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• A Petrov-Galerkin formulation for advection-reaction-diffusion problems

  S. Idelsohn, N. Nigro, M. Storti, G. Buscaglia

  Comput. Methods Appl. Mech. Engrg., (1996). Vol. 136 (1–2), pp. 27-46

  
  

  Abstract

  In this work we present a new method called (SU + C)PG to solve advection-reaction-diffusion scalar equations by the Finite Element Method (FEM). The SUPG (for Streamline Upwind Petrov-Galerkin) method is currently one of the most popular methods for advection-diffusion problems due to its inherent consistency and efficiency in avoiding the spurious oscillations obtained from the plain Galerkin method when there are discontinuities in the solution. Following this ideas, Tezduyar and Park treated the more general advection-reaction-diffusion problem and they developed a stabilizing term for advection-reaction problems without significant diffusive boundary layers. In this work an SUPG extension for all situations is performed, covering the whole plane represented by the Peclet number and the dimensionless reaction number. The scheme is based on the extension of the super-convergence feature through the inclusion of an additional perturbation function and a corresponding proportionality constant. Both proportionality constants (that one corresponding to the standard perturbation function from SUPG, and the new one introduced here) are selected in order to verify the ‘super-convergence’ feature, i.e. exact nodal values are obtained for a restricted class of problems (uniform mesh, no source term, constant physical properties). It is also shown that the (SU + C)PG scheme verifies the Discrete Maximum Principle (DMP), that guarantees uniform convergence of the finite element solution. Moreover, it is shown that super-convergence is closely related to the DMP, motivating the interest in developing numerical schemes that extend the super-convergence feature to a broader class of problems.

  Abstract
  In this work we present a new method called (SU + C)PG to solve advection-reaction-diffusion scalar equations by the Finite Element Method (FEM). The SUPG (for Streamline [...]

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• Steady state incompressible flows using explicit schemes with an optimal local preconditioning

  M. Storti, N. Nigro, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (1995). Vol. 124 (3), pp. 231-252

  
  

  Abstract

  Solving large systems of equations from CFD problems by the explicit pseudo-temporal scheme requires a very low amount of memory and is highly parallelizable, but the CPU time largely depends on the conditioning of the system. For advective systems it is shown that the rate of convergence depends on a condition number defined as the ratio of the maximum and the minimum group velocities of the continuum system. If the objective is to reach the steady state, the temporal term can be modified in order to reduce this condition number. Another possibility consists in the addition of a local preconditioning mass matrix. In this paper an optimal preconditioning for incompressible flow is presented, also applicable to compressible ones with locally incompressible zones, like stagnation points, in contrast with the artificial compressibility method. The preconditioned system has a rate of convergence independent from Mach number. Moreover, the discrete solution is highly improved, eliminating spurious oscillations frequently encountered in incompressible flows.

  Abstract
  Solving large systems of equations from CFD problems by the explicit pseudo-temporal scheme requires a very low amount of memory and is highly parallelizable, but the CPU [...]

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• A Petrov-Galerkin technique for the solution of transonic and supersonic flows

  C. Baumann, M. Storti, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (1992). Vol. 95 (1), pp. 49-70

  
  

  Abstract

  This paper is both the description of a streamline-upwind/Petrov-Galerkin (SUPG) formulation and the documentation of the development of a code for the finite element solution of transonic and supersonic flows. The aim of this work is to present a formulation to be able to treat domains of any configuration and to use the appropriate physical boundary conditions, which are the major stumbling blocks of the finite difference schemes. The implemented code has the following features: the Hughes' SUPG-type formulation with an oscillation-free shock-capturing operator, adaptive refinement, explicit integration with local time-step and hourglassing control. An automatic scheme for dealing with slip boundary conditions and a boundary-augmented lumped mass matrix for speeding up convergence. The theoretical background of the SUPG formulation is described briefly. How the foregoing formulation was used in the finite element code and which are the appropriate boundary conditions to be used are also described. Finally some results obtained with this code are discussed.

  Abstract
  This paper is both the description of a streamline-upwind/Petrov-Galerkin (SUPG) formulation and the documentation of the development of a code for the finite element solution [...]

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• Multigrid methods and adaptive refinement techniques in elliptic problems by finite element methods

  M. Storti, N. Nigro, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (1991). Vol. 93(1), pp. 13-30

  
  

  Abstract

  Multigrid and adaptive refinement techniques are powerful tools in the resolution of problems arising from computational mechanics. In structured grids the number of numerical operations needed in the resolution via the multigrid method is of order N, the number of degrees of freedom, regardless the dimensionality of the problem. In this work a multigrid technique, which works on unstructured meshes, is described. These meshes are obtained from a user defined tesselation through bisection refinement upon any element. Original techniques for nodes numbering, irregular nodes detection and nodes classification are developed in view of multigrid method implementation. The error indicator used in the refinement process is based on truncation error estimates. The refinement strategy is oriented toward error reduction at constant computational effort. Finally several numerical examples are presented.

  Abstract
  Multigrid and adaptive refinement techniques are powerful tools in the resolution of problems arising from computational mechanics. In structured grids the number of numerical [...]

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• An efficient tangent scheme for solving phase-change problems

  M. Storti, L. Crivelli, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (1988). Vol. 66(1), pp. 65-86

  
  

  Abstract

  Using weak formulations and finite elements to solve heat-conduction problems with phase change finally leads to the solution, at each time step, of a nonlinear system of equations in the nodal temperatures. The Newton-Raphson scheme is an effective procedure to cope with this type of problems; however, the choice of a good approximation to the tangent matrix is critical when the latent heat is comparatively large. In this work we derive an exact expression for the tangent matrix and analyze the behavior of its terms for different values of the physical parameters of the system. We demonstrate that this method has good convergence properties. In fact, the rate of convergence is quadratic when the trial approximation is sufficiently close to the solution. Finally, several numerical examples are given.

  Abstract
  Using weak formulations and finite elements to solve heat-conduction problems with phase change finally leads to the solution, at each time step, of a nonlinear system of [...]

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• A reduction method for nonlinear structural dynamic analysis

  S. Idelsohn, A. Cardona

  Comput. Methods Appl. Mech. Engrg., (1985). Vol. 49(3), pp. 253-279

  
  

  Abstract

  A computational algorithm for predicting the nonlinear dynamic response of a structure is presented. The nonlinear system of ordinary differential equations resulting from the finite element discretization is highly reduced by means of a Rayleigh-Ritz analysis. The basis vectors are chosen to be the current tangent eigenmodes together with some modal derivatives that indicate the way in which the spectrum is changing. Only a few basis updatings are required during the whole time integration. The truncation error introduced at every change of basis is pointed out as the cause for a divergence-type behaviour, and some means for eliminating it are discussed. Results for examples involving large displacements are shown and compared to the results obtained by integrating the complete system of equations.

  Abstract
  A computational algorithm for predicting the nonlinear dynamic response of a structure is presented. The nonlinear system of ordinary differential equations resulting from [...]

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• Pre- and post-degradation analysis of composite materials with different moduli in tension and compression

  S. Idelsohn, G. Laschet, C. Nyssen

  Comput. Methods Appl. Mech. Engrg., (1982). Vol. 30 (2), pp. 133-149

  
  

  Abstract

  An isoparametric finite element tor the analysis of multi-layer composite materials is presented. Several linear and nonlinear stress-strain relations are discussed. Special attention is given to the composite materials with different moduli in tension and compression, for which a new mathematical model is presented and tested. Different failure criteria for the matrix degradation are incorporated in the element and several post-degradation behaviors are also considered. Finally we discuss the role of the Newton-Raphson method in composite materials, especially in the presence of geometrical nonlinearities.

  Abstract
  An isoparametric finite element tor the analysis of multi-layer composite materials is presented. Several linear and nonlinear stress-strain relations are discussed. Special [...]

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• On the use of deep, shallow or flat shell finite elements for the analysis of thin shell structures

  S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (1981). Vol. 26 (3), pp. 321-330

  
  

  Abstract

  This paper is confined to the study of thin shells. The aim is to summarize the different theories used and to examine the assumptions upon which each of them is based. The intention is to show when it is more suitable to use a particular approximation and to indicate the errors it introduces. Beginning with the general deep shell theory, some simplifications are introduced to obtain the shallow shell theories. The special implications of this theory for the finite element method are also examined. Finally the particular case of flat elements is discussed.

  Abstract
  This paper is confined to the study of thin shells. The aim is to summarize the different theories used and to examine the assumptions upon which each of them is based. The [...]

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• Incompressible lagrangian fluid flow with thermal coupling

  R. Aubry, S. Idelsohn, E. Oñate

  Monograph CIMNE (2006). M95

  
  

  Abstract

  A method is presented for the solution of an incompressible viscous fluid flow with heat transfer and solidification using a fully Lagrangian description of the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation. First of all, the Navier-Stokes equations of motion coupled with the Boussinesq approximation must be reformulated in the Lagrangian framework, whereas they have been mostly derived in an Eulerian context. Secondly, the Lagrangian formulation implies to follow the material particles during their motion, which means to convect the mesh in the case of the Finite Element Method (FEM), the spatial discretisation method chosen in this work. This provokes various difficulties for the mesh generation, mainly in three dimensions, whereas it eliminates the classical numerical difficulty to deal with the convective term, as much in the Navier-Stokes equations as in the energy equation. Even without the discretization of the convective term, an efficient iterative solver, which constitutes the only viable alternative for three dimensional problems, must be designed for the class of Generalized Stokes Problems (GSP), which could be able to behave well independently of the mesh Reynolds number, as it can vary greatly for coupled fluid-thermal analysis. Moreover, it offers a natural framework to treat free-surface problems like wave breaking and rough fluid-structure contact. On one hand, the convection of the mesh during one time step after the resolution of the non-linear system provides explicitly the locus of the domain to be considered. On the other hand, fluid-to-fluid and fluid-to-wall contact, as well as the update of the domain due to the remeshing, must be accurately and efficiently performed. Finally, the solidification of the fluid coupled with its motion through a variable viscosity is considered An efficient overall algorithm must be designed to bring the method effective, particularly in a three dimensional context, which is the ambition of this monograph. Various numerical examples are included to validate and highlight the potential of the method.

  Abstract
  A method is presented for the solution of an incompressible viscous fluid flow with heat transfer and solidification using a fully Lagrangian description of the motion. [...]

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• The challenge of mass conservation in the solution of free‐surface flows with the fractional‐step method: Problems and solutions

  S. Idelsohn, E. Oñate

  Int. J. Numer. Meth. Biomed. Engng (2010). Vol. 26 (10), pp. 1313-1330

  
  

  Abstract

  The purpose of this paper is to put in evidence that the fractional‐step method (FSM) used to solve the incompressible transient Euler and Navier–Stokes equations for free‐surface flows has a problem inherent to the method that may produce unacceptable variations of the domain volume. A simple modification of the free‐surface boundary term is introduced in order to reduce considerably the volume loss and preserve the computational advantages of the FSM.

  Abstract
  The purpose of this paper is to put in evidence that the fractional‐step method (FSM) used to solve the incompressible transient Euler and Navier–Stokes equations [...]

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