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Geometry Independent Field approximaTion error-driven local adaptivity in elasto -dynamics

This paper discusses the use of Geometry Independent Field Approximation (GIFT) for error-driven local adaptivity in elasto-dynamics. The authors compare the performance of NURBS-IGA and FEM in dynamics and propose an adaptive hierarchical local refinement approach using GIFT. Numerical examples are provided to validate the effectiveness of the proposed method.

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Geometry Independent Field approximaTion error-driven local adaptivity in elasto -dynamics

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  1. Geometry Independent Field approximaTion error-driven local adaptivityin elasto-dynamics Peng Yu, Pierre Kerfriden, Stéphane P.A. Bordas

  2. Outline • Motivation • Problem statement • Methodology • Numerical examples • Conclusions and Following work

  3. Motivation • NURBS-IGA performs better than FEM in dynamics. (High order continuity) • IGA--- IsoGeometric Analysis Cottrell, J. Austin, et al. "Isogeometric analysis of structural vibrations."  Computer methods in applied mechanics and engineering 195.41 (2006): 5257-5296.

  4. Motivation • NURBS are limited to global refinement. (Tensor product) • PHT are with local refinement. (Hierarchical ) • PHT ---Polynomial splinesover Hierarchical T-meshes PHT NURBS Local refinement Global refinement • But PHT will lose information of geometry for curves because it is not rational

  5. Motivation • RHT can describe curves but limited to . • RHT ---Rationalsplinesover Hierarchical T-meshes Nguyen-Thanh, N., et al. "An adaptive three-dimensional RHT-splines formulation in linear elasto-statics and elasto-dynamics." Computational Mechanics 53.2 (2014): 369-385. • Multiple-patch NURBS can achieve local refinement • But over-accuracy and lose continuity at boundary of patch Chemin, Alexandre, Thomas Elguedj, and Anthony Gravouil. "Isogeometric local h-refinement strategy based on multigrids." Finite Elements in Analysis and Design 100 (2015): 77-90.

  6. Motivation • GIFT---Geometry-Independent Field approximaTion IGA GIFT • GIFT: NURBS + PHT exact geometry local refinement

  7. Problem statement • Variational equation for vibration problem • Kirchoff plate theory (thin plate) is independent variable

  8. Eigenvector Natural frequency Problem statement • GIFT form of discrete governing equation NURBS • Mapping PHT • Separation of solution for static state • Eigenvalue problem

  9. Methodology • Refinement by IGA for vibration Nguyen-Thanh, N., et al. "An adaptive three-dimensional RHT-splines formulation in linear elasto-statics and elasto-dynamics." Computational Mechanics 53.2 (2014): 369-385. Shojaee, S., et al. "Free vibration analysis of thin plates by using a NURBS-based isogeometric approach." Finite Elements in Analysis and Design 61 (2012): 23-34. NURBS refinement (over-accurate and waste CPU time) RHT prior refinement (not practical) • Error-driven adaptivity for vibration

  10. Solution in coarse mesh Solution in reference mesh Methodology • Error estimation (not limited to vibration) Error indicator L2 norm of error indicator for element If tolerance, refine element

  11. Methodology • Adaptive hierarchical local refinement (Algorithm 1)

  12. th natural frequency in reference mesh th natural frequency in coarse mesh • Call Algorithm 1 to process local refinement at mode ? Know mode Locate mode Methodology • Error-driven local adaptivity for vibration Error indicator for natural frequency If , define error indicator for normalized eigenvector

  13. Methodology • Modal Assurance Criterion (MAC) Consistent correspondence check between mode shapes • MAC matrix values

  14. Methodology • Adaptivity by sweeping modes from low to high ……… 3th mode 2th mode Initial mesh 4th mode 1th mode Vibration mode

  15. Numerical examples • Heterogeneous platehole

  16. Numerical examples

  17. Numerical examples • Heterogeneous Lshaped-bracket non-symmetric boundary condition

  18. Numerical examples

  19. Conclusions and Following work • Contributions • GIFT for dynamics • Error-driven adaptive hierarchical local refinement • Modal analysis • Following work • 3D vibration • Space-time adaptivity

  20. Thank you!

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