A residual correction method based on finite calculus

Abstract

In this paper, a residual correction method based upon an extension of the finite calculus concept is presented. The method is described and applied to the solution of a scalar convection‐diffusion problem and the problem of elasticity at the incompressible or quasi‐incompressible limit. The formulation permits the use of equal interpolation for displacements and pressure on linear triangles and tetrahedra as well as any low order element type. To add additional stability in the solution, pressure gradient corrections are introduced as suggested from developments of sub‐scale methods. Numerical examples are included to demonstrate the performance of the method when applied to typical test problems.

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

“…The FIC method can also be applied to derive a modified equation relating the pressure and the volumetric strain change over a finite size domain as [16,18] …”

“…Once more, for consistency, the Neumann boundary conditions should incorporate a FIC stabilization term as in Eq. (24a) [16,18].…”

“…The FIC formulation in conjunction with the FEM has been successfully applied to the static and dynamic solution of quasi-incompressible and full incompressible solids using 3-noded triangles and 4-noded quadrilaterals and tetrahedra with equal order interpolation for the displacements and the pressure [16,18,45].…”

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

“…The FIC procedure has been interpreted in Reference [28] as a general residual correction method where a numerical solution is sought to a modified system of governing differential equations. In the modified equations not only the original residuals but also the derivatives of these residuals multiplied by characteristic length distances appear.…”

“…The form of the resulting equation system will be identical to Equation (32) (as long as the last two integrals of Equations (26) are still neglected). Application of the FIC method to the formulation of a four-node quadrilateral element with linear interpolation for all variables adequate for quasi and fully incompressible situations is reported in Reference [28].…”