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  • A domain decomposition strategy for reduced order models. Application to the incompressible Navier-Stokes equations

     Baiges Aznar, Joan; Codina Rovira, Ramon; Idelsohn Barg, Sergio Rodolfo
    Computer methods in applied mechanics and engineering
    Vol. 267, p. 23-42
    DOI: 10.1016/j.cma.2013.08.001
    Date of publication: 2013-12
    Journal article

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    In this work, a domain decomposition strategy for non-linear hyper-reduced-order models is presented. The basic idea consists of restricting the reduced-order basis functions to the nodes belonging to each of the subdomains into which the physical domain is partitioned. An extension of the proposed domain decomposition strategy to a hybrid full-order/reduced-order model is then described. The general domain decomposition approach is particularized for the reduced-order finite element approximation of the incompressible Navier-Stokes equations with hyper-reduction. When solving the reduced incompressible Navier-Stokes equations, instabilities in the form of large gradients of the recovered reduced-order unknown at the subdomain interfaces may appear, which is the motivation for the design of additional stability terms giving rise to penalty matrices. Numerical examples illustrate the behavior of the proposed method for the simulation of the reduced-order systems, showing the capability of the approach to adapt to configurations which are not present in the original snapshot set.

  • Explicit reduced-order models for the stabilized finite element approximation of the incompressible Navier–Stokes equations

     Baiges Aznar, Joan; Codina Rovira, Ramon; Idelsohn Barg, Sergio Rodolfo
    International journal for numerical methods in fluids
    Vol. 72, num. 12, p. 1219-1243
    DOI: 10.1002/fld.3777
    Date of publication: 2013-08
    Journal article

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    In this paper, we present an explicit formulation for reduced-order models of the stabilized finite element approximation of the incompressible Navier–Stokes equations. The basic idea is to build a reduced-order model based on a proper orthogonal decomposition and a Galerkin projection and treat all the terms in an explicit way in the time integration scheme, including the pressure. This is possible because the reduced model snapshots do already fulfill the continuity equation. The pressure field is automatically recovered from the reduced-order basis and solution coefficients. The main advantage of this explicit treatment of the incompressible Navier–Stokes equations is that it allows for the easy use of hyper-reduced order models, because only the right-hand side vector needs to be recovered by means of a gappy data reconstruction procedure. A method for choosing the optimal set of sampling points at the discrete level in the gappy procedure is also presented. Numerical examples show the performance of the proposed strategy.

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

     Badia Rodriguez, Santiago I.; Baiges Aznar, Joan
    SIAM journal on scientific computing
    Vol. 35, num. 1, p. A491-A516
    DOI: 10.1137/120880732
    Date of publication: 2013-02
    Journal article

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    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.

    In this work we design hybrid continuous-discontinuous finite element spaces that permit discontinuities on non-matching element interfaces of non-conforming 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 SUPGtype continuous Galerkin formulations and the jump stabilization of discontinuous Galerkin formulations. Optimal stability and convergence results are obtained. For the adaptive setting, we use an standard error estimator and marking strategy. Numerical experiments show the optimal accuracy of the hybrid algorithm both for uniformly and adaptively refined non-conforming 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.

  • A variational multiscale method with subscales on the element boundaries for the Helmholtz equation

     Baiges Aznar, Joan; Codina Rovira, Ramon
    International journal for numerical methods in engineering
    Vol. 93, num. 6, p. 664-684
    DOI: 10.1002/nme.4406
    Date of publication: 2013-02
    Journal article

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    In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low-order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least-squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method.

  • A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes

     Baiges Aznar, Joan; Codina Rovira, Ramon; Henke, Florian; Shahmiri, S.; Wall, W. A.
    International journal for numerical methods in engineering
    Vol. 90, num. 5, p. 636-658
    DOI: 10.1002/nme.3339
    Date of publication: 2012-05
    Journal article

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    In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic parameter of the formulation which allows to ensure stability is independent of the numerical approximation, relatively small, and can be fixed a priori. Moreover, the formulation is symmetric for symmetric problems. An additional element-discontinuous stress field is used to enforce the boundary conditions in the Poisson problem. Additional terms are required in order to guarantee stability in the convection–diffusion equation and the Stokes problem. The proposed method is then easily extended to the transient Navier–Stokes equations.

  • The Fixed-Mesh ALE approach for the numerical simulation of floating solids

     Baiges Aznar, Joan; Codina Rovira, Ramon; Coppola Owen, Angel H.
    International journal for numerical methods in fluids
    Vol. 67, num. 8, p. 1004-1023
    DOI: 10.1002/fld.2403
    Date of publication: 2011-11
    Journal article

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    In this paper, we propose a method to solve the problem of floating solids using always a background mesh for the spatial discretization of the fluid domain. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian–Eulerian framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. We pay special attention to the tracking of the various interfaces and their intersections, and to the approximate imposition of coupling conditions between the solid and the fluid.

  • Finite element approximation of transmission conditions in fluids and solids introducing boundary subgrid scales

     Codina Rovira, Ramon; Baiges Aznar, Joan
    International journal for numerical methods in engineering
    Vol. 87, num. 1-5, p. 386-411
    DOI: 10.1002/nme.3111
    Date of publication: 2011-08
    Journal article

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    Terms involving jumps of stresses on boundaries are proposed for the finite element approximation of the Stokes problem and the linear elasticity equations. These terms are designed to improve the transmission conditions between subdomains at three different levels, namely, between the element domains, between the interfaces in homogeneous domain interaction problems and at the interface between the fluid and the solid in fluid–structure interaction problems. The benefits in each case are respectively the possibility of using discontinuous pressure interpolations in a stabilized finite element approximation of the Stokes problem, a stronger enforcement of the stress continuity in homogeneous domain decomposition problems and a considerable improvement of the behavior of iterative schemes to couple the fluid and the solid in fluid–structure integration algorithms. The motivation to introduce these terms stems from a decomposition of the unknown into a conforming and a non-conforming part, a hybrid formulation for the latter and a simple approximation for the unknowns involved in the hybrid problem.

  • The fixed-Mesh ALE method applied to multiphysics problems using stabilized formulations  Open access  awarded activity

     Baiges Aznar, Joan
    Universitat Politècnica de Catalunya
    Theses

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    The finite element method is a tool very often employed to deal with the numerical simulation of multiphysics problems.Many times each of these problems can be attached to a subdomain in space which evolves in time. Fixed grid methods appear in order to avoid the drawbacks of remeshing in ALE (Arbitrary Lagrangian-Eulerian) methods when the domain undergoes very large deformations. Instead of having one mesh attached to each of the subdomains, one has a single mesh which covers the whole computational domain. Equations arising from the finite element analysis are solved in an Eulerian manner in this background mesh. In this work we present our particular approach to fixed mesh methods, which we call FM-ALE (Fixed-Mesh ALE). Our main concern is to properly account for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian-Eulerian framework, the distinctive feature being that at each time step results are projected onto a fixed, background mesh, that is where the problem is actually solved.We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. When dealing with certain physical problems, and depending on the finite element space used, the standard Galerkin finite element method fails and leads to unstable solutions. The variational multiscale method is often used to deal with this instability. We introduce a way to approximate the subgrid scales on the boundaries of the elements in a variational twoscale finite element approximation to flow problems. The key idea is that the subscales on the element boundaries must be such that the transmission conditions for the unknown, split as its finite element contribution and the subscale, hold. We then use the subscales on the element boundaries to improve transmition conditions between subdomains by introducing the subgrid scales between the interfaces in homogeneous domain interaction problems and at the interface between the fluid and the solid in fluid-structure interaction problems. The benefits in each case are respectively a stronger enforcement of the stress continuity in homogeneous domain decomposition problems and a considerable improvement of the behaviour of the iterative algorithm to couple the fluid and the solid in fluid-structure interaction problems. We develop FELAP, a linear systems of equations solver package for problems arising from finite element analysis. The main features of the package are its capability to work with symmetric and unsymmetric systems of equations, direct and iterative solvers and various renumbering techniques. Performance is enhanced by considering the finite element mesh graph instead of the matrix graph, which allows to perform highly efficient block computations.

  • The fixed-mesh ALE approach applied to solid mechanics and fluid-structure interaction problems

     Baiges Aznar, Joan; Codina Rovira, Ramon
    International journal for numerical methods in engineering
    Vol. 81, num. 12, p. 1529-1557
    DOI: 10.1002/nme.2740
    Date of publication: 2010-03
    Journal article

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    In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an Arbitrary Lagrangian–Eulerian (ALE) framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. For solid mechanics problems subject to large strains, the fixed-mesh (FM)-ALE method avoids the element stretching found in fully Lagrangian approaches. For FSI problems, FM-ALE allows for the use of a single background mesh to solve both the fluid and the structure.

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    A numerical strategy to compute optical parameters in turbulent flow: application to telescopes  Open access

     Codina Rovira, Ramon; Baiges Aznar, Joan; Pérez Sánchez, Daniel; Collados, Manuel
    Computers and fluids
    Vol. 39, num. 1, p. 87-98
    DOI: 10.1016/j.compfluid.2009.07.005
    Date of publication: 2010-01
    Journal article

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    We present a numerical formulation to compute optical parameters in a turbulent air flow. The basic numerical formulation is a large eddy simulation (LES) of the incompressible Navier–Stokes equations, which are approximated using a finite element method. From the time evolution of the flow parameters we describe how to compute statistics of the flow variables and, from them, the parameters that determine the quality of the visibility. The methodology is applied to estimate the optical quality around telescope enclosures.

    We present a numerical formulation to compute optical parameters in a turbulent air flow. The basic numerical formulation is a large eddy simulation (LES) of the incompressible Navier–Stokes equations, which are approximated using a finite element method. From the time evolution of the flow parameters we describe how to compute statistics of the flow variables and, from them, the parameters that determine the quality of the visibility. The methodology is applied to estimate the optical quality around telescope enclosures.

    Postprint (author’s final draft)

  • Fixed mesh methods in computational mechanics

     Codina Rovira, Ramon; Baiges Aznar, Joan
    Date of publication: 2010
    Book chapter

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  • Solving finite element linear systems by a multilevel incomplete factorization algorithm

     Principe Rubio, Ricardo Javier; Baiges Aznar, Joan
    Argentinean Congress on Computacional Mechanics - South American Congress on Computacional Mechanics
    p. 7681
    Presentation of work at congresses

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  • Approximate imposition of boundary conditions in immersed boundary methods

     Codina Rovira, Ramon; Baiges Aznar, Joan
    International journal for numerical methods in engineering
    Vol. 80, num. 11, p. 1379-1405
    DOI: 10.1002/nme.2662
    Date of publication: 2009-12
    Journal article

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    We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non-symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition.

  • GRUP DE RESISTÈNCIA DE MATERIALS I ESTRUCTURES A L'ENGINYERIA

     Bugeda Castelltort, Gabriel; Cervera Ruiz, Miguel; Chiumenti, Michele; Suarez Arroyo, Benjamin; Miquel Canet, Juan; Baiges Aznar, Joan; Codina Rovira, Ramon; Weyler Perez, Rafael; González Lopez, Jose Manuel; Badia Rodriguez, Santiago I.; Agelet de Saracibar Bosch, Carlos; Cante Teran, Juan Carlos; Oller Martinez, Sergio Horacio; Hernandez Ortega, Joaquin Alberto; Barbat Barbat, Horia Alejandro; Davalos Chargoy, Cesar Emilio; Pelà, Luca; Oliver Olivella, Fco. Javier
    Competitive project

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  • The fixed-mesh ALE approach for the numerical approximation of flows in moving domains

     Codina Rovira, Ramon; Houzeaux, G; Coppola Owen, Angel H.; Baiges Aznar, Joan
    Journal of computational physics
    Vol. 228, num. 5, p. 1591-1611
    DOI: 10.1016/j.jcp.2008.11.004
    Date of publication: 2009-03
    Journal article

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    In this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization, and therefore the formulation to be presented falls within the category of fixed-grid methods. Even though the imposition of boundary conditions is a key ingredient that is very often used to classify the fixed-grid method, our approach can be applied together with any technique to impose approximately boundary conditions, although we also describe the one we actually favor. Our main concern is to properly account for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian–Eulerian framework, the distinctive feature being that at each time step results are projected onto a fixed, background mesh, that is where the problem is actually solved.

  • Subscales on the element boundaries in the variational two-scale finite element method

     Codina Rovira, Ramon; Principe Rubio, Ricardo Javier; Baiges Aznar, Joan
    Computer methods in applied mechanics and engineering
    Vol. 198, num. 5-8, p. 838-852
    DOI: 10.1016/j.cma.2008.10.020
    Date of publication: 2009-01
    Journal article

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    In this paper, we introduce a way to approximate the subscales on the boundaries of the elements in a variational two-scale finite element approximation to flow problems. The key idea is that the subscales on the element boundaries must be such that the transmission conditions for the unknown, split as its finite element contribution and the subscale, hold. In particular, we consider the scalar convection–diffusion–reaction equation, the Stokes problem and Darcy’s problem. For these problems the transmission conditions are the continuity of the unknown and its fluxes through element boundaries. The former is automatically achieved by introducing a single valued subscale on the boundaries (for the conforming approximations we consider), whereas the latter provides the effective condition for approximating these values. The final result is that the subscale on the interelement boundaries must be proportional to the jump of the flux of the finite element component and the average of the subscale calculated in the element interiors.

  • Premis CELSA 2007

     Baiges Aznar, Joan
    Award or recognition

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