Possibilities of finite calculus in computational mechanics

Oñate, Eugenio

doi:10.1002/nme.961

Cited by 93 publications

(119 citation statements)

References 41 publications

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“…A simple and effective procedure to derive a stabilized formulation for incompressible flows is based on the so-called Finite Calculus formulations [19][20][21].…”

Section: Stabilization Of the Incompressibility Conditionmentioning

confidence: 99%

The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

Idelsohn

Oñate

2004

Numerical Meth Engineering

426

345

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SUMMARYParticle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied to it. Particle Methods may be used for both, discrete and continuous problems. In this paper, a Particle Method is used to solve the continuous fluid mechanics equations. To evaluate the external applied forces on each particle, the incompressible Navier-Stokes equations using a Lagrangian formulation are solved at each time step. The interpolation functions are those used in the Meshless Finite Element Method and the time integration is introduced by an implicit fractional-step method. In this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the Lagrangian formulation. Once the forces are evaluated, the particles move independently of the mesh. All the information is transmitted by the particles. Fluid-structure interaction problems including free-fluid-surfaces, breaking waves and fluid particle separation may be easily solved with this methodology.

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“…A simple and effective procedure to derive a stabilized formulation for incompressible flows is based on the so-called Finite Calculus formulations [19][20][21].…”

Section: Stabilization Of the Incompressibility Conditionmentioning

confidence: 99%

The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

Idelsohn

Oñate

2004

Numerical Meth Engineering

426

345

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“…In our work we use a stabilized form of the momentum mass balance equations obtained via the Finite Calculus (FIC) technique [12,16,19,37] written as…”

Section: Discretization Of the Fluid Equationsmentioning

confidence: 99%

A FEM-DEM technique for studying the motion of particles in non-Newtonian fluids. Application to the transport of drill cuttings in wellbores

et al. 2015

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We present a procedure for coupling the finite element method (FEM) and the discrete element method (DEM) for analysis of the motion of particles in non-Newtonian fluids. Particles are assumed to be spherical and immersed in the fluid mesh. A new method for computing the drag force on the particles in a non-Newtonian fluid is presented. A drag force correction for non-spherical particles is proposed. The FEM-DEM coupling procedure is explained for Eulerian and Lagrangian flows and the basic expressions of the discretized solution algorithm are given. The usefulness of the FEM-DEM technique is demonstrated in its application to the transport of drill cuttings in wellbores.

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“…The FIC-based stabilized formulation for the Navier-Stokes equations is obtained by writing the mass balance equation in a space-time domain of finite size using higher order Taylor series expansions as [30,33,35] Mass Balance…”

Section: Fic Scheme For Navier-stokes Equationsmentioning

confidence: 99%

“…Other applications of the FIC scheme to incompressible flows and convection-diffusion problems are presented in [31][32][33][34][35].…”

Section: Fic Scheme For Navier-stokes Equationsmentioning

confidence: 99%

“…Within the family of stabilization techniques, the Finite Increment Calculus (FIC, or Finite Calculus in short) formulation has been successfully implemented for the stabilization of advectivediffusive transport and incompressible fluid flow problems by Oñate and co-workers [30][31][32][33][34][35][36][37]. In this paper, a FIC-based stabilized formulation for the numerical solution of the compressible Navier-Stokes equations is considered in the context of Galerkin FEM using an implicit scheme.…”

Section: Introductionmentioning

confidence: 99%

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An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus

Kouhi

Oñate

2015

Comput Mech

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A new implicit stabilized formulation for the numerical solution of the compressible NavierStokes equations is presented. The method is based on the Finite Calculus (FIC) scheme using the Galerkin finite element method (FEM) on triangular grids. Via the FIC formulation, two stabilization terms, called streamline term and transverse term, are added to the original conservation equations in the space-time domain. The non-linear system of equations resulting from the spatial discretization is solved implicitly using a damped Newton method benefiting from the exact Jacobian matrix. The matrix system is solved at each iteration with a preconditioned GMRES method. The efficiency of the proposed stabilization technique is checked out in the solution of 2D inviscid and laminar viscous flow problems where appropriate solutions are obtained especially near the boundary layer and shock waves. The work presented here can be considered as a follow up of a previous work of the authors [24]. In that paper, the stabilized Galerkin FEM based on the FIC formulation was derived for the Euler equations together with an explicit scheme. In the present paper, the extension of this work to the Navier-Stokes equations using an implicit scheme is presented.

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Possibilities of finite calculus in computational mechanics

Cited by 93 publications

References 41 publications

The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

A FEM-DEM technique for studying the motion of particles in non-Newtonian fluids. Application to the transport of drill cuttings in wellbores

An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus

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