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• Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization

  S. Badia, J. Bonilla

  Computer Methods in Applied Mechanics and Engineering (2017). Vol. 313, pp. 133-158 (Preprint)

  
  

  Abstract

  In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).

  Abstract
  In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, [...]

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• Segregated Runge–Kutta time integration of convection-stabilized mixed finite element schemes for wall-unresolved LES of incompressible flows

  O. Colomés, S. Badia

  Computer Methods in Applied Mechanics and Engineering (2017). Vol. 313, pp. 189-215 (Preprint)

  
  

  Abstract

  In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system is carried out using mixed finite element (FE) schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint. These additional terms introduced at the discrete level have been proved to act as implicit LES models. In order to perform meaningful wall-unresolved simulations, we consider a weak imposition of the boundary conditions using a Nitsche’s-type scheme, where the tangential component penalty term is designed to act as a wall law. Next, segregated Runge–Kutta (SRK) schemes (recently proposed by the authors for laminar flow problems) are applied to the LES simulation of turbulent flows. By the introduction of a penalty term on the trace of the acceleration, these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms. SRK schemes are excellent for large-scale simulations, since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level, leading to two uncoupled systems. The pressure system is a Darcy-type problem that can easily be preconditioned using a traditional block-preconditioning scheme that only requires a Poisson solver. At the end, only coercive systems have to be solved, which can be effectively preconditioned by multilevel domain decomposition schemes, which are both optimal and scalable. The framework is applied to the Taylor–Green and turbulent channel flow benchmarks in order to prove the accuracy of the convection-stabilized mixed FEs as LES models and SRK time integrators. The scalability of the preconditioning techniques (in space only) has also been proven for one step of the SRK scheme for the Taylor–Green flow using uniform meshes. Moreover, a turbulent flow around a NACA profile is solved to show the applicability of the proposed algorithms for a realistic problem.

  Abstract
  In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system [...]

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• Balancing domain decomposition by constraints and perturbation

  S. Badia, H. Nguyen

  SIAM journal on numerical anaysis (2016). Vol. 54 (6), pp. 3436-3467

  
  

  Abstract

  In this paper, we formulate and analyze a perturbed formulation of the balancing domain decomposition by constraints (BDDC) method. We prove that the perturbed BDDC has the same polylogarithmic bound for the condition number as the standard formulation. Two types of properly scaled zero-order perturbations are considered: one uses a mass matrix, and the other uses a Robin-type boundary condition, i.e, a mass matrix on the interface. With perturbation, the wellposedness of the local Neumann problems and the global coarse problem is automatically guaranteed, and coarse degrees of freedom can be defined only for convergence purposes but not well-posedness. This allows a much simpler implementation as no complicated corner selection algorithm is needed. Minimal coarse spaces using only face or edge constraints can also be considered. They are very useful in extreme scale calculations where the coarse problem is usually the bottleneck that can jeopardize scalability. The perturbation also adds extra robustness as the perturbed formulation works even when the constraints fail to eliminate a small number of subdomain rigid body modes from the standard BDDC space. This is extremely important when solving problems on unstructured meshes partitioned by automatic graph partitioners since arbitrary disconnected subdomains are possible. Numerical results are provided to support the theoretical findings. -

  Abstract
  In this paper, we formulate and analyze a perturbed formulation of the balancing domain decomposition by constraints (BDDC) method. We prove that the perturbed BDDC [...]

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• Shock capturing techniques for hp-adaptive finite elements

  A. Hierro, S. Badia, P. Kus

  Computer Methods in Applied Mechanics and Engineering (2016). Vol. 309, pp. 532-553 (Preprint)

  
  

  Abstract

  The aim of this work is to propose an hp-adaptive algorithm for discontinuous Galerkin methods that is capable to detect the discontinuities and sharp layers and avoid the spurious oscillation of the solution around them. In order to control the spurious oscillations, artificial viscosity is used with the particularity that it is only applied around the layers where the solution changes abruptly. To do so, a novel troubled-cell detector has been developed in order to mark the elements around those layers and to impose linear order in them. The detector takes advantage of the evolution of the value of the gradient through the adaptive process.

  Abstract
  The aim of this work is to propose an hp-adaptive algorithm for discontinuous Galerkin methods that is capable to detect the discontinuities and sharp layers and avoid the [...]

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• Segregated Runge-Kutta methods for the incompressible Navier-Stokes equations

  O. Colomés, S. Badia

  International Journal for Numerical Methods in Engineering (2015). Vol. 105 (5), pp. 372-400 (Preprint)

  
  

  Abstract

  In this work, we propose Runge-Kutta time integration schemes for the incompressible Navier-Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. The segregated Runge-Kutta methods are motivated as an implicit-explicit Runge-Kutta time integration of the projected Navier-Stokes system onto the discrete divergence-free space, and its re-statement in a velocity-pressure setting using a discrete pressure Poisson equation. We have analysed the preservation of the discrete divergence constraint for segregated Runge-Kutta methods and their relation (in their fully explicit version) with existing half-explicit methods. We have performed a detailed numerical experimentation for a wide set of schemes (from first to third order), including implicit and IMEX integration of viscous and convective terms, for incompressible laminar and turbulent flows. Further, segregated Runge-Kutta schemes with adaptive time stepping are proposed

  Abstract
  In this work, we propose Runge-Kutta time integration schemes for the incompressible Navier-Stokes equations with two salient properties. First, velocity and pressure computations [...]

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• An approach to verification and validation of MHD codes for fusion applications

  S. Smolentsev, S. Badia, R. Bhattacharyay, L. Bühler, L. Chen, Q. Huang, H. Jin, D. Krasnov, D. Lee, E. Valls, C. Mistrangelo, R. Munipalli, M. Ni, D. Pashkevich, A. Patel, G. Pulugundla, P. Satyamurthy, A. Snegirev, V. Sviridov, P. Swain, T. Zhou, O. Zikanov

  Fusion Engineering and Design (2015). Vol. 100 (11/2015), pp. 65-72 (Preprint)

  
  

  Abstract

  We propose a new activity on verification and validation (V&V) of MHD codes presently employed by the fusion community as a predictive capability tool for liquid metal cooling applications, such as liquid metal blankets. The important steps in the development of MHD codes starting from the 1970s are outlined first and then basic MHD codes, which are currently in use by designers of liquid breeder blankets, are reviewed. A benchmark database of five problems has been proposed to cover a wide range of MHD flows from laminar fully developed to turbulent flows, which are of interest for fusion applications: (A) 2D fully developed laminar steady MHD flow, (B) 3D laminar, steady developing MHD flow in a non-uniform magnetic field, (C) quasi-two-dimensional MHD turbulent flow, (D) 3D turbulent MHD flow, and (E) MHD flow with heat transfer (buoyant convection). Finally, we introduce important details of the proposed activities, such as basic V&V rules and schedule. The main goal of the present paper is to help in establishing an efficient V&V framework and to initiate benchmarking among interested parties. The comparison results computed by the codes against analytical solutions and trusted experimental and numerical data as well as code-to-code comparisons will be presented and analyzed in companion paper/papers

  Abstract
  We propose a new activity on verification and validation (V&V) of MHD codes presently employed by the fusion community as a predictive capability tool for liquid metal [...]

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• On discrete maximum principles for discontinuous Galerkin methods

  S. Badia, A. Hierro

  Computer Methods in Applied Mechanics and Engineering (2015). Vol. 286, pp. 107-122 (Preprint)

  
  

  Abstract

  The aim of this work is to propose a monotonicity-preserving method for discontinuous Galerkin (dG) approximations of convection–diffusion problems. To do so, a novel definition of discrete maximum principle (DMP) is proposed using the discrete variational setting of the problem, and we show that the fulfilment of this DMP implies that the minimum/maximum (depending on the sign of the forcing term) is on the boundary for multidimensional problems. Then, an artificial viscosity (AV) technique is designed for convection-dominant problems that satisfies the above mentioned DMP. The noncomplete stabilized interior penalty dG method is proved to fulfil the DMP property for the one-dimensional linear case when adding such AV with certain parameters. The benchmarks for the constant values to satisfy the DMP are calculated and tested in the numerical experiments section. Finally, the method is applied to different test problems in one and two dimensions to show its performance.

  Abstract
  The aim of this work is to propose a monotonicity-preserving method for discontinuous Galerkin (dG) approximations of convection–diffusion problems. To do so, a novel [...]

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• Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

  S. Badia, A. Martín, R. Planas

  Journal of Computational Physics (2014). Vol. 274, pp. 562-591

  
  

  Abstract

  The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling. This system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the different physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires efficient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 x 2 block matrix, and consider an LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknowns, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a flexible and general software design for the code implementation of this type of preconditioners.

  Abstract
  The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic [...]

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• Convergence towards weak solutions of the Navier-Stokes equations for a finite element approximation with numerical subgrid-scale modelling

  S. Badia, J. Gutiérrez-Santacreu

  IMA Journal of Numerical Analysis (2014). Vol. 34 (3), pp. 1193-1221

  
  

  Abstract

  Residual-based stabilized finite element (FE) techniques for the Navier-Stokes equations lead to numerical discretizations that provide convection stabilization as well as pressure stability without the need to satisfy an inf-sup condition. They can be motivated by using a variational multiscale (VMS) framework, based on the decomposition of the fluid velocity into a resolvable FE component plus a modelled subgrid-scale component. The subgrid closure acts as a large eddy simulation turbulence model, leading to accurate under-resolved simulations. However, even though VMS formulations are increasingly used in the applied FE community, their numerical analysis has been restricted to a priori estimates and convergence to smooth solutions only, via a priori error estimates. In this work, we prove that some versions of these methods (based on dynamic and orthogonal closures) also converge to weak (turbulent) solutions of the Navier-Stokes equations. These results are obtained by using compactness results in Bochner-Lebesgue spaces.

  Abstract
  Residual-based stabilized finite element (FE) techniques for the Navier-Stokes equations lead to numerical discretizations that provide convection stabilization as well as [...]

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• On monotonicity-preserving stabilized finite element approximations of transport problems

  S. Badia, A. Hierro

  SIAM Journal on Scientific Computing (2014). Vol. 36 (6), pp. A2673-A2697

  
  

  Abstract

  The aim of this work is to design monotonicity-preserving stabilized finite element techniques for transport problems as a blend of linear and nonlinear (shock-capturing) stabilization. As linear stabilization, we consider and analyze a novel symmetric projection stabilization technique based on a local Scott--Zhang projector. Next, we design a weighting of the aforementioned linear stabilization such that, when combined with a finite element discretization enjoying a discrete maximum principle (usually attained via nonlinear stabilization), it does not spoil these monotonicity properties. Then, we propose novel nonlinear stabilization schemes in the form of an artificial viscosity method where the amount of viscosity is proportional to gradient jumps at either finite element boundaries or nodes. For the nodal scheme, we prove a discrete maximum principle for time-dependent multidimensional transport problems. Numerical experiments support the numerical analysis and we show that the resulting methods provide excellent results. In particular, we observe that the proposed nonlinear stabilization techniques do an excellent job eliminating oscillations around shocks

  Abstract
  The aim of this work is to design monotonicity-preserving stabilized finite element techniques for transport problems as a blend of linear and nonlinear (shock-capturing) [...]

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