Enhanced parametric shape descriptions in PGD-based space separated representations

Space separation within the Proper Generalized Decomposition—PGD—rationale allows solving high dimensional problems as a sequence of lower dimensional ones. In our former works, different geometrical transformations were proposed for addressing complex shapes and spatially non-separable domains. Efficient implementation of separated representations needs expressing the domain as a product of characteristic functions involving the different space coordinates. In the case of complex shapes, more sophisticated geometrical transformations are needed to map the complex physical domain into a regular one where computations are performed. This paper aims at proposing a very efficient route for accomplishing such space separation. A NURBS-based geometry representation, usual in computer aided design—CAD—, is retained and combined with a fully separated representation for allying efficiency (ensured by the fully separated representations) and generality (by addressing complex geometries). Some numerical examples are considered to prove the potential of the proposed methodology.

[1]  A. Huerta,et al.  Parametric solutions involving geometry: A step towards efficient shape optimization , 2014 .

[2]  L. Gallimard,et al.  Explicit solutions for the modeling of laminated composite plates with arbitrary stacking sequences , 2014 .

[3]  Wei Ji,et al.  Characterization of the proper generalized decomposition method for fixed-source diffusion problems , 2019, Annals of Nuclear Energy.

[4]  Francisco Chinesta,et al.  On the Proper Generalized Decomposition applied to microwave processes involving multilayered components , 2019, Math. Comput. Simul..

[5]  Adrien Leygue,et al.  The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer , 2013 .

[6]  Simona Perotto,et al.  Model reduction by separation of variables: a comparison between Hierarchical Model reduction and Proper Generalized Decomposition , 2018 .

[7]  A. Huerta,et al.  NURBS-Enhanced Finite Element Method (NEFEM) , 2011 .

[8]  L. Gallimard,et al.  Assessment of variable separation for finite element modeling of free edge effect for composite plates , 2015 .

[9]  F. Chinesta,et al.  On the effective conductivity and the apparent viscosity of a thin rough polymer interface using PGD‐based separated representations , 2020, International Journal for Numerical Methods in Engineering.

[10]  B. Schrefler,et al.  Towards a framework for non-linear thermal models in shell domains , 2013 .

[11]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[12]  Francisco Chinesta,et al.  Spurious-free interpolations for non-intrusive PGD-based parametric solutions: Application to composites forming processes , 2020, International Journal of Material Forming.

[13]  Adrien Leygue,et al.  Separated representations of 3D elastic solutions in shell geometries , 2014, Adv. Model. Simul. Eng. Sci..

[14]  Juan Casado-Díaz,et al.  A New Algorithm of Proper Generalized Decomposition for Parametric Symmetric Elliptic Problems , 2018, SIAM J. Math. Anal..

[15]  F. Chinesta,et al.  Parametric 3D elastic solutions of beams involved in frame structures , 2015 .

[16]  Ramon Codina,et al.  Reduced order models for thermally coupled low Mach flows , 2018, Adv. Model. Simul. Eng. Sci..

[17]  Adrien Leygue,et al.  On the space separated representation when addressing the solution of PDE in complex domains , 2016 .

[18]  F. Chinesta,et al.  Recent advances on the use of separated representations , 2009 .

[19]  Alessandro Reali,et al.  Duality and unified analysis of discrete approximations in structural dynamics and wave propagation : Comparison of p-method finite elements with k-method NURBS , 2008 .

[20]  T. Hughes,et al.  ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES , 2006 .

[21]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[22]  L. Heltai,et al.  Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils , 2015 .

[23]  Mohammad Javad Kazemzadeh-Parsi,et al.  Isogeometric analysis in solution of unconfined seepage problems , 2019, Comput. Math. Appl..

[24]  John A. Evans,et al.  Robustness of isogeometric structural discretizations under severe mesh distortion , 2010 .

[25]  Chady Ghnatios,et al.  Advanced separated spatial representations for hardly separable domains , 2019 .

[26]  A. Ammar,et al.  PGD-Based Computational Vademecum for Efficient Design, Optimization and Control , 2013, Archives of Computational Methods in Engineering.

[27]  David Ryckelynck,et al.  Hyper‐reduced direct numerical simulation of voids in welded joints via image‐based modeling , 2020, International Journal for Numerical Methods in Engineering.

[28]  John A. Evans,et al.  Discrete spectrum analyses for various mixed discretizations of the Stokes eigenproblem , 2012 .

[29]  P. Ladevèze,et al.  The LATIN multiscale computational method and the Proper Generalized Decomposition , 2010 .

[30]  L. Gallimard,et al.  Shell finite element based on the Proper Generalized Decomposition for the modeling of cylindrical composite structures , 2014 .

[31]  Suresh G. Advani,et al.  3D modeling of squeeze flow of multiaxial laminates , 2016 .

[32]  Etienne Pruliere,et al.  3D simulation of laminated shell structures using the Proper Generalized Decomposition , 2014 .

[33]  Alessandro Reali,et al.  Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems , 2014 .

[34]  Alessandro Reali,et al.  Studies of Refinement and Continuity in Isogeometric Structural Analysis (Preprint) , 2007 .

[35]  F. Chinesta,et al.  Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity , 2012 .