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• The ALE/Lagrangian Particle Finite Element Method: A new approach to computation of free-surface flows and fluid–object interactions

  F. Del Pin, S. Idelsohn, E. Oñate, R. Aubry

  Computers and Fluids (2007). Vol. 36 (1), pp. 27-38

  
  

  Abstract

  The Particle Finite Element Method (PFEM) is a well established numerical method [Aubry R, Idelsohn SR, Oñate E, Particle finite element method in fluid mechanics including thermal convection–diffusion, Comput Struct 2004;83:1459–75; Idelsohn S, Oñate E, Del Pin F, A Lagrangian meshless finite element method applied to fluid–structure interaction problems, Comput Struct 2003;81:655–71; Idelsohn SR, Oñate E, Del Pin F, The particle finite element method a powerful tool to solve incompressible flows with free-surfaces and breaking waves, Int J Num Methods Eng 2004;61:964–84] where critical parts of the continuum are discretized into particles. The nodes treated as particles transport their momentum and physical properties in a Lagrangian way while the rest of the nodes may move in an Arbitrary Lagrangian–Eulerian (ALE) frame. In order to solve the governing equations that represent the continuum, the particles are connected by means of a Delaunay Triangulation [Idelsohn SR, Oñate E, Calvo N, Del Pin F, The meshless finite element method, Int J Num Methods Eng 2003;58(4)]. The resulting partition is a mesh where the Finite Element Method is applied to solve the equations of motion. The application of a fully Lagrangian formulation on the particles provides a natural and simple way to track free surfaces as well as to compute contacts in an accurate and robust fashion. Furthermore, the usage of an ALE formulation allows large mesh deformation with larger time steps than the full Lagrangian scheme.

  Abstract
  The Particle Finite Element Method (PFEM) is a well established numerical method [Aubry R, Idelsohn SR, Oñate E, Particle finite element method in fluid mechanics [...]

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• Modeling bed erosion in free surface flows by the particle finite element method

  E. Oñate, M. Celigueta, S. Idelsohn

  Acta Geotechnica (2006). Vol. 1 (4), pp. 237-252

  
  

  Abstract

  We present a general formulation for modeling bed erosion in free surface flows using the particle finite element method (PFEM). The key feature of the PFEM is the use of an updated Lagrangian description to model the motion of nodes (particles) in domains containing fluid and solid subdomains. Nodes are viewed as material points (called particles) which can freely move and even separate from the fluid and solid subdomains representing, for instance, the effect of water drops or soil/rock particles. A mesh connects the nodes defining the discretized domain in the fluid and solid regions where the governing equations, expressed in an integral form, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility of the fluid is introduced via the finite calculus (FIC) method. An incremental iterative scheme for the solution of the nonlinear transient coupled fluid-structure problem is described. The erosion mechanism is modeled by releasing the material adjacent to the bed surface according to the frictional work generated by the fluid shear stresses. The released bed material is subsequently transported by the fluid flow. Examples of application of the PFEM to solve a number of bed erosion problems involving large motions of the free surface and splashing of waves are presented.

  Abstract
  We present a general formulation for modeling bed erosion in free surface flows using the particle finite element method (PFEM). The key feature of the PFEM is the use of [...]

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• An unstructured grid-based, parallel free surface solver

  R. Löhner, C. Yang, E. Oñate, S. Idelsohn

  Applied Numerical Mathematics (1999). Vol. 31 (3), pp. 271-293

  
  

  Abstract

  An unstructured grid-based, parallel-free surface solver is presented. The overall scheme combines a finite-element, equal-order, projection-type 3-D incompressible flow solver with a finite element, 2-D advection equation solver for the free surface equation. For steady-state applications, the mesh is not moved every timestep, in order to reduce the cost of geometry recalculations and surface repositioning. A number of modifications required for efficient processing on shared-memory, cache-based parallel machines are discussed, and timings are shown that indicate scalability to a modest number of processors. The results show good quantitative comparison with experiments and the results of other techniques. The present combination of unstructured grids (enhanced geometrical flexibility) and good parallel performance (rapid turnaround) should make the present approach attractive to hydrodynamic design simulations.

  Abstract
  An unstructured grid-based, parallel-free surface solver is presented. The overall scheme combines a finite-element, equal-order, projection-type 3-D incompressible flow solver [...]

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• Variational Framework for FIC Formulations in Continuum Mechanics: High Order Tensor-Derivative Transformations and Invariants

  C. Felippa, E. Oñate, S. Idelsohn

  Archives of Comp. Meths. Engng. (2018). Vol. 25 (4), pp. 919–963

  
  

  Abstract

  This is part of an article series on a variational framework for continuum mechanics based on the Finite Increment Calculus (FIC). The formulation utilizes high order derivatives of the classical fields of continuum mechanics integrated over control regions to construct stabilizing modification terms. Fields may include displacements, body forces, strains, stresses, pressure and volumetric strains. To support observer-invariant FIC formulations, we have catalogued field transformation equations as well as sets of linear and quadratic invariants of fields and of their derivatives up to appropriate order. Attention is focused on the two-dimensional case of a body in plane strain under the drilling-rotation transformation group. Results are presented for displacement and body-force derivatives of orders up to 4, and for stress, strain, pressure and volumetric strain derivatives of order up to 3. The material assembled here is self-contained because this catalog is believed to be useful beyond FIC applications; for example gradient-based, nonlocal constitutive models of multiscale mechanics and physics that involve finite characteristic dimensions analogous to FIC steplengths.

  Abstract
  This is part of an article series on a variational framework for continuum mechanics based on the Finite Increment Calculus (FIC). The formulation utilizes high order derivatives [...]

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• Analysis of multifluid flows with large time steps using the particle finite element method

  S. Idelsohn, J. Marti, P. Becker, E. Oñate

  Int. J. Numer. Meth. Fluids (2014). Vol. 75 (9), pp. 621-644

  
  

  Abstract

  Multifluids are those fluids in which their physical properties (viscosity or density) vary internally and abruptly forming internal interfaces that introduce a large nonlinearity in the Navier–Stokes equations. For this reason, standard numerical methods require very small time steps in order to solve accurately the internal interface position. In a previous paper, the authors developed a particle‐based method (named particle finite element method (PFEM)) based on a Lagrangian formulation and FEM for solving the fluid mechanics equations for multifluids. PFEM was capable of achieving accurate results, but the limitation of small time steps was still present. In this work, a new strategy concerning the time integration for the analysis of multifluids is developed allowing time steps one order of magnitude larger than the previous method. The advantage of using a Lagrangian solution with PFEM is shown in several examples. All kind of heterogeneous fluids (with different densities or viscosities), multiphase flows with internal interfaces, breaking waves, and fluid separation may be easily solved with this methodology without the need of small time steps.

  Abstract
  Multifluids are those fluids in which their physical properties (viscosity or density) vary internally and abruptly forming internal interfaces that introduce a large nonlinearity [...]

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• Application of the finite point method to high‐Reynolds number compressible flow problems

  R. Ortega, E. Oñate, S. Idelsohn, R. Flores

  Int. J. Numer. Meth. Fluids (2014). Vol. 74 (10), pp. 732-748

  
  

  Abstract

  In this work, the finite point method is applied to the solution of high‐Reynolds compressible viscous flows. The aim is to explore this important field of applications focusing on two main aspects: the easiness and automation of the meshless discretization of viscous layers and the construction of a robust numerical approximation in the highly stretched clouds of points resulting in such domain areas. The flow solution scheme adopts an upwind‐biased scheme to solve the averaged Navier–Stokes equations in conjunction with an algebraic turbulence model. The numerical applications presented involve different attached boundary layer flows and are intended to show the performance of the numerical technique. The results obtained are satisfactory and indicative of the possibilities to extend the present meshless technique to more complex flow problems.

  Abstract
  In this work, the finite point method is applied to the solution of high‐Reynolds compressible viscous flows. The aim is to explore this important field of applications [...]

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• A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

  R. Ortega, E. Oñate, S. Idelsohn, R. Flores

  Int. J. Numer. Meth. Fluids (2013). Vol. 73 (4), pp. 323-343

  
  

  Abstract

  A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind‐biased discretization of the Euler equations, written in arbitrary Lagrangian–Eulerian form and integrated in time by means of a dual‐time steeping technique. In order to exploit the meshless potential of the method, a domain deformation approach based on the spring network analogy is implemented, and h‐adaptivity is also employed in the computations. Typical movable boundary problems in transonic flow regime are solved to assess the performance of the proposed technique. In addition, an application to a fluid–structure interaction problem involving static aeroelasticity illustrates the capability of the method to deal with practical engineering analyses. The computational cost and multi‐core performance of the proposed technique is also discussed through the examples provided.

  Abstract
  A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind‐biased [...]

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• Numerical simulations of negatively buoyant jets in an immiscible fluid using the Particle Finite Element Method

  M. Mier-Torrecilla, A. Geyer, J. Phillips, S. Idelsohn, E. Oñate

  Int. J. Numer. Meth. Fluids (2012). Vol. 69 (5), pp. 1016-1030

  
  

  Abstract

  Negatively buoyant jets consist in a dense fluid injected vertically upward into a lighter ambient fluid. The numerical simulation of this kind of buoyancy‐driven flows is challenging as it involves multiple fluids with different physical properties. In the case of immiscible fluids, it requires, in addition, to track the motion of the interface between fluids and accurately represent the discontinuities of the flow variables. In this paper, we investigate numerically the injection of a negatively buoyant jet into a homogenous immiscible ambient fluid using the Particle Finite Element Method and compare the two‐dimensional numerical results with experiments on the injection of a jet of dyed water through a nozzle in the base of a cylindrical tank containing rapeseed oil. In both simulations and experiments, the fountain inlet flow velocity and nozzle diameter have been varied to cover a wide range of Froude Fr and Reynolds Re numbers ( 0.1 < Fr < 30, 8 < Re < 1350), reproducing both weak and strong laminar fountains. The flow behaviors observed for the different numerical simulations fit in the regime map based on the Re and Fr values of the experiments, and the maximum fountain height is in good agreement with the experimental observations, suggesting that particle finite element method is a useful tool for the study of immiscible two‐fluid systems.

  Abstract
  Negatively buoyant jets consist in a dense fluid injected vertically upward into a lighter ambient fluid. The numerical simulation of this kind of buoyancy‐driven flows [...]

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• A family of residual‐based stabilized finite element methods for Stokes flows

  E. Oñate, P. Nadukandi, S. Idelsohn, J. García-Espinosa, C. Felippa

  Int. J. Numer. Meth. Fluids (2011). Vol. 65 (1-3), pp. 106-134

  
  

  Abstract

  We present a collection of stabilized finite element (FE) methods derived via first‐ and second‐order finite calculus (FIC) procedures. It is shown that several well known existing stabilized FE methods such as the penalty technique, the Galerkin Least Square (GLS) method, the Pressure Gradient Projection (PGP) method and the orthogonal sub‐scales (OSS) method are recovered from the general residual‐based FIC stabilized form. A new family of stabilized Pressure Laplacian Stabilization (PLS) FE methods with consistent nonlinear forms of the stabilization parameters are derived. The distinct feature of the family of PLS methods is that they are residual‐based, i.e. the stabilization terms depend on the discrete residuals of the momentum and/or the incompressibility equations. The advantages and disadvantages of the different stabilization techniques are discussed and several examples of application are presented.

  Abstract
  We present a collection of stabilized finite element (FE) methods derived via first‐ and second‐order finite calculus (FIC) procedures. It is shown that several well known [...]

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• A finite point method for adaptive three‐dimensional compressible flow calculations

  E. Ortega, E. Oñate, S. Idelsohn

  Int. J. Numer. Meth. Fluids (2008). Vol. 60 (9), pp. 937-971

  
  

  Abstract

  The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. 

  Abstract
  The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation [...]

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