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 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.

Abstract

A set of numerical model experiments has been conducted to simulate the circulation driven by oscillatory forcing over a theoretical continental slope configuration used previously in laboratory experiments. The test case considered was the numerical simulation of the flow over a model of a submarine canyon, and the numerical model used in the analysis was a coastal ocean model version based on an adaptation of the finite-calculus–finite-element method (FIC-FEM) approach implemented in the commercial package Tdyn. Two cases were analyzed involving changes in fluid density. Structured and unstructured finite-element spatial discretizations were generated for the same study domain to compare the resulting velocity field with outputs from the laboratory experiments and to assess which mesh provided a better representation of the complex geometry of the channel model and the water circulation process. The comparison between the laboratory results from the reference article and the output of the numerical model showed good agreement in the structure and magnitude of the phaseaveraged and residual velocity fields.

Abstract

In this paper we present a stabilized FIC-FEM formulation for the multidimensional transient advection-diffusion-absorption equation. The starting point is the non-local form of the governing equations for the multidimensional transient advection-diffusion-absorption problems obtained via the Finite Increment Calculus (FIC) procedure. The FIC governing equations have a residual form that introduces a characteristic length vector that depends on streamline, absorption and shock capturing stabilization parameters, as well as on a characteristic element size that ensures a stabilized numerical solution using a standard Galerkin FEM. The value of the stabilization parameters is obtained as an extension of the steady-state form. The accuracy of the FIC-FEM formulation is verified in the solution of several transient advection-diffusion-absorption problems using regular meshes of 3-noded triangles and 4-noded quadrilaterals.

Abstract

We present a procedure for coupling the uid and transport equations to model the distribution of a pollutant in a street canyon, in this case, black carbon (BC). The fluid flow is calculated with a stabilized finite element method using the Quasi-Static Variational Multiscale (QS-VMS) technique. For the temperature and pollutant transport we use a semi-Lagrangian procedure, based on the Particle Finite Element Method (PFEM) combined with an Eulerian method based on a Finite Increment Calculus (FIC) formulation. Both methods are implemented on the open-source KRATOS Multiphysics platform. The coupled numerical formulation is applied to the prediction of the transport of BC in a street canyon, which can be a useful tool to lessen the impact of pollutants on pedestrians. Two test cases have been studied: a 2D simplified case and a more complex 3D one. The main goal of this study is to propose a useful tool to study the effect of pollution on pedestrians in a street-level scale. Good comparison with experimental results is obtained.

Abstract

We present a stable finite element formulation for the shallow water equations using the finite increment calculus (FIC) procedure. This research is focused on the stability properties of the FIC technique and uses linear triangles for the spatial discretization with an equal order of interpolation for all the variables. The extension to higher order polynomial interpolation functions and different geometries is straightforward. The present FIC-FEM procedure is also able to introduce artificial viscosity for an adequate shock capturing. An implicit time integration has been used. Special attention has been payed to the dry domain in order to solve the moving boundaries with a fixed mesh Eulerian approach. Three academic examples are included in order to test the capabilities of the FIC-FEM procedure: the global stabilization, the shock capturing technique and the dry-wet interface. An experimental benchmark tests the overall accuracy of the present formulation.

Abstract

In this paper we present an accurate stabilized FIC-FEM formulation for the 1D advection-diffusion-reaction equation in the exponential and propagation regimes using two stabilization parameters. Both the steady-state and transient solutions are considered. The stabilized formulation is based on the standard Galerkin FEM solution of the governing differential equations derived via the Finite Increment Calculus (FIC) method. The steady-state problem is considered first. The optimal value of the two stabilization parameters ensuring an exact (nodal) FEM solution using uniform meshes of linear 2-noded elements is obtained. In the absence of the absorption term the formulation simplifies to the standard one-parameter Petrov-Galerkin method for the advection-diffusion problem. For the diffusion-reaction case one stabilization parameter is just needed and the diffusion-type stabilization term is identical to that obtained by Felippa and O˜ nate [16] using a variational FIC approach. A procedure for computing the stabilization parameters for the transient problem is proposed. The accuracy of the new FIC-FEM formulation is demonstrated in the solution of steady-state and transient 1D advection-diffusion-radiation problems for a the range of physical parameters and boundary conditions. Finally we outline the procedure to extend the 1D FIC-FEM formulation to multidimensions.