15 Aprile, 2014 15:00 oclock

An Introduction to the Virtual Element Method

LOURENCO BEIRAO DA VEIGA, Universita di Milano

Aula Seminari Saleri del VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

The Virtual Element Method (VEM) is a very recent technology for the discretization of partia differential equationsThe VEM can be interpreted as a novel approach that shares the same variational background as the Finite Element Method but enjoys also a connection with modern Mimetic schemes.

By avoiding the explicit integration of the shape functions that span the discrete Galerkin space and introducing a novel construction of the associated stiffness matrixes, the VEM acquires very interesting properties and advantages with respect to more standard Galerkin methods, yet still keeping the same coding complexity. For instance, the VEM easily allows for polygonal/polyhedral meshes (even non-conforming) with non-convex elements and possibly with curved faces; it allows for discrete solutions of arbitrary $C^k$ regularity, defined on unstructured meshes.

The present talk is an introduction to the VEM, aiming at showing the main ideas of the method. After introducing the method on a simple model elliptic problem, we will present an (optimal) convergence result and some numerical tests. We will moreover address the practical construction of the scheme and also some interesting possibilities such as that of using high regularity discrete spaces and, if time allows, extension to (almost-incompressible) elasticity problems.

contact: marco.verani@polimi.it

15 Aprile, 2014 14:00 oclock

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

On the parabolic equation with the nonlinear Laplacian

Pelin G. Geredeli, Department of Mathematics, Hacettepe University, Ankara

Aula seminari III piano

abstract

abstract

We consider a nonlinear evolution equation of parabolic type
having the p-Laplacian as leading operator.
Under very general conditions on the nonlinearity,
we prove the existence of a regular global attractor.
When the nonlinearity is monotone, and in absence of external source terms,
we give an explicit estimate of the decay rate to zero
of the solution.

15 Aprile, 2014 11:00 oclock

Seminario Matematico e Fisico di Milano

Second order necessary optimality condistions in optimal control

Helene Frankowska, Université Pierre et Marie Curie (Paris 6)

Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9 - Sala Consiglio 7° piano

abstract

abstract

Optimal control theory is an active area of research in both mathematics and engineering sciences. Still, many mathematically challenging problems related to second order optimality conditions remain open, mostly because the structural properties of control systems do not fit well the framework of contemporary optimization literature. In this talk I will discuss second-order necessary optimality conditions in deterministic optimal control from the point of view of second order variational analysis. The new approach introduces second order jets of set valued maps and second order approximations of differential equations and, more generally, control systems and differential inclusions. Then a second order variational differential inclusion leads to a second order Fermat like rule of optimality. In particular, it allows to obtain a new pointwise second order condition verified by the adjoint state of the maximum principle and opens new questions to be investigated.

14 Aprile, 2014 15:00 oclock

Dipartimento di Matematica del Politecnico di Milano

A model of a reactive transport in porous media

ABRAMO AGOSTI, Dipartimento di Matematica, Politecnico di Milano

Aula Seminari Saleri del VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

In this talk I will introduce a model that describes the flow in a porous medium of a mobile species which precipitates into an immobile precipitate, leading to the formation/degradation of crystals inside a porous matrix. The flow is governed by a Darcy problem, where the permeability is changed according to the solute concentration. The dissolution rate is described by a set valued function. The problem has been solved numerically by a treatment of the discontinuity based on event-driven methods and on the Filippov theory. I will present some results about the convergence analysis of a finite element discretization of the problem, and I will compare the results obtained by the application of event-driven methods against regularization methods.

11 Aprile, 2014 11:15 oclock

Sezione di Fisica Matematica

Ideas from statistical mechanics for the notion of infinite mixing.

Marco Lenci, Dipartimento di Matematica, Universita di Bologna

Aula Seminari III Piano, Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

Roughly speaking, mixing is the property of a dynamical system whereby
the dynamics tends to mix up , or randomize, trajectories. When a
finite reference measure exists, the natural formulation of mixing
coincides with the decay of correlations (in time) of the observables
of the system.
In the case of an infinite reference measure, the standard
definition--a.k.a. finite mixing--is inapplicable. Finding an
effective replacement--infinite mixing--is a fundamental, and debated,
question.
Virtually all the definitions that have been attempted so far use
local observables , that is, functions that essentially only see
finite portions of the phase space. We introduce the concept of
global observable , a function that gauges a certain quantity
throughout the phase space. This concept is based on the notion of
infinite-volume average, which is inspired by statistical mechanics
(which is, after all, the field of mathematical physics that has most
successfully dealt with extended systems).
Endowed with the notions of global and local observables, we give a
few definitions of infinite mixing. These fall in two categories:
global-global mixing, which expresses the decorrelation of two
global observables, and global-local mixing, where a global and a
local observable are considered instead.
We will discuss these definitions and, time permitting, see how they
function on some examples of infinite-measure-preserving dynamical
systems.

10 Aprile, 2014 11:30 oclock

Sezione di Fisica Matematica

Introduction to Geological Basin and Probabilistic Petroleum System Analysis

PAOLO RUFFO, ENI S.p.A.

Room “Fausto Saleri”, Dept. of Mathematics, VI floor

abstract

abstract

The modelling of the Basin Evolution requires to understand the key elements of Basin structural architecture and evolution during geological time. The course will introduce some basic notions of basin structural reconstruction with a special focus on the study of hydrodynamic behaviour of faults and fractures and their role in basin and reservoir studies. Starting for the knowledge of Basin evolution the second part of the course will introduce the basic principles of a Petroleum System Modelling with some reference to the main equations involved in its numerical description. The third part of the course will address the probabilistic approach focused in the risk evaluation to support hydrocarbon exploration.

CONTACT: edie.miglio@polimi.it

10 Aprile, 2014 09:30 oclock

Sezione di Fisica Matematica

Introduction to Geological Basin and Probabilistic Petroleum System Analysis

CORRADO MAGISTRONI, ENI S.p.A.

Room “Fausto Saleri”, Dept. of Mathematics, VI floor

abstract

abstract

The modelling of the Basin Evolution requires to understand the key elements of Basin structural architecture and evolution during geological time. The course will introduce some basic notions of basin structural reconstruction with a special focus on the study of hydrodynamic behaviour of faults and fractures and their role in basin and reservoir studies. Starting for the knowledge of Basin evolution the second part of the course will introduce the basic principles of a Petroleum System Modelling with some reference to the main equations involved in its numerical description. The third part of the course will address the probabilistic approach focused in the risk evaluation to support hydrocarbon exploration.

CONTACT: edie.miglio@polimi.it

9 Aprile, 2014 14:30 oclock

Sezione di Fisica Matematica

Adaptive inexact Newton methods and their application to two-phase flows

MARTIN VOHRALIK, Inria - France

Aula Seminari Saleri del VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

We consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations. To solve these sys-tems, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. We derive adaptive stopping criteria for both iterative solvers. Our criteria are based on an a posteriori error estimate which distinguishes the different error components, namely the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error does no longer affect the overall error sig-nificantly. Our estimates also yield a guaranteed upper bound on the overallerror at each step of the nonlinear and linear solvers. We prove the (local) efficiency and robustness of the estimates with respect to the size of the non-linearity owing, in particular, to the error measure involving the dual normof the residual. Our developments hinge on equilibrated flux reconstructions and yield a general framework.
We show how to apply this framework to vari-ous discretization schemes like finite elements, nonconforming finite elements,discontinuous Galerkin, finite volumes, and mixed finite elements; to different linearizations like fixed point and Newton; and to arbitrary iterative linear solvers. Numerical experiments for the p-Laplacian illustrate the tight overall error control and important computational savings achieved in our approach.
In the second part of the talk, we present extensions to the two-phase flow in porous media, where again all error components (spatial, temporal, lineariza-tion, algebraic) are distinguished and fully adaptive algorithms are designed and numerically tested. Details can be found in [1, 2, 3].
References
[1] A. Ern and M. Vohral´?k, Adaptive inexact Newton methods: a posteriori error control and speedup of calculations, SIAM News, 46 (2013), pp. 1,4.
[2] , Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput., 35 (2013), pp. A1761–A1791.
[3] M. Vohral´?k and M. F. Wheeler, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows, Comput. Geosci., 17 (2013), pp. 789–812.
Key words and phrases. nonlinear PDE, a posteriori error estimate, adaptive linearization,
adaptive algebraic solution, adaptive mesh refinement, stopping criterion, two-phase flow.

CONTACT: luca.formaggia@polimi.it

8 Aprile, 2014 09:30 oclock

Sezione di Fisica Matematica

Numerical Models in the Oil Industry

ALBERTO COMINELLI, ENI S.p.A.

Room “Fausto Saleri”, Dept. of Mathematics, VI floor

abstract

abstract

In this talk I will present an overview of the state of the art simulation technology used now-a-days in the industry.

Reservoir simulation is used in the oil industry to support strategic economic decisions by running numerical reservoir models to predict the results of various depletion strategies. This technology has evolved in the last 50 years combining mathematics and computers to assist the oil industry in the development of more and more complex fields using increasingly sophisticated recovery processes.

A numerical reservoir simulation model consists essentially of 1) a representation of the reservoir geology and petrophysics in terms of cells and interfaces 2) a flow model and 3) a well model implemented to reflect the possible depletion processes.

In this presentation we first cast reservoir simulation models as the result of a complex process where information from various sources and scales is collected to provide a quantitative, digital, representation of the subsurface. This includes a geophysical framework, flow relevant geological and petrophysical features, constitutive relations between fluids and porous media, and a fit for purpose representation of the phase behaviour of reservoir fluids. For computational purpose, the reservoir is represented by means of a geo-cellular grid consisting of cells filling the rock volume included inside a structural framework. In this grid information from different scales are integrated, and both upscaling and down-scaling play key roles.

Then we present the flow model, that is to say the set of partial differential equations suited to describe the behaviour of reservoir fluids during depletions. In the numerical simulator these flow equations are discretised in a mass conservative manner using cells as control volumes and interfaces as connections. Wells are included by means of dedicated models connected in a fully implicit manner with the reservoir equations.

The numerical model is advanced in time by means of a proper time marching scheme, balancing efficiency and accuracy. In this process the computational burden is given by the solution of large scale sparse linear systems, deriving from the discretisation of the mass balance equations, which represent a mix of parabolic and hyperbolic equations. In a typical reservoir simulation run, sparse linear systems are solved tens of thousands of times, and they are often ill conditioned because of extreme permeability contrasts, non linearity in the flow models or in the constitutive relations, time-variable degree of implicitness, critical well models or complex grid developed to account for, e.g., highly faulted/fractured reservoirs.

The numerical technology is now evolving from solvers suited for logically cartesian grids towards scalable solvers based on the decoupling of parabolic and hyperbolic components in the linearised system. Nonetheless, grid developed to capture heterogeneity and structural features may still be too detailed to suit numerical simulation, and this may require an upscaling towards coarser grids.

The talk will covers the main components of the simulation technology using some real life cases as examples. A view on the future, with emphasis on modelling of heterogeneous and fractured reservoirs, will finally be provided.


CONTACT: edie.miglio@polimi.it

7 Aprile, 2014 10:00 oclock

Sezione di Fisica Matematica

Numerical methods for flow in fractured porous media

ANNA SCOTTI, Politecnico di Milano

Room “Fausto Saleri”, Dept. of Mathematics, VI floor

abstract

abstract

The simulation of multiphase flow in fractured porous media is a computational challenge. A key issue is the efficient and effective coupling between flow in the porous matrix and in the fracture network: the presence of fractures indeed can influence fluid flow at a variety of space scales. Due to their high aspect ratio fractures are usually represented as lower dimensional interfaces coupled with the flow in the surrounding medium by suitable conditions. However, even in this hybrid dimensional approach, geometric conformity can represent a strict constraint for the generation of the computational grid which, with standard numerical discretization techniques, should honour as much as possible the geometry of the fractures without too degenerated/distorted elements. For instance, finite volumes with two point flux approximation (TPFA) demand a good quality of the mesh to mitigate discretization error.
Such standard techniques are here compared with modern computational methods that, even though more complex, are more robust with respect to mesh geometry and can be beneficial for the problem at hand: the extended finite elements and mimetic finite differences. The former allow us to have fractures that are completely irrespective of the underlying grid thanks to suitable enrichments of the finite element spaces. The latter, coupled with a suitable solver for the fracture network, are a promising alternative to finite volumes in the context of matching discretization, since very general grids, with distorted and non-convex elements can be handled while maintaining a good accuracy.


CONTACT: anna.scotti@polimi.it