COLLECTIONS

Documents published in Scipedia

• Improving the convergence rate of the Petrov‐Galerkin techniques for the solution of transonic and supersonic flows

  C. Baumann, M. Storti, S. Idelsohn

  Int. J. Numer. Meth. Engng. (1992). Vol. 34 (2), pp. 453-568

  
  

  Abstract

  This paper report progress on a technique to accelerate the convergence to steady solutions when the streamline‐upwind/Petrov‐Galerkin (SUPG) technique is used. Both the description of a SUPG formulation and the documentation of the development of a code for the finite element solution of transonic and supersonic flows are reported. 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, together with an appropriate convergence rate to the steady solution. 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. It is shown that the velocities at which the error is absorbed in and ejected from the domain (that is damping and group velocities respectively) are strongly affected by the time step used, and that damping gives an O(N2) algorithm contrasting with the O(N) one given by absorption at the boundaries. Nonetheless, the absorbing effect is very low when very different eigenvalues are present, such as in the transonic case, because the stability condition imposes a too slow group velocity for the smaller eigenvalues. To overcome this drawback we present a new mass matrix that provides us with a scheme having the highest group velocity attainable in all the components. In Section 1 we will describe briefly the theoretical background of the SUPG formulation. In Section 2 it is described how the foregoing formulation was used in the finite element code and which are the appropriate boundary conditions to be used. Finally in Section 3 we will show some results obtained with this code

  Abstract
  This paper report progress on a technique to accelerate the convergence to steady solutions when the streamline‐upwind/Petrov‐Galerkin (SUPG) technique is used. Both the [...]

  • 14
  •  read

• Upwind techniques via variational principles

  S. Idelsohn

  Int. J. Numer. Meth. Engng. (1989). Vol. 28 (4), pp. 769-784

  
  

  Abstract

  The amount of upwind or magnitude of off‐centering needed in the numerical solution of second order differential equations with significant first derivatives is justified more rigorously in this paper by requiring the satisfaction of a variational principle. It will be shown that, for a discrete solution, a set of variational principles can be found which allow the problem to be solved as a self‐adjoint system. The technique presented here will give a clear indication of those regions where (i) upwind techniques are not required, (ii) upwind techniques are necessary and sufficient and (iii) upwind techniques are insufficient and artificial viscosity is required.

  Abstract
  The amount of upwind or magnitude of off‐centering needed in the numerical solution of second order differential equations with significant first derivatives is justified [...]

  • 9
  •  read

• Making curved interfaces straight in phase‐change problems

  M. Storti, L. Crivelli, S. Idelsohn

  Int. J. Numer. Meth. Engng. (1987). Vol. 24 (2), pp. 375-392

  
  

  Abstract

  This paper presents a method for straightening curved interfaces arising in phase‐change problems. The method works on isoparametric finite elements, performing a second transformation which maps the master element onto a new one in which the interface looks like a straight line. This allows using the current Gaussian integration technique for squares to evaluate the integrals over each phase. The method provides a better estimation of the contribution of latent heat effects to the residual vector, compared to those obtained by using the assumption that the interface is straight. Appropriate guidelines for solving the non‐linear system of equations arising in this kind of problem are also given. Several numerical examples are presented to show the performance of the method.

  Abstract
  This paper presents a method for straightening curved interfaces arising in phase‐change problems. The method works on isoparametric finite elements, performing a second [...]

  • 13
  •  read

• Solution of non‐linear thermal transient problems by a reduction method

  A. Cardona, S. Idelsohn

  Int. J. Numer. Meth. Engng. (1986). Vol. 23 (6), pp. 1023-1042

  
  

  Abstract

  A new algorithm for solving transient thermal problems in a reduced subspace of the original space of discretization is described. The basis of the subspace is formed by using the system response at the first time step (or an approximation to it) and a set of orthogonal vectors obtained by the algorithm of Lanczos. Derivatives of these vectors are included when treating non‐linear cases. The method allows one to handle the sharp gradients that appear in thermally loaded structures, and the response is accurately predicted by using only a small number of degrees of freedom in the reduced system. The algorithm is specially well suited for treating large‐scale problems. Examples dealing with one‐, two‐ and three‐dimensional cases of linear and non‐linear conduction problems are presented.

  Abstract
  A new algorithm for solving transient thermal problems in a reduced subspace of the original space of discretization is described. The basis of the subspace is formed by using [...]

  • 8
  •  read

• A temperature‐based finite element solution for phase‐change problems

  L. Crivelli, S. Idelsohn

  Int. J. Numer. Meth. Engng. (1986). Vol. 23 (1), pp. 99-119

  
  

  Abstract

  A finite element procedure for solving multidimensional phase change problems is described. The algorithm combines a temperature formulation with a finite element treatment of the differential equation and discontinuous integration within the two‐phase elements to avoid the necessity of regularization. A new criterion for the computation of the iteration matrix is proposed. It is based on a quasi‐Newton correction of the Jacobian matrix for conduction problems without change of phase. A set of test problems with exact solution is analysed and demonstrates that the procedure can accurately evaluate the front position and temperature history with a reasonable computational effort.

  Abstract
  A finite element procedure for solving multidimensional phase change problems is described. The algorithm combines a temperature formulation with a finite element treatment [...]

  • 14
  •  read

• A family of conforming finite elements for deep shell analysis

  G. Sander, S. Idelsohn

  Int. J. Numer. Meth. Engng. (1982). Vol. 18 (3), pp. 363-380

  
  

  Abstract

  The problem related to the derivation of conforming deep shell finite elements is examined in the light of the thin shell theory and using the classical Loves strain energy formulation. A family of quadrangular finite elements allowing for variable curvature is developed. It is shown how an exact conformity of the displacements can be ensured in a large number of cases. Various static and dynamic applications are used to illustrate the advantages of these elements. 

  Abstract
  The problem related to the derivation of conforming deep shell finite elements is examined in the light of the thin shell theory and using the classical Loves strain energy [...]

  • 16
  •  read

• Global–Local ROM for the solution of parabolic problems with highly concentrated moving sources

  A. Cosimo, A. Cardona, S. Idelsohn

  Comput. Methods Appl. Mech. Engrg., (2017). Vol. 326, pp. 739-756

  
  

  Abstract

  Problems characterised by steep moving gradients are challenging for any numerical technique and even more for the successful formulation of Reduced Order Models (ROMs). The aim of this work is to study the numerical solution of problems with steep moving gradients, by placing the focus on parabolic problems with highly concentrated moving sources. More specifically, a Global–Local scheme well-suited for reduction methods is formulated. With this Global–Localscheme, the local nature of the steep moving gradients is exploited by modelling the neighbourhood of the heat source with a moving local domain. This domain is coupled to the global domain without requiring any re-mesh and preserving the meshes of both domains during the whole simulation. Then, a ROM based on the Proper Orthogonal Decomposition (POD) technique is developed for the moving local domain. The proposed technique establishes a valid approach for tackling the non-separability of the space and time dimensions of these problems. In order to assess the numerical performance of the proposed numerical techniques, several evaluation tests are performed.

  Abstract
  Problems characterised by steep moving gradients are challenging for any numerical technique and even more for the successful formulation of Reduced Order Models (ROMs). [...]

  • 13
  •  read

• 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 [...]

  • 8
  •  read

• 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 [...]

  • 34
  •  read

• 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 [...]

  • 12
  •  read