Certified real‐time shape optimization using isogeometric analysis, PGD model reduction, and a posteriori error estimation

Chamoin, Ludovic; Thai, Hieu

doi:10.1002/nme.6045

Cited by 17 publications

(8 citation statements)

References 81 publications

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“…Finally, let us notice that even if the analytical sensitivity enables to get rid of the perturbation step, there is still the choice of the refinement level of the analysis model that needs to be done when setting up the optimization problem (see previous discussion for the plate with a hole). It can be interesting to adapt the refinement level during the optimization by using, for example, advanced tools as error estimators [2,4,14].…”

Section: Typementioning

confidence: 99%

A New Lighting on Analytical Discrete Sensitivities in the Context of IsoGeometric Shape Optimization

Hirschler

Bouclier

Duval

et al. 2020

Arch Computat Methods Eng

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Isogeometric shape optimization has been now studied for over a decade. This contribution aims at compiling the key ingredients within this promising framework, with a particular attention to sensitivity analysis. Based on all the researches related to isogeometric shape optimization, we present a global overview of the process which has emerged. The principal feature is the use of two refinement levels of the same geometry: a coarse level where the shape updates are imposed and a fine level where the analysis is performed. We explain how these two models interact during the optimization, and especially during the sensitivity analysis. We present new theoretical developments, algorithms, and quantitative results regarding the analytical calculation of discrete adjoint-based sensitivities. In order to highlight the versatility of this sensitivity analysis method, we perform eight benchmark optimization examples with different types of objective functions (compliance, displacement field, stress field, and natural frequencies), different types of isogeometric element (2D and 3D standard solids, and a Kirchhoff-Love shell), and different types of structural analysis (static and vibration). The numerical performances of the analytical sensitivities are compared with approximate sensitivities. The results in terms of accuracy and numerical cost make us believe that the presented method is a viable strategy to build a robust framework for shape optimization.

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Section: Typementioning

confidence: 99%

A New Lighting on Analytical Discrete Sensitivities in the Context of IsoGeometric Shape Optimization

Hirschler

Bouclier

Duval

et al. 2020

Arch Computat Methods Eng

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show abstract

“…We can now combine Equation (12) with the constrains (14). Substituting the approximations (17) and (16) into (12) and (16) into (14), and minimizing the resulting energy leads to the following equations:…”

Section: Hybrid Equilibrium Finite Element Formulationsmentioning

confidence: 99%

“…The approximated solutions (17) obtained fromŝ defines the generalized stress field which strongly satisfy the equilibrium conditions. The approximations (16) obtained fromv define generalized boundary displacements which weakly satisfy compatibility.…”

Section: Hybrid Equilibrium Finite Element Formulationsmentioning

confidence: 99%

“…12 There are recent applications of this technique, namely the work by Chamoin and Thai, exploring isogeometric analysis and PGD solutions to deal with geometric parameters and applying the constitute relation error as an error estimator and indicator for mesh refinement. [13][14][15][16] Another approach using the dual error analysis concept was performed by Moitinho, using the extended formulation from Prager to general problems proposed by Debongnie and using an independent PGD solution, based on hybrid equilibrium finite elements, to define the equilibrated stress field. [17][18][19][20] When analyzing the errors of a solutions, it is often desirable to have information of specific quantities of interest (QoI).…”

Section: Introductionmentioning

confidence: 99%

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Error estimation for proper generalized decomposition solutions: Dual analysis and adaptivity for quantities of interest

Reis

Almeida

Dı́ez

et al. 2020

Numerical Meth Engineering

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When designing a structure or engineering a component, it is crucial to use methods that provide fast and reliable solutions, so that a large number of design options can be assessed. In this context, the proper generalized decomposition (PGD) can be a powerful tool, as it provides solutions to parametric problems, without being affected by the “curse of dimensionality.” Assessing the accuracy of the solutions obtained with the PGD is still a relevant challenge, particularly when seeking quantities of interest with guaranteed bounds. In this work, we compute compatible and equilibrated PGD solutions and use them in a dual analysis to obtain quantities of interest and their bounds, which are guaranteed. We also use these complementary solutions to compute an error indicator, which is used to drive a mesh adaptivity process, oriented for a quantity of interest. The corresponding solutions have errors that are much lower than those obtained using a uniform refinement or an indicator based on the global error, as the proposed approach focuses on minimizing the error in the quantity of interest.

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“…A similar approach is proposed in [19] to efficiently reduce the computational cost of parametrised fluid models. In [20] an isogeometric analysis framework is used to deal with the domain parametrisation and build a reduced-basis method to speed up shape optimisation problems. A similar approach is proposed in [21].…”

Section: Introductionmentioning

confidence: 99%

State Estimation with Model Reduction and Shape Variability. Application to biomedical problems

Galarce,

Lombardi,

Mula

2021

Preprint

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We develop a mathematical and numerical framework to solve state estimation problems for applications that present variations in the shape of the spatial domain. This situation arises typically in a biomedical context where inverse problems are posed on certain organs or portions of the body which inevitably involve morphological variations. If one wants to provide fast reconstruction methods, the algorithms must take into account the geometric variability. We develop and analyze a method which allows to take this variability into account without needing any a priori knowledge on a parametrization of the geometrical variations. For this, we rely on morphometric techniques involving Multidimensional Scaling, and couple them with reconstruction algorithms that make use of reduced model spaces precomputed on a database of geometries. We prove the potential of the method on a synthetic test problem inspired from the reconstruction of blood flows and quantities of medical interest with Doppler ultrasound imaging.

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Certified real‐time shape optimization using isogeometric analysis, PGD model reduction, and a posteriori error estimation

Cited by 17 publications

References 81 publications

A New Lighting on Analytical Discrete Sensitivities in the Context of IsoGeometric Shape Optimization

A New Lighting on Analytical Discrete Sensitivities in the Context of IsoGeometric Shape Optimization

Error estimation for proper generalized decomposition solutions: Dual analysis and adaptivity for quantities of interest

State Estimation with Model Reduction and Shape Variability. Application to biomedical problems

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