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

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
1). IGA VS FEM: 1.CP vs Nodes; 2) basis high order vs co; 3) one more space
Cottrell, J. Austin, et al. "Isogeometric analysis of structural vibrations." 
Computer methods in applied mechanics and engineering  (2006): 3

4. Motivation NURBS PHT Global refinement Local refinement
NURBS are limited to global refinement. (Tensor product)
PHT are with local refinement. (Hierarchical )
PHT --- Polynomial splines over Hierarchical T-meshes
NURBS
PHT
1). IGA VS FEM: 1.CP vs Nodes; 2) basis high order vs co; 3) one more space
Global refinement
Local refinement
But PHT will lose information of geometry for curves because it is not rational 4

5. Motivation RHT can describe curves but limited to .
RHT --- Rational splines over 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):
Multiple-patch NURBS can achieve local refinement
But over-accuracy and lose continuity at boundary of patch
1). IGA VS FEM: 1.CP vs Nodes; 2) basis high order vs co; 3) one more space
Chemin, Alexandre, Thomas Elguedj, and Anthony Gravouil. "Isogeometric local h-refinement strategy based on multigrids." Finite Elements in Analysis and Design 100 (2015): 5

6. Motivation GIFT--- Geometry-Independent Field approximaTion IGA GIFT
1). IGA VS FEM: 1.CP vs Nodes; 2) basis high order vs co; 3) one more space
GIFT
GIFT: NURBS PHT
exact geometry
local refinement 6

7. Problem statement is independent variable
Variational equation for vibration problem
Kirchoff plate theory (thin plate)
1. 2D annular clamped ; 1. NM –semidiscrete equation of motion;
is independent variable 7

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

9. Methodology Refinement by IGA for vibration NURBS refinement
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):
Nguyen-Thanh, N., et al. "An adaptive three-dimensional RHT-splines formulation in linear elasto-statics and elasto-dynamics." Computational Mechanics 53.2 (2014):
NURBS refinement
(over-accurate and waste CPU time)
1. 2D annular clamped ; 1. NM –semidiscrete equation of motion;
RHT prior refinement
(not practical)
Error-driven adaptivity for vibration 9

10. Methodology Error estimation (not limited to vibration)
Error indicator
Solution in coarse mesh
Solution in reference mesh
L2 norm of error indicator for element
1. 2D annular clamped ; 1. NM –semidiscrete equation of motion; If
tolerance, refine element 10

11. Methodology Adaptive hierarchical local refinement (Algorithm 1)
Hilight point 3,4 – related to knot vetor right. 11

12. ? Methodology Error-driven local adaptivity for vibration
Error indicator for natural frequency
th natural frequency in reference mesh
th natural frequency in coarse mesh If
, define error indicator for normalized eigenvector
1. See diference in mesh; 2.PHT works (blackboard,local support),detail..paper; 3. Bezier extraction (not limit to PHT)
Call Algorithm to process local refinement at mode ?
Know mode
Locate mode 12

13. Methodology Modal Assurance Criterion (MAC)
Consistent correspondence check between mode shapes
MAC matrix values
1. See diference in mesh; 2.PHT works (blackboard,local support),detail..paper; 3. Bezier extraction (not limit to PHT) 13

14. Methodology Adaptivity by sweeping modes from low to high ………
Initial mesh
1th mode
2th mode
3th mode
4th mode
1. See diference in mesh; 2.PHT works (blackboard,local support),detail..paper; 3. Bezier extraction (not limit to PHT)
Vibration mode 14

15. Numerical examples Heterogeneous platehole
1. 2D annular clamped ; 1. NM –semidiscrete equation of motion; 15

16. Numerical examples
1. 2D annular clamped ; 1. NM –semidiscrete equation of motion; 16

17. Numerical examples
Heterogeneous Lshaped-bracket non-symmetric boundary condition
1. 2D annular clamped ; 1. NM –semidiscrete equation of motion; 17

18. Numerical examples
1. 2D annular clamped ; 1. NM –semidiscrete equation of motion; 18

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!