Parametric solutions involving geometry: A step towards efficient shape optimization

Ammar, Amine; Huerta, Antonio; Chinesta, Francisco; Cueto, Elías; Leygue, Adrien

doi:10.1016/j.cma.2013.09.003

Cited by 81 publications

(97 citation statements)

References 24 publications

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“…The absorption coe cient ↵, see (6b), is a natural candidate to evaluate possible remediation actions in physical boundaries. More challenging implementations can include geometrical parameters, see [42,43].…”

Section: The Parameterized Wave Propagation Weak Formmentioning

confidence: 99%

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Modesto

Zlotnik

Huerta

2015

Computer Methods in Applied Mechanics and Engineering

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116

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Solving the Helmholtz equation for a large number of input data in an heterogeneous media and unbounded domain still represents a challenge. This is due to the particular nature of the Helmholtz operator and the sensibility of the solution to small variations of the data. Here a reduced order model is used to determine the scattered solution everywhere in the domain for any incoming wave direction and frequency. Moreover, this is applied to a real engineering problem: water agitation inside real harbors for low to mid-high frequencies.The Proper Generalized Decomposition (PGD) model reduction approach is used to obtain a separable representation of the solution at any point and for any incoming wave direction and frequency. Here, its applicability to such a problem is discussed and demonstrated. More precisely, the contributions of the paper include the PGD implementation into a Perfectly Matched Layer framework to model the unbounded domain, and the separability of the operator which is addressed here using an e cient higher-order projection scheme.Then, the performance of the PGD in this framework is discussed and improved using the higher-order projection and a Petrov-Galerkin approach to construct the separated basis. Moreover, the e ciency of the higherorder projection scheme is demonstrated and compared with the higher-order singular value decomposition.

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Section: The Parameterized Wave Propagation Weak Formmentioning

confidence: 99%

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Modesto

Zlotnik

Huerta

2015

Computer Methods in Applied Mechanics and Engineering

Self Cite

116

View full text Add to dashboard Cite

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“…The PGD methodology allows to compute efficiently a multi-parametric solution defined in a high-dimensional space (spatial coordinates, frequency and other parameters). Multi-parametric models are of great interest in science and engineering because they make possible real-time simulation, optimization and inverse analysis, as illustrated in [4,44].…”

Section: Multi-parametric Modelsmentioning

confidence: 99%

Real‐time monitoring of thermal processes by reduced‐order modeling

Aguado

Huerta

Chinesta

et al. 2014

Numerical Meth Engineering

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SUMMARYThis work presents a simple technique for real-time monitoring of thermal processes. Real-time simulationbased control of thermal processes is a big challenge because high-fidelity numerical simulations are costly and cannot be used, in general, for real-time decision making. Very often, processes are monitored or controlled with a few measurements at some specific points. Thus, the strategy presented here is centered on fast evaluation of the response only where it is needed. To accomplish this, classical harmonic analysis is combined with recent model reduction techniques. This leads to an advanced harmonic methodology, which solves in real-time the transient heat equation at the monitored point. In order to apply the reciprocity principle, harmonic analysis is used in the space-frequency domain. Then, Proper Generalized Decomposition, a reduced order approach, pre-computes a transfer function able to produce the output response for a given excitation. This transfer function is computed offline and only once. The response at the monitoring point can be recovered performing a computationally inexpensive post-processing step. This last step can be performed online for real-time monitoring of the thermal process. Examples show the applicability of this approach for a wide range of problems ranging from fast temperature evaluation to inverse problems.

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“…Figure 1 shows a parameter dependant geometry for an airfoil and the objective is to find the air flow around it. The method applied was proposed in [3] and later extended in [4]. It is based on the idea of having a reference domain T and a mapping function that relates all possible geometries to the reference domain.…”

Section: Motivation Examplesmentioning

confidence: 99%

Effect of the separated approximation of input data in the accuracy of the resulting PGD solution

Zlotnik

Dı́ez

González

et al. 2015

Adv. Model. and Simul. in Eng. Sci.

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The proper generalized decomposition (PGD) requires separability of the input data (e.g. physical properties, source term, boundary conditions, initial state). In many cases the input data is not expressed in a separated form and it has to be replaced by some separable approximation. These approximations constitute a new error source that, in some cases, may dominate the standard ones (discretization, truncation…) and control the final accuracy of the PGD solution. In this work the relation between errors in the separated input data and the errors induced in the PGD solution is discussed. Error estimators proposed for homogenized problems and oscillation terms are adapted to asses the behaviour of the PGD errors resulting from approximated input data. The PGD is stable with respect to error in the separated data, with no critical amplification of the perturbations. Interestingly, we identified a high sensitiveness of the resulting accuracy on the selection of the sampling grid used to compute the separated data. The separation has to be performed on the basis of values sampled at integration points: sampling at the nodes defining the functional interpolation results in an important loss of accuracy. For the case of a Poisson problem separated in the spatial coordinates (a complex diffusivity function requires a separable approximation), the final PGD error is linear with the truncation error of the separated data. This relation is used to estimate the number of terms required in the separated data, that has to be in good agreement with the truncation error accepted in the PGD truncation (tolerance for the stoping criteria in the enrichment procedure). A sensible choice for the prescribed accuracy of the PGD solution has to be kept within the limits set by the errors in the separated input data.

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Parametric solutions involving geometry: A step towards efficient shape optimization

Cited by 81 publications

References 24 publications

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Real‐time monitoring of thermal processes by reduced‐order modeling

Effect of the separated approximation of input data in the accuracy of the resulting PGD solution

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