153 research outputs found
A Reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations
In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the
Galerkin projection on the reduced spaces does not necessarily preserved the
inf-sup stability even if the snapshots were generated through a stable full
order method. Therefore, in this work we aim at building a stabilized Reduced
Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes
problems in parametric reduced order settings. This work extends the results
presented for parametrized steady Stokes and Navier-Stokes problems in a work
of ours \cite{Ali2018}. We apply classical residual-based stabilization
techniques for finite element methods in full order, and then the RB method is
introduced as Galerkin projection onto RB space. We compare this approach with
supremizer enrichment options through several numerical experiments. We are
interested to (numerically) guarantee the parametrized reduced inf-sup
condition and to reduce the online computational costs.Comment: arXiv admin note: text overlap with arXiv:2001.0082
Space-time POD-Galerkin approach for parametric flow control
In this contribution we propose reduced order methods to fast and reliably
solve parametrized optimal control problems governed by time dependent
nonlinear partial differential equations. Our goal is to provide a tool to deal
with the time evolution of several nonlinear optimality systems in many-query
context, where a system must be analysed for various physical and geometrical
features. Optimal control can be used in order to fill the gap between
collected data and mathematical model and it is usually related to very time
consuming activities: inverse problems, statistics, etc. Standard
discretization techniques may lead to unbearable simulations for real
applications. We aim at showing how reduced order modelling can solve this
issue. We rely on a space-time POD-Galerkin reduction in order to solve the
optimal control problem in a low dimensional reduced space in a fast way for
several parametric instances. The proposed algorithm is validated with a
numerical test based on environmental sciences: a reduced optimal control
problem governed by viscous Shallow Waters Equations parametrized not only in
the physics features, but also in the geometrical ones. We will show how the
reduced model can be useful in order to recover desired velocity and height
profiles more rapidly with respect to the standard simulation, not losing
accuracy
A monolithic and a partitioned Reduced Basis Method for Fluid-Structure Interaction problems
The aim of this work is to present an overview about the combination of the
Reduced Basis Method (RBM) with two different approaches for Fluid-Structure
Interaction (FSI) problems, namely a monolithic and a partitioned approach. We
provide the details of implementation of two reduction procedures, and we then
apply them to the same test case of interest. We first implement a reduction
technique that is based on a monolithic procedure where we solve the fluid and
the solid problems all at once. We then present another reduction technique
that is based on a partitioned (or segregated) procedure: the fluid and the
solid problems are solved separately and then coupled using a fixed point
strategy. The toy problem that we consider is based on the Turek-Hron benchmark
test case, with a fluid Reynolds number Re = 100
A Weighted POD Method for Elliptic PDEs with Random Inputs
In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to assess the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and high dimensional problems. \ua9 2018, Springer Science+Business Media, LLC, part of Springer Nature
ATHENA: Advanced Techniques for High Dimensional Parameter Spaces to Enhance Numerical Analysis
ATHENA is an open source Python package for reduction in parameter space. It
implements several advanced numerical analysis techniques such as Active
Subspaces (AS), Kernel-based Active Subspaces (KAS), and Nonlinear Level-set
Learning (NLL) method. It is intended as a tool for regression, sensitivity
analysis, and in general to enhance existing numerical simulations' pipelines
tackling the curse of dimensionality. Source code, documentation, and several
tutorials are available on GitHub at https://github.com/mathLab/ATHENA under
the MIT license
Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid--structure interaction problems
In this work we focus on reduced order modelling for problems for which the
resulting reduced basis spaces show a slow decay of the Kolmogorov -width,
or, in practical calculations, its computational surrogate given by the
magnitude of the eigenvalues returned by a proper orthogonal decomposition on
the solution manifold. In particular, we employ an additional preprocessing
during the offline phase of the reduced basis method, in order to obtain
smaller reduced basis spaces. Such preprocessing is based on the composition of
the snapshots with a transport map, that is a family of smooth and invertible
mappings that map the physical domain of the problem into itself. Two test
cases are considered: a fluid moving in a domain with deforming walls, and a
fluid past a rotating cylinder. Comparison between the results of the novel
offline stage and the standard one is presented.Comment: 26 pages, 11 figure
Hierarchical Model Reduction Techniques for Flow Modeling in a Parametrized Setting
In this work we focus on two different methods to deal with parametrized
partial differential equations in an efficient and accurate way. Starting from
high fidelity approximations built via the hierarchical model reduction
discretization, we consider two approaches, both based on a projection model
reduction technique. The two methods differ for the algorithm employed during
the construction of the reduced basis. In particular, the former employs the
proper orthogonal decomposition, while the latter relies on a greedy algorithm
according to the certified reduced basis technique. The two approaches are
preliminarily compared on two-dimensional scalar and vector test cases
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