Computation of the stabilization parameter for the finite element solution of advective-diffusive problems

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

“…These terms are essential in order to introduce the necessary stabilization for the discrete solution of some problems using whatever numerical technique. Details of the derivation of Equation (4) for steady state and transient advective-diffusive and fluid flow problems can be found in References [19][20][21][22].…”

“…The underlined term in Equation (3) introduces a non-locality effect in the standard equilibrium equations (dN/dx = 0). Distance h is the characteristic length of the discrete problem and its value depends on the material properties and the parameters of the discretization method chosen (such as the grid size) [19][20][21][22][23][24][25][26]. Note that for h → 0 the standard infinitesimal form of the balance equation (dN/dx = 0) is recovered.…”

“…Consequently, this term will be neglected in the subsequent derivation and only the term involving the derivatives of r i will be retained in Equation (21). Note that this term can take high values in zones where sharp gradients of the numerical solution error occur, despite the fact that the actual value of r i may be relatively low.…”

“…In advective-diffusive and fluid flow problems it is usual to accept that the characteristic length vector has the direction of the velocity vector (this is the so-called streamline upwind Petrov-Galerkin or SUPG assumption [26,31,32]). A method for computing the characteristic lengths in terms of the element residuals for advective-diffusive and fluid flow problems is described in References [21][22][23][24][25]. In the quasistatic example presented in this paper we have obtained good results using a simpler definition of the characteristic lengths with h i = h j = [ (e) ] 1/n d where (e) is the element area or volume for 2D and 3D problems, respectively.…”

“…The FIC approach has been successfully used to derive stabilized finite element and meshless methods for a wide range of advective-diffusive and fluid flow problems [19][20][21][22][23][24][25][26]. The same ideas were applied in Reference [27] to derive a stabilized formulation for quasi-incompressible and incompressible solids allowing the use of linear triangles and tetrahedra.…”