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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]

The inverse problem for radiative transfer is important in many applications, such as optical tomography and remote sensing. Major challenges include large memory requirements and computational expense, which arise from high-dimensionality and the need for iterations in solving the inverse problem. Here, to alleviate these issues, we propose adaptive-mesh inversion: a goal-oriented hp -adaptive mesh refinement method for solving inverse radiative transfer problems. One novel aspect here is that the two optimizations (one for inversion, and one for mesh adaptivity) are treated simultaneously and blended together. By exploiting the connection between duality-based mesh adaptivity and adjoint-based inversion techniques, we propose a goal-oriented error estimator, which is cheap to compute, and can efficiently guide the mesh-refinement to numerically solve the inverse problem. We use discontinuous Galerkin spectral element methods to discretize the forward and the adjoint problems. Then, based on the goal-oriented error estimator, we propose an hp -adaptive algorithm to refine the meshes. Numerical experiments are presented at the end and show convergence speed-up and reduced memory occupation by the goal-oriented mesh adaptive method. [ABSTRACT FROM AUTHOR]

In this article we present the nonconforming spectral element method (NSEM) for linear second-order ordinary differential equations with boundary conditions having analytic solutions, and the interface problem. A fully discrete spectral element method for one-dimensional parabolic problems with smooth solutions and parabolic interface problem also have been considered. The stability and error estimates of NSEM for a two-point boundary value problem with analytic solutions are derived. The numerical formulation is described and it is essentially a least-squares formulation and spectral elements are nonconforming. The normal equations which arise in the minimization of the functional are solved using preconditioned conjugate gradient method without storing the stiffness matrix and load vector. The method is exponentially accurate. In the fully discrete formulation for the parabolic problems, Crank–Nicolson scheme is used in time variable and higher-order spectral elements are used in spatial variable. This method is second-order accurate in time and exponential accurate in spatial variable. Various numerical examples are presented to show the accuracy of the method. Finally, we review the nonconforming spectral element method for various problems in higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the approximation of abstract thermoelastic models. It is by now well known that approximated systems are not in general uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal in this paper is to study the uniform exponential/polynomial stability of a sequence of a system of weakly coupled thermoelastic models. We prove that when 0 ≤ β < 1 2 , the total energy of solutions is not uniformly exponentially stable, but it decays uniformly polynomially to zero. Finally, the results are applied to space semi-discretizations of thermoelastic beam equation in a bounded interval with homogeneous Dirichlet boundary conditions. We consider finite element, spectral element and finite difference semi-discretizations. Finally, we illustrate the mathematical results with several numerical experiments. [ABSTRACT FROM AUTHOR]

This paper proposes a new shell element formulation using the scaled boundary finite element (SBFE) method. A shell element is treated as a three-dimensional continuum. Its bottom surface is approximated with a quadrilateral spectral element and the shell geometry is represented through normal scaling of the bottom surface. Neumann expansion is applied to approximate the inversions of the matrix polynomials of the thickness coordinate ξ, including the Jacobian matrix and the coefficient of the second-order term in the SBFE equation. This permits the solution along the thickness to be expressed as a matrix exponential function whose exponent is a high-order matrix polynomial of ξ. After introducing the boundary conditions on the top and bottom surfaces and evaluating the resulting matrix exponential via Padé expansion, we derive the element stiffness and mass matrices. Poisson thickness locking is avoided fundamentally. Numerical examples demonstrate the applicability and efficiency of the formulation. [ABSTRACT FROM AUTHOR]

We consider a model coupling the Darcy equations in a porous medium with the Navier-Stokes equations in the cracks, for which the coupling is provided by the pressure's continuity on the interface. We discretize the coupled problem by the spectral element method combined with a nonoverlapping domain decomposition method. We prove the existence of solution for the discrete problem and establish an error estimation. We conclude with some numerical tests confirming the results of our analysis. [ABSTRACT FROM AUTHOR]

The non-destructive testing (NDT) plays a crucial role in geotechnical engineering and geophysical applications, especially in the design of earthquake-resistant foundations, geotechnical field investigation, and material characterization and detection of underground anomaly. Currently, the existing signal interpretation methods in NDT measurements still predominantly rely on empirical relations or subjective judgements. In this paper, we present the GeoNDT software, which is developed to provide an advanced physics-based signal interpretation method for NDT characterization of multiphase geomaterials. GeoNDT is able to model the propagation of stress waves and dispersion relations in dry (elastodynamic), saturated (two-phase poroelastodynamic), and three-phase frozen (multiphase poroelastodynamic) geomaterials using the meshless spectral element method. GeoNDT is flexible, general-purpose, and can be used seamlessly for advanced signal interpretation in geophysical laboratory testing including the bender element and ultrasonic pulse velocity tests, characterization of complex multiphase geomaterials, and in situ shallow seismic geophysics including the falling weight deflectometer and multichannel analysis of surface waves tests. The advanced physics-based signal interpretation feature of GeoNDT allows the quantitative characterization of geophysical and geomechanical properties of geomaterials and multilayered geosystems independently without making any simplified assumptions as common in the current practice. [ABSTRACT FROM AUTHOR]

The topological optimization technique is an effective way to provide the optimal design for the polymeric stents. Conventional topology optimization procedure for polymeric stent was all based on classical theory of elasticity and usually only used one layer of shell element in structural analysis. It will lead to that the simulated stent structural stiffness deviates from the real case. Thus, a novel 3D topology optimization procedure based on Cosserat theory is proposed to maximize the radial stiffness of polymeric stent by taking advantage of the size effect of polymer materials. The stent structural analysis is implemented by a proposed high-order Cosserat spectral element method, ensuring that the analysis and optimization results can achieve high accuracy with only one layer of element for the polymeric stent. Cosserat characteristic length of polymeric material is obtained by the three-point bending test, and a parametric analysis is conducted to identify the effect of Cosserat characteristic length, volume fraction, and unit cell size on the optimized stent structure, and the optimal results based on the Cosserat theory are compared with the conventional ones. It is found that the size effect has a significant impact on the topology optimization results of polymeric stent. Compared with conventional commercial stent, the optimal polymeric stent obtained by the Cosserat theory has lower volume fraction and higher radial stiffness. Our results provide an effective way to further improve the mechanical performance of the polymeric stent. [ABSTRACT FROM AUTHOR]

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. • First efficient multiphysics spectral structural element for smart composite strips with delaminations. • Enforces nonlinear displacement field continuity at delamination and piezo patch intersections. • Employs virtual electric node to satisfy equipotential conditions on piezo surfaces. • Yields faster computation and better accuracy compared to the standard FE counterpart. • It will benefit guided wave-based structural health monitoring of delaminated structures. [ABSTRACT FROM AUTHOR]

A nonconforming spectral element method for the Stokes problem on nonsmooth domains has been proposed in Mohapatra et al. (J Comput Appl Math 372:112696, 2020). The main focus of this article is to study the performance of this method for Stokes problems on smooth curvilinear domains and Stokes problem with mixed boundary conditions. Various test cases are considered including the generalized Stokes problem in R 2 and R 3 to verify the exponential accuracy of the method. [ABSTRACT FROM AUTHOR]

In this work, we study the material and geometric uncertainty effects on the static, free vibration and dynamic behaviour of sandwich beam structures. A higher-order sandwich panel theory is considered for the analysis. The elastic properties of the sandwich beam are considered as 1-D non-Gaussian random field which causes local variation in mass and stiffness matrices of the beam. The discretization of the non-Gaussian random fields is performed using the expansion optimal linear estimation. To perform the numerical analysis, Monte-Carlo simulation along with the computationally efficient time-domain spectral element method is proposed. Numerical results are obtained for different boundary conditions as well as for different materials in the sandwich face sheet and core. Results obtained in this work quantify the effects of material and geometric uncertainty in the response behaviour of a sandwich beam. The individual effect of core thickness and Poisson's ratio of the core on the static, free vibration and dynamic response is quantified. It is observed that the uncertainties in material and geometric properties along with loading and boundary conditions influence the static, free vibration and dynamic response. [ABSTRACT FROM AUTHOR]

Multiphase flow can be difficult to simulate with high accuracy due to the wide range of scales associated with various multiphase phenomena. These scales may range from the size of individual particles to the entire domain of interest. Traditionally, large scale systems can only be simulated using averaging approaches that filter out the locations of individual particles. In this work, the Euler–Lagrange method is used to simulate large-scale dense particle systems in which each individual particle is tracked. In order to accomplish this, the highly scalable spectral element code nek5000 has been extended to handle the multiple levels of multiphase coupling in these systems. These levels include what has been called one-, two-, and four-way coupling. Here, each level has been separated to detail the computational impact of each stage. A binned ghost particle algorithm has also been developed to efficiently handle the challenges of two- and four-way coupling in a parallel processing context. The algorithms and their implementations are then shown to scale to 65,536 Message Passing Interface (MPI) ranks in both the strong and weak limits. After this, validation is performed through simulation of a small-scale fluidized bed. Lastly, a large-scale fluidized bed is simulated with 65,536 MPI ranks and is able to capture the unique physics of the onset of fluidization. [ABSTRACT FROM AUTHOR]

The approach of the next-generation computing platforms offers a tremendous opportunity to advance the state-of-the-art in global atmospheric dynamical models. We detail our incremental approach to utilize this emerging technology by enhancing concurrency within the High-Order Method Modeling Environment (HOMME) atmospheric dynamical model developed at the National Center for Atmospheric Research (NCAR). The study focused on improvements to the performance of HOMME which is a Fortran 90 code with a hybrid (MPIOpenMP) programming model. The article describes the changes made to the use of message passing interface (MPI) and OpenMP as well as single-core optimizations to achieve significant improvements in concurrency and overall code performance. For our optimization studies, we utilize the "Cori" system with an Intel Xeon Phi Knights Landing processor deployed at the National Energy Research Supercomputing Center and the "'Cheyenne" system with an Intel Xeon Broadwell processor installed at the NCAR. The results from the studies, using "workhorse" configurations performed at NCAR, show that these changes have a transformative impact on the computational performance of HOMME. Our improvements have shown that we can effectively increase potential concurrency by efficiently threading the vertical dimension. Further, we have seen a factor of two overall improvement in the computational performance of the code resulting from the single-core optimizations. Most notably from the work is that our incremental approach allows for high-impact changes without disrupting existing scientific productivity in the HOMME community. [ABSTRACT FROM AUTHOR]

The climate of stratosphere was simulated with a spectral element model under two different vertical level distributions, one had 28 levels with 0.3 hPa at the top of the atmosphere and another 66 levels with 4.5×10-6 hPa at the top boundary. Simulation results were validated against ERA-Interim reanalysis dataset from ECMWF (European Center for Medium-Range Weather Forecasts) and the influence of vertical level distributions on the numerical simulation of stratospheric temporal mean state had been analyzed. It is shown that the lift of upper boundary can reduce the temperature error above the height of 100 hPa level significantly. It can also improve the simulation of seasonal variation of zonal mean zonal wind on 50 hPa in the northern high latitudes. In both configurations, cold biases of 10-year averaged zonal mean of temperature are mainly discerned at levels around 75 hPa at low latitudes and levels around 175 hPa at high southern latitudes. The simulated seasonal variation of westerly at 50 hPa in the northern hemisphere with 66-layer model coincides with results of ERA-Interim better. [ABSTRACT FROM AUTHOR]

The under integration of the volume terms in the discontinuous Galerkin spectral element approximation introduces errors at non-conforming element faces that do not cancel and lead to free-stream preservation errors. We derive volume and face conditions on the geometry under which a constant state is preserved. From those, we catalog eight constraints on the geometry that preserve a constant state. Numerical examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

The application of the composite materials in the aeronautical and aerospace industries has been increasing over the last several decades. Compared to conventional metallic materials, they present better strength to weight and stiffness to weight ratio. However, they can also present a high level of uncertainty, mainly associated with the manufacturing processes. Besides the uncertainty in the composite material parameters, which can play a role in the structural dynamic response, randomness can also be associated with boundary condition and external excitation sources. This paper treats the dynamic analysis of a composite beam under random excitation and uncertainties in the boundary condition. The beam is modelled by the spectral element method, a wave propagation technique. Some numerical examples are used to study the influence of random source on the dynamic behaviour of the composite structure. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in constructing a scheme solving compressible Navier–Stokes equations, with desired properties including high order spatial accuracy, conservation, and positivity-preserving of density and internal energy under a standard hyperbolic type CFL constraint on the time step size, e.g., Δ t = O (Δ x). Strang splitting is used to approximate convection and diffusion operators separately. For the convection part, i.e., the compressible Euler equation, the high order accurate postivity-preserving Runge–Kutta discontinuous Galerkin method can be used. For the diffusion part, the equation of internal energy instead of the total energy is considered, and a first order semi-implicit time discretization is used for the ease of achieving positivity. A suitable interior penalty discontinuous Galerkin method for the stress tensor can ensure the conservation of momentum and total energy for any high order polynomial basis. In particular, positivity can be proven with Δ t = O (Δ x) if the Laplacian operator of internal energy is approximated by the Q k spectral element method with k = 1 , 2 , 3. So the full scheme with Q k (k = 1 , 2 , 3) basis is conservative and positivity-preserving with Δ t = O (Δ x) , which is robust for demanding problems such as solutions with low density and low pressure induced by high-speed shock diffraction. Even though the full scheme is only first order accurate in time, numerical tests indicate that higher order polynomial basis produces much better numerical solutions, e.g., better resolution for capturing the roll-ups during shock reflection. • The hyperbolic subproblem is solved by a positivity-preserving RKDG method. • The parabolic subproblem is solved by classical schemes to achieve positivity. • A proper choice of the IPDG method achieves total conservation. • Numerically a spatially higher order scheme produces better solutions. [ABSTRACT FROM AUTHOR]

Exact evaluation of high-order modes of a multi-crack Euler–Bernoulli beam using the conventional spectral element method (SEM) suffers from the deficiency of numerical ill-conditioning, mainly due to round-off errors in floating-point computations involved in evaluating the spectral element stiffness matrix, therefore, the modeling of the multiple cracks intensively increases the complexity of the matrix. To address this limitation, an enhanced SEM with the spectral element stiffness matrix constructed in a set of local coordinate systems is developed. With the method, two sets of local coordinate systems are created to derive two types of spectral element stiffness matrices for a multi-crack beam. The proposed spectral element stiffness matrices are of improved capacities to characterize the high-order modes of a multi-crack beam by largely eliminating the round-off errors involved in the conventional SEM. Moreover, the function of evaluating high-order modes endows the enhanced SEM with distinctive capability to reveal the effect of multiple cracks on the modal property of a multi-crack beam. The effectiveness of the proposed method in evaluating high-order modes and revealing the effects of multiple cracks on the modal property of a beam is investigated on various damage cases. [ABSTRACT FROM AUTHOR]

This paper develops a unified and efficient simulation technique based on the scaled boundary finite element method (SBFEM) for transient structural–acoustic problems in two dimensions. The structural component can be an assembly of thin to moderately thick unidirectional plates. Each of them is treated as a plane strain problem and the principle of virtual work involving the inertial contribution is applied to derive the scaled boundary finite element equation. Only the longitudinal dimension is discretized with line elements and the solution through the thickness is expressed analytically as a Padé expansion. A variable transformation process together with certain structural assumptions leads to the static stiffness and mass matrices. The acoustic field can be a semi-infinite region, which is first divided by an artificial boundary into a near field and an exterior domain. The former is analyzed using the improved continued-fraction approach and the latter is modeled by the improved doubly-asymptotic open boundary. These formulations are based on velocity potential and established consistently in the SBFEM framework. High-order spectral elements are employed for both the structural and acoustic domains, to which independent discretization schemes are applied. Numerical examples are presented to demonstrate the excellent performances of the proposed technique. [ABSTRACT FROM AUTHOR]

Vibration-based methods are among the most widely studied in structural health monitoring (SHM). It is well known, however, that the low-order modes, characterizing the global dynamic behaviour of structures, are relatively insensitive to local damage. Such local damage may be easier to detect by methods based on wave propagation which involve local high frequency behaviour. The present work considers the numerical analysis of wave propagation in cables. A two-step approach is proposed which allows taking into account the cable sag and the distribution of the axial forces in the wave propagation analysis. In the first step, the static deformation and internal forces are obtained by the finite element method (FEM), taking into account geometric nonlinear effects. In the second step, the results from the static analysis are used to define the initial state of the dynamic analysis which is performed by means of the spectral element method (SEM). The use of the SEM in the second step of the analysis allows for a significant reduction in computational costs as compared to a FE analysis. This methodology is first verified by means of a full FE analysis for a single stretched cable. Next, simulations are made to study the effects of damage in a single stretched cable and a cable-supported truss. The results of the simulations show how damage significantly affects the high frequency response, confirming the potential of wave propagation based methods for SHM. [ABSTRACT FROM AUTHOR]

A spectral element method-based formulation is developed to obtain the frequency-dependent effective Young's and shear modulus of a two-dimensional square grid lattice. The methodology presented in this work is a simplified approach to obtain effective parameters. The lattice is assumed to be constituted of Euler–Bernoulli beam elements which are arranged in a specific order to obtain the said geometry. Spectral element formulation of the beam element considering axial and flexural deformation is presented, which in due course is employed to obtain the global spectral element matrix of the unit cell. A detailed procedure to obtain the frequency-dependent stiffness matrix of the unit cell is presented on which proper boundary conditions are applied using specific transformation matrices for each case. The frequency-dependent properties are also calculated using the software package COMSOL Multiphysics to validate the results. Frequency responses of the unit cell are obtained using the proposed methodology as well as the software package, and the results are compared. It is observed that the proposed methodology predicts the peaks at nearly the same frequency range as obtained through the finite element solution obtained in COMSOL Multiphysics. Finally, the results are presented which show the frequency-dependent effective parameters for the unit cell. The methodology presented here opens up a new way to calculate the effective properties of two-dimensional lattice structures and with some modifications can also be extended to three-dimensional lattice structures. • A spectral element method-based homogenization for square grid 2D lattice. • Frequency-dependent effective Young's and shear modulus of grid structure. • Exact closed form dynamic solution for the displacement field. • Transmittance is compared. [ABSTRACT FROM AUTHOR]

The propagation of elastic waves in heterogeneous media is of interest for impact dynamics and non-destructive detection. This work presents a refined spectral element model (RSEM) to study the wave propagation in multiscale hybrid composite (MHC) shells subjected to impulsive loadings. The doubly-curved MHC shell consists of epoxy, carbon fibers, and graphene platelets (GPLs). The GPLs are functionally distributed along the thickness of the shell. For the three-phase MHC, the Halpin-Tsai micromechanical model in conjunction with the Mori-Tanaka approach is exploited to determine the effective material properties. In the framework of four-variable shear deformation theory, the governing equations along with the natural boundary conditions are derived using Hamilton's principle. A two-node spectral shell element is developed according to the closed-from solutions. The accuracy of the RSEM is verified by comparison with published results in aspects of the natural frequency and transient responses. The wave dispersion characteristics, including the wave number, phase velocity, and group velocity are examined. In the context of high frequency and short wavelength, the proposed RSEM achieves high computational efficiency benefiting from its independence of mesh structure. The time domain responses clearly indicate the wave-boundary interactions, e.g., wave reflection, dispersion, and interference. It is revealed that the present model can well capture the fundamental wave modes of the MHC shell. Moreover, the inclusion of GPLs plays a significant role in improving transverse moduli and mitigating the discontinuities of inter-laminar shear stress. [ABSTRACT FROM AUTHOR]

Slope uncertainty predominantly originates from the imperfect analysis model and the inaccuracy and imprecision of the observations. The strength reduction method (SRM) is widely used to attain the safety factor (SF) of the slopes, which is similar to interpretation of the limit state (LS). In this paper, the spectral element method (SEM), using an elasto-plastic Mohr⁻Coulomb failure criterion, is employed to project the plausible LS of the soil slopes. An iterative SRM search method is proposed to evaluate the SF of the slopes regardless of the LS interpretation. The proposed SRM paradigm encompasses the design trigger to trace the uncertain parameters in decision-making. This method is applied to three numerical examples: (1) a homogeneous dry slope, (2) a dry slope with a weak layer, and (3) a partially-wet slope with a weak layer. It is shown that for the case study examples, the proposed SRM reasonably converges to the required precision. Results further are compared and contrasted with some of the conventional and standard techniques in slope stability. This hybrid procedure paves the road for fast and safe stability analysis of man-made and natural slopes.

When the shock wave of underwater explosion propagates to the surfaces of different boundaries, it gets reflected. Then, a negative pressure area is formed by the superposition of the incident wave and reflected wave. Cavitation occurs when the value of the negative pressure falls below the vapor pressure of water. An improved numerical model based on the spectral element method is applied to investigate the cavitation effect of underwater shock near different boundaries, mainly including the feature of cavitation effect near different boundaries and the influence of different parameters on cavitation effect. In the implementation of the improved numerical model, the bilinear equation of state is used to deal with the fluid field subjected to cavitation, and the field separation technique is employed to avoid the distortion of incident wave propagating through the mesh and the second-order doubly asymptotic approximation is applied to simulate the non-reflecting boundary. The main results are as follows. As the peak pressure and decay constant of shock wave increases, the range of cavitation domain increases, and the duration of cavitation increases. As the depth of water increases, the influence of cavitation on the dynamic response of spherical shell decreases. [ABSTRACT FROM AUTHOR]

This study investigates the pattern of flow past two staggered array cylinders using the spectral element method by varying the distance between the cylinders and the angle of incidence (α) at low Reynolds numbers (Re = 100-800). Six flow patterns are identified as Shear layer reattachment (SLR), Induced separation (IS), Vortex impingement (VI), Synchronized vortex shedding (SVS), Vortex pairing and enveloping (VPE), and Vortex pairing splitting and enveloping (VPSE). These flow patterns can be transformed from one to another by changing the distance between the cylinders, the angle of incidence, or Re. SLR, IS and VI flow patterns appear in regimes with small angles of incidence (i.e., α ≤ 30° ) and hold only a single von Karman vortex shedding in a wake with one shedding frequency. SVS, VPE and VPSE flow patterns appear in regimes with large angles of incidence (i.e., 30° ≤ α ≤ 50° ) and present two synchronized von Karman vortices. Quantitative analyses and physical interpretation are also conducted to determine the generation mechanisms of the said flow patterns. [ABSTRACT FROM AUTHOR]

A new time-domain spectral element with nine degrees of freedom per node is formulated based on higher-order sandwich panel theory, incorporating the flexible behaviour of the core with composite face sheets. Static, free vibrations and wave propagation analysis are carried out using the formulated element. Results obtained using this element are compared with those available in the literature and with commercial finite element codes. The fast convergence of the spectral element method is demonstrated by solving the high-frequency wave propagation problem. A method of computing the wave characteristics, namely wavenumbers and group velocities, in a higher-order sandwich panel is developed using the formulated element. The effect of core damping is studied in detail with different core types, which can be used effectively in sandwich beam design. [ABSTRACT FROM AUTHOR]

The spectral element marker particle (SEMP) method is a high-order numerical scheme for modelling multiphase flow where the governing equations are discretised using the spectral element method and the (compressible) fluid phases are tracked using marker particles. Thus far, the method has been successfully applied to two-phase problems involving the collapse of a two-dimensional bubble in the vicinity of a rigid wall. In this article, the SEMP method is extended to include a third fluid phase before being applied to bubble collapse problems near a fluid-fluid interface. Two-phase bubble collapse near a rigid boundary (where a highly viscous third phase approximates the rigid boundary) is considered as validation of the method. A range of fluid parameter values and geometric configurations are studied before a bioengineering application is considered. A simplified model of (micro)bubble-cell interaction is presented, with the aim of gaining initial insights into the flow mechanisms behind sonoporation and microbubble-enhanced targeted drug delivery. Results from this model indicate that the non-local cell membrane distortion (blebbing) phenomenon often observed experimentally may result from stress propagation along the cell surface and so be hydrodynamical in origin. [ABSTRACT FROM AUTHOR]

The most common boundary condition (BC) imposed at the model top in many nonhydrostatic, height-based models is a rigid lid (no mass-flux) complemented by an absorbing sponge layer. This BC is unphysical and not appropriate for large-scale heating and cooling processes present in the upper atmosphere with a high model top. To address this problem, we have derived a physics-based open atmosphere BC from first principles that allows fluid to smoothly exit and enter the model domain during heating and cooling processes, respectively. The new BC is derived from both the compressible and hydrostatic continuity equation and accounts for bulk and surface heating as well as horizontal divergence by modeling the velocity normal to the model top (vertical velocity). This physics-based BC was implemented in two height-based, spectral element nonhydrostatic atmospheric models and tested on both 1D and 3D atmospheric test cases using two idealized heating tests targeted toward high altitude applications. The numerical results indicate that the BC is stable using explicit, fully implicit, and horizontally explicit, vertical implicit (HEVI) time integrators. Unlike the rigid BC, the proposed BC produces stable numerical results without producing spurious oscillations near the model top. Unlike sponge layers, the proposed BC does not require any tuning parameters since all parameters are consistent with the BCs used in many hydrostatic models. • A new open boundary condition (BC) has been derived for high altitude (HA) applications. • This physics-based BC was implemented in two height-based nonhydrostatic atmospheric models: NUMA and NEPTUNE. • Unlike the rigid BC and sponge BC, the proposed BC does not produce spurious oscillations. [ABSTRACT FROM AUTHOR]

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

Accurate, efficient and versatile computational methods for functionally graded piezoelectric (FGP) shells are still restricted. This contribution proposes new FGP shell finite elements with superior comprehensive performance. In the formulation, a shell element is treated as a three-dimensional continuum and its middle surface is represented with a quadrilateral spectral element. Along the thickness, the shell geometry is described by scaling the middle surface while the displacements and electric potential are approximated via a consistent quadratic Lagrange interpolation. The electric potential is scaled by a factor λ to avoid numerical instability and the assumed natural strain method is applied to eliminate numerical locking. The piezoelectric material properties are assumed to vary continuously through the shell thickness obeying a simple power-law function. Binomial theorem is utilized to perform analytical integral along the thickness in evaluating the stiffness and mass matrices and nodal load vector. The validity of the developed approach is demonstrated by solving piezoelectric or FGP plate problems with reference solutions. A simply supported FGP square plate, a fully clamped FGP cylindrical shell and a partly clamped FGP hyperbolic paraboloid shell are systematically analyzed. The influence of the power-law index and span-to-thickness ratio on the static and free vibration behaviors of the FGP structures is investigated. The optimal value of λ for general FGP shells is also determined. • New functionally graded piezoelectric shell elements are developed. • The full three-dimensional piezoelectric constitutive law is adopted. • Both the displacements and electric potential are quadratic along the thickness. • The electric potential is properly scaled to avoid numerical instability. • Functionally graded piezoelectric plates and shells are systematically analyzed. [ABSTRACT FROM AUTHOR]

In this work we present a robust and accurate arbitrary order solver for the fixed-boundary plasma equilibria in toroidally axisymmetric geometries. To achieve this we apply the mimetic spectral element formulation presented in [56] to the solution of the Grad–Shafranov equation. This approach combines a finite volume discretization with the mixed finite element method. In this way the discrete differential operators (∇, ∇×, ∇⋅) can be represented exactly and metric and all approximation errors are present in the constitutive relations. The result of this formulation is an arbitrary order method even on highly curved meshes. Additionally, the integral of the toroidal current J ϕ is exactly equal to the boundary integral of the poloidal field over the plasma boundary. This property can play an important role in the coupling between equilibrium and transport solvers. The proposed solver is tested on a varied set of plasma cross sections (smooth and with an X-point) and also for a wide range of pressure and toroidal magnetic flux profiles. Equilibria accurate up to machine precision are obtained. Optimal algebraic convergence rates of order p + 1 and geometric convergence rates are shown for Soloviev solutions (including high Shafranov shifts), field-reversed configuration (FRC) solutions and spheromak analytical solutions. The robustness of the method is demonstrated for non-linear test cases, in particular on an equilibrium solution with a pressure pedestal. [ABSTRACT FROM AUTHOR]

A numerical method is proposed to compute high-frequency low-leakage modes in structural waveguides surrounded by infinite solid media. In order to model arbitrary shape structures, a waveguide formulation is used, which consists of applying to the elastodynamic equilibrium equations a space Fourier transform along the waveguide axis and then a discretization method to the cross-section coordinates. However several numerical issues must be faced related to the unbounded nature of the cross-section, the number of degrees of freedom required to achieve an acceptable error in the high-frequency regime as well as the number of modes to compute. In this paper, these issues are circumvented by applying perfectly matched layers (PML) along the cross-section directions, a high-order spectral element method for the discretization of the cross-section, and an eigensolver shift suited for the computation of low-leakage modes. First, computations are performed for an embedded cylindrical bar, for which literature results are available. The proposed PML waveguide formulation yields good agreement with literature results, even in the case of weak impedance contrast. Its performance with high-order spectral elements is assessed in terms of convergence and accuracy and compared to traditional low-order finite elements. Then, computations are performed for an embedded square bar. Dispersion curves exhibit strong similarities with cylinders. These results show that the properties of low-leakage modes observed in cylindrical bars can also occur in other types of geometry. [ABSTRACT FROM AUTHOR]

A combination of block-Jacobi and deflation preconditioning is used to solve a high-order discontinuous element-based collocation discretization of the Schur complement of the Poisson–Neumann system as arises in the operator splitting of the incompressible Navier–Stokes equations. The preconditioners and deflation vectors are chosen to mitigate the effects of ill-conditioning due to highly-elongated domains typical of simulations of strongly non-hydrostatic environmental flows, and to achieve Generalized Minimum RESidual method (GMRES) convergence independent of the size of the number of elements in the long direction. The ill-posedness of the Poisson–Neumann system manifests as an inconsistency of the Schur complement problem, but it is shown that this can be accounted for with appropriate projections out of the null space of the Schur complement matrix without affecting the accuracy of the solution. The block-Jacobi preconditioner is shown to yield GMRES convergence independent of the polynomial order and only weakly dependent on the number of elements within a subdomain in the decomposition. The combined deflation and block-Jacobi preconditioning is compared with two-level non-overlapping block-Jacobi preconditioning of the Schur problem, and while both methods achieve convergence independent of the grid size, deflation is shown to require half as many GMRES iterations and 25 % less wall-clock time for a variety of grid sizes and domain aspect ratios. The deflation methods shown to be effective for the two-dimensional Poisson–Neumann problem are extensible to the three-dimensional problem assuming a Fourier discretization in the third dimension. A Fourier discretization results in a two-dimensional Helmholtz problem for each Fourier component that is solved using deflated block-Jacobi preconditioning on its Schur complement. Here again deflation is shown to be superior to two-level non-overlapping block-Jacobi preconditioning, requiring about half as many GMRES iterations and 15 % less time. [ABSTRACT FROM AUTHOR]

Spectral element method is proposed to analyze the mechanical response of transversely isotropic elastic multi-layered pavement structure subjected to axisymmetric loading. Based on the basic constitutive equations and modified Love’s function for transversely isotropic elastic media, the governing state equation of a multi-layered transversely isotropic medium was deduced. From the equation, s spectral layer element for a single layer (i.e., a stiffness matrix) was acquired. The global stiffness matrix was obtained by assembling the interrelated layer elements based on the principle of the finite element method and the boundary conditions, and the solution for the corresponding problem was obtained by solving the algebraic equations of the global stiffness matrix. The solutions in the physical domain were acquired by means of the Fourier–Bessel superposition. The validity of the proposed solution was examined by comparing with an existing exact solution and finite element method respectively. Subsequent to solution validation, a parametric study by varying n -values of unbound layer was carried to investigate the influence of the transversal isotropy on pavement response. The proposed analytical solution demonstrated the ability of solving the mechanical response of transversely isotropic elastic multi-layered pavement structures subjected to axisymmetric loading. [ABSTRACT FROM AUTHOR]

A spectral element formulation of the immersed boundary method (IBM) is presented. The spectral element formulation (SE-IBM) is a generalisation of the finite element immersed boundary method (FE-IBM) based on high-order approximations of the fluid variables. Several schemes for tracking the movement of the immersed boundary are considered and a semi-implicit Euler scheme is shown to offer advantages in terms of accuracy and efficiency. High-order spectral element approximations provide improved area conservation properties of the IBM due to the incompressibility constraint being more accurately satisfied. Superior orders of convergence are obtained for SE-IBM compared with FE-IBM in both L 2 and H 1 norms. The area conservation and convergence properties of the scheme are demonstrated on a series of benchmark problems. [ABSTRACT FROM AUTHOR]

We develop a controllability strategy for the computation of frequency-domain solutions of the 3D visco-elastic wave equation, in the perspective of seismic imaging applications. We generalize the controllability results for such equations beyond the sound-soft scattering (obstacle) problem. We detail the conjugate gradient implementation and show how an inner elliptic problem needs to be solved to compute the Riesz representative of the gradient at each iteration. We select a spectral-element spatial discretization and a fourth-order Runge-Kutta time discretization. We implement the controllability method in the framework of the SEM46 full waveform modeling and inversion software, to inherit for its excellent scalability which relies on an efficient domain decomposition algorithm. We perform a series of numerical experiments to validate the approach and illustrate its scalability up to more than fifteen hundred cores. In this case, with an elapsed time of less than 50 minutes, we solve a problem on a cubic domain containing up to 160 wavelengths in each direction, involving more than 1.7 billion unknowns. • Efficient numerical solution of frequency-domain visco-elastic wave equation. • Controllability methods (CM) extended to general visco-elastic problem beyond sound-soft scattering problems. • CM combined with spectral element method, explicit time scheme and parallelization through domain decomposition. • CM offer robust, non-intrusive approach for time-harmonic wave equations. • CM achieve strong scalability on massively parallel architectures. [ABSTRACT FROM AUTHOR]

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 C 1 -continuous Lobatto shape functions interpolate the deflection, while the axial displacement and shear rotation are interpolated using the C 0 -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. [Display omitted] • Time-domain spectral element based on a zigzag laminate theory developed for the first time. • Considers interfacial slope discontinuities of the in-plane displacement. • Employs C1-continuous Lobatto basis functions for interpolating deflection. • Yields excellent accuracy with a multi-fold reduction in computing time from standard FE. • Useful for guided wave-based structural health monitoring of multi-material laminated structures. [ABSTRACT FROM AUTHOR]

A frequency domain spectral finite element formulation is presented for the wave propagation analysis of laminated composite curved beams using the first order shear deformation theory (FSDT) and the classical laminate theory (CLT). The elements are derived from the exact solution of the governing equation of motion in frequency domain, obtained through Fourier transformation of the time domain equation. The formulation is validated by comparing the results for natural frequencies with the published results. The new elements are then employed to perform dispersion and wave propagation analyses of curved composite beams. The numerical results reveal that the wavenumbers predicted by the CLT show large deviation from those of the FSDT even for thin beams, and the deviation increases and occurs at lower frequencies with the increase in the thickness to radius ratio. The orthotropy ratio of the composite has a significant effect on the wavenumbers for tangential and mid-surface rotation modes. The wave propagation response predicted by the CLT differs widely from the FSDT prediction, for thin and thick, and shallow and deeply curved beams at both low and high frequencies. Thus, the CLT should not be used for wave propagation analysis of even thin curved laminated beams. [ABSTRACT FROM AUTHOR]

In this paper, spectral finite elements (SFEs) are developed for wave propagation analysis of isotropic curved beams using three different beam models: (1) the refined third-order shear deformation theory (TOT), (2) the first-order shear deformation theory (FSDT), and (3) the classical shell theory (CST). The formulation is validated by comparing the results for the wavenumber dispersion relations and natural frequencies with the published results based on the FSDT. The numerical study reveals that even for a very thin curved beam with radius-to-thickness ratio of 1000, the wavenumbers predicted by the CST at high frequencies show significant deviation from those of the shear deformable theories, FSDT and TOT. The FSDT results for the wavenumber of the flexural displacement mode differ significantly from the TOT results at high frequencies even for thin beams. The deviation increases and occurs at lower frequencies with the decrease in the radius-tothickness ratio. The results for wave propagation response show that the CST yields highly erroneous response for flexural mode wave propagation even for thin beams and at a relatively low frequency of 20 kHz. The FSDT results too differ by unacceptably high margin from the TOT results for flexural wave response of thin beams at frequencies greater than 100 kHz, which are typically used for structural health monitoring (SHM) applications. For thick beams, FSDT results for the tangential wave response also show large deviation from the TOT results. [ABSTRACT FROM AUTHOR]

We present a generalized form of open boundary conditions, and an associated numerical algorithm, for simulating incompressible flows involving open or outflow boundaries. The generalized form represents a family of open boundary conditions, which all ensure the energy stability of the system, even in situations where strong vortices or backflows occur at the open/outflow boundaries. Our numerical algorithm for treating these open boundary conditions is based on a rotational pressure correction-type strategy, with a formulation suitable for C 0 spectral-element spatial discretizations. We have introduced a discrete equation and associated boundary conditions for an auxiliary variable. The algorithm contains constructions that prevent a numerical locking at the open/outflow boundary. In addition, we have developed a scheme with a provable unconditional stability for a sub-class of the open boundary conditions. Extensive numerical experiments have been presented to demonstrate the performance of our method for several flow problems involving open/outflow boundaries. We compare simulation results with the experimental data to demonstrate the accuracy of our algorithm. Long-time simulations have been performed for a range of Reynolds numbers at which strong vortices or backflows occur at the open/outflow boundaries. We show that the open boundary conditions and the numerical algorithm developed herein produce stable simulations in such situations. [ABSTRACT FROM AUTHOR]

This study presents the vortex structure and numerical instability increase occurring when the level of elasticity is enhanced in inertial flows in planar contraction configuration for finitely extensible nonlinear elastic model by Peterlin (FENE-P) fluid . The re-entrant corner effect on corner vortices is also considered. The calculations are performed using extended matrix logarithm formulation described in a previous paper: A. Jafari et al. A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements. Computer & Fluids 2010; 39(9):1425-1438. In that reference, the proposed algorithm has been tested for simple geometry such as Poiseuille flow. In this study, we are interested in the capability of this algorithm for more complex geometry. This formulation helps to reach higher values of the Weissenberg number when compared with the classical one. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

For obtaining dynamic response of structure to high frequency shock excitation spectral elements have several advantages over conventional methods. At higher frequencies transverse shear and rotary inertia have a predominant role. These are represented by the First order Shear Deformation Theory (FSDT). But not much work is reported on spectral elements with FSDT. This work presents a new spectral element based on the FSDT/Mindlin Plate Theory which is essential for wave propagation analysis of sandwich plates. Multi-transformation method is used to solve the coupled partial differential equations, i.e., Laplace transforms for temporal approximation and wavelet transforms for spatial approximation. The formulation takes into account the axial-flexure and shear coupling. The ability of the element to represent different modes of wave motion is demonstrated. Impact on the derived wave motion characteristics in the absence of the developed spectral element is discussed. The transient response using the formulated element is validated by the results obtained using Finite Element Method (FEM) which needs significant computational effort. Experimental results are provided which confirms the need to having the developed spectral element for the high frequency response of structures. [ABSTRACT FROM AUTHOR]

Efficient numerical methods are essential in the analysis of finite hydrodynamic bearings with surface texturing. This is especially evident in optimization and parametric studies where the discretization and integration methods are used to solve the governing two-dimensional Reynolds equation multiple times. Performance comparison studies of the methods are thus required to select the method that is most suitable for a particular bearing geometry. In this work, we conduct a comprehensive and systematic comparison of typical implementations of the finite difference, finite volume, finite element and spectral element discretization methods together with the Newton–Cotes formula and Gauss quadratures for hydrodynamic bearings governed by the two-dimensional Reynolds equation. The methods were evaluated by comparing the approximation errors in the calculation of the maximum pressure, load capacity, coefficient of friction, and minimum film thickness for parallel and convergent bearings textured with elliptical grooves or trapezoidal dimples. The number of degrees of freedom required by the methods to achieve the error cut-off values of 5%, 1% and 0.1% were calculated. Our results demonstrate that the spectral element method uses up to 72 times fewer degrees of freedom than the other methods for the same cut-off values. Also, our study revealed that the shape of the groove/dimple and the bearing convergence ratio can have a significant effect on the approximation errors of the numerical methods used. Specifically, for piecewise-linear texture features (e.g. trapezoidal dimples) and positive convergence ratio it is easier, for the methods, to accurately approximate the solution. In such cases, the finite volume and finite element methods are reasonable choices and provide a good trade-off between the ease of implementation and approximation errors. The worse performance was observed for the finite difference and thus this method is not recommended when the computational efficiency and the accuracy of results are of importance. [ABSTRACT FROM PUBLISHER]

SUMMARY We are concerned with the numerical simulation of wave motion in arbitrarily heterogeneous, elastic, perfectly-matched-layer-(PML)-truncated media. We extend in three dimensions a recently developed two-dimensional formulation, by treating the PML via an unsplit-field, but mixed-field, displacement-stress formulation, which is then coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient hybrid scheme. The hybrid treatment leads to, at most, third-order in time semi-discrete forms. The formulation is flexible enough to accommodate the standard PML, as well as the multi-axial PML. We discuss several time-marching schemes, which can be used à la carte, depending on the application: (a) an extended Newmark scheme for third-order in time, either unsymmetric or fully symmetric semi-discrete forms; (b) a standard implicit Newmark for the second-order, unsymmetric semi-discrete forms; and (c) an explicit Runge-Kutta scheme for a first-order in time unsymmetric system. The latter is well-suited for large-scale problems on parallel architectures, while the second-order treatment is particularly attractive for ready incorporation in existing codes written originally for finite domains. We compare the schemes and report numerical results demonstrating stability and efficacy of the proposed formulations. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]