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 1249 prodotti
-
15/2015 - 26/03/2015
Taffetani, M.; de Falco, C.; Penta, R.; Ambrosi, D.; Ciarletta, P.
Biomechanical modelling in nanomedicine: multiscale approaches and future challenges | Abstract | | Nanomedicine is the branch of nanotechnology devoted to the miniaturization of devices and to the functionalization of processes for the diagnosis and the design of tools of clinical use. In the perspective to develop patient-specific treatments and effective therapies against currently incurable diseases, biomechanical modelling plays a key role in enabling their translation to clinical practice. Establishing a dynamic interaction with experiments, a modelling approach is expected to allow investigating problems with lower economic burden, evaluating a larger range of conditions. Since biological systems have a wide range of typical characteristic length and timescales, a multiscale modelling approach is necessary both for providing a proper description of the biological complexity at the single scales and for keeping the largest amount of functional interdependence among them. This work starts with a survey both of the common frameworks for modelling a biological system, at scales from atoms to a continuous distribution of matter, and of the available multiscale methods that link the different levels of investigation. In the following, we define an original approach for dealing with the specific case of transport and diffusion of nanoparticles and/or drug-delivery carriers from the systemic circulation to a target tissue microstructure. Using a macro–micro viewpoint, we discuss the existing multiscale approaches and we propose few original strategies for overcoming their limitations in bridging scales. In conclusion, we highlight and critically discuss the future challenges of multiscale modelling for achieving the long-term objective to assist the nanomedical research in proposing more accurate clinical approaches for improved medical benefit. |
-
14/2015 - 18/03/2015
Canuto, C.; Nochetto, R.H.; Stevenson R,; Verani, M.
Convergence and Optimality of hp-AFEM | Abstract | | We design and analyze an adaptive hp-finite element method
(hp-AFEM) in dimensions $n=1,2$.
The algorithm consists of iterating two routines:
HP-NEARBEST finds a near-best hp-approximation of the current
discrete solution and data to a desired accuracy, and
REDUCE improves the discrete solution to a finer but comparable accuracy.
The former hinges on a recent algorithm by Binev for adaptive
hp-approximation, and acts as a coarsening step. We prove
convergence and instance optimality.
|
-
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. |
-
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. |
-
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. |
-
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. |
-
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. |
-
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. |
|
