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The mathematics achievement discrepancy between girls and boys, with its subsequent occupational consequences, is an issue that has received considerable attention in the literature. It is often referred to as the 'math-gap' and favours boys. A major component of the explanation of this gap resides in determining its age of onset. We analyse here data from more than 10,000 (cross-sectional study) and 2000 (longitudinal study) French students aged 4-7 years, tested in the framework of the Etude Longitudinale Française depuis l'Enfance (ELFE). The results allow to precisely determine the age of onset, since the gender difference, non-existent (or even slightly in favour of girls) in kindergarten (4-5 years), is clearly in favour of boys in first grade (6-7 years). They could therefore provide an important element in the controversial debate on the origin of gender-differentiated performance in mathematics.

We present a filter-based approach to computing artificial viscosities for spectral element methods. A number of applications for this approach are presented.

Low-order finite element (FE) systems are considered as preconditioners for spectral element (SE) discretizations of the Poisson problem in canonical and complex domains. The FE matrices are based ...

In this paper, we present a characteristic-based numerical procedure for simulating incompressible flows in domains with moving boundaries. Our approach utilizes an operator-integration-factor splitting technique to help produce an efficient and stable numerical scheme. Using the spectral element method and an arbitrary Lagrangian–Eulerian formulation, we investigate flows where the convective acceleration effects are non-negligible. Several examples, ranging from laminar to turbulent flows, are considered. Comparisons with a standard, semi-implicit time-stepping procedure illustrate the improved performance of our scheme.

Laplacian- and optimization-based mesh-improvement methods are developed for high-order finite- and spectral-element based on 2D quadrilateral and 3D hexahedral meshes in general domains. A robust high-order interpolation library is used during the mesh smoothing process to improve the quality of the surface mesh while retaining the integrity of the original surface approximation. Boundary layer resolution in the original mesh is preserved through various controls in the smoothing process, including weighted interpolation between the optimized and original mesh. All mesh motion and gradient evaluations are performed on an element-by-element basis to ensure that all elements in a large mesh can be smoothed in parallel with minimum communication between different processors. Mesh quality improvements are shown to reduce the condition number of the preconditioned linear systems governing the numerical solution of the discretized partial differential equations, with corresponding reductions in iteration counts. The mesh smoother is tested on various meshes and is found to significantly improve the computational efficiency of calculations.

We analyze the potential performance benefits of estimating expected quantities in large eddy simulations of turbulent flows using true ensembles rather than ergodic time averaging. Multiple realizations of the same flow are simulated in parallel, using slightly perturbed initial conditions to create unique instantaneous evolutions of the flow field. Each realization is then used to calculate statistical quantities. Provided each instance is sufficiently de-correlated, this approach potentially allows considerable reduction in the time to solution beyond the strong scaling limit for a given accuracy. This paper focuses on the theory and implementation of the methodology in Nek5000, a massively parallel open-source spectral element code.

We develop a multirate timestepper for semi-implicit solutions of the unsteady incompressible Navier-Stokes equations (INSE) based on a recently-developed multidomain spectral element method (SEM). For {\em incompressible} flows, multirate timestepping (MTS) is particularly challenging because of the tight coupling implied by the incompressibility constraint, which manifests as an elliptic subproblem for the pressure at each timestep. The novelty of our approach stems from the development of a stable overlapping Schwarz method applied directly to the Navier-Stokes equations, rather than to the convective, viscous, and pressure substeps that are at the heart of most INSE solvers. Our MTS approach is based on a predictor-corrector (PC) strategy that preserves the temporal convergence of the underlying semi-implicit timestepper. We present numerical results demonstrating that this approach scales to an arbitrary number of overlapping grids, accurately models complex turbulent flow phenomenon, and improves computational efficiency in comparison to singlerate timestepping-based calculations., Comment: 34 pages, 13 figures

Detecting structural breaks is an essential task for the statistical analysis of time series, for example, for fitting parametric models to it. In short, structural breaks are points in time at which the behaviour of the time series substantially changes. Typically, no solid background knowledge of the time series under consideration is available. Therefore, a black-box optimization approach is our method of choice for detecting structural breaks. We describe a genetic algorithm framework which easily adapts to a large number of statistical settings. To evaluate the usefulness of different crossover and mutation operations for this problem, we conduct extensive experiments to determine good choices for the parameters and operators of the genetic algorithm. One surprising observation is that use of uniform and one-point crossover together gave significantly better results than using either crossover operator alone. Moreover, we present a specific fitness function which exploits the sparse structure of the break points and which can be evaluated particularly efficiently. The experiments on artificial and real-world time series show that the resulting algorithm detects break points with high precision and is computationally very efficient. A reference implementation with the data used in this paper is available as an applet at the following address: http://www.imm.dtu.dk/~pafi/TSX/. It has also been implemented as package SBRect for the statistics language R.

An overlapping mesh methodology that is spectrally accurate in space and up to third-order accurate in time is developed for solution of unsteady incompressible flow equations in three-dimensional domains. The ability to decompose a global domain into separate, but overlapping, subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries. The methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally-accurate spatial interpolation and high-order accurate temporal extrapolation, and requires few, if any, iterations, yet maintains the global accuracy and stability of the underlying flow solver. The overlapping mesh methodology is thoroughly validated using two-dimensional and three-dimensional benchmark problems in laminar and turbulent flows. The spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver. The influence of long integration times, as well as inflow-outflow global boundary conditions on the performance of the overlapping grid solver is assessed. In a turbulent benchmark of fully-developed turbulent pipe flow, the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data. Scaling tests are presented that show near linear strong scaling, even for moderately large processor counts. A novel overlapping grid method for incompressible flow equations is developed.The overlapping grid method is based on spectral-element discretization.The method maintains global spectral accuracy with polynomial refinement.An explicit interface extrapolation scheme for temporal coupling is proposed.The scheme allows for high-order temporal accuracy with only a few iterations.

The authors develop basic components of a parametric model-order reduction (pMOR) procedure targeting high Rayleigh-number buoyancy-driven flows. The pMOR is based on Galerkin formulation of the governing Boussinesq equations using eigenfunction bases derived from proper orthogonal decomposition. The advantages of pMOR over parametric interpolation are demonstrated for quantities of interest (QOIs) with linear and nonlinear dependencies on the solution in low Rayleigh-number cases. Constraint-base and Leray-type regularizations are shown to offer significant advantages over the standard reduced-order Galerkin formulation in a variety of examples including 3D Rayleigh–Benard flow at Rayleigh number 10 7 and turbulent flow in a half-pipe at Reynolds number 5300.

Large eddy simulation (LES) of an experimental benchmark investigating buoyant mixing among two fluids is developed for verification and validation. Flow physics involve a completely closed system where after a valve is opened, two initially stationary fluids with different densities are allowed to mix. The Boussinesq approximation and low-Mach equations are coded into spectral element code Nek5000 to predict the buoyancy forces, and miscibility is accounted for using the advection-diffusion equation for concentration. Due to computational costs of higher order solutions, current analysis consists of initial flow development after the valve is opened. Minor differences between the models and Schmidt number are found within the turbulent spectrum and time integrated quantities. Moreover, solutions for higher polynomial orders are compared to quantify error due to spatial discretization. Error estimates using a posteriori spectral derivation of truncation and quadrature error are also calculated. Validation with particle image velocimetry (PIV) reveals good prediction of both average and fluctuating components of the primary flow within the cold leg but less accurate results in the downcomer. It is concluded that instabilities due to resolving more of the interface using the high molecular Schmidt number is essential for accurately predicting this type of flow.

We investigate convergence behavior of a spectral element method based on Legendre polynomial shape functions solving linear elasticity equations for a range of Poisson's ratios of a material. We document uniform convergence rates independent of Poisson's ratio for a wide class of problems with both straight and curved elements in two and three dimensions, demonstrating locking-free properties of the spectral element method with nearly incompressible materials. We investigate computational efficiency of the current method without a preconditioner and with a simple mass-matrix preconditioner, however no attempt to optimize a choice of a preconditioner was made.

We present scalable implementations of spectral-element-based Schwarz overlapping (overset) methods for the incompressible Navier-Stokes (NS) equations. Our SEM-based overset grid method is implemented at the level of the NS equations, which are advanced independently within separate subdomains using interdomain velocity and pressure boundary-data exchanges at each timestep or sub-timestep. Central to this implementation is a general, robust, and scalable interpolation routine, {\em gslib-findpts}, that rapidly determines the computational coordinates (processor $p$, element number $e$, and local coordinates $(r,s,t) \in \hat{\Omega} := [-1,1]^3$) for any arbitrary point $\mathbf{x}^* =(x^*,y^*,z^*) \in \Omega \subset {\rm I\!R}^3$. The communication kernels in $gslib$ execute with at most $\log P$ complexity for $P$ MPI ranks, have scaled to $P > 10^6$, and obviate the need for development of any additional MPI-based code for the Schwarz implementation. The original interpolation routine has been extended to account for multiple overlapping domains. The new implementation discriminates the possessing subdomain by distance to the domain boundary, such that the interface boundary data is taken from the inner-most interior points. We present application of this approach to several heat transfer and fluid dynamic problems, discuss the computation/communication complexity and accuracy of the approach, and present performance measurements for $P > 12,000$., Comment: 12 pages, 10 figures, submitted to Computers & Fluids for special issue for the 30th International Conference on Parallel Computational Fluid Dynamics

We investigate efficient algorithms and a practical implementation of an explicit-type high-order timestepping method based on Krylov subspace approximations, for possible application to large-scale engineering problems in electromagnetics. We consider a semi-discrete form of the Maxwell's equations resulting from a high-order spectral-element discontinuous Galerkin discretization in space whose solution can be expressed analytically by a large matrix exponential of dimension $$\kappa \times \kappa $$ ? × ? . We project the matrix exponential into a small Krylov subspace by the Arnoldi process based on the modified Gram---Schmidt algorithm and perform a matrix exponential operation with a much smaller matrix of dimension $$m\times m$$ m × m ( $$m\ll \kappa $$ m ? ? ). For computing the matrix exponential, we obtain eigenvalues of the $$m\times m$$ m × m matrix using available library packages and compute an ordinary exponential function for the eigenvalues. The scheme involves mainly matrix-vector multiplications, and its convergence rate is generally $$O(\Delta t^{m-1})$$ O ( Δ t m - 1 ) in time so that it allows taking a larger timestep size as $$m$$ m increases. We demonstrate CPU time reduction compared with results from the five-stage fourth-order Runge---Kutta method for a certain accuracy. We also demonstrate error behaviors for long-time simulations. Case studies are also presented, showing loss of orthogonality that can be recovered by adding a low-cost reorthogonalization technique.

We investigate stability properties of the spectral element method for advection dominated incompressible flows. In particular, properties of the widely used convective form of the nonlinear term are studied. We remark that problems which are usually associated with the nonlinearity of the governing Navier---Stokes equations also arise in linear scalar transport problems, which implicates advection rather than nonlinearity as a source of difficulty. Thus, errors arising from insufficient quadrature of the convective term, commonly referred to as `aliasing errors', destroy the skew-symmetric properties of the convection operator. Recovery of skew-symmetry can be efficiently achieved by the use of over-integration. Moreover, we demonstrate that the stability problems are not simply connected to underresolution. We combine theory with analysis of the linear advection-diffusion equation in 2D and simulations of the incompressible Navier---Stokes equations in 2D of thin shear layers at a very high Reynolds number and in 3D of turbulent and transitional channel flow at moderate Reynolds number. For the Navier---Stokes equations, where the divergence-free constraint needs to be enforced iteratively to a certain accuracy, small divergence errors can be detrimental to the stability of the method and it is therefore advised to use additional stabilization (e.g. so-called filter-based stabilization, spectral vanishing viscosity or entropy viscosity) in order to assure a stable spectral element method.

In this paper, a spectral-element discontinuous Galerkin (SEDG) lattice Boltzmann discretization and an exponential time-marching scheme are used to study the flow field past two circular cylinders in tandem arrangement. The basic idea is to discretize the streaming step of the lattice Boltzmann equation by using the SEDG method to get a system of ordinary differential equations (ODEs) whose exact solutions are expressed by using a large matrix exponential. The approximate solution of the resulting ODEs are obtained from a projection method based on a Krylov subspace approximation. This approach allows us to approximate the matrix exponential of a very large and sparse matrix by using a matrix of much smaller dimension. The exponential time integration scheme is useful especially when computations are carried out at high Courant-Friedrichs-Lewy (CFL) numbers, where most explicit time-marching schemes are inaccurate. Simulations of flow were carried out for a circular cylinder at Re=20 and for two circular cylinders in tandem at Re=40 and a spacing of 2.5D, where D is the diameter of the cylinders. We compare our results with those from a fourth-order Runge-Kutta scheme that is restricted by the CFL number. In addition, important flow parameters such as the drag coefficients of the two cylinders and the wake length behind the rear cylinder were calculated by using the exponential time integration scheme. These results are compared with results from our simulation using the RK scheme and with existing benchmark results.

We investigate the stability of a temporal discretization of interface terms in grid overlapping methods. A matrix stability analysis is performed on a model problem of the one-dimensional diffusion equation on overlapping grids. The scheme stability is first analyzed theoretically, and a proof of the unconditional stability of the first-order interface extrapolation scheme with the first- and second-order time integration for any overlap size is presented. For the higher-order schemes, we obtain explicit estimates of the spectral radius of the corresponding discrete matrix operator and document the values of the stability threshold depending on the number of grid points and the size of overlap. The influence of iterations on stability properties is also investigated. Numerical experiments are then presented relating the obtained stability bounds to the observed numerical values. Semidiscrete analysis confirms the derived scaling for the stability bounds.

This paper presents analytical and numerical results for two new anisotropic modifications of the Rational and Clark-α LES models. The main difference from their standard form is that in this study horizontal (as opposed to isotropic) spatial filtering is used, which is appropriate for turbulent mixing in stratified flows. We present several mathematical results regarding the horizontal Rational and Clark-α LES models. We also present numerical experiments that support the analytical developments and show that both horizontal LES models perform better than their standard, isotropic counterparts in approximating mixing in a 3D lock-exchange problem at Reynolds number Re=10,000.

In the present contribution the concept of isogeometric analysis is extended towards the numerical solution of the problem of gradient elasticity in two dimensions. In gradient elasticity the strain energy becomes a function of the strain and its derivative. This assumption results in a governing differential equation which contains fourth order derivatives of the displacements. The numerical solution of this equation with a displacement-based finite element method requires the use of C 1-continuous elements, which are mostly limited to two dimensions and simple geometries. This motivates the implementation of the concept of isogeometric analysis for gradient elasticity. This NURBS based interpolation scheme naturally includes C 1 and higher order continuity of the approximation of the displacements and the geometry. The numerical approach is implemented for two-dimensional problems of linear gradient elasticity and its convergence behavior is studied.

We present a natural element method to treat higher-order spatial derivatives in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear partial differential equation that allows to model phase separation in binary mixtures. Standard classical \({{\mathcal{C}}^0}\)-continuous finite element solutions are not suitable because primal variational formulations of fourth-order operators are well-defined and integrable only if the finite element basis functions are piecewise smooth and globally \({{\mathcal{C}}^1}\)-continuous. To ensure \({{\mathcal{C}}^1}\)-continuity, we develop a natural-element-based spatial discretization scheme. The \({{\mathcal{C}}^1}\)-continuous natural element shape functions are achieved by a transformation of the classical Farin interpolant, which is basically obtained by embedding Sibsons natural element coordinates in a Bernstein–Bezier surface representation of a cubic simplex. For the temporal discretization, we apply the (second-order accurate) trapezoidal time integration scheme supplemented with an adaptively adjustable time step size. Numerical examples are presented to demonstrate the efficiency of the computational algorithm in two dimensions. Both periodic Dirichlet and homogeneous Neumann boundary conditions are applied. Also constant and degenerate mobilities are considered. We demonstrate that the use of \({{\mathcal{C}}^1}\)-continuous natural element shape functions enables the computation of topologically correct solutions on arbitrarily shaped domains.

We propose a novel approach for bridging the Boussinesq equations and the primitive equations. This approach uses spatio-temporal filtering as an alternative to traditional scaling arguments.

In this work we describe a polynomial regression approach that uses derivative information for analyzing the performance of a complex system that is described by a mathematical model depending on s...

In gradient elasticity, the appearance of strain gradients in the free energy density leads to the need of C 1 continuous discretization m ethods. In the present work, the performances of C 1 finite elements and the C 1 Natural Element Method (NEM) are compared. The triangular Argyris and Hsieh-Clough-Tocher finite elements are reparametrized in terms of the Bernstein polynomials. The quadrilateral Bogner-Fox- Schmidt element is used in an isoparametric framework, for which a preprocessing algorithm is presented. Additionally, the C 1 -NEM is applied to non-linear gradient elasticity. Several numerical examples are analyzed to compare the convergence behavior of the different methods. It will be illustrated that the isoparametric elements and the NEM show a significantly better performance than the triangular elements.

Turbulent mixing of water masses of different temperatures and salinities is an important process for both coastal and large-scale ocean circulation. It is, however, difficult to capture computationally. One of the reasons is that mixing in the ocean occurs at a wide range of complexity, with the Reynolds number reaching Re = O ( 10 6 ) , or even higher. In this study, we continue to investigate whether large eddy simulation ( LES ) can be a reliable computational tool for stratified mixing in turbulent oceanic flows. LES is attractive because it can be O ( 1000 ) times faster than a direct numerical simulation ( DNS ) of stratified mixing in turbulent flows. Before using the LES methodology to compute mixing in realistic oceanic flows, however, a careful assessment of the LES sensitivity with respect to Re needs to be performed first. The main objectives of this study are: (i) to investigate the performance of different LES models at high Re, such as those encountered in oceanic flows; and (ii) to study how mixing varies as a function of Re. To this end, as a benchmark we use the lock-exchange problem, which is described by unambigous and simple initial and boundary conditions. The background potential energy, which accurately quantifies irreversible mixing in an enclosed system, is used as the main criterion in a posteriori testing of LES . This study has two main achievements. The first is that we investigate the accuracy of six combinations of two different classes of LES models, namely eddy-viscosity and approximate deconvolution types, for 3 × 10 3 ⩽ Re ⩽ 3 × 10 4 , for which DNS data is computed. We find that all LES models almost always provide significantly more accurate results than cases without LES models. Nevertheless, no single LES model that is persistently superior to others over this Re range could be identified. Then, an ensemble of the four best performing LES models is selected in order to estimate mixing taking place in this system at Re = 10 5 and 10 6 , for which DNS is presently not feasible. Thus the second achievement of this study is to quantify mixing taking place in this system over an Re range that changes by three orders of magnitude. We find that the background potential energy increases by about 67% when Re is increased from Re = 10 3 to Re = 10 6 , within the computation period, with the most significant increase taking place from Re = 3 × 10 3 to Re = 10 5 .

We describe a spectral element solution strategy for simulation of weakly turbulent vascular flows. Novel approaches for treatment of outflow boundary conditions and flow division are presented. We demonstrate that several million gridpoints are required to obtain converged rms statistics when uniform refinement is used and discuss possible approaches to cost reduction.

We present a high-order discontinuous Galerkin discretization of the unsteady incompressible Navier-Stokes equations in convection-dominated flows using triangular and tetrahedral meshes. The scheme is based on a semi-explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the Stokes operator. The nonlinear term is discretized in divergence form by using the local Lax-Friedrichs fluxes; thus, local conservativity is inherent. Spatial discretization of the Stokes operator has employed both equal-order (P"k-P"k) and mixed-order (P"k-P"k"-"1) velocity and pressure approximations. A second-order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic Helmholtz equation for each component of the velocity and a consistent Poisson equation for the pressure. The consistent Poisson operator is replaced by an equivalent (in stability and convergence) operator, namely that arising from the interior penalty discretization of the standard Poisson operator with appropriate boundary conditions. This yields a simpler and more efficient method, characterized by a compact stencil size. We show the temporal and spatial behavior of the method by solving some popular benchmarking tests. For an unsteady Stokes problem, second-order temporal convergence is obtained, while for the Taylor vortex test problem on both semi-structured and fully unstructured triangular meshes, spectral convergence with respect to the polynomial degree k is obtained. By studying the Orr-Sommerfeld stability problem, we demonstrate that the P"k-P"k method yields a stable solution, while the P"k-P"k"-"1 formulation leads to unphysical instability. The good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder. We conclude that the proposed discontinuous Galerkin method with the P"k-P"k formulation is an efficient scheme for accurate and stable solution of the unsteady Navier-Stokes equations in convection-dominated flows.

An object-oriented geophysical and astrophysical spectral-element adaptive refinement (GASpAR) code is introduced. Like most spectral-element codes, GASpAR combines finite-element efficiency with spectral-method accuracy. It is also designed to be flexible enough for a range of geophysics and astrophysics applications where turbulence or other complex multiscale problems arise. The formalism accommodates both conforming and non-conforming elements. Several aspects of this code derive from existing methods, but here are synthesized into a new formulation of dynamic adaptive refinement (DARe) of non-conforming h-type. As a demonstration of the code, several new 2D test cases are introduced that have time-dependent analytic solutions and exhibit localized flow features, including the 2D Burgers equation with straight, curved-radial and oblique-colliding fronts. These are proposed as standard test problems for comparable DARe codes. Quantitative errors are reported for 2D spatial and temporal convergence of DARe.

We describe Fourier pseudospectral time-domain simulations, carried out in order to study light interacting with a metallic nanoscale object. The difficulty of using Fourier methods to accurately predict the electromagnetic scattering in such problems arises from the discontinuity in the dielectric function along the surface of the metallic object. Standard Fourier methods lead to oscillatory behavior in approximating solutions that are nonsmooth or that have steep gradients. By applying the Gegenbauer reconstruction technique as a postprocessing method to the Fourier pseudospectral solution, we successfully reduce the oscillations after postprocessing. Our computational results, including comparison with finite-difference time-domain simulations, demonstrate the efficiency and accuracy of the method.

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.

A spectral element semi-Lagrangian (SESL) method for the shallow water equations on the sphere is presented. The sphere is discretized using a hexahedral grid although any grid imaginable can be used as long as it is comprised of quadrilaterals. The equations are written in Cartesian coordinates to eliminate the pole singularity which plagues the equations in spherical coordinates. In a previous paper [Int. J. Numer. Methods Fluids 35 (2001) 869] we showed how to construct an explicit Eulerian spectral element (SE) model on the sphere; we now extend this work to a semi-Lagrangian formulation. The novelty of the Lagrangian formulation presented is that the high order SE basis functions are used as the interpolation functions for evaluating the values at the Lagrangian departure points. This makes the method not only high order accurate but quite general and thus applicable to unstructured grids and portable to distributed memory computers. The equations are discretized fully implicitly in time in order to avoid having to interpolate derivatives at departure points. By incorporating the Coriolis terms into the Lagrangian derivative, the block LU decomposition of the equations results in a symmetric positive-definite pseudo-Helmholtz operator which we solve using the generalized minimum residual method (GMRES) with a fast projection method [Comput. Methods Appl. Mech. Eng. 163 (1998) 193]. Results for eight test cases are presented to confirm the accuracy and stability of the method. These results show that SESL yields the same accuracy as an Eulerian spectral element semi-implicit (SESI) while allowing for time-steps 10 times as large and being up to 70% more efficient.

A spectral element formulation of the atmospheric two-dimensional shallow water equations on the cubed-sphere is described. The equations are written in tensor form using the contravariant and covariant velocity components. A semi-implicit time discretization results in a reduced Schur complement system for the pressure. The Laplacian operator is approximated by the L2 pseudo-Laplacian arising in the PN/PN-2 spectral element formulation of the incompressible Stokes problem. The overlapping Schwarz preconditioner of Fischer, Miller, and Tufo [An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, in Parallel Solution of Partial Differential Equations, IMA Vol. Math. Appl. 120, Springer, New York, 2000, pp. 159--180] based on the fast diagonalization method, is extended to generalized curvilinear coordinates. To obtain a separable operator for the linear finite element tensor-product approximation within each spectral element, extrema of the inverse metric tensor and its determinant are employed. Convergence rates and parallel CPU timings are compared against a block-Jacobi preconditioner.

We describe the development and implementation of an efficient spectral element code for simulating transitional flows in complex three-dimensional domains. Critical to this effort is the use of geometrically nonconforming elements that allow localized refinement in regions of interest, coupled with a stabilized high-order time-split formulation of the semi-discrete Navier–Stokes equations. Simulations of transition in a model of an arteriovenous graft illustrate the potential of this approach in biomechanical applications.

The accurate evaluation of fuel and cladding peak temperatures is of prime importance for nuclear reactor design and safety. The Global Threat Reduction Initiative reactor conversion program often encounters exotic flow geometries in its mission to aid in converting reactors from high-enriched to low-enriched fuel. These geometries can pose modeling challenges. Analysis presented here concerns a reactor with twisted fuel pins that are in direct contact with each other in a large, hexagonal-pitch lattice. The Reynolds number for a unit cell is only 7500. Such flow conditions can present difficulties for standard approaches based on Reynolds-Averaged Navier-Stokes (RANS). Moreover there are no available experimental data and a small expected margin to the limiting cladding surface temperature. Given some of the geometric uncertainties, reducing the turbulence model uncertainty is thus important for meaningful calculations. A computational fluid dynamics model of a full-length unit cell was built using the commercial code STAR-CCM+. Multiple RANS models were employed, which gave disparate results. To provide higher-fidelity data for comparison, given the lack of experimental data, a periodic single-helical-pitch simulation with a Large Eddy Simulation (LES) approach was performed using Nek5000, a massively-parallel spectral-element code. This was compared with single-pitch RANS simulations from STAR-CCM+. Stream-wise velocity profile shape was generally well-represented by RANS. Cross-velocities and peak turbulent kinetic energy (TKE) were underestimated for most of the turbulence models with respect to LES, while mean flow TKE was universally underestimated. The overall results suggest that the Realizable k-e Two-Layer model, which was the best at reproducing the LES TKE distribution, would likely be the most appropriate turbulence model choice for this flow. Future work includes full conjugate heat transfer simulations of 1/6 sectors of fuel assemblies featuring this type of pin lattice.Copyright © 2014 by ASME

A system F of functions {1, 2, …, n}→{1, 2, …, k} has Natarajan dimension at most d if no (d+1)-element subset A⊂X is 2-shattered. A is 2-shattered if for each x∈A there is a 2-element set Vx⊆{1, 2, …, k} such that for any choice of elements cx∈Vx, a function f∈F exists with f(x)=cx for all x∈A. We improve a lower bound of cdkdnd (due to Haussler and Long) for the maximum size of F of Natarajan dimension at most d by a factor somewhat smaller than k (e.g., by k for d=1). The problem of obtaining a tight bound is related to interesting questions in extremal graph theory.

We present a simple filtering procedure for stabilizing the spectral element method (SEM) for the unsteady advection–diffusion and Navier–Stokes equations. A number of example applications are presented, along with basic analysis for the advection–diffusion case.

We consider the modified conjugate gradient procedure for solving A\underline{x}e\underline{b} in which the approximation space is based upon the Krylov space associated with A1/p and \underline{b}, for any integer p. For the square-root MCG (pe2) we establish a sharpened bound for the error at each iteration via Chebyshev polynomials in \sqrt{A}. We discuss the implications of the quickly accumulating effect of an error in \sqrt{A}e\underline{b} in the initial stage, and find an error bound even in the presence of such accumulating errors. Although this accumulation of errors may limit the usefulness of this method when \sqrt{A}e\underline{b} is unknown, it may still be successfully applied to a variety of small, “almost-SPD” problems, and can be used to jump-start the conjugate gradient method. Finally, we verify these theoretical results with numerical tests.

The following problem is known as group testing problem for n objects. Each object can be essential (defective) or non-essential (intact). The problem is to determine the set of essential objects by asking queries adaptively. A query can be identified with a set Q of objects and the query Q is answered by 1 if Q contains at least one essential object and by 0 otherwise. In the statistical setting the objects are essential, independently of each other, with a given probability p < 1 while in the combinatorial setting the number k < n of essential objects is known. The cut-off point of statistical group testing is equal to p∗ = 12 (3 − √5), i.e., the strategy of testing each object individually minimizes the average number of queries iff p ⩾ p∗ or n = 1. In the combinatorial setting the worst case number of queries is of interest. It has been conjectured that the cut-off point of combinatorial group testing is equal to x∗ = 13 i.e., the strategy of testing n − 1 objects individually minimizes the worst case number of queries iff kn ⩾ α∗ and k < n. Some results in favor of this conjecture are proved.

We develop a class of filters based upon the numerical solution of high-order elliptic problems in ℝd which allow for independent determination of order and cut-off wave number and which default to classical Fourier-based filters in homogeneous domains. However, because they are based on the solution of a PDE, the present filters are not restricted to applications in tensor-product based geometries as is generally the case for Fourier-based filters. The discrete representation of the filtered output is constructed from a Krylov space generated in solving a well-conditioned system arising from a low-order PDE. Copyright © 1999 John Wiley & Sons, Ltd.

We present two projection techniques for computing approximate solutions to linear systems of the form Ax^n=b^n, for a sequence n=1,2,..., e.g., such as arises from time discretization of a partial differential equation. The inexpensive approximate solutions can be used as initial guesses for iterative solution of the system, resulting in significantly reduced computational expense. Examples of two- and three-dimensional incompressible Navier-Stokes calculations are presented in which x represents the pressure, and A is a discrete Poisson operator. In flows containing significant dynamic activity, these projection techniques lead to as much as a two-fold reduction in solution time.

In this note, we examine a modified conjugate gradient procedure for solving A {\underline x} = {\underline b} in which the approximation space is based upon the Krylov space (({\mathscr K}^k_{\sqrt {A}, {\underline b}})) associated with sqrt {A} and {\underline b}. We show that, given initial vectors {\underline b} and \sqrt {A} \, {\underline b} (possibly computed at some expense), the best fit solution in {\mathscr K}^k_{\sqrt {A}, {\underline b}} can be computed using a finite-term recurrence requiring only one multiplication by A per iteration. The initial convergence rate appears, as expected, to be twice as fast as that of the standard conjugate gradient method, but stability problems cause the convergence to be degraded.

Efficient solution of the Navier?Stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems. For unsteady incompressible flows, the pressure operator is the leading contributor to stiffness, as the characteristic propagation speed is infinite. In the context of operator splitting formulations, it is the pressure solve which is the most computationally challenging, despite its elliptic origins. We examine several preconditioners for the consistent L2Poisson operator arising in the PN? PN?2spectral element formulation of the incompressible Navier?Stokes equations. We develop a finite element-based additive Schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior Gauss points. For large two-dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation.

This paper investigates the problem where one is given a finite set of n points in the plane each of which is labeled either “positive” or “negative”. We consider bounded convex polygons, the vertices of which are positive points and which do not contain any negative point. It is shown how such a polygon which is maximal with respect to area can be found in time O(n3logn). With the same running time one can also find such a polygon which contains a maximum number of positive points. If, in addition, the number of vertices of the polygon is restricted to be at most M, then the running time becomes O(Mn3logn). It is also shown how to find a maximum convex polygon which contains a given point in time O(n3logn). Two parallel algorithms for the basic problem are also presented. The first one runs in time O(nlogn) using O(n2) processors, the second one has polylogarithmic time but needs O(n7) processors. Instead of using the area or the number of positive points contained in the polygon as the quantity to be maximized one may also use other measures fulfilling a certain additive property, however, this may affect the running time.

We demonstrate the use of advanced linear stability tools developed for the spectral-element code Nek5000 to investigate the dynamics of nonlinear flows in moderately complex geometries. The aim of stability calculations is to identify the driving mechanism as well as the region most sensitive to the instability: the wavemaker. We concentrate on global linear stability analysis, which considers the linearised Navier–Stokes equations and searches for growing small disturbances, i.e. so-called linear global modes. In the structural sensitivity analysis these modes are associated to the eigenmodes of the direct and adjoint linearised Navier–Stokes operators, and the wavemaker is defined as the overlap of the strongest direct and adjoint eigenmodes. The large eigenvalue problems are solved using matrix-free methods adopting the time-stepping Arnoldi approach. We present here our implementation in Nek5000 with the ARPACK library on a number of test cases.

Contents: J. Bideaud, Introduction. Part I:Encounters with Number and First Acquisitions. C. Sophian, Learning About Numbers: Lessons for Mathematics Education From Preschool Number Development. R. Brissiaud, A Toll for Number Construction: Finger Symbol Sets. S. Parrat-Dayan, J. Von che, Conservation and the Notion of "Half." L.P. Steffe, Learning Stages in the Construction of the Number Sequence. A.J. Baroody, The Development of Preschoolers' Counting Skills and Principles. K.C. Fuson, Relationships Between Counting and Cardinality From Age 2 to Age 8. Part II:Structure of the Numerical Domain. L. Tollefsrud-Anderson, R.L. Campbell, P. Starkey, R.G. Cooper, Jr., Number Conservation: Distinguishing Quantifier from Operator Solutions. R. Gelman, E.B. Meck, Early Principles Aid Initial but Not Later Conceptions of Number. J-P. Fischer, Subitizing: The Discontinuity After Three. M. Fayol, From Number to Numbers in Use: Solving Arithmetic Problems. G. Vergnaud, The Appropriation of the Concept of Number: A Lengthy Process. Part III:Theories and Methods: Critical Approaches. R. Droz, The Multiple Roots of Natural Numbers and Their Multiple Interpretations. X. S ron, G. Deloche, M.P. No l, Number Transcribing by Children: Writing Arabic Numbers Under Dictation. M. Saada-Robert, Understanding the Microgenesis of Number: Sequence Analysis. K.C. Fuson, Y. Kwon, Learning Addition and Subtraction: Effects of Number Words and Other Cultural Tools. Part IV:Difficulties and Remediation. A.J. Baroody, Remedying Common Counting Difficulties. C. Meljac, Conclusion.Some Unforeseen Variants on the Number Construction Scenario. J. Bideaud, Assessment and Perspectives.

A pac-learning algorithm isd-space bounded, if it stores at mostd examples from the sample at any time. We characterize thed-space learnable concept classes. For this purpose we introduce the compression parameter of a concept classb and design our Trial and Error Learning Algorithm. We show: b isd-space learnable if and only if the compression parameter ofb is at mostd. This learning algorithm does not produce a hypothesis consistent with the whole sample as previous approaches e.g. by Floyd, who presents consistent space bounded learning algorithms, but has to restrict herself to very special concept classes. On the other hand our algorithm needs large samples; the compression parameter appears as exponent in the sample size. We present several examples of polynomial time space bounded learnable concept classes: We further relate the compression parameter to the VC-dimension, and discuss variants of this parameter.

In this paper, we investigate the problem of classifying objects which are given by feature vectors with Boolean entries. Our aim is to “(efficiently) learn probably almost optimal classifications” from examples. A classical approach in pattern recognition uses empirical estimations of the Bayesian discriminant functions for this purpose. We analyze this approach for different classes of distribution functions of Boolean features:kth order Bahadur–Lazarsfeld expansions andkth order Chow expansions. In both cases, we obtain upper bounds for the required sample size which are small polynomials in the relevant parameters and which match the lower bounds known for these classes. Moreover, the learning algorithms are efficient.