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 1239 products
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61/2015 - 11/21/2015
Tagliabue, A.; Dedè, L.; Quarteroni, A.
Fluid dynamics of an idealized left ventricle: the extended Nitsche’s method for the treatment of heart valves as mixed time varying boundary conditions | Abstract | | In this work, we study the blood flow dynamics in idealized left ventricles (LV) of the human heart modelled by the Navier-Stokes equations with mixed time varying boundary conditions (BCs). The latter are introduced for simulating the functioning of the aortic and mitral valves. Based on the extended Nitsche’s method firstly presented in [Juntunen and Stenberg, Mathematics of Computation, 2009], we propose a formulation allowing an efficient and straightforward numerical treatment of the opening and closing phases of the heart valves which are associated to different kind of BCs, namely natural and essential. Moreover, our formulation includes terms preventing the numerical instabilities associated to backflow divergence, i.e. nonphysical reinflow at the valves. We present and discuss numerical results for the LV obtained by means of Isogeometric Analysis for the spatial approximation with the aim of both analysing the formulation and showing the effectiveness of the approach. In particular, we show that the formulation allows to reproduce meaningful results even in idealized LV. |
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60/2015 - 11/12/2015
Perotto, S.; Reali, A.; Rusconi, P.; Veneziani, A.
HIGAMod: A Hierarchical IsoGeometric Approach for MODel reduction in curved pipes | Abstract | | In computational hemodynamics we typically need to solve incompressible fluids in domains given by curved pipes or network of pipes. To reduce the computational costs, or conversely to improve models based on a pure 1D (axial) modeling, an approach called ``Hierarchical Model reduction'' (HiMod) was recently proposed. It consists of a diverse numerical approximation of the axial and of the transverse components of the dynamics. The latter are properly approximated by spectral methods
with a few degrees of freedom, while classical finite elements were used for the main dynamics to easily fit any morphology. However affine elements for curved geometries are generally inaccurate.
In this paper we conduct a preliminary exploration of IsoGeometric Analysis (IGA) applied to the axial discretization. With this approach, the centerline is approximated by Non Uniform Rational B-Splines (NURBS).
The same functions are used to represent the axial component of the solution. In this way we obtain an accurate representation of the centerline as well as an accurate representation of the solution with few axial degrees of freedom.
This paper provides preliminary promising results of the combination of HiMod with IGA - referred to as HIGAMod approach - to be applied in any field involving computational fluid dynamics in generic pipe-like domains. |
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59/2015 - 11/12/2015
Menafoglio, A.; Guadagnini, A.; Secchi, P.
Stochastic Simulation of Soil Particle-Size Curves in Heterogeneous Aquifer Systems through a Bayes space approach | Abstract | | We address the problem of stochastic simulation of soil particle-size curves (PSCs) in heterogeneous aquifer systems. Unlike traditional approaches that focus solely on a few selected features of PSCs (e.g., selected quantiles), our approach is conducive to stochastic realizations of the spatial distribution of the entire particle-size distribution which can optionally be conditioned on available measured data. We
model PSCs as cumulative distribution functions, and their densities as functional compositions in a Bayes Hilbert space. This enables us to employ an appropriate geometry to deal with the data dimensionality and constraints, and to develop a simulation method for particle-size densities (PSDs) based upon a suitable and well defined projection procedure.
The new theoretical framework enables us to represent and reproduce the complete information content embedded in PSC data. As a first field application, we test the quality of unconditional and conditional simulations obtained with our methodology by considering as a test bed a set of particle-size curved collected within a shallow alluvial aquifer in the Neckar river valley, Germany.
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58/2015 - 11/06/2015
Iapichino, L.; Rozza, G.; Quarteroni, A.
Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries | Abstract | | The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems.
Thanks to this feature, it allows dealing with arbitrarily complex network and features
more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed. |
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57/2015 - 11/02/2015
Wilhelm, M.; Dedè, L.; Sangalli, L.M.; Wilhelm, P.
IGS: an IsoGeometric approach for Smoothing on surfaces | Abstract | | We propose an Isogeometric approach for smoothing on surfaces, namely estimating a function starting from noisy and discrete measurements. More precisely, we aim at estimating functions lying on a surface represented by NURBS, which are geometrical representations commonly used in industrial applications. The estimation is based on the minimization of a penalized least-square functional. The latter is equivalent to solve a 4th-order Partial Differential Equation (PDE). In this context, we use Isogeometric Analysis (IGA) for the numerical approximation of such surface
PDE, leading to an IsoGeometric Smoothing (IGS) method for fitting data spatially distributed on a surface. Indeed, IGA facilitates encapsulating the exact geometrical representation of the surface in the analysis and also allows the use of at least globally C1?continuous NURBS basis functions for which the 4th-order PDE can be solved using the standard Galerkin method. We show the performance of the proposed IGS method by means of numerical simulations and we apply it to the estimation of the pressure coefficient, and associated aerodynamic force on a winglet
of the SOAR space shuttle. |
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56/2015 - 11/02/2015
Bonaventura, L.; Della Rocca, A.
Monotonicity, positivity and strong stability of the TR-BDF2 method and of its SSP extensions | Abstract | | We analyze the one-step method TR-BDF2 from the point of view
of monotonicity, strong stability and positivity. All these properties
are strongly related and reviewed in the common framework of abso-
lute monotonicity. The radius of absolute monotonicity is computed
and it is shown that the parameter value which makes the method
L-stable is also the value which maximizes the radius of monotonicity.
Two hybrid variants of TR-BDF2 are proposed, that reduce the for-
mal order of accuracy and maximize the absolute monotonicity radius,
while keeping the native L-stability useful in stiff problems. Numeri-
cal experiments compare these different hybridization strategies with
other methods commonly used in the presence of stiff and mildly stiff
source terms. The results show that both strategies provide a good
compromise between accuracy and robustness at high CFL numbers,
without suffering from the limitations of alternative approaches al-
ready available in literature.
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55/2015 - 11/02/2015
Fumagalli, A.; Zonca, S.; Formaggia, L.
Advances in computation of local problems for a flow-based upscaling in fractured reservoirs | Abstract | | In this article we present some advances to increase the efficiency and applicability of a flow-based upscaling procedure to solve single and multi-phase flows in natural fractured reservoirs. These geological formations may be characterized by hundreds up to hundreds of thousands of fractures, ranging from small to medium scales, which spread all the reservoir. An explicit representation of all the fractures in real scenarios make soon unfeasible performing numerical simulations. An upscaling procedure is thus required. We assume that the reservoir can be modeled with a coarse corner-point grid where the fractures are geometrically uncoupled, by using an embedded discrete fracture model. To describe the scaled up problem, we consider a flow-based upscaling procedure where multiple sub-regions are used to derive transmissibilities, mean depths and pore volumes related to the coarse degrees of freedom associated with fractures and rock matrix. Our focus is to further enhance the upscaling process by allowing the splitting of unconnected rock matrix regions and to compare two ways of setting up the local problems used to compute the transmissibility between coarse cells. Numerical examples confirm the effectiveness of the proposed approach. |
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54/2015 - 11/02/2015
Canuto, C.; Nochetto, R. H.; Stevenson, R.; Verani, M.
Adaptive Spectral Galerkin Methods with Dynamic Marking | Abstract | | The convergence and optimality theory of adaptive Galerkin methods is
almost exclusively based on the Dorfler marking. This entails a fixed
parameter and leads to a contraction constant bounded below away from
zero. For spectral Galerkin methods this is a severe limitation which affects performance. We present a dynamic marking strategy that allows for a super-linear relation between consecutive discretization errors, and show exponential convergence with linear computational complexity whenever the solution belongs to a Gevrey approximation class.
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