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• Fractional step like schemes for free surface problems with thermal coupling using the Lagrangian PFEM

  R. Aubry, E. Oñate, S. Idelsohn

  Computational Mechanics (2006). Vol. 38 (4-5), pp. 294-309

  
  

  Abstract

  The method presented in Aubry et al. (Comput Struc 83:1459–1475, 2005) for the solution of an incompressible viscous fluid flow with heat transfer using a fully Lagrangian description of motion is extended to three dimensions (3D) with particular emphasis on mass conservation. A modified fractional step (FS) based on the pressure Schur complement (Turek 1999), and related to the class of algebraic splittings Quarteroni et al. (Comput Methods Appl Mech Eng 188:505–526, 2000), is used and a new advantage of the splittings of the equations compared with the classical FS is highlighted for free surface problems. The temperature is semi-coupled with the displacement, which is the main variable in a Lagrangian description. Comparisons for various mesh Reynolds numbers are performed with the classical FS, an algebraic splitting and a monolithic solution, in order to illustrate the behaviour of the Uzawa operator and the mass conservation. As the classical fractional step is equivalent to one iteration of the Uzawa algorithm performed with a standard Laplacian as a preconditioner, it will behave well only in a Reynold mesh number domain where the preconditioner is efficient. Numerical results are provided to assess the superiority of the modified algebraic splitting to the classical FS.

  Abstract
  The method presented in Aubry et al. (Comput Struc 83:1459–1475, 2005) for the solution of an incompressible viscous fluid flow with heat transfer using a fully Lagrangian [...]

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• FIC/FEM Formulation with Matrix Stabilizing Terms for Incompressible Flows at Low and High Reynolds Numbers

  E. Oñate, A. Valls, J. García-Espinosa

  Computational Mechanics (2006). Vol. 38 (4-5), pp. 440-455

  
  

  Abstract

  We present a general formulation for incompressible fluid flow analysis using the finite element method. The necessary stabilization for dealing with convective effects and the incompressibility condition are introduced via the Finite Calculus method using a matrix form of the stabilization parameters. This allows to model a wide range of fluid flow problems for low and high Reynolds numbers flows without introducing a turbulence model. Examples of application to the analysis of incompressible flows with moderate and large Reynolds numbers are presented.

  Abstract
  We present a general formulation for incompressible fluid flow analysis using the finite element method. The necessary stabilization for dealing with convective effects and [...]

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• Direct computation of instability points for contact problems

  H. Tschöpe, E. Oñate, P. Wriggers

  Computational Mechanics (2003). Vol. 31 (1-2), pp. 173–178

  
  

  Abstract

  The extended system is known as a reliable algorithm for the direct computation of instability points on the equilibrium path of mechanical structures. This article describes the application of the extended system as critical point computation method to mechanical contact problems. In this type of problems inequality constraints have to be considered. Moreover a prediction method based on the extended system algorithm is presented which allows the detection of favorable starting values for a critical point computation on the equilibrium path.

  Abstract
  The extended system is known as a reliable algorithm for the direct computation of instability points on the equilibrium path of mechanical structures. This article describes [...]

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• A mesh-free finite point method for advective-diffusive transport and fluid flow problems

  E. Oñate, S. Idelsohn

  Computational Mechanics (1998). Vol. 21 (4-5), pp. 283-292

  
  

  Abstract

  The finite point method (FPM) is a gridless numerical procedure based on the combination of weighted least square interpolations on a cloud of points with point collocation for evaluating the approximation integrals. In the paper, details of a procedure for stabilizing the numerical solution for advective-diffusive transport and fluid flow problems using the FPM are given. The method is based on a consistent introduction of the stabilizing terms in the governing differential equations. One example showing the applicability of the FPM is given.

  Abstract
  The finite point method (FPM) is a gridless numerical procedure based on the combination of weighted least square interpolations on a cloud of points with point collocation [...]

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• On the derivation and possibilities of the secant stiffness matrix for non linear finite element analysis

  E. Oñate

  Computational Mechanics (1995). Vol. 15 (6), pp. 572-593

  
  

  Abstract

  In this paper the general non symmetric parametric form of the incremental secant stiffness matrix for non linear analysis of solids using the finite element metod is derived. A convenient symmetric expression for a particular value of the parameters is obtained. The geometrically non linear formulation is based on a Generalized Lagrangian approach. Detailed expressions of all the relevant matrices involved in the analysis of 3D solids are obtained. The possibilities of application of the secant stiffness matrix for non linear structural problems including stability, bifurcation and limit load analysis are also discussed. Examples of application are given for the non linear analysis of pin joined frames.

  Abstract
  In this paper the general non symmetric parametric form of the incremental secant stiffness matrix for non linear analysis of solids using the finite element metod is derived. [...]

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• New assumed strain triangles for non linear shell analysis

  F. Flores, E. Oñate, F. Zarate

  Computational Mechanics (1995). Vol. 17 (1-2), pp. 107-114

  
  

  Abstract

  A comparison between new and existing triangular finite elements based on the shell theory proposed by Juan Carlos Simo and co-workers is presented. Particular emphasis is put on the description of new triangles which show a promising behaviour for linear and non linear shell analysis.

  Abstract
  A comparison between new and existing triangular finite elements based on the shell theory proposed by Juan Carlos Simo and co-workers is presented. Particular emphasis is [...]

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• Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

  A. Franci, E. Oñate, J. Puigbó

  Int. J. Numer. Meth. Engng. (2016). Vol. 107 (11), pp. 970-990

  
  

  Abstract

  We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics.

  Abstract
  We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are [...]

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• A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

  P. Nadukandi, E. Oñate, J. García-Espinosa

  Int. J. Numer. Meth. Engng. (2012). Vol. 89 (11), pp. 1367-1391

  
  

  Abstract

  A new Petrov–Galerkin (PG) method involving two parameters, namely α1 and α2, is presented, which yields the following schemes on rectangular meshes: (i) a compact stencil obtained by the linear interpolation of the Galerkin FEM and the classical central finite difference method (FDM), should the parameters be equal, that is, α1 = α2 = α; and (ii) the nonstandard compact stencil presented in (Int. J. Numer. Meth. Engng 2011; 86:18–46) for the Helmholtz equation if the parameters are distinct, that is, α1 ≠ α2. The nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms (specified by α1) and the mass terms (specified by α2) that appear in the stencils obtained by the standard Galerkin FEM and the classical central FDM, respectively. On square meshes, these two schemes were shown to provide solutions to the Helmholtz equation that have a dispersion accuracy of fourth and sixth order, respectively (Int. J. Numer. Meth. Engng 2011; 86:18–46). The objective of this paper is to study the performance of this PG method for the Helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions.

  Abstract
  A new Petrov–Galerkin (PG) method involving two parameters, namely α1 and α2, is presented, which yields the following schemes on rectangular [...]

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• A four‐noded quadrilateral element for composite laminated plates/shells using the refined zigzag theory

  A. Eijo, E. Oñate, S. Oller

  Int. J. Numer. Meth. Engng. (2013). Vol. 95 (8), pp. 631-660

  
  

  Abstract

  A new bilinear four‐noded quadrilateral element (called quadrilateral linear refined zigzag) for the analysis of composite laminated and sandwich plates/shells based on the refined zigzag theory is presented. The element has seven kinematic variables per node. Shear locking is avoided by introducing an assumed linear shear strain field. The performance of the element is studied in several examples where the reference solution is the 3D finite element analysis using 20‐noded hexahedral elements.

  Abstract
  A new bilinear four‐noded quadrilateral element (called quadrilateral linear refined zigzag) for the analysis of composite laminated and sandwich plates/shells based on [...]

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• Improving mass conservation in simulation of incompressible flows

  P. Ryzhakov, E. Oñate, R. Rossi, S. Idelsohn

  Int. J. Numer. Meth. Engng. (2012). Vol. 90 (12), pp. 1435-1451

  
  

  Abstract

  We propose a technique for improving mass‐conservation features of fractional step schemes applied to incompressible flows. The method is illustrated by using a Lagrangian fluid formulation, where the mass loss effects are particularly apparent. However, the methodology is general and could be used for fixed grid approaches. The idea consists in reflecting the incompressibility condition already in the intermediate velocity. This is achieved by predicting the end‐of‐step pressure and using this prediction in the fractional momentum equation. The resulting intermediate velocity field is thus much closer to the final incompressible one than that of the standard fractional step scheme. In turn, the predicted pressure can be used as the boundary condition necessary for the solution of the pressure Poisson equation in case a continuous Laplacian matrix is employed. Using this approximation of the end‐of‐step incompressible pressure as the essential boundary condition considerably improves the conservation of mass, specially for the free surface flows of fluids with low viscosity. The pressure prediction does not require the resolution of any additional equations system. The efficiency of the method is shown in two examples. The first one shows the performance of the method with respect to mass conservation. The second one tests the method in a challenging fluid–structure interaction benchmark, which can be naturally resolved by using the presented approach.

  Abstract
  We propose a technique for improving mass‐conservation features of fractional step schemes applied to incompressible flows. The method is illustrated by using a Lagrangian [...]

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