MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1275 products
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05/2007 - 02/18/2007
Alì, G.; Culpo, M.; Micheletti, S.
Domain decomposition techniques for microelectronic modeling | Abstract | | This paper is meant to be the continuation of the previous work [1] where
a coupled ODE/PDE method for the simulation of semiconductor devices
was introduced. From a strictly mathematical viewpoint, analytical results
on coupled PDE/ODE systems (as arising in integrated circuit simulation)
can be found in [2]. In particular, in the present paper, we numerically
investigate an algorithm of Domain Decomposition type for the simulation
of circuits containing distributed devices (x 1) as well as semiconductors
in which some part is modeled with lumped parameters (x 2). It is worth
noticing the original employment of the Domain Decomposition technique
within the con_nes of a heterogeneous PDE/ODE coupling, versus its
typical use in a homogeneous full-PDE context. The results presented
here have been studied in the seminal work [3], while a more thorough
analysis is ongoing [4].
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04/2007 - 02/15/2007
Nobile, Fabio; Tempone, Raul; Clayton G., Webster
An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data | Abstract | | This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The method consists of a Galerkin approximation in the space variables and
a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled
deterministic problems, just as in the Monte Carlo method. This work includes a priori and a posteriori procedures to adapt the anisotropy of
the sparse grids to each given problem. These procedures seem to be very effective for the problems under study. The proposed method combines
the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh approximately equally in the solution, while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables, as in the case where input random variables are
Karhunen-Loève truncations of “smooth” random fields. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)-exponential convergence in the asymptotic regime and algebraic convergence in the pre-asymptotic regime, with respect to the total number of collocation points. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo. In particular, for moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.
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03/2007 - 01/23/2007
Santiago, Badia; Quaini, Annalisa; Quarteroni, Alfio
Splitting methods based on algebraic factorization for fluid-structure interaction | Abstract | | We discuss in this paper the numerical approximation of fluid-structure interaction (FSI) problems dealing with strong added-mass effect. We propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, at each iteration the fluid velocity is computed separately from the coupled pressure-structure velocity system, reducing the computational cost.We investigate explicit-implicit decomposition through algebraic splitting techniques originally designed for the FSI problem. This approach leads to two different families of methods which extend to FSI the algebraic pressure correction method and the Yosida method, two schemes that were previously adopted for pure fluid problems. Furthermore, we have considered the inexact factorization of the fluid-structure system as a preconditioner. The numerical properties of these methods have been tested on a model problem representing a blood-vessel system.
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02/2007 - 01/18/2007
Nochetto, Ricardo H.; Veeser, Andreas; Verani, Marco
A safeguarded dual weighted residual method | Abstract | | The dual weighted residual (DWR) method yields reliable a posteriori error bounds for linear output functionals provided that the error incurred by the numerical approximation of the dual solution is negligible. In that case its performance is generally superior than that of global energy norm
error estimators which are
unconditionally reliable. We present a simple numerical example for which neglecting the approximation error leads to severe underestimation of the functional error, thus showing that the DWR method may be unreliable. We propose a remedy that preserves the original performance, namely a DWR method safeguarded by additional asymptotically higher order a posteriori terms. In particular, the enhanced estimator is unconditionally reliable and asymptotically coincides with the original DWR method. These properties are illustrated via the aforementioned example. |
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01/2007 - 01/10/2007
Nobile, Fabio; Vergara, Christian
An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions | Abstract | | In this work we focus on the modelling and numerical simulation of the fluid-structure interaction mechanism in vascular dynamics. We first propose a simple membrane model to describe the deformation of the arterial wall, which is derived from the Koiter s shell equations and is
applicable to an arbitrary geometry. Secondly, we consider a reformulation of the fluid-structure problem, in which the newly derived membrane model, thanks to its simplicity, is embedded into the fluid equations and will appear as a generalized Robin boundary condition. The original problem is then reduced to the solution of
subsequent fluid equations defined on a moving domain and may be achieved with a fluid solver, only. We also derive a stability estimate for the resulting numerical scheme. Finally, we propose new outflow absorbing boundary conditions, which
are easy to implement and allow to reduce significantly the spurious pressure wave reflections that typically appear in artificially
truncated computational domains. We present several numerical results showing the effectiveness of the proposed approaches. |
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MOX 97 - 12/22/2006
Badea, Lori; Discacciati, Marco; Quarteroni, Alfio
Mathematical analysis of the Navier-Stokes/Darcy coupling | Abstract | | We consider a differential system based on the coupling of the Navier Stokes and Darcy equations for modeling the interaction between surface and subsurface flows. We formulate the problem as an interface equation, we analyze the associated (nonlinear) Steklov-Poincaré operators, and we prove its wellposedness. |
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MOX 96 - 12/21/2006
Massimi, Paolo; Quarteroni, Alfio; Saleri, Fausto; Scrofani, Giovanni
Modeling of Salt Tectonics | Abstract | | In this work a general framework for the simulation of sedimentary basins in presence of salt structures is addressed. Sediments and evaporites are modeled as non-Newtonian fluids and the thermal effects induced by the presence of salt are taken into account. The computational strategy is based on a Lagrangian methodology
with intensive grid adaptivity, together with a kinematic modeling of faults and different kinds of boundary conditions representing sedimentation, erosion, basement evolution, lithospheric compression and extension. The proposed methodology is applied to simple test cases as well as to a realistic geological reconstruction of industrial interest.
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MOX 95 - 12/12/2006
Babuška, Ivo; Nobile, Fabio; Tempone Raul
Reliability of Computational Science | Abstract | | Today’s computers allow us to simulate large, complex physical problems.
Many times the mathematical models describing such problems are
based on a relatively small amount of available information such as experimental
measurements. The question arises whether the computed data
could be used as the basis for decision in critical engineering, economic,
medicine applications. The representative list of engineering accidents occurred
in the past years and their reasons illustrates the question. The
paper describes a general framework for Verification and Validation which
deals with this question. The framework is then applied to an illustrative
engineering problem, in which the basis for decision is a specific quantity of
interest, namely the probability that the quantity does not exceed a given
value. The V&V framework is applied and explained in detail. The result
of the analysis is the computation of the failure probability as well as
a quantification of the confidence in the computation, depending on the
amount of available experimental data.
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