaptive spectral mesh. We introduce two sets of error indicators for time and space, respectively, and derive optimal estimates. [ABSTRACT FROM AUTHOR]

To simulate the ground motions of 2D arbitrary sites subjected to plane SV waves, we put forward a numerical method combining the multi-transmitting formula (MTF) and spectral element method (SEM). The investigation results demonstrate a high simulation accuracy of the presented method. Most importantly, the MTF exhibits a notably superior stability performance in the SEM than in traditional finite-element method (FEM). Without any additional treatment, high-frequency instability does not occur when controlling the Courant number value C and the ratio between the MTF parameter S and C; the upper limit value of this ratio is significantly smaller than that of the SH-wave problem. In addition, a preliminary stability criterion for the MTF in SEM is given based on the homogeneous half-space model. For low-frequency stability, the occurrence time of instability is considerably lagged when no interventions are adopted. Moreover, the low-frequency instability can be readily eliminated by using smaller modification parameter in the SEM compared with the FEM. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell's transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the derivatives of eigenvalues. [ABSTRACT FROM AUTHOR]

This article aims to provide a comprehensive review of the latest advancements in numerical methods and practical implementations in the field of fractional stochastic partial differential equations (FSPDEs). This type of equation integrates fractional calculus, stochastic processes, and differential equations to model complex dynamical systems characterized by memory and randomness. It introduces the foundational concepts and definitions essential for understanding FSPDEs, followed by a comprehensive review of the diverse numerical methods and analytical techniques developed to tackle these equations. Then, this article highlights the significant expansion in numerical methods, such as spectral and finite element methods, aimed at solving FSPDEs, underscoring their potential for innovative applications across various disciplines. [ABSTRACT FROM AUTHOR]

This article proposes a novel method for calculating radar cross-sections (RCSs) that combines the spectral element method and the integral method, allowing for RCS calculations at any position in a free space or a half-space. This approach replaces the field source with an incident field using the scattered field equation of the spectral element method, enabling the arbitrary placement of the field source without being limited by the computational domain. By applying the superposition theorem and the volume equivalence principle, the scattered field of the objects at any position is obtained through integral equations, eliminating limitations on the computation points imposed by the computational domain. Based on Green's function's important role throughout the calculation process and its symmetry properties, the RCS calculation of symmetric models will be more advantageous. Finally, several examples, including symmetry models, are provided to validate both the feasibility and accuracy of this proposed method. [ABSTRACT FROM AUTHOR]

A Mw 6.6 earthquake struck Luding county, in Sichuan province of China on September 5, 2022, causing severe damage of nearby infrastructures. The earthquake indicates an awakening of southeast branch of Xianshuihe fault since the last 1973 Mw 7.6 Luhuo earthquake in Sichuan, thus attracting attention from the public. A physics-based method, combining spectral element method (SEM) and finite element method (FEM), was presented in this paper to investigate the correlation between near-fault motions and response of a road tunnel, which has a closest distance of about 6.0 km to the fault. This method allows to incorporate the rupture details, geological irregularity, 3D velocity structure, as well as the underground tunnel. The ground motions simulated by SEM were validated by comparison to specific station records and empirical ground motion model (GMM), which were subsequently input into the tunnel to obtain its axial deformation and sectional inner forces. Parametric study on tunnel alignment was conducted, and wavelet transformation was used to extract dynamic pulses from the ground motions. The results show that the tunnel response is remarkably affected by dynamic pulse directivity, and the axial forces could be dominated by permanent fault displacement for tunnel of the strike-normal alignment. [ABSTRACT FROM AUTHOR]