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• Elemental enriched spaces for the treatment of weak and strong discontinuous fields

  S. Idelsohn, J. Gimenez, J. Marti, N. Nigro

  Comput. Methods Appl. Mech. Engrg., (2017). Vol. 313, pp. 535-559

  
  

  Abstract

  This paper presents a finite element that incorporates weak, strong and both weak plus strong discontinuities with linear interpolations of the unknown jumps for the modeling of internal interfaces. The new enriched space is built by subdividing each triangular or tetrahedral element in several standard linear sub-elements. The new degrees of freedom coming from the assembly of the sub-elements can be eliminated by static condensation at the element level, resulting in two main advantages: first, an elemental enrichment instead of a nodal one, which presents an important reduction of the compunting time when the internal interface is moving all around the domain and second, an efficient implementation involving minor modifications allowing to reuse existing finite element codes. The equations for the internal interface are constructed by imposing the local equilibrium between the stresses in the bulk of the element and the tractions driving the cohesive law, with the proper equilibrium operators to account for the linear kinematics of the discontinuity. To improve the continuity of the unknowns on both sides of the elements on which a static condensation is done, a contour integral has been added. These contour integrals named inter-elemental forces can be interpreted as a “do nothing” boundary condition (Coppola-Owen and Codina, 2011) published in another context, or as the usage of weighting functions that ensure convergence of the approach as proposed by J.C. Simo (Simo and Rifai, 1990). A series of numerical tests for scalar unknowns as a simple representation of more general numerical simulations are presented to illustrate the performance of the enriched elemental space.

  Abstract
  This paper presents a finite element that incorporates weak, strong and both weak plus strong discontinuities with linear interpolations of the unknown [...]

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• An embedded strategy for the analysis of fluid structure interaction problems

  S. Costarelli, L. Garelli, M. Cruchaga, M. Storti, R. Ausensi, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (2016). Vol. 300, pp. 106-128

  
  

  Abstract

  A Navier–Stokes solver based on Cartesian structured finite volume discretization with embedded bodies is presented. Fluid structure interaction with solid bodies is performed with an explicit partitioned strategy. The Navier–Stokes equations are solved in the whole domain via a Semi-Implicit Method for Pressure Linked Equations (SIMPLE) using a colocated finite volume scheme, stabilized via the Rhie–Chow discretization. As uniform Cartesian grids are used, the solid interface usually do not coincide with the mesh, and then a second order Immersed Boundary Method is proposed, in order to avoid the loss of precision due to the staircase representation of the surface. This fact also affects the computation of fluid forceson the solid wall and, accordingly, the results in the fluid–structure analysis. In the present work, first and second order approximations for computing the fluid forces at the interface are studied and compared. The solver is specially oriented to General Purpose Graphic Processing Units (GPGPU) hardware and the efficiency is discussed. Moreover, a novel submerged buoy experiment is also reported. The experiment consists of a sphere with positive buoyancy fully submerged in a cubic tank, subject to harmonic displacements imposed by a shake table. The sphere is attached to the bottom of the tank with a string. The position of the buoy is determined from video records with a Motion Capture algorithm. The obtained amplitude and phase curves allow a precise determination of the added mass and drag forces. Due to this aspect the experimental data can be of interest for comparison with other fluid–structure interaction codes. Finally, the numerical results are compared with the experiments, and allow the confirmation of the numerically predicted drag and added mass of the body.

  Abstract
  A Navier–Stokes solver based on Cartesian structured finite volume discretization with embedded bodies is presented. Fluid structure interaction with [...]

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• Reduced-order subscales for POD models

  J. Baiges, R. Codina, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (2015). Vol. 291; pp. 173-196

  
  

  Abstract

  In this work the Reduced-Order Subscales for Proper Orthogonal Decomposition models are presented. The basic idea consists in splitting the full-order solution into the part which can be captured by the reduced-order model and the part which cannot, the subscales, for which a model is required. The proposed model for the subscales is defined as a linear function of the solution of the reduced-order model. The coefficients of this linear function are obtained by comparing the solution of the full-order model with the solution of the reduced-order model for the same initial conditions, which, for convenience, are evaluated in the snapshots used to train the original reduced-order-model. The difference between both solutions are the subscales, for which a model can be built using a least-squares procedure. The subscales are then introduced as a correction in the reduced-order model, resulting in an important improvement in accuracy. The enhanced reduced-order model is tested in several numerical examples. These practical cases show that the use of the subscales leads to more accurate solutions, successfully corrects errors introduced by hyper-reduction, and allows to solve complex flow problems using a reduced number of degrees of freedom.

  Abstract
  In this work the Reduced-Order Subscales for Proper Orthogonal Decomposition models are presented. The basic idea consists in splitting the full-order solution into the part [...]

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• Improving the k-compressibility of Hyper Reduced Order Models with moving sources: Applications to welding and phase change problems

  A. Cosimo, A. Cardona, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (2014). Vol. 274, pp. 237-263

  
  

  Abstract

  The simulation of engineering problems is quite often a complex task that can be time consuming. In this context, the use of Hyper Reduced Order Models (HROMs) is a promising alternative for real-time simulations. In this work, we study the design of HROMs for non-linear problems with a moving source. Applications to nonlinear phase change problems with temperature dependent thermophysical properties are particularly considered; however, the techniques developed can be applied in other fields as well. A basic assumption in the design of HROMs is that the quantities that will be hyper-reduced are k-compressible in a certain basis in the sense that these quantities have at most k non-zero significant entries when expressed in terms of that basis. To reach the computational speed required for a real-time application, k must be small. This work examines different strategies for addressing hyper-reduction of the nonlinear terms with the objective of obtaining k-compressible signals with a notably small k. To improve performance and robustness, it is proposed that the different contributing terms to the residual are separately hyper-reduced. Additionally, the use of moving reference frames is proposed to simulate and hyper-reduce cases that contain moving heat sources. Two application examples are presented: the solidification of a cube in which no heat source is present and the welding of a tube in which the problem posed by a moving heat source is analysed.

  Abstract
  The simulation of engineering problems is quite often a complex task that can be time consuming. In this context, the use of Hyper Reduced Order Models (HROMs) is a promising [...]

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

  J. Baiges, R. Codina, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (2013). Vol. 267, pp. 23-42

  
  

  Abstract

  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.

  Abstract
  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 [...]

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• Objectivity tests for Navier–Stokes simulations: The revealing of non-physical solutions produced by Laplace formulations

  A. Limache, R. Sánchez, L. Dalcín, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (49–50), pp. 4180-4192

  
  

  Abstract

  Laplace formulations are weak formulations of the Navier–Stokes equations commonly used in computational fluid dynamics. In these schemes, the viscous terms are given as a function of the Laplace diffusion operator only. Despite their popularity, recently, it has been proven that they violate a fundamental principle of continuum mechanics, the principle of objectivity. It is remarkable that such flaw has not being noticed before, neither detected in numerical experiments. In this work, a series of objectivity tests have been designed with the purpose of revealing such problem in real numerical experiments. Through the tests it is shown how, for slip boundaries or free-surfaces, Laplace formulations generate non-physical solutions which widely depart from the real fluid dynamics. These tests can be easily reproduced, not requiring complex simulation tools. Furthermore, they can be used as benchmarks to check consistency of developed or commercial software. The article is closed with a discussion of the mathematical aspects involved, including the issues of boundary conditions and objectivity.

  Abstract
  Laplace formulations are weak formulations of the Navier–Stokes equations commonly used in computational fluid dynamics. In these schemes, the viscous terms are given [...]

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• All-hexahedral element meshing: Generation of the dual mesh by recurrent subdivision

  N. Calvo, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (2000). Vol. 182 (3–4), pp. 371-378

  
  

  Abstract

  The domain geometry is defined by means of a closed all-quadrilateral mesh. The outer mesh imposes very strong restrictions on the possible connectivities between the inner hexahedral elements. Following the guidelines of the outer topology, the inner one is almost entirely defined. Several ways may be decided for certain configurations, some of them requiring special considerations in order to achieve a valid FEM mesh. The process is entirely performed by constructing the (graph theoretical) dual of the hexahedral mesh, this means no metric information is handled until the final (positioning and smoothing) steps. The essential steps of this scheme are described by means of examples.

  Abstract
  The domain geometry is defined by means of a closed all-quadrilateral mesh. The outer mesh imposes very strong restrictions on the possible connectivities between the inner [...]

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• The DNL absorbing boundary condition: applications to wave problems

  M. Storti, J. D'ELIA, R. Bonet Chaple, N. Nigro, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (2000). Vol. 182 (3–4), pp. 483-498

  
  

  Abstract

  A general methodology for developing absorbing boundary conditions is presented. For planar surfaces, it is based on a straightforward solution of the system of block difference equations that arise from partial discretization in the directions transversal to the artificial boundary followed by discretization on a constant step 1D grid in the direction normal to the boundary. This leads to an eigenvalue problem of the size of the number of degrees of freedom in the lateral discretization. The eigenvalues are classified as right- or left-going and the absorbing boundary condition consists in imposing a null value for the ingoing modes, leaving free the outgoing ones. Whereas the classification is straightforward for operators with definite sign, like the Laplace operator, a virtual dissipative mechanism has to be added in the mixed case, usually associated with wave propagation phenomena, like the Helmholtz equation. The main advantage of the method is that it can be implemented as a black-box routine, taking as input the coefficients of the linear system, obtained from standard discretization (FEM or FDM) packages and giving on output the absorption matrix. We present the application of the DNL methodology to typical wave problems, like Helmholtz equations and potential flow with free surface (the ship wave resistance and sea-keeping problems).

  Abstract
  A general methodology for developing absorbing boundary conditions is presented. For planar surfaces, it is based on a straightforward solution of the system of block difference [...]

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• Physics based GMRES preconditioner for compressible and incompressible Navier-Stokes equations

  N. Nigro, M. Storti, S. Idelsohn, T. Tezduyar

  Comput. Methods Appl. Mech. Engrg., (1998). Vol. 154 (3–4), pp. 203-228

  
  

  Abstract

  This paper presents the implementation of a local physics preconditioning mass matrix [8] for an unified approach of 3D compressible and incompressible Navier-Stokes equations using an SUPG finite element formulation and GMRES implicit solver. During the last years a lot of effort has been dedicated to finding a unified approach for compressible and incompressible flow in order to treat fluid dynamic problems with a very wide range of Mach and Reynolds numbers [10,26,37]. On the other hand, SUPG finite element formulation and GMRES implicit solver is one of the most robust combinations to solve state of the art CFD problems [1,6,9,22,29,30,31]. The selection of a good preconditioner and its performance on parallel architecture is another open problem in CFD community. The local feature of the preconditioner presented here means that no communication among processors is needed when working on parallel architectures. Due to these facts we consider that this research can make some contributions towards the development of a unified fluid dynamic model with high rates of convergence for any combination of Mach and Reynolds numbers, being very suitable for massively parallel computations. Finally, it is important to remark that while this kind of preconditioning produces stabilized results in nearly incompressible regimes the standard version exhibits some numerical drawbacks that lead to solutions without physical meaning.

  Abstract
  This paper presents the implementation of a local physics preconditioning mass matrix [8] for an unified approach of 3D compressible and incompressible Navier-Stokes equations [...]

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• Equal-order interpolations: a unified approach to stabilize the incompressible and advective effects

  M. Storti, N. Nigro, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (1997). Vol. 143 (3–4), pp. 317-331

  
  

  Abstract

  In this paper we present a new SUPG formulation for compressible and near incompressible Navier-Stokes equations [5]. It introduces an extension of the exact solution for one-dimensional systems to the multidimensional case, in a similar way to that arising in the scalar problem. It is important to note that this formulation satisfies both the one-dimensional advective-diffusive system limit case and the advection-dominated multidimensional system case presented by Mallet et al. Another interesting feature of this formulation is that it introduces naturally a stabilizing term for the incompressibility condition, in a similar way to that found by other authors [1–4]. However, in our formulation the stabilization is introduced to the whole system of equations, while other authors introduce a term to stabilize the incompressibility condition and another one for the advective term. In Section 1 we present Navier-Stokes equations for compressible flow and, then, we pass to detail several topics related to the numerical discretization of such advective-diffusive multidimensional systems of PDEs, in the Petrov-Galerkin context. The method is applicable and described for the general Re > 0 laminar flow, but the nature of the stabilizing effect of the artificial diffusion matrix introduced is discussed in depth for the simpler Stokes (Re = 0) flow. Several numerical results are shown in Section 5, taking the well-known test problem of the square-cavity and a variant of this, namely a multiply connected square-cavity, as a validation for this code

  Abstract
  In this paper we present a new SUPG formulation for compressible and near incompressible Navier-Stokes equations [5]. It introduces an extension of the exact solution for [...]

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