Computational Reduction for Parametrized PDEs: Strategies and Applications

Manzoni, Andrea; Quarteroni, Alfio; Rozza, Gianluigi

doi:10.1007/s00032-012-0182-y

Cited by 38 publications

(27 citation statements)

References 56 publications

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“…Proper orthogonal decomposition techniques reduce the dimension of a system by transforming the original variables onto a new set of uncorrelated variables such that the total norm of the spectrum present in all of the original variables is captured well by a few of the uncorrelated variables [22]. This spectral decomposition allows construction of a reduced basis in which the solution is sought.…”

Section: Review Of Classical Podmentioning

confidence: 99%

“…Proper orthogonal decomposition techniques have been applied to numerous science and engineering applications [13][14][15][16][17][18][19]. They often have good approximation properties [20,21] and are naturally applied to nonaffine and nonlinear problems [22]. However, reduced basis methods for the solution of parameterized PDEs based on POD are not without their drawbacks [22].…”

Section: Introductionmentioning

confidence: 99%

“…They often have good approximation properties [20,21] and are naturally applied to nonaffine and nonlinear problems [22]. However, reduced basis methods for the solution of parameterized PDEs based on POD are not without their drawbacks [22]. POD techniques form the reduced basis by sampling the parameter domain and then computing the full-order model (FOM) solutions, called snapshots.…”

Section: Introductionmentioning

confidence: 99%

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Exploration of efficient reduced‐order modeling and a posteriori error estimation

Chaudhry

Estep

Gunzburger

2017

Numerical Meth Engineering

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Efficient algorithms are considered for the computation of a reduced-order model based on the proper orthogonal decomposition methodology for the solution of parameterized elliptic partial differential equations. The method relies on partitioning the parameter space into subdomains based on the properties of the solution space and then forming a reduced basis for each of the subdomains. This yields more efficient offline and online stages for the proper orthogonal decomposition method. We extend these ideas for inexpensive adjoint based a posteriori error estimation of both the expensive finite element method solutions and the reducedorder model solutions, for a single and multiple quantities of interest. Various numerical results indicate the efficacy of the approach. KEY WORDS: Error estimation; reduced-order modeling; a posteriori analysis; quantity of interest; proper orthogonal decomposition for a given 2 L 2 . /. The QoI in (2) is a linear functional of the solution u . Nonlinear QoIs require special treatment and are often dealt by linearization of the QoI [1, 2]. EFFICIENT REDUCED-ORDER MODELING ¹w i ; i D 1; : : : ; M º D arg min 1 KHere, . ; / X and k k X with X D 0 and 1 respectively denote the L 2 and H 1 inner product and norm. The reduced basis vectors w i are determined by computing the SVD of the matrix having

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Section: Review Of Classical Podmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exploration of efficient reduced‐order modeling and a posteriori error estimation

Chaudhry

Estep

Gunzburger

2017

Numerical Meth Engineering

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“…In many applications, it is of particular interest to investigate different material properties and shapes. However, it is unaffordable to compute a new solution for each input parameter and so we consider a reduced order model [14,19,20,23] to reduce those computational costs. It is for instance very important if we want to perform optimization on the material properties and more particularly structural-acoustic design.…”

Section: Reduced Basis Approximation For the Energy Finite Element Mementioning

confidence: 99%

Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA)

Devaud

Rozza

2015

ESAIM: Proc.

Self Cite

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Abstract. In many engineering applications, the investigation of the vibro-acoustic response of structures is of great interest. Hence, great effort has been dedicated to improve methods in this field in the last twenty years. Classical techniques have the main drawback that they become unaffordable when high frequency impact waves are considered. In that sense, the Energy Finite Element Analysis (EFEA) is a good alternative to those methods. Based on an approximate model, EFEA gives time and locally space-averaged energy densities and has been proven to yield accurate results.However, when dealing with structural-acoustic design, it is necessary to obtain the energy density varying a large number of parameters. It is computationally unaffordable and too expensive to compute such solutions for each set of parameters. To prevent this drawback, we introduce a reduced order model which allows to drastically decrease those computational costs, while yielding a reliable and accurate approximation. In this paper, we present an approximation of the EFEA solution considering the Reduced Basis (RB) method. The RB method has already been applied successfully to many different problems. A complete development of this procedure in the context of EFEA is introduced here. Numerical tests and examples are provided for both geometrical and physical parameters.

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“…This means that we employ manifold learning techniques over a data base of previously computed CFD results. While there is a vast corps of literature on (linear) model reduction of flow problems, these techniques rarely achieve real‐time performance. For this particular application, by real‐time we mean a code able to provide feedback response at some 30 to 60 Hz (ie, the usual number of frames per second in modern smartphone cameras).…”

Section: Introductionmentioning

confidence: 99%

An augmented reality platform for interactive aerodynamic design and analysis

Badías

Curtit

González

et al. 2019

Numerical Meth Engineering

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Summary While modern CFD tools are able to provide the user with reliable and accurate simulations, there is a strong need for interactive design and analysis tools. State‐of‐the‐art CFD software employs massive resources in terms of CPU time, user interaction, and also GPU time for rendering and analysis. In this work, we develop an innovative tool able to provide a seamless bridge between artistic design and engineering analysis. This platform has three main ingredients: computer vision to avoid long user interaction at the pre‐processing stage, machine learning to avoid costly CFD simulations, and augmented reality for an agile and interactive post‐processing of the results.

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Computational Reduction for Parametrized PDEs: Strategies and Applications

Cited by 38 publications

References 56 publications

Exploration of efficient reduced‐order modeling and a posteriori error estimation

Exploration of efficient reduced‐order modeling and a posteriori error estimation

Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA)

An augmented reality platform for interactive aerodynamic design and analysis

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