Finite increment calculus (FIC): a framework for deriving enhanced computational methods in mechanics
 Oñate Eugenio^{1}Email author
https://doi.org/10.1186/s4032301600659
© Eugenio 2016
Received: 19 March 2016
Accepted: 30 March 2016
Published: 28 April 2016
Abstract
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 advectiondiffusionreaction, fluid mechanics and fluidstructure 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.
Keywords
Background
The finite increment calculus (FIC) (sometimes called finite calculus, in short) was proposed by Oñate [1] as a conceptual framework for deriving stabilized numerical methods [mainly the finite element method (FEM)] for solving advective–diffusive transport and fluid compressible flow problems in mechanics for situations where numerical methods typically fail (i.e., high Peclet/Reynolds numbers and incompressible situations) [2, 3].
The essence of the FIC approach lays in solving the governing differential equations in mechanics written in a modified form, using any discretization method such as the Galerkin finite element (FE) method, any standard finite difference (FD) scheme, the finite volume (FV) method, meshless methods, etc. The FIC modified governing equations are obtained by writing the equations for balance of heat, momentum and mass in a spacetime domain of finite incremental size, and not in a domain of infinitesimal size, as it is usually done.
Accounting for the finiteness of the balance spacetime domain introduces naturally additional terms in the classical differential equations of continuum mechanics, which how become a function of the balance domain dimensions. The merit of the modified governing equations derived via FIC is that they are a natural starting point for deriving stabilized numerical schemes. Moreover, the different stabilized FE, FD and FV methods typically used in practice can be recovered using FIC.
The FIC technique has been used in conjunction with the finite element method (FEM) for deriving stabilized FIC/FEM procedures for a wide range of problems in advective–diffusivereactive transport, fluid mechanics, incompressible solids and fluidstructure interaction (FSI) situations, among others [1, 4–28]. The FIC approach has also been used in conjunction with the meshless finite point method for solving some of these problems [13, 29–31].
The layout of the paper is the following. In the next section the main concepts of the FIC method are introduced. Details of the FIC approach for advection–diffusionreaction problems solved with the FEM are presented. Then the possibilities of the FIC/FEM technique in fluid and solid mechanics and FSI problems solved with the particle finite element method (PFEM) are outlined. Examples of applications are given for some of the problems considered. Finally, a brief outline of the possibilities of the FIC/FEM approach for error estimation and mesh adaptivity is given.
Basic ideas of the FIC approach
Standard calculus theory assumes that the balance domain d is of infinitesimal size. Hence, the underlined term in Eq. (3) can be neglected and the resulting flux balance equation is simply \({dq\over dx}=0\). In FIC we will relax this assumption and allow the space balance domain to have a finite size. The new balance Eq. (3) incorporates the underlined term which introduces the characteristic length h. Obviously, accounting for higher order terms in the Taylor expansions of Eq. (2) would lead to additional terms in Eq. (3) incorporating higher powers of h.
Distance h in Eq. (3) can be interpreted as a free parameter depending on the location of point C within the balance domain. Note that \(d\le h\le d\) and, hence, h can take a negative value. At the discrete solution level the domain d should be replaced by the balance domain around a node. This gives for an equal size discretization \(l^e\le h \le l^e\) where \(l^e\) is the element or cell dimension. The fact that Eq. (3) is the exact flux balance equation (up to second order terms) for any 1D domain of finite size and that the position of point C is arbitrary, can be used to derive numerical schemes with enhanced properties. 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].
Applications to the 1D convection–diffusion problems
Boundary conditions
The FIC governing equations are completed with the standard Dirichlet condition prescribing the value of \(\phi \) at the boundary \(\Gamma _\phi \) (i.e., \(\phi = \phi ^p =0\) at \(\Gamma _\phi \)).
We emphasize that the relevant feature of the FIC procedure is that the underlined terms in Eqs. (5) and (6) introduce the necessary stabilization in the discrete solution using whatever numerical scheme.
The FIC method in spacetime
In Eq. (7) \(s_g\) is a sign parameter that can take the values \(+1\) or \(1\). The value of \(s_g\) leads to FIC time integration schemes with distinct numerical properties [33].
The usual sum convention for repeated indexes is used in Eq. (7) and in the following, unless otherwise specified.
A conceptual interpretation of FIC
The problem arises when for some (typically coarse) discretizations the numerical solution provides non physical or very inaccurate values of \( {\hat{\phi }}\). The numerical method is then said to be unstable. A situation of this kind is represented by curves \(M_1\) and \(M_2\) of the left hand side of Fig. 2. These unstabilities disappear by an appropriate mesh refinement (curves \(M_3,M_4, \ldots \)) at the obvious increase of the computational cost.
In the FIC formulation the starting point are the modified differential equations of the problem in \(\Omega \) and the corresponding FIC Neumann boundary condition as previously described. The FIC governing equations can not be used to find an analytical solution, \(\phi (x)\), for the physical problem. However, the numerical solution of the FIC equations can be readily found. What is interesting and useful is that, by adequately choosing the values of the characteristic length parameters, the numerical solution of the FIC equations will be always stable for any discretization chosen.
This process is schematically represented in Fig. 2 where it is shown that the numerical oscillations for the coarser discretizations \(M_1\) and \(M_2\) disappear when using the FIC procedure.
In summary the FIC approach allows us to obtain a better numerical solution for a given discretization. Indeed, as for the standard infinitesimal case, the FIC numerical solution for \(M_\infty \) will yield the (unreachable) exact analytical solution and this ensures the consistency of the method.
The FIC method has been classified in [6] as a particular case of “modified equations methods” where the standard differential equations are first augmented using physical concepts and then discretized using a numerical technique. An interpretation of the FIC/FEM equations as a residual correction method is presented in [16].
FIC/FEM solution of advection–diffusionreaction problems
The boundary conditions at the Dirichlet and Neumann boundary conditions are prescribed as explained in a previous section. Recall that the FIC form of the Neumann boundary conditions as defined in Eq. (6) should be used for consistency of the FIC procedure.
 (i)
Advection–diffusion (\(s=0\))
 (ii)
Helmholtz (\(v=0\), \(s<0\))
 (iii)
Advectionreaction (\(k=0\))
 (iv)
Diffusionreaction (\(v=0\))
Finite element discretization
The last integral in Eq. (11) has been expressed as a sum of the element contributions to allow for interelement discontinuities in the term \({\pmb \nabla }{\hat{r}}\).
Note that the residual terms have disappeared at the Neumann boundary integral in Eq. (11). This is due to taking into account the FIC term in Eq. (6).
This FIC/FEM formulation yields stabilized numerical solutions for the advection–diffusionreaction equation for a wide range of the physical parameters.
For instance, FIC/FEM solutions for 1D and 2D steadystate advection–diffusionabsorption problems have been respectively obtained by Oñate et al. [21, 22] using a nonlinear expression for the stabilization parameters. 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 steadystate and transient 1D advection–diffusionreaction equation using two (linear) stabilization parameters.
A FIC/FEM technique for the transient diffusion equation
The solution of transient problems in mechanics using large time steps has been attempted by different researchers. One example, is the LATIN method proposed by Ladevèze et al. [34, 35]. This is a general nonincremental iterative non linear solution scheme for timedependent problems in mechanics which works globally over the entire timespace domain.
Idelsohn et al. [36, 37] have proposed a Lagrangian numerical procedure for solving convectivediffusive transport and fluidflow problems using large time steps.

Stable explicit solution schemes allowing larger time steps.

Accurate implicit solution schemes allowing larger time steps.
The FIC approach in fluid mechanics
Mass balance equation
Boundary conditions
In the discretized problem the characteristic distances \(h_{ij}\) and \(h_{i}\) become of the order of the typical element dimension. The standard differential equations of momentum and mass balance in fluid mechanics are recovered by making these distances equal to zero.
Eqs. (19), (23) and (24) are the starting point for deriving stabilized FEM for solving the incompressible NavierStokes equations. The underlined FIC terms in Eqs. (19) and (24a) suffice for overcoming the numerical instabilities occurring for high Reynolds numbers, whereas the underlined terms in Eq. (23) take care of the eventual instabilities due to the incompressibility constraint.
The discretized system of finite element equations is obtained by interpolating the velocities and the pressure in terms of nodal values using a mixed FEM formulation and then applying the Galerkin method to the FIC governing equations.
An important feature of the FIC/FEM formulation is that it leads to a stabilized set of equations when using equal order FEM interpolations for the velocity and pressure variables [1, 12].
Applications to the FIC/FEM procedure in fluid mechanics
The FIC/FEM procedure has been successfully applied for solving the NavierStokes equations for incompressible flow problems using the standard Eulerian formulation presented above [12, 19, 38, 39]. A mixed FEM with a linear interpolation for the velocities and the pressure was used in both cases. FIC/FEM formulations for Stokes flows were reported in [26, 40]. Applications of the FIC/FEM technique to fluidstructure interaction (FSI) problems were reported in [14] (for Eulerian flows) and in [41–44] (for Lagrangian flows). The FIC/FEM procedure has been also applied to viscous and inviscid compressible flows as reported in [7–9].
Oñate, Valls and García [23, 24] showed that the stabilization terms introduced by the FIC approach in the momentum equation in Eulerian flows provide enough stability for solving high Reynolds number flows with the FEM without the need of resorting to a turbulence model. This important feature of the FIC method has been recently extended and exploited by Cotela [4] and Cotela, Oñate and Rossi [5] for the FIC/FEM analysis of a range of incompressible turbulent flow problems.
The FIC/FEM computation used a structured mesh of some 5 millions linear tetrahedral elements. The thickness of the layer of elements around the cylinder is 0.001. The time step was set to 0.025 s. The timeaveraged drag coefficient computed is 1.07 and compares well with the experimental value of 1.12. The Strouhal number computed is 2.02 and also agrees with experimental measurements.
Figures 7c1–2 show the isosurfaces of the vorticity vector modulus for three different vorticity values. Figure 7c3 shows streamlines behind the cylinder within the recirculation area. Details of this problem can be found in [23].
A particle finite element method via FIC
The good performance of the FIC/FEM formulation in terms of mass conservation and accuracy for analysis of Lagrangian flows and FSI problems are reported in [44].
A FIC/FEM formulation for solid mechanics
The underlined terms in Eqs. (25) and (26) result from the FIC assumptions and, as usual, \(h_{ij}\) and \(h_k\) are the characteristic length parameters. The governing equations are completed with the adequate boundary conditions. Once more, for consistency, the Neumann boundary conditions should incorporate a stabilization term identical (with a minus sign in front of the FIC term) to that of Eq. (24b) [16, 18].
The FIC formulation in conjunction with the FEM has been successfully applied to the static and dynamic solution of quasiincompressible and full incompressible solids using 3noded triangles and 4noded quadrilaterals with equal order interpolation for the displacements and the pressure [16, 18, 45].
Error estimation and mesh adaption procedures via FIC
The FIC approach can also be used for deriving a procedure for estimating the error in the numerical solution. The FIC contributions to the discretized form of the modified governing equations can be interpreted as a residual error term that depends on the element size and that tends to zero for very fine meshes and, indeed, it vanishes for the exact solution.
The application of the FIC technique to the estimation of the numerical error can be viewed as a particular class of the socalled residualbased error estimation procedures widely used in computational mechanics [46]. The fact that the residual error estimation terms emerge naturally in the FIC formulation is a distinct feature of the method.
Oñate et al. [20] have used the FIC method for formulating a residualbased error estimation technique and the corresponding mesh adaption scheme for analysis of incompressible flows with the FEM.
Concluding remarks
The acceptance that the spacetime domain where the balance equations are established in mechanics has a finite size leads to a modified set of FIC governing differential equations that incorporate the space and time dimensions of the balance domain. The FIC governing equations can be taken as the starting point for deriving a wide range of numerical schemes with improved features in terms of stability and accuracy for solving steadystate and transient problems of advective–diffusivereactive transport, fluid dynamics and incompressible solids, among many others.
Declarations
Acknowledgements
E. Oñate thanks Profs. S.R. Idelsohn, C. Felippa and J. Miquel and Dr. P. Nadukandi for many useful comments.
Dedicated to Prof. Pierre Ladevèze on the occasion of his 70th birthday.
Competing interests
The author declares no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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