An accurate FIC-FEM formulation for the 1D advection–diffusion–reaction equation

Oñate, Eugenio; Miquel, Juan Carlos; Nadukandi, Prashanth

doi:10.1016/j.cma.2015.09.022

Cited by 14 publications

(28 citation statements)

References 63 publications

(106 reference statements)

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“…Figure 3 shows FIC/FEM results for two advection-diffusion-absorption problems solved with 2-noded linear elements. Exact nodal values are found [21,28]. Figure 4 shows the FIC/FEM solution for two Helmholtz problems using 2-noded elements.…”

Section: Finite Element Discretizationmentioning

confidence: 99%

“…Also, for M ∞ the numerical solutionφ will coincide with the exact one φ at all points in . Indeed in some problems the M ∞ solution can be found at the nodes by a "clever" choice of the FIC parameters in the discretized problem [1,6,10,21,28,32].…”

Section: A Conceptual Interpretation Of Ficmentioning

confidence: 99%

“…This goal can be reached by computing the characteristic length parameter h using an adequate "optimality" condition; for instance, in the FEM context we can look for the elemental or nodal value of h that ensures a prescribed (small) error in the numerical solution [1,10,32]. In some cases, the optimal value of h for each element leading to an exact nodal solution can be found [1,6,21,28,32].…”

Section: Basic Ideas Of the Fic Approachmentioning

confidence: 99%

“…Nodally, exact FIC/FEM solutions for the 1D diffusion-absorption case and the Helmholz equation using a single (linear) stabilization parameter were reported by Felippa and Oñate [6] using a variational approach. Oñate, Miquel and Nadukandi [28] have recently presented a general FIC/FEM framework for accurately solving the steady-state and transient 1D advection-diffusion-reaction equation using two (linear) stabilization parameters. Figure 3 shows FIC/FEM results for two advection-diffusion-absorption problems solved with 2-noded linear elements.…”

Section: Finite Element Discretizationmentioning

confidence: 99%

“…The FIC form of the advection-diffusion-reaction equation is written (neglecting the FIC term in time) as [21,28] …”

Section: Fic/fem Solution Of Advection-diffusion-reaction Problemsmentioning

confidence: 99%

See 4 more Smart Citations

Finite increment calculus (FIC): a framework for deriving enhanced computational methods in mechanics

Oñate

2016

Adv. Model. and Simul. in Eng. Sci.

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In this paper we present an overview of the possibilities of the finite increment calculus (FIC) approach for deriving computational methods in mechanics with improved numerical properties for stability and accuracy. The basic concepts of the FIC procedure are presented in its application to problems of advection-diffusion-reaction, fluid mechanics and fluid-structure interaction solved with the finite element method (FEM). Examples of the good features of the FIC/FEM technique for solving some of these problems are given. A brief outline of the possibilities of the FIC/FEM approach for error estimation and mesh adaptivity is given.

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Section: Finite Element Discretizationmentioning

confidence: 99%

Section: A Conceptual Interpretation Of Ficmentioning

confidence: 99%

Section: Basic Ideas Of the Fic Approachmentioning

confidence: 99%

Section: Finite Element Discretizationmentioning

confidence: 99%

“…The FIC form of the advection-diffusion-reaction equation is written (neglecting the FIC term in time) as [21,28] …”

Section: Fic/fem Solution Of Advection-diffusion-reaction Problemsmentioning

confidence: 99%

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Finite increment calculus (FIC): a framework for deriving enhanced computational methods in mechanics

Oñate

2016

Adv. Model. and Simul. in Eng. Sci.

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A stabilization technique for coupled convection‐diffusion‐reaction equations

Hernández

Massart

Peerlings

et al. 2018

Numerical Meth Engineering

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Partial differential equations having diffusive, convective, and reactive terms appear in the modeling of a large variety of processes in several branches of science. Often, several species or components interact with each other, rendering strongly coupled systems of convection-diffusion-reaction equations. Exact solutions are available in extremely few cases lacking practical interest due to the simplifications made to render such equations amenable by analytical tools. Then, numerical approximation remains the best strategy for solving these problems. The properties of these systems of equations, particularly the lack of sufficient physical diffusion, cause most traditional numerical methods to fail, with the appearance of violent and nonphysical oscillations, even for the single equation case. For systems of equations, the situation is even harder due to the lack of fundamental principles guiding numerical discretization. Therefore, strategies must be developed in order to obtain physically meaningful and numerically stable approximations. Such stabilization techniques have been extensively developed for the single equation case in contrast to the multiple equations case. This paper presents a perturbation-based stabilization technique for coupled systems of one-dimensional convection-diffusion-reaction equations. Its characteristics are discussed, providing evidence of its versatility and effectiveness through a thorough assessment. KEYWORDS convection-diffusion equation, differential equations, finite element methods, stability Int J Numer Methods Eng. 2018;116:43-65.wileyonlinelibrary.com/journal/nme

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Fast explicit time integration schemes for parabolic problems in mechanics

Oñate,

Zárate,

Gimenez

et al. 2024

Numerical Meth Engineering

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We present a family of fast explicit time integration schemes of first, second and third order accuracy for parabolic problems in mechanics solved via standard numerical methods that have considerable higher computational efficiency versus existing explicit methods of the same order. The derivation of the new explicit schemes is inspired on the finite increment calculus (FIC) procedure used for obtaining stabilized numerical schemes in fluid and solid mechanics. The new (so‐called) explicit FIC‐Time (EFT) schemes allow considerable larger time steps than the standard first order forward Euler (FE) scheme and the second and third order Adams–Bashforth schemes. The comparison with Runge–Kutta schemes also favors the FIC‐Time schemes in terms of the limit time step size (for second order schemes) and the total number of matrix‐vector multiplications per time step (for second and third order schemes). The new first order explicit schemes have a faster convergence to steady‐state than the FE scheme. The accuracy and efficiency of the new EFT schemes are verified in examples of application to the transient heat conduction equation using the finite element method.

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An accurate FIC-FEM formulation for the 1D advection–diffusion–reaction equation

Cited by 14 publications

References 63 publications

Finite increment calculus (FIC): a framework for deriving enhanced computational methods in mechanics

Finite increment calculus (FIC): a framework for deriving enhanced computational methods in mechanics

A stabilization technique for coupled convection‐diffusion‐reaction equations

Fast explicit time integration schemes for parabolic problems in mechanics

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