COLLECTIONS

Documents published in Scipedia

• Adaptive finite element simulation of incompressible flows by hybrid continuous-discontinuous Galerkin formulations

  S. Badia, J. Baiges

  SIAM journal on scientific computing (2013). Vol. 35 (1), pp. A491-A516 (Preprint)

  
  

  Abstract

  In this work we design hybrid continuous-discontinuous finite element spaces that permit discontinuities on nonmatching element interfaces of nonconforming meshes. Then we develop an equal-order stabilized finite element formulation for incompressible flows over these hybrid spaces, which combines the element interior stabilization of SUPG-type continuous Galerkin formulations and the jump stabilization of discontinuous Galerkin formulations. Optimal stability and convergence results are obtained. For the adaptive setting, we use a standard error estimator and marking strategy. Numerical experiments show the optimal accuracy of the hybrid algorithm for both uniformly and adaptively refined nonconforming meshes. The outcome of this work is a finite element formulation that can naturally be used on nonconforming meshes, as discontinuous Galerkin formulations, while keeping the much lower CPU cost of continuous Galerkin formulations.

  Abstract
  In this work we design hybrid continuous-discontinuous finite element spaces that permit discontinuities on nonmatching element interfaces of nonconforming meshes. Then we [...]

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• Unconditionally stable operator splitting algorithms for the incompressible magnetohydrodynamics (MHD) system discretized by a stabilized finite element formulation based on projections

  S. Badia, R. Planas, J. Gutiérrez-Santacreu

  International Journal for Numerical Methods in Engineering (2013). Vol. 93 (3), pp. 302-328

  
  

  Abstract

  In this article, we propose different splitting procedures for the transient incompressible magnetohydrodynamics (MHD) system that are unconditionally stable. We consider two levels of splitting, on one side we perform the segregation of the fluid pressure and magnetic pseudo‐pressure from the vectorial fields computation. At the second level, the fluid velocity and induction fields are also decoupled. This way, we transform a fully coupled indefinite multi‐physics system into a set of smaller definite ones, clearly reducing the CPU cost. With regard to the finite element approximation, we stick to an unconditionally convergent stabilized finite element formulation because it introduces convection stabilization, allows to circumvent inf‐sup conditions (clearly simplifying implementation issues), and is able to capture non‐smooth solutions of the magnetic subproblem. However, residual‐based finite element formulations are not suitable for segregation, because they lose the skew‐symmetry of the off‐diagonal blocks. Therefore, in this work, we have proposed a novel term‐by‐term stabilization of the MHD system based on projections that is still unconditionally convergent.

  Abstract
  In this article, we propose different splitting procedures for the transient incompressible magnetohydrodynamics (MHD) system that are unconditionally stable. We consider [...]

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• On stabilized finite element methods based on the Scott-Zhang projector: circumventing the inf-sup condition for the Stokes problem

  S. Badia

  Computer Methods in Applied Mechanics and Engineering (2012). Vol. 247-248, pp. 65-72

  
  

  Abstract

  In this work we propose a stabilized nite element method that permits us to circumvent discrete inf-sup conditions, e.g. allowing equal order interpolation. The type of method we propose belongs to the family of symmetric stabilization techniques, which are based on the introduction of additional terms that penalize the di erence between some quantities, i.e. the pressure gradient in the Stokes problem, and their nite element projections. The key feature of the formulation we propose is the de nition of the projection to be used, a non-standard Scott-Zhang projector that is well-de ned for L1() functions. The resulting method has some appealing features: the projector is local and nested meshes or enriched spaces are not required.

  Abstract
  In this work we propose a stabilized nite element method that permits us to circumvent discrete inf-sup conditions, e.g. allowing equal order interpolation. The type [...]

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• An Overview on Numerical Analyses of Nematic Liquid Crystal Flows

  S. Badia, F. Guillén-González, J. Gutiérrez-Santacreu

  Archives of Computational Methods in Engineering (2011). Vol. 18 (3), pp. 285-313

  
  

  Abstract

  The purpose of this work is to provide an overview of the most recent numerical developments in the field of nematic liquid crystals. The Ericksen-Leslie equations govern the motion of a nematic liquid crystal. This system, in its simplest form, consists of the Navier-Stokes equations coupled with an extra anisotropic stress tensor, which represents the effect of the nematic liquid crystal on the fluid, and a convective harmonic map equation. The sphere constraint must be enforced almost everywhere in order to obtain an energy estimate. Since an almost everywhere satisfaction of this restriction is not appropriate at a numerical level, two alternative approaches have been introduced: a penalty method and a saddle-point method. These approaches are suitable for their numerical approximation by finite elements, since a discrete version of the restriction is enough to prove the desired energy estimate.

  Abstract
  The purpose of this work is to provide an overview of the most recent numerical developments in the field of nematic liquid crystals. The Ericksen-Leslie equations govern [...]

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• Finite element approximation of nematic liquid crystal flows using a saddle-point structure

  S. Badia, F. Guillén-González, J. Gutiérrez-Santacreu

  Journal of Computational Physics (2011). Vol. 230 (4), pp. 1686-1706

  
  

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• Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction

  S. Badia, A. Quaini, A. Quarteroni

  Journal of Computational Physics (2009). Vol. 228 (21), pp. 7986-8014

  
  

  Abstract

  The interaction between a fluid and a poroelastic structure is a complex problem that couples the Navier–Stokes equations with the Biot system. The finite element approximation of this problem is involved due to the fact that both subproblems are indefinite. In this work, we first design residual-based stabilization techniques for the Biot system, motivated by the variational multiscale approach. Then, we state the monolithic Navier–Stokes/Biot system with the appropriate transmission conditions at the interface. For the solution of the coupled system, we adopt both monolithic solvers and heterogeneous domain decomposition strategies. Different domain decomposition methods are considered and their convergence is analyzed for a simplified problem. We compare the efficiency of all the methods on a test problem that exhibits a large added-mass effect, as it happens in hemodynamics applications.

  Abstract
  The interaction between a fluid and a poroelastic structure is a complex problem that couples the Navier–Stokes equations with the Biot system. The finite element [...]

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• Robin-Robin preconditioned Krylov methods for fluid-structure interaction problems

  S. Badia, F. Nobile, C. Vergara

  Computer Methods in Applied Mechanics and Engineering (2009). Vol. 198 (33-36), pp. 2768-2784

  
  

  Abstract

  In this work, we propose a Robin–Robin preconditioner combined with Krylov iterations for the solution of the interface system arising in fluid–structure interaction (FSI) problems. It can be seen as a partitioned FSI procedure and in this respect it generalizes the ideas introduced in [S. Badia, F. Nobile, C. Vergara, J. Comput. Phys. 227 (2008) 7027–7051]. We analyze the convergence of GMRES iterations with the Robin–Robin preconditioner on a model problem and compare its efficiency with some existing algorithms. The method is shown to be very efficient for many challenging fluid–structure interaction problems, such as those characterized by a large added-mass effect or by enclosed fluids. In particular, the possibility to solve balloon-type problems without any special treatment makes this algorithm very appealing compared to the computationally intensive existing approaches.

  Abstract
  In this work, we propose a Robin–Robin preconditioner combined with Krylov iterations for the solution of the interface system arising in fluid–structure [...]

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• Modular vs non-modular preconditioners for fluid-structure systems with large added-mass effect

  S. Badia, A. Quaini, A. Quarteroni

  Computer Methods in Applied Mechanics and Engineering (2008). Vol. 197 (49-50), pp. 4216-4232

  
  

  Abstract

  In this article we address the numerical simulation of fluid–structure interaction (FSI) problems featuring large added-mass effect. We analyze different preconditioners for the coupled system matrix obtained after space–time discretization and linearization of the FSI problem. The classical Dirichlet–Neumann preconditioner has the advantage of “modularity” because it allows to reuse existing fluid and structure codes with minimum effort (simple interface communication). Unfortunately, its performance is very poor in case of large added-mass effects. Alternatively, we consider two non-modular approaches. The first one consists in preconditioning the coupled system with a suitable diagonal scaling combined with an ILUT preconditioner. The system is then solved by a Krylov method. The drawback of this procedure is that the combination of fluid and structure codes to solve the coupled system is not straightforward. The second non-modular approach we consider is a splitting technique based on an inexact block-LU factorization of the linear FSI system. The resulting algorithm computes the fluid velocity separately from the coupled pressure–structure system at each iteration, reducing the computational cost. Independently of the preconditioner, the efficiency of semi-implicit algorithms (i.e., those that treat geometric and fluid nonlinearities in an explicit way) is highlighted and their performance compared to the one of implicit algorithms. All the methods are tested on three-dimensional blood-vessel systems. The algorithm combining the non-modular ILUT preconditioner with Krylov methods proved to be the fastest.

  Abstract
  In this article we address the numerical simulation of fluid–structure interaction (FSI) problems featuring large added-mass effect. We analyze different preconditioners [...]

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• Fluid-structure partitioned procedures based on Robin transmission conditions

  S. Badia, F. Nobile, C. Vergara

  Journal of Computational Physics (2008). Vol. 227 (14), pp. 7027-7051

  
  

  Abstract

  In this article we design new partitioned procedures for fluid–structure interaction problems, based on Robin-type transmission conditions. The choice of the coefficient in the Robin conditions is justified via simplified models. The strategy is effective whenever an incompressible fluid interacts with a relatively thin membrane, as in hemodynamics applications. We analyze theoretically the new iterative procedures on a model problem, which represents a simplified blood-vessel system. In particular, the Robin–Neumann scheme exhibits enhanced convergence properties with respect to the existing partitioned procedures. The theoretical results are checked using numerical experimentation

  Abstract
  In this article we design new partitioned procedures for fluid–structure interaction problems, based on Robin-type transmission conditions. The choice of the coefficient [...]

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• On atomistic-to-continuum coupling by blending

  S. Badia, M. Parks, P. Bochev, M. Gunzburger, R. Lehoucq

  Multiscale modeling and simulation (2008). Vol. 7 (1), pp. 381-406

  
  

  Abstract

  A mathematical framework for the coupling of atomistic and continuum models by blending them over a subdomain subject to a constraint is developed. Using the framework, four classes of atomistic-to-continuum (AtC) blending methods are established, their consistency is studied, and their relative merits are discussed. In addition, the framework helps clarify the origin of ghost forces and formalizes the notion of a patch test. Numerical experiments with the AtC methods are used to illustrate the theoretical results.

  Abstract
  A mathematical framework for the coupling of atomistic and continuum models by blending them over a subdomain subject to a constraint is developed. Using the framework, [...]

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