Maria Morandi Cecchi

Finite-element modelling of simple shear flow in Newtonian and non-Newtonian fluids around a circular rigid particle

Abstract The flow perturbation around a circular rigid particle during simple shear deformation has been investigated for both Newtonian and non-Newtonian (power-law) fluids by finite-element modelling. If the particle is rotating under the applied shear couple and no-slip occurs at the particle–fluid interface, a ‘bow-tie-shaped’ streamline pattern results for both Newtonian and power-law fluids and the shape of streamlines does not change noticeably if the stress exponent ( n ) is changed. In contrast, if the fluid is allowed to separate from the particle, a ‘double-bulge-shaped’ flow develops in the case of Newtonian fluids, and the type of streamline pattern is influenced by n . We suggest that both stair-stepping and non-stair-stepping geometries of porphyroclast tails may be produced in mylonites, depending on the degree of coherence between the porphyroclasts and the embedding matrix. A different behaviour of the fluid–particle interface may occur as the result of changing fluid rheology, owing to the contrasting stress fields developed for Newtonian and non-Newtonian fluids.

Shallow water theory and its application to the Venice lagoon

Abstract We analyze the computational methods for the shallow water equations. First we derive the shallow water equations from the Navier-Stokes equations and then we discuss the weak formulation of a differential problem and its discretization by the finite element method. Parallel computations are performed by domain decomposition methods. Finally, we present a finite element model to solve the motion problem of the shallow water due to the tides into the Venice lagoon. A comparison has been made between the numerical simulation and the real data on the internal measurement stations of the lagoon. The experimental and simulated data present a good agreement in all the measurement stations.

The Reverse Bordering Method

The bordering method allows recursive computation of the solution of a system of linear equations by adding one new row and one new column at each step of the procedure. When some of the intermediate systems are nearly singular, it is possible, by the block bordering method, to add several new rows and columns simultaneously. However, in that case, the solutions of some of the intermediate systems are not computed. The reverse bordering method allows computation of the solutions of these systems afterwards. Such a procedure has many applications in numerical analysis, that include orthogonal polynomials, Pade approximation, and the progressive forms of extrapolation processes.

A projection method for shallow water equations

The dynamics of shallow water has been studied and an algorithm for this dynamics has been developed. Results have been obtained with data of the Venice lagoon using a model made expressively by a semi‒implicit method based on a finite element method in space. Comparison has been made between field data and the results of the simulation. Very good agreement is shown over a long period of simulation.

Computing the coefficients of a recurrence formula for numerical integration by moments and modified moments

Abstract To evaluate the class of integrals ∫1−1e−αxƒ(x) dx, where R † and the function f(x) is known only approximately in a tabular form, we wish to use a Gaussian quadrature formula. Nodes and weights have to be computed using the family of monic orthogonal polynomials, with respect to the weight function e−αx, obtained through the three-term recurrence relation Pk+1(x) = (x + Bk+1)Pk(x) − Ck+1Pk−1(x). To guarantee a good precision, we must evaluate carefully the values for the coefficients Bk+1 and Ck+1. Such evaluations are made completely formally through a Mathematica program to obtain great precision. A comparison between various methods, starting from moments and modified moments, is shown. Numerical results are also presented.

The polynomial approximation in the finite element method

Dipartimento di Matematica Pura e Applicata, Universitd di Padova, Via Belzoni 7, 1-35131 Padova, Italy Received 20 October 1992 Abstract In the analysis of a finite element method (FEM) we can describe the shape of a given element by a set of elementary functions known as shape functions. The approaches describing these functions are quite different ones. In the plane (x 1, x2), these functions are a product of Lagrangian polynomials when the coordinate system can be chosen with the axes parallel to the sides of the element, otherwise a system of barycentric coordinates (sometimes called area coordinates) could be introduced. The aim of this paper is the description and the representation of shape functions when the element has trianoular shape (the simplest). The representation has been done by using two algorithmic schemes: Neville-Aitken and De Casteljau. For these schemes we have deduced very important properties. Keywords: Barycentric coordinates; Shape functions; Neville-Aitken scheme; De Casteljau scheme; Characteristic space

The Best-Approximation Weighted-Residuals method for the steady convection-diffusion-reaction problem

Abstract: In this paper we present an analytical, parameter-free, Petrov-Galerkin method that gives stable solutions of convection dominated boundary-value problems. We call it the Best Approximation Weighted Residuals (BAWR) method since it gives the best approximation in the norm induced by the inner-product used to build the weighted-residuals approximation. The method computes the optimal weighting functions by solving suitable adjoint problems. Moreover, through a localization technique it becomes computationally efficient without loosing accuracy. The analysis is confirmed by numerical results.

A package for representing C 1 interpolating surfaces: Application to the lagoon of Venice's bed

The paper deals with the description of a method and the accompanying software, the package LABSUP, for representing C1 interpolating surfaces. The application to the lagoon of Venices bed is also proposed. The surfaces are built over the Delaunay triangulation and the polynomial patches used for the representation can be chosen among the Q18 element, the Clough-Tocher or the Powell–Sabin finite elements or simply using global Bézier methods. The first three patches require the knowledge of the gradients at the nodes, or at least a suitable estimation of them. Therefore, interesting in itself is the derivative estimation process based on the minimization of the energy functional associated with the interpolant. For the representation of the lagoon of Venices bed we only used the reduced Clough-Tocher finite element, due to the high number of points involved for which one needs to compute the Delaunay triangulation, and simply the partial derivatives of first order. A brief description of the software modules together with some graphical results of parts of the lagoon of Venices bed are also presented.

An anisotropic unstructured triangular adaptive mesh algorithm based on error and error gradient information

In this paper, an algorithm based on unstructured triangular meshes using standard refinement patterns for anisotropic adaptive meshes is presented. It consists of three main actions: anisotropic refinement, solution-weighted smoothing and patch unrefinement. Moreover, a hierarchical mesh formulation is used. The main idea is to use the error and error gradient on each mesh element to locally control the anisotropy of the mesh. The proposed algorithm is tested on interpolation and boundary-value problems with a discontinuous solution.

Adaptivity in space and time for shallow water equations

In this paper, adaptive algorithms for time and space discretizations are added to an existing solution method previously applied to the Venice Lagoon Tidal Circulation problem. An analysis of the interactions between space and time discretizations adaptation algorithms is presented. In particular, it turns out that both error estimations in space and time must be present for maintaining the adaptation efficiency. Several advantages, for adaptivity and for time decoupling of the equations, offered by the operator-splitting adopted for shallow water equations solution are presented. Copyright