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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]

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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]