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A detailed description of GENE-3D, a newly developed global stellarator version of the well established gyrokinetic turbulence code GENE, is provided – along with some initial simulation results. First, the underlying gyrokinetic equations and the use of field-aligned coordinates in non-axisymmetric magnetohydrodynamic equilibria are discussed. Then, various aspects regarding the numerical implementation in GENE-3D are described. Finally, the code is validated and its parallel performance is assessed, along with the influence of numerical precision.

The impact of dynamic models for applications to LES of compressible flows is assessed in the framework of a numerical model based on high order discontinuous finite elements. The projections onto lower dimensional subspaces associated with lower degree basis functions are used as LES filter, along the lines proposed in Variational Multiscale templates. Comparisons with DNS results available in the literature for plane and constricted channel flows at Mach numbers 0.2, 0.7 and 1.5 show clearly that the dynamic models are able to improve the prediction of most key features of the flow with respect to the Smagorinsky models employed so far in a VMS-DG context.

We perform in this paper the numerical analysis of some penalty stabilized solvers for the unsteady Navier---Stokes equations. We consider low-order and high-order methods. The low-order method is a pure penalty method, while the high-order one is a projection-stabilized method. We perform their numerical analysis (stability and convergence) for solutions that only need to bear the natural regularity. In this analysis, the stability is based upon specific inf-sup conditions. No local orthogonality properties are needed for the projection-interpolation operator. The convergence is based upon the representation of the stabilizing terms by means of bubble finite element spaces. We include some numerical tests for realistic flows that confirm the theoretical expectations.

A dynamic model for a fixed-bed reactor for methanol synthesis is presented. The model is compared with its steady state version. The analysis points out that the numerical stability of the dynamic model is improved by opportunely increasing the level of detail. It is appropriate to introduce the diffusion terms, to work with mass fractions, to select good discretization methods for each term of the model equations. Since these aspects are usually neglected in steady state analysis, this paper investigates step-by-step their implementation, emphasizing their importance (I) in the transformation of an original hyperbolic PDE system into a parabolic PDE system; (II) in removing non-physical oscillations generated by first-order systems that may lead to relevant model prediction errors; and (III) in the approximation of the convection terms using the forward formulation, which is more stable and provides more realistic solutions.

We investigate the possibility of reducing the computational burden of LES models by employing local polynomial degree adaptivity in the framework of a high order DG method. A novel degree adaptation technique especially featured to be effective for LES applications is proposed and its effectiveness is compared to that of other criteria already employed in the literature. The resulting locally adaptive approach allows to achieve significant reductions in computational cost of representative LES computations., 39 pages, 10 figures, 5 tables

The Hybrid RANS/LES method proposed is based on the hybrid filter approach introduced by Germano in 2004. In this work, instead of using two explicit models respectively for LES and RANS, we use only a model for LES and we reconstruct the RANS field exploiting the properties of hybrid filter. The reconstruction is obtained from the resolved velocity and the LES subgrid stress tensor. The model has been implemented using a variational approach in a DG-FEM framework and tested for the turbulent channel flow test case. Different configurations for hybrid terms have been compared with DNS data and with pure LES results.

We consider the finite element discretization of a convection-diffusion equation, where the convection term is handled via a fluctuation splitting algorithm. We prove a posteriori error estimates which allow us to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach. Nous considérons une discrétisation par éléments finis d’une équation de convection-diffusion, où un algorithme de décentrage est utilisé pour traiter le terme de convection. Nous prouvons des estimations d’erreur a posteriori qui permettent d’adapter le maillage pour optimiser la discrétisation de ces équations. Des résultats numériques confirment l’intérêt d’une telle approche. Junta de Andalucía

This paper proposes and compares different steady-state models for the methanol synthesis fixed-bed reactor. The main goal is to indicate the pros and cons of these alternative models, to highlight their peculiarities in providing accurate and reliable estimations, and to warn the reader about possible risks involved in certain modeling assumptions. The Lurgi-type shell and tube boiler-reactor mathematical modeling is also laid out, before more complex issues and research, such as dynamic simulation or PDE-based predictive control, are broached.

In this article, we propose a novel discontinuous Galerkin method for convection-diffusion-reaction problems, characterized by three main properties. The first is that the method is hybridizable; t...

In this article, we devise a simulation tool of the dynamics of a tethered buoy, which is a mooring system consisting of a rigid floating body (buoy) connected by an elastic cable (mooring line) to the bottom of the fluid environment. The accuracy and robustness of the overall solution algorithm are based on an appropriate choice of three main ingredients. The first ingredient is the use of a novel mixed finite element formulation for the mooring line, which allows a robust modeling of the cable even in the limit of an infinite value of the Young modulus. The second ingredient is the use of quaternions in the equations of motion of the floating body, instead of Euler angles, which avoids angle singularities in the presence of large rotations. The third ingredient is the use of an implicit method (Backward Euler) for time discretization, in conjunction with a damped Newton method for the solution of the resulting nonlinear system, which avoids computational inefficiency of explicit time integration schemes in the presence of quasi-inextensible cables and a fine spatial grid size. The physical soundness of the mooring system model and the accuracy and robustness of the proposed numerical procedure are validated in the study of a tethered buoy operating under static and dynamic industrial-like working conditions.

A semi-implicit and semi-Lagrangian discontinuous Galerkin method for the shallow water equations is proposed, for applications to geophysical scale flows. A non conservative formulation of the advection equation is employed, in order to achieve a more treatable form of the linear system to be solved at each time step. The method is equipped with a simple p-adaptivity criterion, that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution. Numerical results show that the method captures well the main features of gravity and inertial gravity waves, as well as reproducing correct solutions in nonlinear test cases with analytic solutions. The accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high Courant numbers and with automatic choice of the local approximation degree.

This work provides the theoretical formulation, implementation directives and validation case for the extension of the nonlinear model predictive control from the time domain to the spatio-temporal domain. Effectiveness and online feasibility are demonstrated for industrial processes of distributed nature. The case of methanol synthesis (control of hot-spot temperature and axial position) is selected.

A two dimensional mesoscale atmospheric model is presented based on two systems of dynamical equations, the non-hydrostatic compressible Euler equations and the corresponding hydrostatic system. Both equation systems are discretized with a discontinuous Galerkin method in space and a linear semi-implicit multistep method in time. To cover elementwise polynomial spaces up to order four, exact quadrature rules are used. The method is applied to a terrain following quadrilateral grid. To validate the models, gravity wave propagation and mountain wave experiments are analyzed. The numerical models exhibit the proper wave propagation characteristics and high order convergence rates.

The Discontinuous Galerkin Coastal Ocean Model (DGCOM) is a two-dimensional shallow water model for simulating coastal ocean processes. The article of record may be found at http://www.doi.org/10.1002/fld.2118 We extend the explicit in time high-order triangular discontinuous Galerkin (DG) method to semi-implicit (SI) and then apply the algorithm to the two-dimensional oceanic shallow water equations; we implement high-order SI time-integrators using the backward difference formulas from orders one to six. The reason for changing the time-integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high-order DG methods. Changing the time- integration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element-type area integrals, but also the finite volume-type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time-integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high-order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time-explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high-order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high-order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high-order (HO) time-integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low-order time discretizations.

We present semi-implicit (implicit-explicit) formulations of the compressible Navier-Stokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur complement form of the linear implicit problem. We present results for five different forms of the compressible NSE and describe in detail how to formulate the semi-implicit time-integration method for these equations. Finally, we compare all five equations and compare the semi-implicit formulations of these equations both using the Schur and No Schur forms against an explicit Runge-Kutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semi-implicit formulations are faster than the explicit Runge-Kutta method for all the tests studied, especially if the Schur form is used. While we have used the spectral element method for discretizing the spatial operators, the semi-implicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semi-implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including inertia-gravity waves, density current (i.e., Kelvin-Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of nonreflecting boundary conditions. Therefore, we show results for all five semi-implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: no-flux (reflecting) and nonreflecting boundary conditions (NRBCs). It is shown that the NRBCs exert a strong impact on the accuracy and efficiency of the models. Office of Naval Research Junta de Andalucía German Research Foundation

A new, conservative semi-Lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form. The approach combines the accuracy and conservation properties of the Discontinuous Galerkin (DG) method with the computational efficiency and robustness of Semi-Lagrangian (SL) techniques. Unconditional stability in the von Neumann sense is proved for the proposed discretization in the one-dimensional case. A monotonization technique is then introduced, based on the Flux Corrected Transport approach. This yields a multi-dimensional monotonic scheme for the piecewise constant component of the computed solution that is characterized by a smaller amount of numerical diffusion than standard DG methods. The accuracy and stability of the method are further demonstrated by two-dimensional tracer advection tests in the case of incompressible flows. The comparison with results obtained by standard SL and DG methods highlights several advantages of the new technique.

In this work we deal with the numerical simulation of thermal oxidation in silicon device technology. This application involves the coupled solution of a diffusion–reaction problem and of a mechanical problem. These two problems are mutually dependent through the exchange of stresses and fluxes that are typically post-processed fields in standard finite element approaches, and as such they may suffer from a lack of accuracy and from physical inconsistencies. In this paper, we propose a novel numerical approach to the simulation of the thermal oxidation process, that is characterized by the use of mixed and hybrid finite elements. The main advantage of such formulations is that stresses and fluxes are directly computed quantities, rather than obtained from post-processing techniques. We also address the procedures and the techniques that must be devised for handling the coupled interaction problem and the presence of a computational grid moving in time. The numerical approach we propose is eventually validated on a realistic example of the thermal oxidation process in a local oxidation structure (LOCOS).