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• A finite point method in computational mechancis. Applications to convective transport and fluid flow.

  E. Oñate, S. Idelsohn, O. Zienkiewicz, R. Taylor

  Int. J. Numer. Meth. Engng. (1996). Vol. 39 (22), pp. 3839-3866

  
  

  Abstract

  The paper presents a fully meshless procedure fo solving partial differential equations. The approach termed generically the ‘finite point method’ is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals. Some examples showing the accuracy of the method for solution of adjoint and non‐self adjoint equations typical of convective‐diffusive transport and also to the analysis of compressible fluid mechanics problem are presented.

  Abstract
  The paper presents a fully meshless procedure fo solving partial differential equations. The approach termed generically the ‘finite point method’ is based on [...]

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• Transient analysis of tube rolling processes by a semi‐analytical formulation

  H. Antúnez, S. Idelsohn

  Int. J. Numer. Meth. Engng. (1994). Vol. 37 (21), pp. 3621-3632

  
  

  Abstract

  A semi‐analytical formulation is presented for transient metal‐forming processes which, being axisymmetric in geometry, are subjected to non‐axisymmetric loads and boundary conditions. The problems are described in terms of the flow formulation, and the pseudo‐concentration method (two material based) is used to account for time integration and non‐axisymmetric free surfaces. Proper Fourier series expansion of both the scalar fields defined to identify the free surfaces and of all the variables of the mechanical problem allows the performance of an advective transport–corresponding to time integration–for a non‐axisymmetric velocity field. A block diagonal matrix is obtained for each harmonic component, as it has been achieved for the constant time solution. The new configuration found as a result of this analysis is taken to calculate the velocity field for the incremented time, which in turn is used to perform the following time step

  Abstract
  A semi‐analytical formulation is presented for transient metal‐forming processes which, being axisymmetric in geometry, are subjected to non‐axisymmetric loads and boundary [...]

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• A preconditioning mass matrix to accelerate the convergence to the steady Euler solutions using explicit schemes

  M. Storti, C. Baumann, S. Idelsohn

  Int. J. Numer. Meth. Engng. (1992). Vol. 34 (2), pp. 519-541

  
  

  Abstract

  When explicit time marching algorithms are used to reach the steady state of problems governed by the Euler equations, the rate of convergence is strongly impaired both in the zones with low Mach number and in the zones with transonic flow, e.g. Mach ⩽ α and | Mach − 1| ⩽ α, with α ⩽ 0·2. The rate of convergence becomes slower as α diminishes. We show in this paper, with analytical and numerical results, how the use of a preconditioning mass matrix accelerates the convergence in the aforementioned ranges of Mach numbers. The preconditioning mass matrix (PMM) we advocate in this paper can be applied to any FEM/FVM that uses an explicit time marching scheme to find the steady state. The method's rate of convergence to the steady state is studied, and results for the one‐ and two‐dimiensional cases are presented. In Sections 1‐3, using the one‐dimensional Euler equations, we first explain why there exists a slow rate of convergence when the plain lumping of mass is used. Then the convergence rate to steady solutions is analysed from its two constituents, that is, convergence by absorption at the boundaries and by damping in the domain. Next we give the natural solution to this problem, and with several examples we show the effectiveness of the proposed mass matrix when compared with the plain scheme. In Sections 4‐8 we give the multidimensional version of the preconditioning mass matrix. We make a stability analysis and compare the group velocities and damping with and without the new mass matrix. To finish, we show the velocity of convergence for a common test problem

  Abstract
  When explicit time marching algorithms are used to reach the steady state of problems governed by the Euler equations, the rate of convergence is strongly impaired both in [...]

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• 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 [...]

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• 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 [...]

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• 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 [...]

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• 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 [...]

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• 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 [...]

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• 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 [...]

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• 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). [...]

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