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• A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

  R. Ortega, E. Oñate, S. Idelsohn, R. Flores

  Int. J. Numer. Meth. Fluids (2013). Vol. 73 (4), pp. 323-343

  
  

  Abstract

  A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind‐biased discretization of the Euler equations, written in arbitrary Lagrangian–Eulerian form and integrated in time by means of a dual‐time steeping technique. In order to exploit the meshless potential of the method, a domain deformation approach based on the spring network analogy is implemented, and h‐adaptivity is also employed in the computations. Typical movable boundary problems in transonic flow regime are solved to assess the performance of the proposed technique. In addition, an application to a fluid–structure interaction problem involving static aeroelasticity illustrates the capability of the method to deal with practical engineering analyses. The computational cost and multi‐core performance of the proposed technique is also discussed through the examples provided.

  Abstract
  A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind‐biased [...]

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

  J. Baiges, R. Codina, S. Idelsohn

  Int. J. Numer. Meth. Fluids (2013). Vol. 72 (12), pp. 1219-1243

  
  

  Abstract

  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

  Abstract
  In this paper, we present an explicit formulation for reduced‐order models of the stabilized finite element approximation of the incompressible Navier–Stokes equations. [...]

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• A new enrichment space for the treatment of discontinuous pressures in multi‐fluid flows

  R. Ausas, G. Buscaglia, S. Idelsohn

  Int. J. Numer. Meth. Fluids (2011). Vol. 70 (7), pp. 829-850

  
  

  Abstract

  In this work, a new enrichment space to accommodate jumps in the pressure field at immersed interfaces in finite element formulations, is proposed. The new enrichment adds two degrees of freedom per element that can be eliminated by means of static condensation. The new space is tested and compared with the classical P1 space and to the space proposed by Ausas et al (Comp. Meth. Appl. Mech. Eng., Vol. 199, 1019–1031, 2010) in several problems involving jumps in the viscosity and/or the presence of singular forces at interfaces not conforming with the element edges. The combination of this enrichment space with another enrichment that accommodates discontinuities in the pressure gradient has also been explored, exhibiting excellent results in problems involving jumps in the density or the volume forces.

  Abstract
  In this work, a new enrichment space to accommodate jumps in the pressure field at immersed interfaces in finite element formulations, is proposed. The new enrichment adds [...]

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• Numerical simulations of negatively buoyant jets in an immiscible fluid using the Particle Finite Element Method

  M. Mier-Torrecilla, A. Geyer, J. Phillips, S. Idelsohn, E. Oñate

  Int. J. Numer. Meth. Fluids (2012). Vol. 69 (5), pp. 1016-1030

  
  

  Abstract

  Negatively buoyant jets consist in a dense fluid injected vertically upward into a lighter ambient fluid. The numerical simulation of this kind of buoyancy‐driven flows is challenging as it involves multiple fluids with different physical properties. In the case of immiscible fluids, it requires, in addition, to track the motion of the interface between fluids and accurately represent the discontinuities of the flow variables. In this paper, we investigate numerically the injection of a negatively buoyant jet into a homogenous immiscible ambient fluid using the Particle Finite Element Method and compare the two‐dimensional numerical results with experiments on the injection of a jet of dyed water through a nozzle in the base of a cylindrical tank containing rapeseed oil. In both simulations and experiments, the fountain inlet flow velocity and nozzle diameter have been varied to cover a wide range of Froude Fr and Reynolds Re numbers ( 0.1 < Fr < 30, 8 < Re < 1350), reproducing both weak and strong laminar fountains. The flow behaviors observed for the different numerical simulations fit in the regime map based on the Re and Fr values of the experiments, and the maximum fountain height is in good agreement with the experimental observations, suggesting that particle finite element method is a useful tool for the study of immiscible two‐fluid systems.

  Abstract
  Negatively buoyant jets consist in a dense fluid injected vertically upward into a lighter ambient fluid. The numerical simulation of this kind of buoyancy‐driven flows [...]

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• A family of residual‐based stabilized finite element methods for Stokes flows

  E. Oñate, P. Nadukandi, S. Idelsohn, J. García-Espinosa, C. Felippa

  Int. J. Numer. Meth. Fluids (2011). Vol. 65 (1-3), pp. 106-134

  
  

  Abstract

  We present a collection of stabilized finite element (FE) methods derived via first‐ and second‐order finite calculus (FIC) procedures. It is shown that several well known existing stabilized FE methods such as the penalty technique, the Galerkin Least Square (GLS) method, the Pressure Gradient Projection (PGP) method and the orthogonal sub‐scales (OSS) method are recovered from the general residual‐based FIC stabilized form. A new family of stabilized Pressure Laplacian Stabilization (PLS) FE methods with consistent nonlinear forms of the stabilization parameters are derived. The distinct feature of the family of PLS methods is that they are residual‐based, i.e. the stabilization terms depend on the discrete residuals of the momentum and/or the incompressibility equations. The advantages and disadvantages of the different stabilization techniques are discussed and several examples of application are presented.

  Abstract
  We present a collection of stabilized finite element (FE) methods derived via first‐ and second‐order finite calculus (FIC) procedures. It is shown that several well known [...]

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• A finite point method for adaptive three‐dimensional compressible flow calculations

  E. Ortega, E. Oñate, S. Idelsohn

  Int. J. Numer. Meth. Fluids (2008). Vol. 60 (9), pp. 937-971

  
  

  Abstract

  The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. 

  Abstract
  The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation [...]

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• The violation of objectivity in Laplace formulations of the Navier–Stokes equations

  A. Limache, S. Idelsohn, R. Rossi, E. Oñate

  Int. J. Numer. Meth. Fluids (2007). Vol. 54 (6-8), pp. 639-664

  
  

  Abstract

  The Navier–Stokes equations written in Laplace form are often the starting point of many numerical methods for the simulation of viscous flows. Imposing the natural boundary conditions of the Laplace form or neglecting the viscous contributions on free surfaces are traditionally considered reasonable and harmless assumptions. With these boundary conditions any formulation derived from integral methods (like finite elements or finite volumes) recovers the pure Laplacian aspect of the strong form of the equations. This approach has also the advantage of being convenient in terms of computational effort and, as a consequence, it is used extensively. However, we have recently discovered that these resulting Laplacian formulations violate a basic axiom of continuum mechanics: the principle of objectivity. In the present article we give an accurate account about these topics. We also show that unexpected differences may sometimes arise between Laplace discretizations and divergence discretizations. 

  Abstract
  The Navier–Stokes equations written in Laplace form are often the starting point of many numerical methods for the simulation of viscous flows. Imposing the natural [...]

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• Iterative solution of panel method discretizations for potential flow problems. The modal multipolar preconditioning

  J. D'ELIA, M. Storti, S. Idelsohn

  Int. J. Numer. Meth. Fluids (2000). Vol. 32 (1), pp. 1-22

  
  

  Abstract

  The iterative solution of linear systems arising from panel method discretization of three‐dimensional (3D) exterior potential problems coming mainly from aero‐hydrodynamic engineering problems, is discussed. An original preconditioning based on an approximate eigenspace decomposition is proposed, which corrects bad conditioning arising from a pair of surfaces that are very close to each other, which is a very common situation in slender wings and other aerodynamic profiles. This preconditioning has been tested with the standard Bi‐conjugate gradient (Bi‐CG) and conjugate gradient squared (CGS) iterative methods.

  Abstract
  The iterative solution of linear systems arising from panel method discretization of three‐dimensional (3D) exterior potential problems coming mainly from aero‐hydrodynamic [...]

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• Discrete non‐local absorbing boundary condition for exterior problems governed by Helmholtz equation

  R. Bonet Chaple, N. Nigro, M. Storti, S. Idelsohn

  Int. J. Numer. Meth. Fluids (1999). Vol. 29 (5), pp. 605-621

  
  

  Abstract

  The finite element method is employed to approximate the solutions of the Helmholtz equation for water wave radiation and scattering in an unbounded domain. A discrete, non‐local and non‐reflecting boundary condition is specified at an artificial external boundary by the DNL method, yielding an equivalent problem that is solved in a bounded domain. This procedure formulates a boundary value problem in a bounded region by imposing a relation in the discrete medium between the nodal values at the two last layers. For plane geometry, this relation can be found by straightforward eigenvalue decomposition. For circular geometry, the plane condition is applied at the external layer and this condition is condensed through a structured annular region, resulting in a condition at an inner radius. Exterior problems with a bounded internal physical obstacle are considered. It is well‐known that these kind of problems are well‐posed, and have a unique solution. Numerical studies based on standard Galerkin methodology examine the dependence of the DNL condition with respect to the circular annular region width. The DNL condition is compared with local boundary conditions of several orders. Numerical examples confirm the important improvement in accuracy obtained by the DNL method over standard conditions. 

  Abstract
  The finite element method is employed to approximate the solutions of the Helmholtz equation for water wave radiation and scattering in an unbounded domain. A discrete, non‐local [...]

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• GMRES physics‐based preconditioner for all Reynolds and Mach numbers: numerical examples

  N. Nigro, M. Storti, S. Idelsohn

  Int. J. Numer. Meth. Fluids (1997). Vol. 25 (12), pp. 1347-1371

  
  

  Abstract

  This paper presents several numerical results using a vectorized version of a 3D finite element compressible and nearly incompressible Euler and Navier–Stokes code. The assumptions were set on laminar flows and Newtonian fluids. The goal of this research is to show the capabilities of the present code to treat a wide range of problems appearing in laminar fluid dynamics towards the unification from incompressible to compressible and from inviscid to viscous flow codes. Several authors with different approaches have tried to attain this target in CFD with relative success. At the beginning the methods based on operator splitting and perturbation were preferred, but lately, with the wide usage of time‐marching algorithms, the preconditioning mass matrix (PMM) has become very popular. With this kind of relaxation scheme it is possible to accelerate the rate of convergence to steady state solutions with the modification of the mass matrix under certain restrictions. The selection of the mass matrix is not an easy task, but we have certain freedom to define it in order to improve the condition number of the system. In this paper we have used a physics‐based preconditioner for the GMRES implicit solver developed previously by us and an SUPG formulation for the semidiscretization of the spatial operator. In sections 2 and 3 we present some theoretical aspects related to the physical problem and the mathematical model, showing the inviscid and viscous flow equations to be solved and the variational formulation involved in the finite element analysis. Section 4 deals with the numerical solution of non‐linear systems of equations, with some emphasis on the preconditioned matrix‐free GMRES solver. Section 5 shows how boundary conditions were treated for both Euler and Navier–Stokes problems. Section 6 contains some aspects about vectorization on the Cray C90. The performance reached by this implementation is close to 1 Gflop using multitasking. Section 7 presents several numerical examples for both models covering a wide range of interesting problems, such as inviscid low subsonic, transonic and supersonic regimes and viscous problems with interaction between boundary layers and shock waves in either attached or separated flows.

  Abstract
  This paper presents several numerical results using a vectorized version of a 3D finite element compressible and nearly incompressible Euler and Navier–Stokes code. [...]

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