Time Domain Spectral Element Research Articles

Efficient layerwise multiphysics spectral element model for delaminated composite strips with PWAS transducers

Efficient structural element-based models are essential for fast simulations of wave propagation in composite structures for model-based and physics-informed data-driven structural health monitoring. This article introduces the first efficient multiphysics time-domain spectral structural element for wave propagation analysis of beam and panel-type composite structures with piezoelectric transducers (patch or full layers) containing delaminations. A general framework is presented to model multiple delaminations and transducer patches located arbitrarily. The intact host and patch transducer-bonded laminates and sub-laminates between delaminations are modelled separately using an electromechanically coupled efficient layerwise zigzag theory (ZIGT) for kinematics and a piecewise quadratic variation for the electric potential across piezoelectric layers. The high-order spectral element (SE) features a virtual electric node to model equipotential surfaces of piezoelectric transducers apart from the usual physical nodes having mechanical and internal electric degrees of freedom. A hybrid point-least squares continuity approach is employed to maintain continuity at the intersections of delaminated sub-laminates or the patch-bonded laminate with the host laminate. The model’s performance in capturing electroelastic waves’ interaction with delamination is examined with reference to the conventional finite element (FE) solution based on the ZIGT, continuum-based FE solutions, and the SE solution based on a non-layerwise version of the laminate theory. Finally, the model is used to examine the impact of interfacial location and size of delaminations on wave propagation behaviour.

Wave Propagation for Structural Health Monitoring (WaveProSHM): open software with GUI

Guided wave–based SHM methods have been explored extensively in recent decades. The wave propagation modelling including piezoelectric transducers and various damage types helps understand intricacies of guided wave behavior and for designing a robust SHM system. However, modelling aspects are still challenging. Moreover, commercially available software for the modelling of guided waves often lacks appropriate high-order elements and code optimization for fast GPU computations. This research aimed to develop and disseminate to the SHM community the open software based on highly efficient vectorised code of the time domain spectral element method which can be run on GPU. The developed code is written in MATLAB with Graphical User Interface (GUI) and communicates with GMSH – open source finite element mesh generator. It can model signals of propagating guided waves in structures which are excited and registered by an array of piezoelectric transducers. The capabilities of the software are shown in an example of a large-scale problem of interacting guided waves with delamination.

Efficient zigzag theory-based spectral element model for guided waves in composite structures containing delaminations

Lamb wave-based structural health monitoring offers excellent potential for real-time delamination detection in laminated structures. It necessitates fast and accurate numerical simulation of guided wave interaction with delaminations. Towards this objective, we present an efficient layerwise zigzag theory (ZIGT) based time-domain spectral element for wave propagation analysis of composite beam- and panel-type structures (strips) containing delamination at arbitrary locations. The ZIGT delivers accuracy like layerwise theories by allowing slope discontinuities in the in-plane displacement field at layer interfaces while maintaining computational efficiency like equivalent single-layer (ESL) theories, making it ideal for such analyses. The high-order elemental nodes at Lobatto points have only four displacement variables, irrespective of the number of layers in the laminate. Delamination is modelled using the region approach, adapting the recently proposed hybrid point-least squares continuity method to satisfy the nonlinear displacement field continuity at the delamination fronts. The model’s efficacy is examined vis-à-vis elasticity-based elements, ESL theory-based elements, and the ZIGT-based standard finite element for free vibration and guided wave propagation responses of composite strips featuring multiple delaminations. The present model shows superior overall efficiency, accuracy, and convergence. An application of the model is also explored for identifying delamination locations from Lamb wave velocity fields.

A high-order perfectly matched layer scheme for second-order spectral-element time-domain elastic wave modelling

In this study, we develop a new high-order perfectly matched layer (PML) formulation for the 3D spectral-element time-domain (SETD) method to solve the second-order elastic wave equations in terms of displacements. The high-order PML consists of multiple well-known frequency-dependent complex stretching functions, leading to the complicated temporal convolution which requires large computational resources. To avoid the evaluation of convolution, we deal with the high-order stretching functions by introducing the recursive auxiliary first-order differential equations which can be easily solved in time domain. In addition, an equivalent concept of PML is adopted for absorbing waves, preserving the original form of wave equations in physical domain. Finally, we obtain a high-order, non-convolutional and unchanged-form PML formulation, which can be easily incorporated into the SETD codes designed for the physical domain, to make them applicable for the second-order wave equations in the unbounded media. Numerical experiments demonstrate the accuracy and stability of the high-order PML through the comparison with the reference solution. It is found that the high-order PML provides a much better absorption performance than the first-order PML, particularly for the grazing-incidence waves and surface waves in the large-scale domain. We also give a fluid-filled borehole model to illustrate that the high-order PML works well for absorbing the complex guided waves in the presence of fluid-solid interaction.

A hybrid implicit-explicit discontinuous Galerkin spectral element time domain (DG-SETD) method for computational elastodynamics

SUMMARY A hybrid implicit-explicit (IMEX) discontinuous Galerkin spectral element time domain (DG-SETD) algorithm is proposed to simulate 3D elastic wave propagation in inhomogeneous media. In this method, the original problem can be divided into a number of well designed subdomains, and the mesh generation of different subdomains is completely independent, thus allowing flexible spatial discretization of complex geometry. The neighboring subdomains are connected by a Riemann transmission condition (RTC), and spectral elements with different interpolation orders can be used in different subdomains to maximize the computational efficiency of multiscale problems to facilitate parallel computing for different subdomains. In particular, the explicit or implicit time iteration scheme can be appropriately selected for a subdomain according to the size of its mesh elements to increase the global time step increment, thus giving higher computational efficiency: For subdomains with coarse meshes, the explicit time integration scheme is used and the time step increment is limited by the Courant−Friedrichs−Lewy (CFL) stability condition; for subdomain with fine structures and thus fine meshes, an implicit time integration scheme is used so that a large time step increment can be used without affecting the stability. In addition, a jump condition of displacement and velocity is introduced to accurately simulating fractures and faults, including lossless and viscous fractures with plane, curve or cross structures. This avoids the volume modeling of the extremely thin fracture structures, and effectively reduces the number of degrees of freedoms (DoFs) in the discretized system without the loss of accuracy. The accuracy, robustness and efficiency of the DG-SETD algorithm are demonstrated by quantitative comparisons of the waveforms with the commercial finite element software COMSOL.

A spectral element time domain (SETD) method based on poroelastic linear slip conditions for 3D wave propagation in fractured porous media

In this study, we present a spectral element time domain (SETD) method for the simulation of wave propagation in fractured porous media. To avoid volumetric meshing of the extremely thin-layer fractures, we incorporate the fractures into spectral element model by treating them as geometrical interfaces, where the interaction between solid deformation and pore fluid flow is described by the poroelastic linear-slip conditions across the fractures. For different kinds of fractures (i.e. permeable, open, and impermeable), the explicit time schemes are presented for solving the stress tensor and fluid pressure over the fracture surfaces, which are subsequently implemented into the SETD framework to involve both elastic scattering and wave-induced fluid flow (WIFF) effects. Through various numerical examples, the proposed algorithm is shown to be able to accurately and effectively simulate and characterize the wave propagation in 3D porous media containing discrete and intersecting fractures with different permeabilities.

A High-Efficiency Spectral Element Method Based on CFS-PML for GPR Numerical Simulation and Reverse Time Migration

Improving the accuracy and efficiency of the numerical simulation of ground penetrating radar (GPR) becomes a pressing need with the rapidly increased amount of inversion data and the growing demand for migration imaging quality. In this article, we present a numerical spectral element time-domain (SETD) simulation procedure for GPR forward modeling and further apply it to the reverse time migration (RTM) with complex geoelectric models. This approach takes into account the flexibility of the finite element methods and the high precision of the spectral methods. Meanwhile, in this procedure, the complex frequency shifted perfectly matched layer (CFS-PML) is loaded to effectively suppress the echo at the truncated boundary, and the per-element GPU parallel framework used can achieve up to 5.7788 times the efficiency compared with the CPU calculation. The experiments on SETD spatial convergence and CFS-PML optimal parameter selection showed that, under the same degree of freedom, the SETD offered substantially better accuracy compared with the traditional FETD. The experiments on RTM of different profiles with different orders of SETD via a complex geoelectric model verify the universality of the algorithm. The results indicate that the RTM imaging effect has been significantly improved with the increase of SETD order. It fully proves the great potential of efficient and high-precision SETD simulation algorithm in the RTM imaging direction and shows certain guiding significance for underground target structure exploration.

Nearly perfectly matched layer implementation for time domain spectral element modelling of wave propagation in 3D heterogeneous and anisotropic porous media

Numerical modelling of wave propagation in the heterogeneous and anisotropic porous media is important for seismic exploration. When modelling time-domain poroelastic waves in an unbounded region, the implementation of perfectly matched layer (PML) is a common technique to absorb the outgoing waves. The PML in terms of first-order equations has been well proposed. However, the first-order PML needs fundamental and nontrivial reconstructions when applied to the second-order systems in displacements, such as finite element method (FEM) and spectral element method (SEM). It is well known SEM is an efficient algorithm for simulating seismic waves for its high accuracy and geometrical flexibility. However, at present, few literature studies the implementation of PML into time-domain SEM for simulating waves in the 3D unbounded heterogeneous porous media. In this work, combining with auxiliary ordinary differential equations (ODEs), we first systematically extend nearly PML (NPML) into time-domain SEM in terms of the second-order poroelastic system. The scheme preserves the original form of wave equations, leading to a simple and flexible incorporation into the existing algorithm. It avoids the temporary convolution through solving a set of first-order ODEs in time domain, resulting in high computational efficiency. Furthermore, to solve the coupling poroelastic/acoustic/elastic waves, we employ domain decomposition technique to handle the different regions independently, and apply the corresponding boundary conditions at the region interfaces to guarantee the coupling. Numerical examples demonstrate the accuracy and stability of the proposed method by comparing the numerical results with the analytical solutions and other independent numerical solutions. Finally, we apply the algorithm to solve wave propagation in a large-scale model involved a fluid-filled borehole embedded in heterogeneous porous formations.

An Electromagnetic-Plasma Fluid Model Simulation of Waveguide Plasma Limiter Filled With Different Easily Ionized Inert Gas

To protect electronic systems against high-power microwave (HPM), a plasma waveguide limiter is presented. In response to the above phenomenon, a self-consistent 3-D multiphysics electromagnetic-plasma fluid model coupling full-wave Maxwell’s equations with plasma fluid equations is established to describe the operating mechanism of a 3-D simplified sandwich structure plasma limiter filled with four easily ionized inert gases. The plasma limiter system is analyzed with the spectral-element time-domain (SETD) method. The choice of filling easily ionizable inert gas, influence of the number of layers, and pressure on the plasma limiter are discussed. Numerical results demonstrate that the gas with low critical breakdown field strength is more suitable for protecting HPMs. In the case of 20 torr, the order of microwave breakdown is xenon, argon, neon, and helium, and the two-layer plasma limiter (plasma–slab–plasma) has better protective characteristics than a one-layer plasma limiter (slab–plasma–slab) with the same length of gas chamber. Our research can provide theoretical guidance for designers and give the complete physical physics process.

Efficient time-domain spectral element with zigzag kinematics for multilayered strips

Efficient structural elements are widely sought for fast computation of wave propagation response of anisotropic multilayered structures for their design, non-destructive testing, and structural health monitoring purposes. Although zigzag theories are known to combine excellent accuracy with high computational efficiency, no spectral element has been developed so far based on these theories, which would enable fast and accurate simulation of guided waves in such structures. In this article, we develop an efficient time-domain spectral strip element for analyzing ultrasonic wave propagation in multi-material laminated beams and panels while accounting for the zigzag profile of the in-plane displacement across the thickness. The theory superimposes a layerwise linear (zigzag) variation of the in-plane displacement of a layerwise theory onto the global third-order variation of an equivalent single layer (ESL) third-order theory while retaining the computational advantage of the latter. The high-order element has an arbitrary number of unequally spaced nodes with only four kinematic variables at each node, regardless of the number of laminas. The recently proposed generalized Hermite-type C1-continuous Lobatto shape functions interpolate the deflection, while the axial displacement and shear rotation are interpolated using the C0-continuous Lobatto shape functions. A comprehensive numerical study establishes high efficiency, excellent accuracy, and fast convergence of the new element in analyzing free vibration and guided wave propagation in single-material composite and multi-material fiber-metal laminate structures. Its combined performance in terms of these parameters is much superior to its standard FE counterpart, the ESL theory-based elements, and the continuum elements in the considered frequency range.

Transient simulation of dielectric breakdown during high-power microwave propagation based on the SETD method

This article is mainly devoted to the research of dielectric breakdown phenomenon during high power microwave propagation. Under the force of the high-intensity fields, bound charges break free and a part of an insulator becomes electrically conductive, which may lead to electrical breakdown and thermal breakdown in high power microwave (HPM) devices or on the output window. A three-dimensional (3D) multi-physics electromagnetic-thermal model is developed and solved by spectral-element time-domain (SETD) method. The nonlinear conductivity caused by HPM is introduced into Maxwell's equations, which makes its nonlinear. The heat flux from transient power loss is utilized to construct the heat source for solving the thermal conduction equation. In the SETD process, the Galerkin's method and central difference scheme are employed for the space discretization and temporal discretization, respectively. The numerical results show that physical mechanism of electrical breakdown, thermal effect and nonlinear effect can be captured accurately during high power microwave propagation. Once the strength of the electric field exceeds the critical value experienced by the dielectric, the current increases rapidly, along with the temperature change is very severe. The validity and accuracy of the proposed method are demonstrated, and our research is beneficial for the HPM devices protection.

Seismic analysis of a I-sectioned steel frame based on an improved time-domain spectral element method

An improved time-domain spectral element method (ITSEM) is developed for seismic analysis of I-sectioned steel structures in this paper. The developed method considers the effect of section type on spectral stiffness using shear correction coefficient. In the calculation process of the time-domain spectral stiffness and mass matrices, the invariant integral operations of the element matrices are extracted and saved in advance. Thus, the global mass and stiffness matrices of the structures can be obtained only by simple addition, subtraction, multiplication and division operations, which greatly improves computational efficiency. Next, the natural frequencies of a I-sectioned beam are computed and compared with the finite element method (FEM) and analytical results to validate the ITSEM. Then, the seismic responses of a ten-story steel frame using ITSEM are obtained and compared with those by FEM, which proves the effectiveness, accuracy and high efficiency of ITSEM in elastic seismic analysis.

Material property identification in composite structures using time domain spectral elements

Lightweight, highly optimized, aerospace structures like spacecraft, aircraft, and launch vehicles, are made up of composite structures. These composite structures as well as the additive manufacturing techniques used in recent times require material property identification. Also, one of the main concerns in composite structures is the degradation of the material properties due to aging of structure and moisture absorption. Identification of material properties of the degraded structure poses an important problem to determine the margins available for the safe performance of the structure. Here we propose a new method to identify the material property (like elastic properties and density estimation) nondestructively, in any general elastic structure with particular emphasis on composite structures. The proposed method requires few measured responses and a good numerical model to determine the material properties. The new procedure requires a simple input force on the structure at some known locations. This generates and propagates high-frequency stress waves in the structure. Very few measured responses are used, to compute the material properties. Time Domain Spectral Finite Element (TSFEM) is used as the numerical model since it has high convergence rates for high-frequency content transient loads. A new anisotropic Timoshenko composite time-domain spectral beam element is formulated for this purpose, which will be used to perform the material property identification for both metallic as well as composite structures. Measured responses are used in this numerical model which is coupled with, the nonlinear least square technique to get the material properties. The proposed approach is demonstrated using both numerically and experimentally obtained responses. These examples show the efficiency of the developed algorithm based on TSFEM in estimating the material property of a structure.

An efficient thin layer equivalent technique of SETD method for thermo-mechanical multi-physics analysis of electronic devices

A coupled thermo-mechanical simulation technique is proposed for reliability analyses of electronic devices based on the spectral element time domain (SETD) method. The transient spatial distribution of temperature, deformation and stress in electronic devices with different spatial scales can be obtained by using the implicit time integration method in the proposed technique. Through the thermo-mechanical equations, thin layers of electronic devices in the coupled fields can be made equivalent to surface boundaries to avoid volumetric meshing, thus reducing the number of degrees of freedom in computation and greatly improving the numerical efficiency of reliability analyses. Finally, through the thermo-mechanical analyses of different electronic devices such as a coaxial through-silicon via (TSV), and an integrated circuit (IC) package, various stress mechanisms related to the failure of electronic devices are successfully predicted to further demonstrate the capabilities of the technique to analyze the reliability of large-scale electronic devices.

Spectral element modeling of ultrasonic guided wave propagation in optical fibers

Recent advancements in fiber optic methods have enabled their use for guided wave sensing. It opens up new possibilities for Structural Health Monitoring. The aim of this paper is to provide insight for the physics related to guided wave propagation and coupling between the optical fiber and solid structure. For this purpose, a new approach for non-matching interface based on Lagrange multipliers and the time domain spectral element method was developed. A parallelized code has been implemented in order to simulate the guided wave propagation in the structure, its coupling into the optical fiber and the propagation in the fiber in a computationally efficient way. The paper presents four studies showing the efficacy of the modeling approach. The paper first shows the improvement in the computation speed through the use of parallelization and a more efficient implementation. Then the results of the simulation of wave propagation in the fiber are compared with results from previous simulation studies using commercially available software. The third study shows that the spectral element method is able to capture the directional sensitivity of optical fiber based sensors. Lastly, the simulation is used for detection of simulated damage using the spectral element method based simulation. The results indicate that indeed the spectral element implementation is able to recreate the wave coupling phenomena, capture the physics of the system including directional sensitivity and reflections from damage.

Numerical Simulation of Streamer Discharge Modeled by Drift-Diffusion Equations Based on SETD Method

Dynamic characteristics of streamer discharge in a parallel-plane electrode system are described within the framework of a drift-diffusion model coupled with Poisson’s equation at atmospheric pressure. The spectral-element time-domain (SETD) method is applied to the solution of streamer discharge problems, and the flux-corrected-transport (FCT) algorithm is also applied to suppress possible numerical divergence or numerical oscillation. The performance of the proposed method is examined in a number of numerical tests. Then the tempo-spatial evolutions of the streamer are obtained in detail under the different situations. Numerical results show that after the initial stage of adjustment, the streamers propagate in a steady-state mode, where the characteristics of the steady state are decided by background ionization frequency which is determined by pressure and background electric field together. And space charge effect is a dominant factor for the distortion of spatial electric field, making the discharge channel expand toward both electrodes faster. These works presented here are instructive for high-voltage insulating device protection and plasma application.

Influence of static strong magnetic field on antenna radiation in hypersonic vehicle

To enhance the radiation performance of the Beidou antenna in the near-space hypersonic vehicle, the static strong magnetic field is used to weaken the electron density in plasma surrounding the antenna. In order to demonstrate the effect of this program, a time-domain multi-physical method is proposed. In the proposed method, what is first analyzed is the reduction of electron concentration in plasma sheath by static strong magnetic field with the spectral element time domain (SETD) method, which has spectral accuracy. Then, the electron density after mitigation is extracted to replace the original electron concentration around the antenna. Hence, the distribution of the manipulated plasma sheath can be obtained. Finally, the radiation characteristics of BeiDou antenna installed in the vehicle are analyzed by the conformal finite difference time domain (CFDTD) method. The simulation results exhibit radiation patterns under different conditions. With the plasma sheath, the radiated electromagnetic waves are greatly attenuated, which will significantly affect the transmission of communication signals. Importantly, the radiation patterns are effectively improved with the external static magnetic field, confirming that it provides an effective tool to mitigate the influence of plasma sheath on the radiation performance of antenna in hypersonic vehicle.

Model-Assisted Guided-Wave-Based Approach for Disbond Detection and Size Estimation in Honeycomb Sandwich Composites.

One of the axioms of structural health monitoring states that the severity of damage assessment can only be done in a learning mode under the supervision of an expert. Therefore, a numerical analysis was conducted to gain knowledge regarding the influence of the damage size on the propagation of elastic waves in a honeycomb sandwich composite panel. Core-skin debonding was considered as damage. For this purpose, a panel was modelled taking into account the real geometry of the honeycomb core using the time-domain spectral element method and two-dimensional elements. The presented model was compared with the homogenized model of the honeycomb core and validated in the experimental investigation. The result of the parametric study is a function of the influence of damage on the amplitude and energy of propagating waves.

Transient Electromagnetic–Thermal Cosimulation for Micrometer-Level Components

In this article, an electromagnetic–thermal coupling algorithm based on the discontinuous Galerkin time-domain (DGTD) and spectral-element time-domain (SETD) methods is proposed, which can adopt two independent grids for thermal simulation and electrical simulation, respectively. It employs time-domain solvers for both the electrical and thermal fields. Compared with the conventional cosimulation scheme (e.g., time-domain solvers for the thermal field and frequency-domain solver for the electrical field), it can provide better simulation precision for problems characterized with small size or drastic field changes. By comparing the results of electromagnetic–thermal coupling simulation with the results of electromagnetic simulation, the effect of temperature change on electromagnetic field propagation is analyzed.

Numerical simulation of multi-carrier microwave breakdown in air-SF6 mixtures

In this paper, the multi-carrier microwave breakdown in air-SF6 mixtures is analyzed with the spectral-element time-domain method. In this process, a three-dimensional multi-physics model coupling Maxwell's equations with electron fluid equations is established. The tail-erosion and frequency shift phenomenon caused by multi-carrier microwave breakdown in different proportions of air-SF6 mixtures can be observed. Numerical results demonstrate that multi-carriers will make the wireless communication system sensitive to microwave breakdown. On the other hand, the different proportions of air-SF6 mixtures can obviously improve the breakdown threshold to inhibit the multi-carrier microwave breakdown, which is favorable to the propagation of multi-carrier microwaves in air. Meanwhile, increasing the pressure can suppress the frequency shift phenomenon, and the proportion of different SF6 in the mixed gas has little effect on the frequency shift of the transmitted wave. Our research provides theoretical guidance for comprehensively exploring the characteristics and physical mechanism of multi-carrier microwave breakdown in air-SF6 mixtures and is helpful for protection of microwave devices and plasma application.