Marco Picasso

An Anisotropic Error Indicator Based on Zienkiewicz-Zhu Error Estimator: Application to Elliptic and Parabolic Problems

The anisotropic error indicator presented in [M. Picasso, Comm. Numer. Methods Engrg., 19 (2003), pp. 13--23.] in the frame of the Laplace equation is extended to elliptic and parabolic problems. Our error indicator is derived using the anisotropic interpolation estimates of [L. Formaggia and S. Perotto, Numer. Math., 89 (2001), pp. 641--667; L. Formaggia and S. Perotto, Numer. Math., (2002), DOI 10.1007/s002110200415], together with a Zienkiewicz--Zhu error estimator to approach the error gradient. A numerical study of the effectivity index is proposed for elliptic, diffusion-convection, and parabolic problems. An adaptive algorithm is implemented, aimed at controlling the relative estimated error.

Variance reduction methods for CONNFFESSIT-like simulations

Abstract The aim of this paper is to present simple but efficient variance reduction methods for CONNFFESSIT-like simulations, extending the ideas of Hulsen et al. [J. Non-Newtonian Fluid Mech. 70 (1997) 79–101] and Ottinger et al. [J. Non-Newtonian Fluid Mech. 70 (1997) 255–261]. Strongly correlated local ensembles of dumbbells were used, and equilibrium ensembles of dumbbells were subtracted. This idea is extended here to non-equilibrium ensembles of dumbbells. The methods are first presented in the frame of the plane Couette flow for Hookean and FENE dumbbells. Extensions to two-dimensional flows are discussed.

Numerical simulation of 3D viscoelastic flows with free surfaces

A numerical model is presented for the simulation of viscoelastic flows with complex free surfaces in three space dimensions. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different.A splitting method is used for the time discretization. The prediction step consists in solving three advection problems, one for the volume fraction of liquid (which allows the new liquid domain to be obtained), one for the velocity field, one for the extra-stress. The correction step corresponds to solving an Oldroyd-B fluid flow problem without advection in the new liquid domain.Two different grids are used for the space discretization. The three advection problems are solved on a fixed, structured grid made out of small cubic cells, using a forward characteristics method. The Oldroyd-B problem without advection is solved using continuous, piecewise linear stabilized finite elements on a fixed, unstructured mesh of tetrahedrons.Efficient post-processing algorithms enhance the quality of the numerical solution. A hierarchical data structure reduces the memory requirements.Convergence of the numerical method is checked for the pure extensional flow and the filling of a tube. Numerical results are presented for the stretching of a filament. Fingering instabilities are obtained when the aspect ratio is large. Also, results pertaining to jet buckling are reported.

Adaptive finite elements for a linear parabolic problem

A posteriori error estimates for the heat equation in two space dimensions are presented. A classical discretization is used, Euler backward in time, and continuous, piecewise linear triangular finite elements in space. The error is bounded above and below by an explicit error estimator based on the residual. Numerical results are presented for uniform triangulations and constant time steps. The quality of our error estimator is discussed. An adaptive algorithm is then proposed. Successive Delaunay triangulations are generated, so that the estimated relative error is close to a preset tolerance. Again, numerical results demonstrate the efficiency of our approach

Numerical simulation of Rhonegletscher from 1874 to 2100

Due to climatic change, many Alpine glaciers have significantly retreated during the last century. In this study we perform the numerical simulation of the temporal and spatial change of Rhonegletscher, Swiss Alps, from 1874 to 2007, and from 2007 to 2100. Given the shape of the glacier, the velocity of ice u is obtained by solving a 3D nonlinear Stokes problem with a nonlinear sliding law along the bedrock-ice interface. The shape of the glacier is updated by computing the volume fraction of ice @f, which satisfies a transport equation. The accumulation due to snow fall and the ablation due to melting is accounted by adding a source term to the transport equation. A decoupling algorithm allows the two above problems to be solved using different numerical techniques. The nonlinear Stokes problem is solved on a fixed, unstructured finite element mesh consisting of tetrahedrons. The transport equation is solved using a fixed, structured grid of smaller cells. The numerical simulation, from 1874 to 2007, is validated against measurements. Afterwards, three different climatic scenarios are considered in order to predict the shape of Rhonegletscher until 2100. A dramatic retreat of Rhonegletscher during the 21st century is anticipated. Our results contribute to a better understanding of the impact of climatic change on mountain glaciers.

Stabilized Finite Elements on Anisotropic Meshes: A Priori Error Estimates for the Advection-Diffusion and the Stokes Problems

Stabilized finite elements on strongly anisotropic meshes are considered. The design of the stability coefficients is addressed for both the advection-diffusion and the Stokes problems when using continuous piecewise linear finite elements on triangles. Using the polar decomposition of the Jacobian of the affine mapping from the reference triangle to the current one, K, and from a priori error estimates, a new definition of the stability coefficients is proposed. Our analysis shows that these coefficients do not depend on the element diameter hK but on a characteristic length associated with K via the polar decomposition. A numerical assessment of the theoretical analysis is carried out.

GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows

In this paper, order one finite elements together with Galerkin least-squares (GLS) methods are used for solving a three-held Stokes problem arising from the numerical study of viscoelastic hows. Stability and convergence results are established, even when the solvent viscosity is small compared to the viscosity due to the polymer chains. An iterative algorithm decoupling velocity-pressure and stress calculations is proposed. The link with the modified elastic viscous split stress (EVSS) method studied in (M. Fortin, R. Guenette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95; R. Guenette, M. Fortin, J. Non-Newtonian Fluid Mech. 60 (1995) 27-52) is presented. Numerical results are in agreement with theoretical predictions, and with those presented in (M. Fortin, R. Guenette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95)

A new algorithm to simulate the dynamics of a glacier: theory and applications

We propose a novel Eulerian algorithm to compute the changes of a glacier geometry for given mass balances. The surface of a glacier is obtained by solving a transport equation for the volume of fluid (VOF). The surface mass balance is taken into account by adding an interfacial term in the transport equation. An unstructured mesh with standard stabilized finite elements is used to solve the non-linear Stokes problem. The VOF function is computed on a structured grid with a high resolution. The algorithm is stable for Courant numbers larger than unity and conserves mass to a high accuracy. To demonstrate the potential of the algorithm, we apply it to reconstructed Late-glacial states of a small valley glacier, Vadret Muragl, in the Swiss Alps.

Calculation of variable-topology free surface flows using CONNFFESSIT

The capability of calculation of non-Newtonian flows: finite elements and stochastic simulation technique (CONNFFESSIT) to compute transient free surface flows is extended to deal with transient viscoelastic flows for domains in which the topology (connectivity) can change in time and also to deal with multivalued free surfaces, i.e. those that cannot be described by a single-valued height function nor be mapped onto an equivalent problem. Such flows are of great practical relevance as they are prevalent in thick-part mold filling, compression molding, multilayer extrusion and thermoforming, peeling of a pressure sensitive adhesive and numerous other processes in which a viscoelastic fluid stream can fold back onto and reconnect with itself. The method is based on an Eulerian approach for the solution of the conservation equations on a fixed grid, while molecular models are used both as stress calculators and to track the position of the free surface. The finite element scheme uses continuous piecewise elements for both velocity and pressure on an unstructured mesh. The scheme is stabilized by a Galerkin/least-squares method and is convergent. The method is quantitatively tested by comparing with previous numerical results

An Anisotropic Error Estimator for the Crank-Nicolson Method: Application to a Parabolic Problem

In this paper we derive two a posteriori upper bounds for the heat equation. A continuous, piecewise linear finite element discretization in space and the Crank-Nicolson method for the time discretization are used. The error due to the space discretization is derived using anisotropic interpolation estimates and a postprocessing procedure. The error due to the time discretization is obtained using two different continuous, piecewise quadratic time reconstructions. The first reconstruction is developed following G. Akrivis, C. Makridakis, and R. H. Nochetto [Math. Comp., 75 (2006), pp. 511-531], while the second one is new. Moreover, in the case of isotropic meshes only, upper and lower bounds are provided as in [R. Verfurth, Calcolo, 40 (2003), pp. 195-212]. An adaptive algorithm is developed. Numerical studies are reported for several test cases and show that the second error estimator is more efficient than the first one. In particular, the second error indicator is of optimal order with respect to both the mesh size and the time step when using our adaptive algorithm.