Quaderni MOX
Pubblicazioni
del Laboratorio di Modellistica e Calcolo Scientifico MOX. I lavori riguardano prevalentemente il campo dell'analisi numerica, della statistica e della modellistica matematica applicata a problemi di interesse ingegneristico. Il sito del Laboratorio MOX è raggiungibile
all'indirizzo mox.polimi.it
Trovati 1239 prodotti
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13/2015 - 17/03/2015
Bartezzaghi, A.; Dedè, L.; Quarteroni, A.;
Isogeometric Analysis of High Order Partial Differential Equations on Surfaces | Abstract | | We consider the numerical approximation of high order Partial Differential Equations (PDEs) defined on surfaces in the three dimensional space, with particular emphasis on closed surfaces. We consider computational domains that can be represented by B-splines or NURBS, as for example the sphere, and we spatially discretize the PDEs by means of NURBS based Isogeometric Analysis in the framework of the standard Galerkin method. We numerically solve benchmark Laplace-Beltrami problems of the fourth and sixth order, as well as the corresponding eigenvalue problems, with the goal of analyzing the role of the continuity of the NURBS basis functions on closed surfaces. In this respect, we show that the use of globally high order continuous basis functions, as allowed by the construction of periodic NURBS, leads to the efficient solution of the high order PDEs. Finally, we consider the numerical solution of high order phase field problems on closed surfaces, namely the Cahn-Hilliard and crystal equations.
Key words. High order Partial Dierential Equations; Surfaces; Isogeometric Analysis; Error estimation; Laplace-Beltrami operators; Phase field models. |
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12/2015 - 16/03/2015
Antonietti, P. F.; Beirao da Veiga, L.; Scacchi, S.; Verani, M.
A $C^1$ virtual element method for the Cahn-Hilliard equation with polygonal meshes | Abstract | | In this paper we develop an evolution of the $C^1$ virtual elements of minimal degree for the approximation of the Cahn-Hilliard equation.
The proposed method has the advantage of being conforming in $H^2$ and making use of a very simple set of degrees of freedom, namely 3 degrees of freedom per vertex of the mesh. Moreover, although the present method is new also on triangles, it can make use of general polygonal meshes. As a theoretical and practical support, we prove the convergence of the semi-discrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests. |
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11/2015 - 05/03/2015
Antonietti, P. F.; Marcati, C.; Mazzieri, I.; Quarteroni, A.
High order discontinuous Galerkin methods on simplicial elements for the elastodynamics equation | Abstract | | In this work apply the discontinuous Galekin (dG) spectral element method on meshes made of simplicial elements for the approximation of the elastodynamics equation. Our approach combines the high accuracy of spectral methods, the geometrical flexibility of simplicial elements and the computational flexibility of dG methods. We analyze the dissipation, dispersion and stability properties of the resulting scheme, with a focus on the choice of different sets of basis functions. Finally, we test the
method on benchmark as well as realistic test cases. |
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10/2015 - 12/02/2015
Antonietti, P. F.; Grasselli, M.; Stangalino, S.; Verani, M.
Discontinuous Galerkin approximation of linear parabolic problems with dynamic boundary conditions | Abstract | | In this paper we propose and analyze a Discontinuous Galerkin method for a
linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree $pgeq 1$ on meshes with granularity $h$ along with a backward Euler time-stepping scheme with time-step $Delta t$, we prove that the fully-discrete solution is bounded by the data and it converges, in a suitable (mesh-dependent) energy norm, to the exact solution with optimal order $h^p + Delta t$. The sharpness of the theoretical estimates are verified through several numerical experiments. |
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09/2015 - 11/02/2015
Ghiglietti, A.; Ieva, F.; Paganoni, A.M.; Aletti, G.
On linear regression models in infinite dimensional spaces with scalar response | Abstract | | In functional linear regression, the parameters estimation involves solving a non necessarily well-posed problem and it has points of contact with a range of methodologies, including statistical smoothing, deconvolution and projection on finite-dimensional subspaces. We discuss the standard approach based explicitly on functional principal components analysis, nevertheless the choice of the number of basis components remains something subjective and not always properly discussed and justified. In this work we discuss inferential properties of least square estimation in this context with different choices of projection subspaces, as well as we study asymptotic behaviour increasing the dimension of subspaces. |
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06/2015 - 11/02/2015
Perotto, S.; Zilio, A.
Space-time adaptive hierarchical model reduction for parabolic equations | Abstract | | We formalize the pointwise HiMod approach in an unsteady setting,
by resorting to a model discontinuous in time, continuous and hierarchically reduced in space.
The selection of the modal distribution and of the space-time discretization is automatically performed
via an a posteriori analysis of the global error.
The results of the numerical verification confirm the robustness of the proposed adaptive procedure
in terms of accuracy as well as of sensitivity with respect to the goal quantity.
The validation results in the groundwater experimental setting are actually more than satisfying,
with an improvement in the concentration predictions by means of the adaptive HiMod approximation. |
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07/2015 - 05/02/2015
Giovanardi, B.; Scotti, A.; Formaggia, L.; Ruffo, P.
A general framework for the simulation of geochemical compaction | Abstract | | We propose a mathematical model and a numerical scheme to describe compaction
processes in a sedimentary rock layer undergoing both mechanical and
geochemical processes. We simulate the sedimentation process by providing a
sedimentation rate and we account for chemical reactions using simplified kinetics
describing either the conversion of a solid matrix into a fluid, as in the case of
kerogen degradation into oil, or the precipitation of a mineral solute on the solid
matrix of the rock. We use a Lagrangian description that enables to recast the
equations in a fixed frame of reference. We present an iterative splitting scheme
that allows solving the set of governing equations efficiently in a sequential manner.
We assess the performances of this strategy in terms of convergence and mass
conservation. Some numerical experiments show the capability of the scheme to
treat two test cases, one concerning the precipitation of a mineral, the other the
dissolution of kerogen. |
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08/2015 - 02/02/2015
Agosti, A.; Formaggia, L.; Giovanardi, B.; Scotti, A.
Numerical simulation of geochemical compaction with discontinuous reactions | Abstract | | The present work deals with the numerical simulation of porous media subject to the coupled effects of mechanical compaction and reactive flows that can significantly alter the porosity due to dissolution, precipitation or transformation of the solid matrix. These chemical processes can be effectively modelled by ODEs with discontinuous right hand side, where the discontinuity depends on time and on the solution itself. Filippov theory can be applied to prove existence and to determine the solution behaviour at the discontinuities. From the numerical point of view, tailored numerical schemes are needed to guarantee positivity, mass conservation and accuracy. In particular, we rely on an event-driven approach such that, if the trajectory crosses a discontinuity, the transition point is localized exactly and integration is restarted accordingly.
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