High-order Galerkin Method Research Articles

A numerical investigation on coupling of conforming and hybridizable interior penalty discontinuous Galerkin methods for fractured groundwater flow problems

The present paper focuses on the numerical modeling of groundwater flows in fractured porous media using the codimensional model description. Therefore, fractures are defined explicitly as a (d−1)-dimensional geometric object immersed in a d-dimensional region and can act arbitrarily as a drain or a barrier. We numerically investigate a novel numerical strategy combining distinctive classes of conforming and nonconforming high-order Galerkin methods, both eligible for static condensation. This procedure is here particularly relevant, leading to a smaller and sparser final system with coupled degrees of freedom solely on the mesh skeleton. Precisely, we combine an inspired hybridizable interior penalty discontinuous Galerkin (HIP) formulation inside the bulk region and a standard continuous Galerkin (CG) approximation on the fracture network. The distinctive discretization of corresponding PDEs and the coupling strategy are rigorously exposed, and the local and global matrix assemblies are detailed. Extensive numerical experiments are then achieved to prove the model’s performances for 2D/3D analytic and realistic benchmarks. Qualitative comparisons are also considered with other discretization methods and commercial software, such as comsol.

Influence investigation of electromagnetic-acoustic transducer parameters on thickness measurement accuracy by numerical modeling methods

The electromagnetic-acoustic method is applied in the control of the electrically conductive products thickness. This method is based on the electrodynamic interaction of eddy currents induced in an electrically conductive material with an external magnetic field. Acoustic waves are generated with multiple reflections from the media interface. The recorded signal reflected allows determining the product thickness. Electromagnetic-acoustic transducer includes a magnetic system, generating and receiving coils. Thickness measurement accuracy of the control object is determined by the geometry of the generating and receiving coils, as well as the size of the gap between them. To assess this effect by the experimental data is rather difficult task. The acoustic wave propagation numerical model in a plate with electromagnetic-acoustic thickness measurement is proposed and developed for the problem solution. The numerical model is implemented in the COMSOL Multiphysics software environment using a discontinuous high-order Galerkin method with time explicit integration scheme. Model adequacy was confirmed using the results of a full-scale experiment, for which a specialized electromagnetic-acoustic thickness gauge with a transducer and a thickness gauge were used. To estimate the uncertainty of thickness measurements, an array of values of the received signal was processed in the MathCad software environment. The adequacy of the model has been confirmed by comparing the simulation results with a full-scale experiment. The influence of the transducer design on the thickness measurement accuracy was estimated. Conclusions are drawn, as well as general recommendations for the development of an electromagnetic-acoustic transducer, and methods for object thickness measuring are given based on the investigation results. The results can be used in the design of an electromagnetic-acoustic transducer and in the development of thickness measurement techniques.

High-order Galerkin method for Helmholtz and Laplace problems on multiple open arcs

We present a spectral Galerkin numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on a finite collection of open arcs in two-dimensional space. A boundary integral method is employed, giving rise to a first kind Fredholm equation whose variational form is discretized using weighted Chebyshev polynomials. Well-posedness of the discrete problems is established as well as algebraic or even exponential convergence rates depending on the regularities of both arcs and excitations. Our numerical experiments show the robustness of the method with respect to number of arcs and large wavenumber range. Moreover, we present a suitable compression algorithm that further accelerates computational times.

Numerical investigation of the combined impact of variable gravity field and throughflow on the onset of convective motion in a porous medium layer

The present article is to examine the joint influence of variable gravity filed and throughflow on the onset of convective motion in a porous medium layer using higher order Galerkin method. The porous layer subjected to a constant upward vertical throughflow and variable downward gravity field which differs with distance across the layer. We discussed the four cases of gravity field variation: (i) G(z) = − z, (ii) G(z) = − z2, (iii) G(z) = − z3 and (iv) G(z) = − (ez − 1). Results indicate that both the throughflow and gravity variation parameters are to postpone the onset of convective motion. The system is observed to be more stable for case (iv), while more unstable for case (iii).

Physics–Dynamics Coupling with Element-Based High-Order Galerkin Methods: Quasi-Equal-Area Physics Grid

Atmospheric modeling with element-based high-order Galerkin methods presents a unique challenge to the conventional physics–dynamics coupling paradigm, due to the highly irregular distribution of nodes within an element and the distinct numerical characteristics of the Galerkin method. The conventional coupling procedure is to evaluate the physical parameterizations ( physics) on the dynamical core grid. Evaluating the physics at the nodal points exacerbates numerical noise from the Galerkin method, enabling and amplifying local extrema at element boundaries. Grid imprinting may be substantially reduced through the introduction of an entirely separate, approximately isotropic finite-volume grid for evaluating the physics forcing. Integration of the spectral basis over the control volumes provides an area-average state to the physics, which is more representative of the state in the vicinity of the nodal points rather than the nodal point itself and is more consistent with the notion of a “large-scale state” required by conventional physics packages. This study documents the implementation of a quasi-equal-area physics grid into NCAR’s Community Atmosphere Model Spectral Element and is shown to be effective at mitigating grid imprinting in the solution. The physics grid is also appropriate for coupling to other components within the Community Earth System Model, since the coupler requires component fluxes to be defined on a finite-volume grid, and one can be certain that the fluxes on the physics grid are, indeed, volume averaged.

Darcy-Brinkman Ferro convection with temperature dependent viscosity

The method of small perturbation is used to examine the variable viscosity effect on Darcy-Brinkman ferroconvection. Takingthe viscosity to be temperature dependent and assuming the fluid and solid matrix to be in local thermal equilibrium, the eigenvalues of the stationary instability are computed using the higher order Galerkin method. The study reveals that viscosity variationeffect is to hasten the threshold of porous mediumferroconvection and its destabilizing effect is enhanced when the magnetic mechanism is effective. Further the convection cell size at the onset of ferroconvection is significantly affected by the variable viscosity effect. The impact of porous and magnetic parameters on the stability is also discussed.

Nonlinear vibration control of a cantilevered fluid-conveying pipe using the idea of nonlinear energy sink

Different from the both-end supported pipe conveying fluid as a conservative system, the cantilevered fluid-transporting pipe is a non-conservative system and its dynamic behavior is more complex with flutter instabilities when the flow velocity is beyond the critical value. Indeed, controlling such a flutter system is always challenging in engineering applications. This study presents nonlinear vibrations of cantilevered pipe conveying fluid passively controlled via a nonlinear energy sink (NES). Based on the Hamilton principle, the nonlinear dynamic equations coupling with the NES are derived and discretized using high-order Galerkin method. It is indicated that increasing the mass and damping of NES results in an increase in critical flow velocity. Importantly, the optimal placed position of NES where the critical flow velocity is highest has a strong relationship with the pipe’s flutter mode. In the following, the nonlinear analysis shows the dynamic controlling effect on vibration amplitude of the pipe can be classified to three suppression regions with increasing the flow velocity. Varying the mass, damping and stiffness of NES is followed by variations of the suppression regions which are associated with controlling effects and dynamic behaviors of the pipe system.

Strong scaling for numerical weather prediction at petascale with the atmospheric model NUMA

Numerical weather prediction (NWP) has proven to be computationally challenging due to its inherent multiscale nature. Currently, the highest resolution global NWP models use a horizontal resolution of 9 km. At this resolution, many important processes in the atmosphere are not resolved. Needless to say, this introduces errors. In order to increase the resolution of NWP models, highly scalable atmospheric models are needed. The non-hydrostatic unified model of the atmosphere (NUMA), developed by the authors at the Naval Postgraduate School, was designed to achieve this purpose. NUMA is used by the Naval Research Laboratory, Monterey as the engine inside its next generation weather prediction system NEPTUNE. NUMA solves the fully compressible Navier–Stokes equations by means of high-order Galerkin methods (both spectral element as well as discontinuous Galerkin methods can be used). NUMA is capable of running middle and upper atmosphere simulations since it does not make use of the shallow-atmosphere approximation. This article presents the performance analysis and optimization of the spectral element version of NUMA. The performance at different optimization stages is analyzed using a theoretical performance model as well as measurements via hardware counters. Machine-independent optimization is compared to machine-specific optimization using Blue Gene (BG)/Q vector intrinsics. The best portable version of the main computations was found to be about two times slower than the best non-portable version. By using vector intrinsics, the main computations reach 1.2 PFlops on the entire IBM Blue Gene supercomputer Mira (12% of the theoretical peak performance). The article also presents scalability studies for two idealized test cases that are relevant for NWP applications. The atmospheric model NUMA delivers an excellent strong scaling efficiency of 99% on the entire supercomputer Mira using a mesh with 1.8 billion grid points. This allows running a global forecast of a baroclinic wave test case at a 3-km uniform horizontal resolution and double precision within the time frame required for operational weather prediction.

Numerical Solution of the Onset of Natural Convection in a Rotating Nanofluid Layer Induced by Purely Internal Heating

In this paper, the influence of rotation on the onset of natural convection made by purely internal heating in a horizontal layer of nanofluid is investigated. The boundaries are taken to be rigid-rigid and the normal flux of volumetric fraction of nanoparticles under the thermophoretic effects is supposed to be zero on the boundaries. The model used for nanofluid combines the effects of Brownian motion and thermophoresis. The purely internal heating problem determines that there is no applied temperature difference across the layer and so the external Rayleigh number is no longer applicable. Therefore, here the important parameter is an internal Rayleigh number, one based on the heat source strength. Linear stability theory based upon normal mode technique is used to find the critical internal Rayleigh number. The influence of rotation, the Lewis number, the modified diffusivity ratio and the nanoparticle Rayleigh number on the onset of convection are observed numerically using the higher order Galerkin Method. It is found that the effect of increasing the Lewis number, the modified diffusivity ratio and the nanoparticle Rayleigh number is to hasten the onset of convection, while rotation has a stabilizing effect on the stability of the system.

Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES

The high order spectral element approximation of the Euler equations is stabilized via a dynamic sub-grid scale model (Dyn-SGS). This model was originally designed for linear finite elements to solve compressible flows at large Mach numbers. We extend its application to high-order spectral elements to solve the Euler equations of low Mach number stratified flows. The major justification of this work is twofold: stabilization and large eddy simulation are achieved via one scheme only.Because the diffusion coefficients of the regularization stresses obtained via Dyn-SGS are residual-based, the effect of the artificial diffusion is minimal in the regions where the solution is smooth. The direct consequence is that the nominal convergence rate of the high-order solution of smooth problems is not degraded. To our knowledge, this is the first application in atmospheric modeling of a spectral element model stabilized by an eddy viscosity scheme that, by construction, may fulfill stabilization requirements, can model turbulence via LES, and is completely free of a user-tunable parameter.From its derivation, it will be immediately clear that Dyn-SGS is independent of the numerical method; it could be implemented in a discontinuous Galerkin, finite volume, or other environments alike. Preliminary discontinuous Galerkin results are reported as well. The straightforward extension to non-linear scalar problems is also described. A suite of 1D, 2D, and 3D test cases is used to assess the method, with some comparison against the results obtained with the most known Lilly–Smagorinsky SGS model.

Influence of magnetic field on the onset of nanofluid convection induced by purely internal heating

This note considers the effect of magnetic field on the onset of convection in a nanofluid layer induced by purely uniform internal heating. The nanofluid layer bounded between two rigid surfaces and also incorporates the effect of Brownian motion along with thermophoresis. The zero nanoparticle flux condition under the thermophoretic effects is considered at the boundaries. The stability condition are found for two sets of thermal boundary conditions namely, case (i) both boundaries isothermal and case (ii) lower insulated and upper isothermal using the higher order Galerkin method. The purely internal heating problem shows that there is no applied temperature difference across the layer and so the external Rayleigh number is no longer appropriate. Therefore, here the relevant parameter is an internal Rayleigh number, one based on the heat source strength. It is found that the critical internal Rayleigh number increases with an increase in the magnetic Chandrasekhar number, while decreases with an increase in the Lewis number, the nanoparticle Rayleigh number and the modified diffusivity ratio. A comparative study between the previously published results and the present results for a special case is found to be in good agreement.

Higher order Galerkin methods in time for free surface flows

AbstractIn many engineering applications, free surface or two‐phase flows are discretized in time with an explicit decoupling of geometry and fluid flow. Such a strategy leads to a capillary CFL condition of the form $\Delta t \le \sqrt{\rm{We}}\,h^{3/2}$ [3]. For the case of surface tension dominated flows (i.e. high Weber number We) this can dictate infeasibly small time steps.As an alternative we suggest a Galerkin method in time based on the discontinuous Galerkin method of first order (dG(1)). For this choice, an energy estimate can be proved [7], so unconditional stability of the method is given. While for ODEs or parabolic PDEs the method is of third order at the discrete points in time tn [4], in the case of free surface flows second order convergence can still be achieved.Numerical examples using the Arbitrary Lagrangian Eulerian (ALE) method for both capillary one‐phase and two‐phase flow demonstrate this convergence order. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

High-order Galerkin methods for scalable global atmospheric models

Three different high-order finite element methods are used to solve the advection problem—two implementations of a discontinuous Galerkin and a spectral element (high-order continuous Galerkin) method. The three methods are tested using a 2D Gaussian hill as a test function, and the relative L 2 errors are compared. Using an explicit Runge–Kutta time stepping scheme, all three methods can be parallelized using a straightforward domain decomposition and are shown to be easily and efficiently scaled across multiple-processor distributed memory machines. The effect of a monotonic limiter on a DG scheme is demonstrated for a non-smooth solution. Additionally, the necessary geometry for implementing these methods on the surface of a sphere is discussed.

Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems

Recently, a discontinuous Galerkin finite element method with plane wave basis functions and Lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of Helmholtz problems in the mid-frequency regime. In this paper, this method is extended to three dimensions and several new elements are proposed. Computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher-order Galerkin method. Copyright © 2005 John Wiley & Sons, Ltd.

On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem

The paper presents a Galerkin numerical method for solving the hyper-singular boundary integral equations for the exterior Helmholtz problem in three dimensions with a Neumann's boundary condition. Previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and Galerkin methods are not available due to the presence of a hypersingular integral operator. This paper proposes a high order Galerkin method by using singularity subtraction technique to reduce the hyper-singular operator to a weakly singular one. Moreover, we show here how to extend the previous work (J. Appl. Numer. Math. 36 (4) (2001) 475–489) on sparse preconditioners to the Galerkin case leading to fast convergence of two iterative solvers: the conjugate gradient normal method and the generalised minimal residual method. A comparison with the collocation method is also presented for the Helmholtz problem with several wavenumbers.

A novel implementation of multilevel fast multipole algorithm for higher order Galerkin's method

A new approach is proposed to reduce the memory requirements of the multilevel fast multipole algorithm (MLFMA) when applied to the higher order Galerkin's method. This approach represents higher order basis functions by a set of point sources such that a matrix-vector multiply is equivalent to calculating the fields at a number of points from given current sources at these points. The MLFMA is then applied to calculate the point-to-point interactions. This permits the use of more levels in MLFMA than applying MLFMA to basis-to-basis interactions directly and, thus, reduces the memory requirements significantly.

Timely Communication: Diagonal Edge Preconditioners in p-version and Spectral Element Methods

Domain decomposition preconditioners for high-order Galerkin methods in two dimensions are often built from modules associated with the decomposition of the discrete space into subspaces of functions related to the interior of elements, individual edges, and vertices. The restriction of the original bilinear form to a particular subspace gives rise to a diagonal block of the preconditioner, and the action of its inverse on a vector has to be evaluated in each iteration. Each block can be replaced by a preconditioner in order to decrease the cost. Knowledge of the quality of this local preconditioner can be used directly in a study of the convergence rate of the overall iterative process. The Schur complement of an edge with respect to the variables interior to two adjacent elements is considered. The assembly and factorization of this block matrix are potentially expensive, especially for polynomials of high degree. It is demonstrated that the diagonal preconditioner of one such block has a condition number that increases approximately linearly with the degree of the polynomials. Numerical results demonstrate that the actual condition numbers are relatively small.

Upwinding of high order Galerkin methods in conduction−convection problems

AbstractUpwinded parabolic and cubic elements are derived on a uniform grid of size h for the finite element Galerkin method applied to the solution of the model conduction−convection problem εu″ — Ku′ = 0, ε, K > 0, subject to the boundary conditions u(0) = 1, u (1) = 0. Extension of the results to more complicated problems is indicated. Finally numerical results are given comparing some of the methods derived for a range of L(=Kh/2ε), the grid Peclet number.

Generalized Collocation Methods for Time-Dependent, Nonlinear Boundary-Value Problems

Abstract This paper briefly describes the development of generalized collocation methods for solving coupled systems of nonlinear, one-dimensional, parabolic, partial differential equations. parabolic, partial differential equations. The methods are applied to several problems that are representative of fluid flow in a porous medium. The numerical results for the problems solved indicate that collocation methods using C piecewise polynomials with Gauss-Legendre collocation points polynomials with Gauss-Legendre collocation points are more efficient than conventional, second-order, finite-difference methods. In particular, for problems whose solutions are smooth, one obtains more accuracy per unit of computer time with increasingly higher-order collocation methods. For problems whose solutions are characterized by a steep, shock-like, transient wave front, one can conclude that increasing the order of the collocation method does not necessarily result in a more efficient method, but high-accuracy collocation methods still appear to be more efficient than conventional, second-order, finite-difference methods. Also, for a given accuracy, collocation methods generally require less computer storage than conventional, second-order, finite-difference methods. Finally, the collocation technique presented here is shown to be not directly applicable to the saturation equation for immiscible flow. Introduction The purpose of this paper is to investigate collocation methods for the numerical solution of nonlinear equations similar to those that describe the flow of a fully compressible fluid in a porous medium and those that describe the process by which one miscible liquid displaces another liquid in a porous medium. The objective is to determine if collocation methods are applicable and economically justified for solving these types of petroleum engineering problems. The conclusions petroleum engineering problems. The conclusions reached are based on the amount of computer time and computer storage that is required to obtain a specified accuracy. Since only two problems are solved, the results may not be extendable to more general problems; however, experience in solving a wide variety of problems indicates that this is the general behavior that might be expected when using collocation methods. Previously published results indicate how Galerkin, collocation, and finite-element methods can be used to solve various problems that arise in petroleum engineering. Most of these papers present a variety of arguments in favor of these present a variety of arguments in favor of these methods. The principal argument is that better answers can be obtained for the same computational effort than by finite-difference methods. Culham and Varga have presented the most convincing evidence in favor of Galerkin methods. However, they only considered piecewise linear and piecewise cubic-basis functions and they only considered one problem. Because of these limitations, it is not problem. Because of these limitations, it is not clear whether their conclusions can be extended to higher-order Galerkin methods or to other types of problems. Also, they did not consider collocation problems. Also, they did not consider collocation methods in any detail. This paper considers the use of increasingly higher-order collocation methods and compares the results with those obtained with the usual second-order, finite-difference method. The Galerkin method is not considered, even though that is the method with which collocation must ultimately compete. This was done for several reasons. A preliminary operation count indicates that Galerkin methods will not be competitive with collocation methods for most problems. Also, it is not clear how to implement the higher-order Galerkin methods to achieve maximum efficiency. These remarks indicate that the relative merits of collocation and Galerkin methods are still uncertain. PROBLEMS SOLVED PROBLEMS SOLVED Two problems are solved to serve as a basis for the comparisons. The first problem, called the "gas flow problem," is the same problem considered by Culham and Varga. This problem is nonlinear and includes a volumetric source term. SPEJ P. 345

Numerical Methods for Time-Dependent, Nonlinear Boundary Value Problems

Abstract This paper presents and examines in detail extensions to the Galerkin method oil solution that make it numerically superior to conventional methods used to solve a certain class of time-dependent, nonlinear boundary value problems. This class of problems includes the equation that describes the flow of a fully compressible fluid in a porous medium. The Galerkin method with several different piecewise polynomial subspaces and a non-Galerkin piecewise polynomial subspaces and a non-Galerkin method specifically employing cubic spline functions are used to approximate the solution of a nonlinear parabolic equation with one spatial variable. With a parabolic equation with one spatial variable. With a known analytic solution of the problem, the accuracies of these approximations are determined and compared with conventional finite-difference approximations. Specially, the various methods are compared on the basis of the amount of computer time necessary to achieve a given accuracy, as well as with respect to the order oil convergence and computer core storage required. These tests indicate that the higher-order Galerkin methods require the least amount of computer time for a given range of accuracy. Introduction The purpose of this paper is to outline in detail the application of the Galerkin method, employing piecewise polynomials, to solve nonlinear piecewise polynomials, to solve nonlinear boundary-value problems and compare the computational efficiency of the Galerkin method with more conventional numerical methods. Numerical methods compared with the Galerkin technique include a non-Galerkin method that utilizes cubic spline interpolation and the conventional finite-difference methods. Four conventional time approximations were also studied in conjunction with the above mentioned space discretization methods. In an earlier paper, Price and Varga showed theoretically that higher-order approximations to certain semilinear convection-diffusion equations were possible by means of Galerkin techniques, but complete numerical results for such approximations were not given. Also, in a paper that introduced the Galerkin method to the petroleum industry, Price et al. demonstrated that higher-order approximations were far superior numerically to the conventional methods used to solve certain linear convection-diffusion type problems. Jennings, Douglas and Dupont and Douglas et al. have considered the application of Galerkin methods to various nonlinear problems, but again complete numerical results, problems, but again complete numerical results, including comprehensive comparisons with existing numerical methods, were not given. Thus, in addition to presenting some new and computationally efficient Galerkin formulations for nonlinear problems and numerically demonstrating their problems and numerically demonstrating their higher order accuracies, it was also desirable to test these. methods to determine if they also exhibited the same superiority in regard to computational efficiency as was demonstrated for the Galerkin methods applied to linear problems. If so, then the Galerkin technique could prove to be an important advancement toward developing faster numerical models for field application. To test and compare each method of solution, a problem involving the nonlinear gas-flow equation problem involving the nonlinear gas-flow equation in one spatial variable with a specific volumetric source term was chosen, for which a closed-form or analytic solution was known. Using this particular problem and its analytic solution, it was possible problem and its analytic solution, it was possible to determine numerically the order of convergence of each method, to compare each method on the basis of computer time expended to obtain a given accuracy, and to compare each method with respect to computer core storage required. In addition, the experimental data were used to define "consistent quadrature" and "consistent interpolation" schemes for the Galerkin methods. Finally, it was possible to formulate conclusions regarding the computational efficiency of the four time approximations investigated. SPEJ P. 374