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 1238 products
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36/2011 - 11/07/2011
Motamed, M.; Nobile, F.; Tempone, R.
A stochastic collocation method for the second order wave equation with a discontinuous random speed | Abstract | | In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the Òprobability errorÓ with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.
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35/2011 - 08/25/2011
Iapichino, L.; Quarteroni, A.; Rozza, G.
A Reduced Basis Hybrid Method for the coupling of parametrized domains represented by fluidic networks | Abstract | | In this paper we propose a reduced basis hybrid method (RBHM) for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent.
The RBHM is applied to Stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled.
The RBHM is built upon the reduced basis element method (RBEM) and it takes advantage from both the reduced basis methods (RB) and the domain decomposition method. We move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes. On the latter, representative solutions, corresponding to the same governing partial differential equations, are computed for different values of some parameters of interest, representing, for example, the deformation of the blocks. A generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration.
The desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across sub-domain interfaces by suitable coupling conditions.
The geometrical parametrization of the domain, by transfinite mapping, induces non-affine parameter dependence: an empirical interpolation technique is used to recover an approximate affine parameter dependence and a sub--sequent offline/online decomposition of the reduced basis procedure. This computational decomposition yields a considerable reduction of the problem complexity. Results computed on some combinations of 2D and 3D geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses, pressure and velocity at sub-domain interfaces. |
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34/2011 - 07/27/2011
Benacchio, T.; Bonaventura, L.
A spectral collocation method for the one dimensional shallow water equations on semi-infinite domains | Abstract | | We introduce a spectral collocation method for the discretization of the shallow water
equations on a one dimensional semi-infinite domain, employing suitably rescaled Laguerre basis functions to obtain an accurate description of the solutions on finite regions of arbitrary size. The time discretization is based on
a semi-implicit, semi-Lagrangian approach that allows to handle the highly inhomogeneous node distribution without loss of efficiency.
The method is first validated on standard test cases and then applied to the implementation of absorbing open boundary conditions by coupling the semi-infinite domain to a finite size domain on which the same equations are discretized by standard finite volume methods. Numerical experiments show that the proposed approach does not produce significant spurious reflections at the interface between the finite and infinite domain, thus providing a reliable tool for absorbing boundary conditions.
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33/2011 - 07/26/2011
Antonietti, P.F.; Beirao da Veiga, L.; Lovadina, C.; Verani, M.
Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems | Abstract | | We present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems. Under a saturation assumption, the global reliability and efficiency of the proposed a posteriori estimator has been proved.
Several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking strategies. Finally, we test an heuristic variant of the proposed error estimator which drastically reduces the overall computational cost of the adaptive procedures. |
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32/2011 - 07/25/2011
Aletti, G; Ghiglietti, A; Paganoni, A.
A modified randomly reinforced urn design | Abstract | | We want to construct a response adaptive design, described in terms of two colors urn model targeting fixed asymptotic allocations. We prove asymptotic results for the process of colors generated by the urn and for the process of its compositions. Applications to sequential clinical trials are considered as well as connections with response-adaptive design of experiments. |
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31/2011 - 07/18/2011
Astorino, M.; Becerra Sagredo, J.; Quarteroni, A.
A modular lattice Boltzmann solver for GPU computing processors | Abstract | | During the last decade, the lattice Boltzmann method (LBM) has been increasingly acknowledged as a valuable alternative to classical numerical techniques (e.g. finite elements,finite volumes, etc.) in fluid dynamics. A distinguishing feature of LBM is undoubtedly its highly parallelizable data structure. In this work we present a general parallel LBM framework for graphic processing units (GPUs). After recalling the essential programming principles of the CUDA C language for GPUs, the details of the implementation will be provided. The modular and generic framework here devised guarantees a flexible use of the code both in two- and three-dimensional problems. In addition, a careful implementation of a memory efficient formulation of the LBM algorithm has allowed to limit the high memory consumption that typically affects this computational method. Numerical examples in two and three dimensions illustrate the reliability and the performance of the code. |
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30/2011 - 07/15/2011
Nobile, F.; Pozzoli, M.; Vergara, C.
Time accurate partitioned algorithms for the solution of fluid-structure interaction problems in haemodynamics | Abstract | | In this work we deal with the numerical solution of the fluid-structure
interaction problem arising in the haemodynamic environment. In particular,
we consider BDF and Newmark time discretization schemes, and we
study different methods for the treatment of the fluid-structure interface
position, focusing on partitioned algorithms for the prescription of the continuity
conditions at the fluid-structure interface. We consider explicit and
implicit algorithms, and new hybrid methods. We study numerically the
performances and the accuracy of these schemes, highlighting the best solutions
for haemodynamic applications. We also study numerically their
convergence properties with respect to time discretization, by introducing
an analytical test case |
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29/2011 - 07/14/2011
Morin, P.; Nochetto, R.H.; Pauletti, S.; Verani, M.
AFEM for Shape Optimization | Abstract | | We examine shape optimization problems in the context of inexact sequential
quadratic programming. Inexactness is a consequence of using adaptive
finite element methods (AFEM) to approximate the state
and adjoint equations (via the dual weighted residual method),
update the boundary, and compute the geometric functional.
We present a novel algorithm that equidistributes the errors due to
shape optimization and discretization, thereby leading to
coarse resolution in the early stages and fine resolution upon
convergence, and thus optimizing the computational effort.
We discuss the ability of the algorithm
to detect whether or not geometric singularities such as corners are
genuine to the problem
or simply due to lack of resolution---a new paradigm in adaptivity.
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