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 1249 products
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45/2014 - 10/24/2014
Pezzuto, S.; Ambrosi,D.; Quarteroni, A.
An orthotropic active-strain model for the myocardium mechanics and its numerical approximation | Abstract | | In the wide literature devoted to the cardiac structural mechanics, the strain energy proposed by Holzapfel and Ogden exhibits a number of interesting features: it has suitable mathematical properties and it is based on few material parameters that can, in principle, be identified by standard laboratory tests. In this work we illustrate the implementation of a numerical solver based on such a model for both the passive and active mechanics of the heart. Moreover we discuss its performance on a few tests that can be regarded as preliminary to the adoption of the Holzapfel-Ogden model for a real cardiac simulation. While the passive behavior of the cardiac muscle is modeled as an orthotropic hyperelastic material, the active contraction is here accounted for a multiplicative decomposition of the deformation gradient, yielding the so-called active strain approach, a formulation that automatically preserves the ellipticity of the stress tensor and introduces just one extra parameter in the model. We adopt the usual volumetric-isochoric decomposition of the stress tensor to obtain a mathematically consistent quasi-incompressible version of the material, then the numerical approximation applies to a classical Hu-Washizu three fields formulation. After introduction of the tangent problem, we select suitable finite element spaces for the representation of the physical fields. Boundary conditions are prescribed by introduction of a Lagrange multiplier. The robustness and performance of the numerical solver are tested versus a novel benchmark test, for which an exact solution is provided. The curvature data obtained from the free contraction of muscular thin films are used to fit the active contraction parameter. |
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44/2014 - 10/23/2014
Pezzuto, S.; Ambrosi,D.
Active contraction of the cardiac ventricle and distortion of the microstructural architecture | Abstract | | The shortening of the myocardial fibers is the microstructural engine that produces the contraction of the cardiac muscle. The complex interplay between fibers shortening and elastic macroscopic strain is functional to the ejection of blood into the pulmonary and arterial networks. Here we address the contraction of the left ventricle in a finite elasticity framework, adopting the “prolate ellipsoid” geometry and the invariants–based strain energy proposed by Holzapfel and Ogden, where the mechanical role of fibers and sheets is accounted for. We show that a microstructurally motivated mathematical model of active strain type reproduces the main indicators of normal cardiac function along the whole PV-loop without introduction of any further ad hoc law. The bare–bones mathematical model depends on one measurable parameter only, i.e. the shortening ratio of the sarcomere units, which we assume to be nearly independent on the prestretch. Strict enforcement of incompressibility and novel treatment of boundary conditions are shown to be crucial to simulate the correct muscle torsion. |
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43/2014 - 10/22/2014
Brugiapaglia, S.; Micheletti, S.; Perotto, S.
Compressed solving: a numerical approximation technique for PDEs based on compressed sensing | Abstract | | We introduce a new numerical method denoted by CORSING (COmpRessed SolvING) to approximate one-dimensional advection-diffusion-reaction problems, motivated by the recent developments in the sparse representation field, and particularly in Compressed Sensing. The object of CORSING is to lighten the computational cost characterizing a Petrov-Galerkin discretization by reducing the dimension of the test space with respect to the trial space. This choice yields an underdetermined linear system which is solved by exploiting optimization procedures, standard in Compressed Sensing, such as the l0- and l1-minimization. A Matlab implementation of the method assesses the robustness and reliability of the proposed strategy, as well as its effectiveness in reducing the computational cost of the corresponding full-sized Petrov- Galerkin problem. Finally, a preliminary extension of CORSING to the two-dimensional setting is checked on the classical Poisson problem. |
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42/2014 - 10/21/2014
Canale, A.; Vantini, S.
Constrained Functional Time Series: an Application to Demand and Supply Curves in the Italian Natural Gas Balancing Platform | Abstract | | In Italy we have assisted to the recent introduction of the natural gas balancing platform, a system in which gas operators virtually sell and buy natural gas in order to balance the common pipelines network. Basically, the operators daily submit demand bids and supply offers which are eventually sorted according to price. Demand and supply curves are hence obtained by cumulating the corresponding quantities. Motivated by market dynamic modeling in the Italian Natural Gas Balancing Platform, we propose a model to analyze time series of bounded and monotonic functions. In detail, we provide the constrained functions with a suitable pre-Hilbert structure and introduce a useful isometric bijective map associating each possible bounded and monotonic function to an unconstrained. We then introduce a functional-to-functional autoregressive model that we use to predict the entire demand/supply function. We estimate the model by minimizing the squared $L^2$ distance between functional data and functional predictions with a penalty term based on the Hilbert-Schmidt squared norm of autoregressive lagged operators. We have proved that the solution always exist, unique and that it is linear on the data with respect to the introduced geometry thus guaranteeing that the plug-in predictions of future entire demand/supply functions satisfy all required constraints. We also provide an explicit expression for estimates and predictions. The approach is of general interest and can be generalized in any situation in which one has to deal with constrained monotonic functions (strictly positive or bounded) which evolve through time (e.g., dose response functions right-censored survival curves or cumulative distribution functions). |
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41/2014 - 10/16/2014
Esfandiar, B.; Porta, G.; Perotto, S.; Guadagnini, A.
Impact of space-time mesh adaptation on solute transport modeling in porous media | Abstract | | We implement a space-time grid adaptation procedure to efficiently improve the accuracy of numerical simulations of solute transport in porous media in the context of model parameter estimation. We focus on the Advection Dispersion Equation (ADE) for the interpretation of non-reactive transport experiments in laboratory-scale heterogeneous porous media. When compared to a numerical approximation based on a fixed space-time discretization, our approach is grounded on a joint automatic selection of the spatial grid and the time step to capture the main (space-time) system dynamics. Spatial mesh adaptation is driven by an anisotropic recovery-based error estimator which enables us to properly select the size, shape and orientation of the mesh elements. Adaptation of the time step is performed through an ad-hoc local reconstruction of the temporal derivative of the solution via a recovery-based approach. The impact of the proposed adaptation strategy on the capability to provide reliable estimates of the key parameters of an ADE model is assessed on the basis of experimental solute breakthrough data measured following tracer injection in a non-uniform porous system. Model calibration is performed in a Maximum Likelihood (ML) framework upon relying on the representation of the ADE solution through a generalized Polynomial Chaos Expansion (gPCE). Our results show that the proposed anisotropic space-time grid adaptation leads to ML parameter estimates and to model results of markedly improved quality when compared to classical inversion approaches based on a uniform space-time discretization. |
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40/2014 - 10/15/2014
Antonietti, P.F.; Mazzieri, I.; Quarteroni, A.
Improving seismic risk protection through mathematical modeling
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39/2014 - 10/14/2014
Ghiglietti, A.; Paganoni, A.M.
Statistical inference for functional data based on a generalization of Mahalanobis distance | Abstract | | In this paper we propose a generalization of Mahalanobis distance that extends the usual multivariate one to functional data generated by stochastic processes. We show that this distance is well defined in L2 and achieves both the goals of (i) considering all the infinite components of data basis expansion and (ii) keeping the same ideas on which is based the Mahalanobis distance. This new mathematical tool is adopted in an inferential context to construct tests on the mean of Gaussian processes for one and two populations. The tests are constructed assuming the covariance structure to be either know or unknown. The power of all the critical regions has been computed analytically. A wide discussion on the behavior of these tests in terms of their power functions is realized, supported by some simulation studies. |
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38/2014 - 10/13/2014
Shen, H.; Truong, Y.; Zanini, P.
Independent Component Analysis for Spatial Stochastic Processes on a Lattice | Abstract | | Independent Component Analysis (ICA) is a widespread data-driven methodology used to solve Blind Source Separation problems. A lot of algorithms have been proposed to perform ICA, but few of them take into account the dependence within the mixtures and not only the dependence between the mixtures. Some algorithms deal with the temporal ICA (tICA) approach exploiting the temporal autocorrelation of the mixtures (and the sources). In particular, colored ICA (cICA), that works in the spectral domain, is an effective method to perform ICA through a Whittle likelihood procedure assuming the sources to be temporal stochastic process. However spatial ICA (sICA) approach is becoming dominant in several field, like fMRI analysis or geo-referred imaging. In this paper we present an extension of cICA algorithm, called spatial colored ICA (scICA), where sources are assumed to be spatial stochastic processes on a lattice. We exploit the Whittle likelihood and a kernel based nonparametric algorithm to estimate the spectral density of a spatial process on a lattice. We illustrate the performance of the proposed method through different simulation studies and a real application using a geo-referred dataset about mobile-phone traffic on the urban area of Milan, Italy. Simulations and the real application showed the improvements provided by scICA method due to take into account the spatial autocorrelation of the mixtures and the sources. |
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