102,495 research outputs found
Non-Intrusive Polynomial Chaos Method Applied to Full-Order and Reduced Problems in Computational Fluid Dynamics: a Comparison and Perspectives
No full textIn this work, Uncertainty Quantification (UQ) based on non-intrusive Polynomial Chaos Expansion (PCE) is applied to the CFD problem of the flow past an airfoil with parameterized angle of attack and inflow velocity. To limit the computational cost associated with each of the simulations required by the non-intrusive UQ algorithm used, we resort to a Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD)-Galerkin approach. A first set of results is presented to characterize the accuracy of the POD-Galerkin ROM developed approach with respect to the Full Order Model (FOM) solver (OpenFOAM). A further analysis is then presented to assess how the UQ results are affected by substituting the FOM predictions with the surrogate ROM ones
Comparison between reduced basis and stochastic collocation methods for elliptic problems
No full textThe stochastic collocation method (Babuška et al. in SIAM J Numer Anal 45(3):1005–1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411–2442, 2008a; SIAM J Numer Anal 46(5):2309–2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118–1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289–294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229–275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41–44):3187–3206, 2009; Arch Comput Methods Eng 17:435–454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions O(1) to moderate dimensions O(10) and to high dimensions O(100) . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs
Shape optimization for viscous flows by reduced basis methods and free-form deformation
No full textIn this paper, we further develop an approach previously introduced in Lassila and Rozza, 2010, for shape optimization that combines a suitable low-dimensional parametrization of the geometry (yielding a geometrical reduction) with reduced basis methods (yielding a reduction of computational complexity). More precisely, free-form deformation techniques are considered for the geometry description and its parametrization, whereas reduced basis methods are used upon a FE discretization to solve systems of parametrized partial differential equations. This allows an efficient flow field computation and cost functional evaluation during the iterative optimization procedure, resulting in effective computational savings with respect to usual shape optimization strategies. This approach is very general and can be applied to a broad variety of problems. In this paper, we apply it to find the optimal shape of aorto-coronaric bypass anastomoses based on vorticity minimization in the down-field region. Blood flows in the coronary arteries are modeled using Stokes equations; afterwards, results have been verified in feedback using Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd
The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: From Laminar to Turbulent Flows
No full textWe present in this double contribution two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases
L’opera scientifica di Leonardo Pisano, detto il Fibonacci, nella cultura italiana del XIII secolo
No full textLeonardo Fibonacci nacque con ogni probabilità a Pisa intorno al 1170, da una
ricca famiglia appartenente al ceto mercantile. Dopo aver condotto i primi studi
verosimilmente presso una scuola d’abaco pisana, intorno al 1185 raggiunse il padre,
Guglielmo dei Bonacci, presso la città Bugia, in Algeria, dove egli esercitava la
professione di pubblico scrivano della Repubblica di Pisa. Qui entrò in contatto con il
sistema posizionale impiegato dagli Arabi per scrivere i numeri, nonché con le tecniche
di calcolo in uso nei paesi islamici. In seguito, il giovane Leonardo condusse una serie
di viaggi in vari paesi del Mediterraneo, quali l’Egitto, la Siria, la Grecia, la Sicilia e la
Provenza, dove ebbe l’opportunità di frequentare altre scuole e di esercitare l’attività
commerciale. Al rientro dai suoi viaggi si dedicò alla composizione dei suoi scritti, dei
quali ci sono pervenuti il Liber abaci, la Pratica geometrie, il Liber quadratorum, il
Flos e l'Epistula ad magistrum Theodorum.
L’intervento si divide in due parti. Nella prima parte G. Germano intende
focalizzare l’attenzione sullo statuto delle opere a carattere matematico e geometrico
all’interno della produzione letteraria italiana del XIII secolo coi loro caratteri
linguistici e stilistici e nei loro rapporti con la tradizione culturale e con le esigenze del
pubblico dei loro committenti e/o fruitori. Trascurate dai filologi e dai critici della
letteratura per le loro peculiarità contenutistiche e formali, che richiedono lo sviluppo di
competenze specifiche, esse meritano una più profonda attenzione per la ricostruzione
di un variegato mondo culturale ancora poco esplorato. Un caso paradigmatico può
essere offerto dall’opera di Leonardo Pisano, detto il Fibonacci, che comprende trattati
di aritmetica e geometria e di cui si ripercorre l’importanza e la fortuna.
Nella seconda parte N. Rozza intende delineare, attraverso degli esempi concreti, una
breve sintesi della lingua e dello stile del Fibonacci, che, pur scrivendo per lo più in
latino, utilizza talvolta anche termini ed espressioni prese in prestito dalla lingua
volgare, forse al fine di rendere le sue argomentazioni più chiare e adatte alle esigenze
di un pubblico vasto e non sempre provvisto di una raffinata formazione letteraria
L’opera scientifica di Leonardo Pisano, detto il Fibonacci, nella cultura italiana del XIII secolo
No full textL’intervento si divide in due parti. Nella prima parte G. Germano intende focalizzare l’attenzione sullo statuto delle opere a carattere tecnico-scientifico all’interno della produzione letteraria italiana del XIII secolo coi loro registri linguistici e stilistici. Trascurate dai filologi e dai critici della letteratura per le loro peculiarità contenutistiche e formali, che richiedono lo sviluppo di competenze specifiche, esse meritano una più profonda attenzione per la ricostruzione di un variegato mondo culturale ancora poco esplorato. Un caso paradigmatico può essere offerto dall’opera di Leonardo Pisano, detto il Fibonacci, che comprende trattati di aritmetica e geometria e di cui si ripercorre l’importanza e la fortuna, con una particolare attenzione alla sorte del Liber abaci. Nella seconda parte N. Rozza intende presentare, attraverso degli esempi concreti, i caratteri della lingua e dello stile del Fibonacci, che, pur scrivendo per lo più in latino, utilizza talvolta anche termini ed espressioni prese in prestito dalla lingua volgare, forse al fine di rendere le sue argomentazioni più chiare e adatte alle esigenze di un pubblico vasto e non sempre provvisto di una raffinata formazione letteraria
Optimal control and shape optimization of Aorto-Coronaric bypass anastomoses
No full textIn this paper we present a new approach in the study of Aorto-Coronaric bypass anastomoses configurations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary. The aim is to provide design indications in the perspective of future development for prosthetic bypasses. With a reduced model based on Stokes equations and a vorticity functional in the down field zone of bypass, a Taylor-like patch is found. A feedback procedure with Navier-Stokes fluid model is proposed based on the analysis of wall shear stress and its related indexes such as OSI
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives
No full textThis chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references
Numerical solution of parametrized Navier-Stokes equations by reduced basis methods
No full textWe apply the reduced basis method to solve Navier-Stokes equations in parametrized domains. Special attention is devoted to the treatment of the parametrized nonlinear transport term in the reduced basis framework, including the case of nonaffine parametric dependence that is treated by an empirical interpolation method. This method features (i) a rapid global convergence owing to the property of the Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in the parameter space, and (ii) the offline/online computational procedures that decouple the generation and projection stages of the approximation process. This method is well suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. Our analysis focuses on: (i) the pressure treatment of incompressible Navier-Stokes problem; (ii) the fulfillment of an equivalent inf-sup condition to guarantee the stability of the reduced basis solutions. The applications that we consider involve parametrized geometries, like e.g. a channel with curved upper wall or an arterial bypass configuration.CMC
A reduced order model for investigating the dynamics of the Gen-IV LFR coolant pool
No full textIn the control field, the study of the system dynamics is usually carried out relying on lumped-parameter or one-dimensional modelling. Even if these approaches are well suited for control purposes since they provide fast-running simulations and are easy to linearize, they may not be sufficient to deeply assess the complexity of the systems, in particular where spatial phenomena have a significant impact on dynamics. Reduced Order Methods (ROM) can offer the proper trade-off between computational cost and solution accuracy. In this work, a reduced order model for the spatial description of the Gen-IV LFR coolant pool is developed for the purpose of being employed in a control-oriented plant simulator of the ALFRED reactor. The spatial modelling of the reactor pool is based on the POD-FV-ROM procedure, previously developed with the aim of extending the literature approach based on Finite Element to the Finite Volume approximation of the Naviera-Stokes equations, and building a reduced order model capable of handling turbulent flows modelled through the RANS equations. The mentioned approach is employed to build a ROM-based component of the ALFRED simulator for the coolant pool. The possibility of varying the input variables of the model has been also undertaken. In particular, the lead velocity at the Steam Generator outlet has been considered as a parametrized boundary condition since it can be a possible control variable. The results have turned out to be very satisfactory in terms of both accuracy and computational time. As a major outcome of the ROM model, it has been proved that its behaviour is more accurate than a 0D-based model without requiring an excessive computational cost
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