the hybridized discontinuous galerkin method for implicit large eddy simulation of transitional turbulent flows abstract we present a high order implicit large eddy simulation iles approach for transitional aerodynamic flows the approach encompasses a hybridized discontinuous galerkin dg method for the discretization of the navier stokes ns equations and a parallel preconditioned newton gmres solver for the resulting nonlinear system of equations the combination of hybridized dg methods with an efficient solution procedure leads to a high order accurate ns solver that is competitive to alternative approaches such as finite volume and finite difference codes in terms of computational cost the proposed approach is applied to transitional flows over the naca 65 18 10 compressor cascade and the eppler 387 wing at reynolds numbers up to 460 000 grid convergence studies are presented and the required resolution to capture transition at different reynolds numbers is investigated numerical results show rapid convergence and excellent agreement with experimental data in short this work aims to demonstrate the potential of high order iles for simulating transitional aerodynamic flows this is illustrated through numerical results and supported by theoretical considerations

weight adjustment rule of neural networks for computing discrete 2 d gabor transforms image processing it is demonstrated that the weight adjustment rule used in the neural network for computing the 2 d gabor transform proposed by j daugman 1988 can be shown to be equivalent to the jacobi iteration scheme for solving simultaneous linear equations it is shown that faster convergence of the algorithm can be achieved by using gauss seidel iteration successive overrelaxation conjugate gradient algorithms and multigrid methods

robust and efficient discontinuous galerkin methods for under resolved turbulent incompressible flows abstract we present a robust and accurate discretization approach for incompressible turbulent flows based on high order discontinuous galerkin methods the dg discretization of the incompressible navier stokes equations uses the local lax friedrichs flux for the convective term the symmetric interior penalty method for the viscous term and central fluxes for the velocity pressure coupling terms stability of the discretization approach for under resolved turbulent flow problems is realized by a purely numerical stabilization approach consistent penalty terms that enforce the incompressibility constraint as well as inter element continuity of the velocity field in a weak sense render the numerical method a robust discretization scheme in the under resolved regime the penalty parameters are derived by means of dimensional analysis using penalty factors of order 1 applying these penalty terms in a postprocessing step leads to an efficient solution algorithm for turbulent flows the proposed approach is applicable independently of the solution strategy used to solve the incompressible navier stokes equations i e it can be used for both projection type solution methods as well as monolithic solution approaches since our approach is based on consistent penalty terms it is by definition generic and provides optimal rates of convergence when applied to laminar flow problems robustness and accuracy are verified for the orr sommerfeld stability problem the taylor green vortex problem and turbulent channel flow moreover the accuracy of high order discretizations as compared to low order discretizations is investigated for these flow problems a comparison to state of the art computational approaches for large eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers

complete discrete 2 d gabor transforms by neural networks for image analysis and compression a three layered neural network is described for transforming two dimensional discrete signals into generalized nonorthogonal 2 d gabor representations for image analysis segmentation and compression these transforms are conjoint spatial spectral representations which provide a complete image description in terms of locally windowed 2 d spectral coordinates embedded within global 2 d spatial coordinates in the present neural network approach based on interlaminar interactions involving two layers with fixed weights and one layer with adjustable weights the network finds coefficients for complete conjoint 2 d gabor transforms without restrictive conditions in wavelet expansions based on a biologically inspired log polar ensemble of dilations rotations and translations of a single underlying 2 d gabor wavelet template image compression is illustrated with ratios up to 20 1 also demonstrated is image segmentation based on the clustering of coefficients in the complete 2 d gabor transform

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

variational multiscale turbulence modelling in a high order spectral element method in the variational multiscale vms approach to large eddy simulation les the governing equations are projected onto an a priori scale partitioning of the solution space this gives an alternative framework for designing and analyzing turbulence models we describe the implementation of the vms les methodology in a high order spectral element method with a nodal basis and discuss the properties of the proposed scale partitioning the spectral element code is first validated by doing a direct numerical simulation of fully developed plane channel flow the performance of the turbulence model is then assessed by several coarse grid simulations of channel flow at different reynolds numbers

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

filtering in legendre spectral methods we discuss the impact of modal filtering in legendre spectral methods both on accuracy and stability for the former we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity computational results confirm that less strict necessary conditions appear to be adequate we proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability the results are illustrated by computational experiments

a low dissipation dg method for the under resolved simulation of low mach number turbulent flows abstract in recent years the use of high order discontinuous galerkin dg methods for the under resolved direct numerical simulations udns of turbulent flows has received special attention the suitability of the approach for this kind of applications is related to the dissipation and dispersion proprieties of the scheme while the dispersion errors are small over a broad range of frequencies a relevant dissipation error mainly acts at the smallest under resolved scales resembling a high frequency filter nevertheless it was recognized flad and gassner 2017 mengaldo et al 2018 that the choice of the interface convective numerical flux strongly affects this dissipation behaviour and ultimately the success of the udns approach in this regard the excess of numerical dissipation caused by some upwind numerical convective fluxes must be avoided in particular when dealing with low speed flows since this behaviour is exacerbated approaching the incompressibility limit fixes for the excess of numerical dissipation of these schemes have been proposed by several authors in the context of different numerical methods see for example weiss and smith 1995 in this work a simple modification of the dissipation term of the low mach preconditioned roe scheme proposed by weiss and smith is considered the aim is to reduce further the amount of numerical dissipation with the intent of improving the results of udns a spatial dg discretization coupled with a linearly implicit rosenbrock type time integrator is here considered as a numerical framework perfectly suited for the assessment and comparison of different numerical flux functions results on canonical turbulent flow problems as the taylor green vortex and the flow in a straight sided channel are presented the improved accuracy of the proposed flux function is demonstrated the new low dissipation flux can be useful also in the context of standard lower order finite volume methods

blowup or no blowup the interplay between theory and numerics the question of whether the 3d incompressible euler equations can develop a finite time singularity from smooth initial data has been an outstanding open problem in fluid dynamics and mathematics recent studies indicate that the local geometric regularity of vortex lines can lead to dynamic depletion of vortex stretching guided by the local non blowup theory we have performed large scale computations of the 3d euler equations on some of the most promising blowup candidates our results show that there is tremendous dynamic depletion of vortex stretching the local geometric regularity of vortex lines and the anisotropic solution structure play an important role in depleting the nonlinearity dynamically and thus prevents a finite time blowup

spatial eigensolution analysis of energy stable flux reconstruction schemes and influence of the numerical flux on accuracy and robustness this study focuses on the dispersion and diffusion characteristics of high order energy stable flux reconstruction esfr schemes via the spatial eigensolution analysis framework proposed in 1 the analysis is performed for five esfr schemes where the parameter c dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of esfr methods also referred to as vcjh schemes proposed in 2 in particular we used five values of c two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high order methods namely the esfr recovering two versions of discontinuous galerkin methods and one recovering the spectral difference scheme the performance of each scheme is assessed when using different numerical intercell fluxes e g different levels of upwinding ranging from under to over upwinding in contrast to the more common temporal analysis the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of fr schemes for problems involving non periodic boundary conditions typically found in open flow problems including turbulence unsteady aerodynamics and aeroacoustics

dealiasing techniques for high order spectral element methods on regular and irregular grids high order methods are becoming increasingly attractive in both academia and industry especially in the context of computational fluid dynamics however before they can be more widely adopted issues such as lack of robustness in terms of numerical stability need to be addressed particularly when treating industrial type problems where challenging geometries and a wide range of physical scales typically due to high reynolds numbers need to be taken into account one source of instability is aliasing effects which arise from the nonlinearity of the underlying problem in this work we detail two dealiasing strategies based on the concept of consistent integration the first uses a localised approach which is useful when the nonlinearities only arise in parts of the problem the second is based on the more traditional approach of using a higher quadrature the main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods thereby reducing aliasing driven instabilities we demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations where in the latter both volumetric and interface approximations must be considered we show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations

discontinuous isogeometric analysis methods for the first order form of the neutron transport equation with discrete ordinate sn angular discretisation in this paper two discontinuous galerkin isogeometric analysis methods are developed and applied to the first order form of the neutron transport equation with a discrete ordinate s n angular discretisation the discontinuous galerkin projection approach was taken on both an element level and the patch level for a given non uniform rational b spline nurbs patch this paper describes the detailed dispersion analysis that has been used to analyse the numerical stability of both of these schemes the convergence of the schemes for both smooth and non smooth solutions was also investigated using the method of manufactured solutions mms for multidimensional problems and a 1d semi analytical benchmark whose solution contains a strongly discontinuous first derivative this paper also investigates the challenges posed by strongly curved boundaries at both the nurbs element and patch level with several algorithms developed to deal with such cases finally numerical results are presented both for a simple pincell test problem as well as the c5g7 quarter core mox uox small light water reactor lwr benchmark problem these numerical results produced by the isogeometric analysis iga methods are compared and contrasted against linear and quadratic discontinuous galerkin finite element dgfem s n based methods

a note on smooth tests of goodness of fit for location scale families summary in this note we show that the test statistic given by rayner best 1986 for neyman s 1937 smooth test of goodness of fit for location scale families does not in general have the alleged chisquared distribution as a result gross error may result from using their test strategy the size of this error is obtained for the logistic distribution smooth test

efficient matrix free high order finite element evaluation for simplicial elements with the gap between processor clock speeds and memory bandwidth speeds continuing to increase the use of arithmetically intense schemes such as high order finite element methods continues to be

on the eddy resolving capability of high order discontinuous galerkin approaches to implicit les under resolved dns of euler turbulence we present estimates of spectral resolution power for under resolved turbulent euler flows obtained with high order discontinuous galerkin dg methods the 1 rule based on linear dispersion diffusion analysis introduced by moura et al 2015 10 is here adapted for 3d energy spectra and validated through the inviscid taylor green vortex problem the 1 rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra as the original rule relies on standard upwinding different riemann solvers are tested very good agreement is found for solvers which treat the different physical waves in a consistent manner relatively good agreement is still found for simpler solvers the latter however displayed spurious features attributed to the inconsistent treatment of different physical waves it is argued that in the limit of vanishing viscosity such features might have a significant impact on robustness and solution quality the estimates proposed are regarded as useful guidelines for no model dg based simulations of free turbulence at very high reynolds numbers

on the use of kinetic energy preserving dg schemes for large eddy simulation abstract recently element based high order methods such as discontinuous galerkin dg methods and the closely related flux reconstruction fr schemes have become popular for compressible large eddy simulation les element based high order methods with riemann solver based interface numerical flux functions offer an interesting dispersion dissipation behavior for multi scale problems dispersion errors are very low for a broad range of scales while dissipation errors are very low for well resolved scales and are very high for scales close to the nyquist cutoff in some sense the inherent numerical dissipation caused by the interface riemann solver acts as a filter of high frequency solution components this observation motivates the trend that element based high order methods with riemann solvers are used without an explicit les model added only the high frequency type inherent dissipation caused by the riemann solver at the element interfaces is used to account for the missing sub grid scale dissipation due to under resolution of vortical dominated structures typical for les type setups element based high order methods suffer from stability issues caused by aliasing errors of the non linear flux terms a very common strategy to fight these aliasing issues and instabilities is so called polynomial de aliasing where interpolation is exchanged with projection based on an increased number of quadrature points in this paper we start with this common no model or implicit les iles dg approach with polynomial de aliasing and riemann solver dissipation and review its capabilities and limitations we find that the strategy gives excellent results but only when the resolution is such that about 40 of the dissipation is resolved for more realistic coarser resolutions used in classical les e g of industrial applications the iles dg strategy becomes quite inaccurate we show that there is no obvious fix to this strategy as adding for instance a sub grid scale models on top doesn t change much or in worst case decreases the fidelity even more finally the core of this work is a novel les strategy based on split form dg methods that are kinetic energy preserving the scheme offers excellent stability with full control over the amount and shape of the added artificial dissipation this premise is the main idea of the work and we will assess the les capabilities of the novel split form dg approach when applied to shock free moderate mach number turbulence we will demonstrate that the novel dg les strategy offers similar accuracy as the iles methodology for well resolved cases but strongly increases fidelity in case of more realistic coarse resolutions

stabilisation of spectral hp element methods through spectral vanishing viscosity application to fluid mechanics modelling in this paper we present a formulation of spectral vanishing viscosity svv for the stabilisation of spectral hp element methods applied to the solution of the incompressible navier stokes equations we construct the svv around a filter with respect to an orthogonal expansions and prove that this methodology provides a symmetric semi positive definite svv operator after providing a few simple one and two dimensional examples to demonstrate the utility of the svv we examine how it can be applied to a spectral hp element discretisation of the navier stokes equations using a velocity correction splitting scheme we provide three fluid flow examples to help illustrate the pros and cons of this approach on stability and accuracy 2005 elsevier b v all rights reserved

de aliasing on non uniform grids algorithms and applications we present de aliasing rules to be used when evaluating non linear terms with polynomial spectral methods on non uniform grids analogous to the de aliasing rules used in fourier spectral methods they are based upon the idea of super collocation followed by a galerkin projection of the non linear terms we demonstrate through numerical simulation that both accuracy and stability can be greatly enhanced through the use of this approach we begin by deriving from the numerical quadrature rules used by galerkin type projection methods the number of quadrature points and weights needed for quadratic and cubic non linearities we then present a systematic study of the effects of super collocation when using both a continuous galerkin and a discontinuous galerkin method to solve the one dimensional viscous burgers equation we conclude by examining three direct numerical simulation flow examples incompressible turbulent flow in a triangular duct incompressible turbulent flow in a channel at re 395 and compressible flow past a pitching airfoil at re 45 000

spatial eigenanalysis of spectral hp continuous galerkin schemes and their stabilisation via dg mimicking spectral vanishing viscosity for high reynolds number flows abstract this study considers the spatial eigensolution analysis of spectral hp continuous galerkin cg schemes complementing a recent work by moura et al 2016 15 which addressed cg s temporal analysis while the latter assumes periodic boundary conditions the spatial approach presumes inflow outflow type conditions and therefore provides insights for a different class of problems the linear advection diffusion problem is considered for a wide range of peclet numbers allowing for viscous effects at different intensities the inviscid linear advection case receives particular attention owing to the manifestation of peculiar characteristics previously observed for discontinuous galerkin dg schemes in the limit of strong over upwinding these effects are discussed in detail due to their potential to negatively affect solution quality and numerical stability of under resolved simulations at high reynolds numbers the spectral vanishing viscosity svv technique is subsequently considered as a natural stabilization strategy in the context of linear advection an optimization procedure is employed to match svv diffusion levels to those of dg at appropriate polynomial orders the resulting cg svv discretisations are tested against under resolved computations of spatially developing vortex dominated flows and display excellent robustness at high reynolds numbers along with superior eddy resolving characteristics at higher polynomial orders this highlights the importance of appropriate stabilization techniques to improve the potential of spectral hp cg methods for high fidelity simulations of transitional and turbulent flows including implicit les under resolved dns approaches

image analysis using multigrid relaxation methods image analysis problems posed mathematically as variational principles or as partial differential equations are amenable to numerical solution by relaxation algorithms that are local iterative and often parallel although they are well suited structurally for implementation on massively parallel locally interconnected computational architectures such distributed algorithms are seriously handi capped by an inherent inefficiency at propagating constraints between widely separated processing elements hence they converge extremely slowly when confronted by the large representations of early vision application of multigrid methods can overcome this drawback as we showed in previous work on 3 d surface reconstruction in this paper we develop multiresolution iterative algorithms for computing lightness shape from shading and optical flow and we examine the efficiency of these algorithms using synthetic image inputs the multigrid methodology that we describe is broadly applicable in early vision notably it is an appealing strategy to use in conjunction with regularization analysis for the efficient solution of a wide range of ill posed image analysis problems

data driven smooth tests for composite hypotheses the classical problem of testing goodness of fit of a parametric family is reconsidered a new test for this problem is proposed and investigated the new test statistic is a combination of the smooth test statistic and schwarz s selection rule more precisely as the sample size increases an increasing family of exponential models describing departures from the null model is introduced and schwarz s selection rule is presented to select among them schwarz s rule provides the right dimension given by the data while the smooth test in the right dimension finishes the job theoretical properties of the selection rules are derived under null and alternative hypotheses they imply consistency of data driven smooth tests for composite hypotheses at essentially any alternative

a spectral vanishing viscosity method for large eddy simulations a new simulation approach for high reynolds number turbulent flows is developed combining concepts of monotonicity in nonlinear conservation laws with concepts of large eddy simulation the spectral vanishing viscosity svv first introduced by e tadmor siam j numer anal 26 30 1989 is incorporated into the navier stokes equations for controlling high wavenumber oscillations unlike hyperviscosity kernels the svv approach involves a second order operator which can be readily implemented in standard finite element codes in the work presented here discretization is performed using hierarchical spectral hp methods accommodating effectively an ab initio intrinsic scale separation the key result is that monotonicity is enforced via svv leading to stable discretizations without sacrificing the formal accuracy i e exponential convergence in the proposed discretization several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddy viscosity spectral les of turbulent channel flow in its current implementation the svv approach for controlling the small scales is decoupled from the large scales but a procedure is proposed that will provide coupling similar to the classical les formulation

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

a low storage curvilinear discontinuous galerkin method for wave problems the low storage curvilinear discontinuous galerkin lsc dg method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous galerkin method we perform a semidiscrete a priori convergence analysis of lsc dg and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind dg method on curvilinear domains computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions additional results show that the sufficient conditions for optimal order convergence of lsc dg may also be necessary conditions

a stabilised nodal spectral element method for fully nonlinear water waves we present an arbitrary order spectral element method for general purpose simulation of non overturning water waves described by fully nonlinear potential theory the method can be viewed as a high order extension of the classical finite element method proposed by cai et al 1998 5 although the numerical implementation differs greatly features of the proposed spectral element method include nodal lagrange basis functions a general quadrature free approach and gradient recovery using global l 2 projections the quartic nonlinear terms present in the zakharov form of the free surface conditions can cause severe aliasing problems and consequently numerical instability for marginally resolved or very steep waves we show how the scheme can be stabilised through a combination of over integration of the galerkin projections and a mild spectral filtering on a per element basis this effectively removes any aliasing driven instabilities while retaining the high order accuracy of the numerical scheme the additional computational cost of the over integration is found insignificant compared to the cost of solving the laplace problem the model is applied to several benchmark cases in two dimensions the results confirm the high order accuracy of the model exponential convergence and demonstrate the potential for accuracy and speedup the results of numerical experiments are in excellent agreement with both analytical and experimental results for strongly nonlinear and irregular dispersive wave propagation the benefit of using a high order possibly adapted spatial discretisation for accurate water wave propagation over long times and distances is particularly attractive for marine hydrodynamics applications

equivalence between the energy stable flux reconstruction and filtered discontinuous galerkin schemes the aim of this paper is to demonstrate the equivalence between filtered discontinuous galerkin dg schemes and the energy stable flux reconstruction esfr schemes expanding on previous demonstrations in 1d 1 and for straight sided elements in 3d 2 we first derive the dg and esfr schemes in strong form and compare the respective flux penalization terms while highlighting the implications of the fundamental assumptions for stability in the esfr formulations notably that all esfr scheme correction fields can be interpreted as modally filtered dg correction fields we present the result in the general context of all higher dimensional curvilinear element formulations through a demonstration that there exists a weak form of the esfr schemes which is both discretely and analytically equivalent to the strong form we then extend the results obtained for the strong formulations to demonstrate that esfr schemes can be interpreted as a dg scheme in weak form where discontinuous edge flux is substituted for numerical edge flux correction theoretical derivations are then verified with numerical results obtained from a 2d euler testcase with curved boundaries given the current choice of high order dg type schemes and the question as to which might be best to use for a specific application the main significance of this work is the bridge that it provides between them clearly outlining the similarities between the schemes results in the important conclusion that it is always less efficient to use esfr schemes as opposed to the weak dg scheme when solving problems implicitly

split form nodal discontinuous galerkin schemes with summation by parts property for the compressible euler equations fisher and carpenter 2013 12 found a remarkable equivalence of general diagonal norm high order summation by parts operators to a subcell based high order finite volume formulation this equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux we show that besides the construction of entropy stable high order schemes a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms thus by changing the subcell finite volume flux to a specific choice we are able to generate in a systematic way all common split forms of the compressible euler advection terms such as the ducros splitting and the kennedy and gruber splitting although these split forms are not entropy stable we present a systematic way to prove which of those split forms are at least kinetic energy preserving with this we construct a unified high order split form dg framework we investigate with three dimensional numerical simulations of the inviscid taylorgreen vortex and show that the new split forms enhance the robustness of high order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows in fact we show that for certain test cases the novel split form discontinuous galerkin schemes are more robust than the discontinuous galerkin scheme with over integration

neural network for quadratic optimization with bound constraints a recurrent neural network is presented which performs quadratic optimization subject to bound constraints on each of the optimization variables the network is shown to be globally convergent and conditions on the quadratic problem and the network parameters are established under which exponential asymptotic stability is achieved through suitable choice of the network parameters the system of differential equations governing the network activations is preconditioned in order to reduce its sensitivity to noise and to roundoff errors the optimization method employed by the neural network is shown to fall into the general class of gradient methods for constrained nonlinear optimization and in contrast with penalty function methods is guaranteed to yield only feasible solutions

simulation of underresolved turbulent flows by adaptive filtering using the high order discontinuous galerkin spectral element method scale resolving simulations of turbulent flows in complex domains demand accurate and efficient numerical schemes as well as geometrical flexibility for underresolved situations the avoidance of aliasing errors is a strong demand for stability for continuous and discontinuous galerkin schemes an effective way to prevent aliasing errors is to increase the quadrature precision of the projection operator to account for the non linearity of the operands polynomial dealiasing overintegration but this increases the computational costs extensively in this work we present a novel spatially and temporally adaptive dealiasing strategy by projection filtering we show this to be more efficient for underresolved turbulence than the classical overintegration strategy for this novel approach we discuss the implementation strategy and the indicator details show its accuracy and efficiency for a decaying homogeneous isotropic turbulence and the transitional taylor green vortex and compare it to the original overintegration approach and a state of the art variational multi scale eddy viscosity formulation

a comparison of the dispersion and dissipation errors of gauss and gauss lobatto discontinuous galerkin spectral element methods we examine the dispersion and dissipation properties of the gauss and gauss lobatto discontinuous galerkin spectral element methods dgsems for linear wave propagation problems we show that the inherent underintegration in the gauss lobatto variant can be interpreted as a modal filtering of the highest polynomial mode this in turn has a drastic impact on the dispersion and dissipation relations of the gauss lobatto dgsem compared to the gauss variant we show that the gauss dgsem is typically more accurate than the gauss lobatto variant needing fewer points per wavelength for a given accuracy while on the other hand being more restricted in the explicit time step choice we show that the spectra of the dgsem operators depend on the boundary conditions applied and that the ratio of the time step restrictions of the two schemes depends on the choice of boundary conditions

a large eddy simulation method for dgsem using non linearly optimized relaxation filters abstract in this paper we apply a specifically designed dissipative spatial filter as sub grid scale model within the increasingly popular discontinuous galerkin methods and the closely related flux reconstruction high order methods for large eddy simulation the parameters of the filter kernel are optimized with data obtained from direct numerical simulation that is filtered and used as a ground truth to fit the overall kinetic energy and dissipation rate over time the optimization is carried out for polynomial degree 3 to 10 the optimal kernels are rigorously tested in the limit of infinite reynolds number flows hit and taylor green vortex flow additionally a brief extension to plane turbulent channel flow is given

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

a dynamic variational multiscale method for large eddy simulations on unstructured meshes abstract a dynamic version of the variational multiscale vms method for large eddy simulation les on unstructured meshes of turbulent compressible flows is presented its key features include two variational analogues of germano s algebraic identity and a computationally efficient agglomeration based numerical procedure agglomeration is used both for separating a priori the scales and computing dynamically a smagorinsky parameter and the turbulent prandtl number on unstructured finite volume or finite element meshes numerical predictions of low speed 20 angle of attack turbulent flows past a 6 1 prolate spheroid and a forward swept scaled wing are presented they are contrasted with counterpart results obtained using similar implementations on unstructured meshes of the vms les traditional les and traditional dynamic les methods they are also compared to experimental data when available

on the behaviour of fully discrete flux reconstruction schemes abstract in this study we employ von neumann analyses to investigate the dispersion dissipation group velocity and error properties of several fully discrete flux reconstruction fr schemes we consider three fr schemes paired with two explicit runge kutta erk schemes and two singly diagonally implicit rk sdirk schemes key insights include the dependence of high wavenumber numerical dissipation relied upon for implicit large eddy simulation iles on the choice of temporal scheme and time step size also the wavespeed characteristics of fully discrete schemes and the relative dominance of temporal and spatial errors as a function of wavenumber and time step size are investigated salient properties from the aforementioned theoretical analysis are then demonstrated in practice using linear advection test cases finally a burgers turbulence test case is used to demonstrate the importance of the temporal discretization when using fr schemes for iles

linear dispersion diffusion analysis and its application to under resolved turbulence simulations using discontinuous galerkin spectral hp methods we investigate the potential of linear dispersion diffusion analysis in providing direct guidelines for turbulence simulations through the under resolved dns sometimes called implicit les approach via spectral hp methods the discontinuous galerkin dg formulation is assessed in particular as a representative of these methods we revisit the eigensolutions technique as applied to linear advection and suggest a new perspective to the role of multiple numerical modes peculiar to spectral hp methods from this new perspective secondary eigenmodes are seen to replicate the propagation behaviour of a primary mode so that dg s propagation characteristics can be obtained directly from the dispersion diffusion curves of the primary mode numerical dissipation is then appraised from these primary eigencurves and its effect over poorly resolved scales is quantified within this scenario a simple criterion is proposed to estimate dg s effective resolution in terms of the largest wavenumber it can accurately resolve in a given hp approximation space also allowing us to present points per wavelength estimates typically used in spectral and finite difference methods although strictly valid for linear advection the devised criterion is tested against 1d burgers turbulence and found to predict with good accuracy the beginning of the dissipation range on the energy spectra of under resolved simulations the analysis of these test cases through the proposed methodology clarifies why and how the dg formulation can be used for under resolved turbulence simulations without explicit subgrid scale modelling in particular when dealing with communication limited hardware which forces one to consider the performance for a fixed number of degrees of freedom the use of higher polynomial orders along with moderately coarser meshes is shown to be the best way to translate available degrees of freedom into resolution power

variational multiscale stabilization of high order spectral elements for the advection diffusion equation one major issue in the accurate solution of advection dominated problems by means of high order methods is the ability of the solver to maintain monotonicity this problem is critical for spectral elements where gibbs oscillations may pollute the solution however typical filter based stabilization techniques used with spectral elements are not monotone in this paper residual based stabilization methods originally derived for finite elements are constructed and applied to high order spectral elements in particular we show that the use of the variational multiscale vms method greatly improves the solution of the transport diffusion equation by reducing over and under shoots and can be therefore considered an alternative to filter based schemes we also combine these methods with discontinuity capturing schemes dc to suppress oscillations that may occur in proximity of boundaries or internal layers additional improvement in the solution is also obtained when a method that we call fos for first order subcells is used in combination with vms and dc in the regions where discontinuities occur fos subdivides a spectral element of order p into p 2 subcells and then uses 1st order basis functions and integration rules on every subcell of the element the algorithms are assessed with the solution of classical steady and transient 1d 2d and pseudo 3d problems using spectral elements up to order 16

neyman type smooth tests for location scale families summary the smooth tests of neyman 1937 are modified to permit composite alternatives and a choice of orthonormal functions in their construction these orthonormal functions may be chosen to improve detection of particular alternatives the tests are easy to apply and have useful orthogonal components a test for exponentiality is examined and shown to perform well against its competitors

eigensolution analysis of spectral hp continuous galerkin approximations to advection diffusion problems this study addresses linear dispersion diffusion analysis for the spectral hp continuous galerkin cg formulation in one dimension first numerical dispersion and diffusion curves are obtained for the advection diffusion problem and the role of multiple eigencurves peculiar to spectral hp methods is discussed from the eigencurves behaviour we observe that cg might feature potentially undesirable non smooth dispersion diffusion characteristics for under resolved simulations of problems strongly dominated by either convection or diffusion subsequently the linear advection equation augmented with spectral vanishing viscosity svv is analysed dispersion and diffusion characteristics of cg with svv based stabilization are verified to display similar non smooth features in flow regions where convection is much stronger than dissipation or vice versa owing to a dependency of the standard svv operator on a local peclet number first a modification is proposed to the traditional svv scaling that enforces a globally constant peclet number so as to avoid the previous issues in addition a new svv kernel function is suggested and shown to provide a more regular behaviour for the eigencurves along with a consistent increase in resolution power for higher order discretizations as measured by the extent of the wavenumber range where numerical errors are negligible the dissipation characteristics of cg with the svv modifications suggested are then verified to be broadly equivalent to those obtained through upwinding in the discontinuous galerkin dg scheme nevertheless for the kernel function proposed the full upwind dg scheme is found to have a slightly higher resolution power for the same dissipation levels these results show that improved cg svv characteristics can be pursued via different kernel functions with the aid of optimization algorithms

summation by parts operators for correction procedure via reconstruction the correction procedure via reconstruction cpr formerly known as flux reconstruction is a framework of high order methods for conservation laws unifying some discontinuous galerkin spectral difference and spectral volume methods linearly stable schemes were presented by vincent et al 2011 2015 but proofs of non linear entropy stability in this framework have not been published yet to the knowledge of the authors we reformulate cpr methods using summation by parts sbp operators with simultaneous approximation terms sats a framework popular for finite difference methods extending the results obtained by gassner 2013 for a special discontinuous galerkin spectral element method this reformulation leads to proofs of conservation and stability in discrete norms associated with the method recovering the linearly stable cpr schemes of vincent et al 2011 2015 additionally extending the skew symmetric formulation of conservation laws by additional correction terms entropy stability for burgers equation is proved for general sbp cpr methods not including boundary nodes

on consistency and rate of convergence of flux reconstruction for time dependent problems this study is directed at a rigorous characterization of the consistency and convergence of discontinuous finite element schemes formulated using flux reconstruction fr we show that the fr formulation is consistent for linear advection and converges to the exact solution for any scheme that is stable in the von neumann sense the numerical solution for a scheme of polynomial order p is composed of p 1p 1 eigenmodes of which one and exactly one is physical such that it exhibits the analytical dispersion behavior in the limit of asymptotic grid resolution the remaining p modes are spurious such that the fraction of energy received by them from the initial condition vanishes in the asymptotic limit on grid refinement the rate of convergence of the numerical solution is a function of time starting from a short time rate at t 0 t 0 associated with interpolation and asymptotically approaching a long time rate as t t associated with numerical differentiation both these rates can be inferred directly from the eigensystem of the numerical derivative operator we verify these analytical expectations using simple experiments in 1 d and 2 d

entropy adjoint p adaptive discontinuous galerkin method for the under resolved simulation of turbulent flows this paper presents an approach to mesh adaptation suitable for scale resolving simulations the methodology is based on the entropy adjoint approach which corresponds to a standard output based a

taylor s frozen flow hypothesis in burgers turbulence by detailed analytical treatment of the shock dynamics in the burgers turbulence with large scale forcing we calculate the velocity structure functions between pairs of points displaced both in time and space our analytical treatment verifies the so called taylor s frozen flow hypothesis without relying on any closure and under very general assumptions we discuss the limitation of the hypothesis and show that it is valid up to time scales smaller than the correlation time scale of temporal velocity correlation function we support the analytical calculation by performing numerical simulation of the periodically kicked burgers equation

analysis of the spectral vanishing viscosity method for periodic conservation laws the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid burgers equation is analyzed it is proved that this kind of vanishing viscosity is responsible for spectral decay of those fourier coefficients located toward the end of the computed spectrum consequently the discretization error is shown to be spectrally small independently of whether or not the underlying solution is smooth this in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments

application of discontinuous galerkin method to mechanical 2d problem with arbitrary polygonal and very high order finite elements abstract discontinuous galerkin with finite difference rules dgfd is applied to mechanical plane stress state problem the considered domain is discretized by polygonal mesh the polygonal elements can be for example a hexagon pentagon or just quadrangle or triangle they do not have to be convex and a fish mesh where the elements have fish shapes is used when the elements are rectangular then the orthogonality of chebyshev basis functions can be utilized in such a case very high order approximate solution can be obtained in this work the approximation order exceeds 10 and reaches 60 which in the latter case means 3600 numbers of degrees of freedom in a single element the paper is illustrated by a benchmark example in which the exact solution is recovered by dgfd method for various meshes in the other example the stress concentration is easily recovered by very high order version of dgfd method

connections between the filtered discontinuous galerkin method and the flux reconstruction approach to high order discretizations the purpose of this paper is to provide new insights on the connections that exist between the discontinuous galerkin method dg the flux reconstruction method fr and the recently identified energy stable flux reconstruction method esfr when solving time dependent conservation laws all these schemes appear to be quite similar and it is important to understand how they are related in this paper we first review results on the stability of the discontinuous galerkin method and extend it to the filtered discontinuous galerkin method we then consider the flux reconstruction approach and show its connections with dg in particular we show how the energy stable flux reconstruction method introduced by vincent et al is equivalent to a filtered dg method hence giving a new proof of its stability also it allows the use of the method without having to know the special form of the flux correction polynomials finally we underline some fundamental differences that exist between fr and dg

a conservative staggered grid chebyshev multidomain method for compressible flows we present a new multidomain spectral collocation method that uses a staggered grid for the solution of compressible flow problems the solution unknowns are defined at the nodes of a gauss quadrature rule the fluxes are evaluated at the nodes of a gauss lobatto rule the method is conservative free stream preserving and exponentially accurate a significant advantage of the method is that subdomain corners are not included in the approximation making solutions in complex geometries easier to compute

stability of energy stable flux reconstruction for the diffusion problem using the interior penalty and bassi and rebay ii numerical fluxes for linear triangular elements abstract the flux reconstruction fr method has gained popularity within the research community the approach has been demonstrated to recover high order methods such as the discontinuous galerkin dg method stability analyses have been conducted for a linear advection problem leading to the energy stable flux reconstruction esfr methods also named vincent castonguay jameson huynh vcjh methods esfr schemes can be viewed as dg schemes with modally filtered correction fields using this class of methods the linear advection diffusion problem has been shown to be stable using the local discontinuous galerkin scheme ldg to compute the viscous numerical flux this stability proof has been extended for linear triangular and tetrahedra elements although the ldg scheme is commonly used it requires on particular meshes a wide stencil which raises the computational cost as a consequence many prefer the compact interior penalty ip or the bassi and rebay ii br2 numerical fluxes this article for the first time derives for both schemes a condition on the penalty term to ensure the stability for the linear diffusion equation the derivations are based on gauss legendre flux points and on periodic boundary conditions moreover the article proposes that for both the ip and br2 numerical fluxes the stability of the esfr scheme is independent of the auxiliary correction field a von neumann analysis is conducted to study the maximal time step of various esfr methods

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

a skew symmetric discontinuous galerkin spectral element discretization and its relation to sbp sat finite difference methods this paper shows that the discontinuous galerkin collocation spectral element method with gauss lobatto points dgsem gl satisfies the discrete summation by parts sbp property and can thus be classified as an sbp sat simultaneous approximation term scheme with a diagonal norm operator in the same way sbp sat finite difference schemes can be interpreted as discontinuous galerkin type methods with a corresponding weak formulation based on an inner product formulation common in the finite element community this relation allows the use of matrix vector notation common in the sbp sat finite difference community to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for dgsem gl and diagonal norm sbp sat based on this result a skew symmetric energy stable discretely conservative dgsem gl formulation applicable to general diagonal norm sbp sat schemes for the nonlinear burgers equation is constructed