An introduction to isogeometric analysis
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The predominant technology that is used by CAD (Com- puter Aided Design) to represent complex geometries is non-uniform rational b-splines (NURBS). This allows cer- tain geometries to be represented exactly including conic and circular sections. There is a vast array of literature focused on NURBS and as a result of several decades f research, many efficient computer algorithms exist for their fast evaluation and refinement. The key concept outlined by Hughes et al. was to employ NURBS not only as a geometry discretisation technology, but also as a discretisation tool for analysis, attributing such methods to the field of ‘isogeometric analysis’ (IGA).
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An introduction to isogeometric analysis
- 1. ! An introduc+on to isogeometric analysis Vinh Phu NGUYEN 1 18 September2014
- 2. 3 How Isogeometric analysis was born? MA ScoF et al, CMAME 2013. CAD Mesh IGA 80% 20%
- 3. 4 B-‐splines basis funcNons Ni,0(⇠) = ( 1 if ⇠i ⇠ < ⇠i+1 0 otherwise Ni,p(⇠) = ⇠ ⇠i ⇠i+p ⇠i Ni,p 1(⇠) + ⇠i+p+1 ⇠ ⇠i+p+1 ⇠i+1 Ni+1,p 1(⇠) ⌅ = {⇠1, ⇠2, . . . , ⇠n+p+1} ⌅ = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} p = 2 ProperNes • ParNNon of Unity • Linear independence • Non-‐negaNvity • Cp-‐m conNnuity • Not interpolants knot vector sigma
- 4. 4 B-‐splines curves/surfaces ⌅ = {⇠1, ⇠2, . . . , ⇠n+p+1} C(⇠) = nX i=1 Ni,p(⇠)Bi S(⇠, ⌘) = nX i=1 mX j=1 Ni,p(⇠)Mj,q(⌘)Bij
- 5. 4 Enriching B-‐splines knot inser+on (h-‐refinement) order eleva+on (p-‐refinement) does not change B-‐splines geometrically/parametrically +
- 6. Knot inser+on to create discon+nui+es p = 2 Crack modeling and composite laminates (Layer wise theory, Z. Gurdal)
- 7. Non Uniform Ra+onal B-‐splines (NURBS)
- 8. One dimensional IGA FEM
- 9. One dimensional IGA FEM Isoparametric
- 10. Two dimensional IGA FEM
- 12. M A MInstitute of Mechanics & Advanced MaterialsI 12 ! ! Examples 1 June 2013
- 13. Tools ! MIGFEM • open source Matlab Isogeometric (X)FEM • 2D/3D solid mechanics with geometry nonlineariNes • 2D XIGA for LEFM and material interfaces • Structural mechanics: beam, plate, shells (large deformaNon) • hFp://sourceforge.net/projects/cmcodes/ ! jem-‐jive (Linux, Mac OS, Windows) • commericial C++ toolkit for PDEs • not a general purpose FE package • tailor made applicaNons, suitable for researchers • apps: XFEM, dG, IGA, DEM, FVM etc. • support parallel compuNng • implements useful concepts available in other programming languages-‐-‐Java, Fortran 90, Matlab and C# • tensor class: useful to evaluaNng complex consNtuNve models • hFp://www.dynaflow.com/en_GB/jive.html -‐ quick prototyping -‐ tutorial codes
- 14. IGA rota+on free nonlinear shell
- 15. IGA rota+on free nonlinear shell
- 16. Isogeometric cohesive elements: 2D example ! •exact geometry by NURBS •It is straigheorward to vary (1) number of plies and (2) # of interface elements: • Suitable for parameter studies/design • Cohesive law: bilinear law of Turon et al. 2006
- 17. Isogeometric cohesive elements: 2D example M. A. GuNerrez. Energy release control for numerical simulaNons of failure in quasi-‐briFle solids. CommunicaNons in Numerical Methods in Engineering, 20(1):19–29, 2004 G. Wimmer and H.E. PeFermann. A semi-‐analyNcal model for the simulaNon of delaminaNon in laminated composites. Composites Science &Technology, 68(12):2332 – 2339, 2008.
- 18. Isogeometric cohesive elements: 3D example with shells ! •RotaNon free B-‐splines shell elements (Kiendl et al. CMAME) • Two shells, one for each lamina • Bivariate B-‐splines cohesive interface elements in between !
- 19. Isogeometric cohesive elements: 3D examples ! • cohesive elements for 3D meshes the same as 2D • large deformaNons • suitable: delaminaNon buckling analysis
- 20. Isogeometric plate and 3D FEM coupling
- 21. M A MInstitute of Mechanics & Advanced MaterialsI 21 ! ! Issues 1 June 2013
- 25. Local refinement Solutions: Hierarchical B-splines and T-splines
- 26. Summary • B-splines/NURBS/T-splines not only for design but also for analysis! • High order continuity: plate/shell theories, gradient elasticity/damage! • Less prone to locking compared to low-order Lagrange elements! • NURBS of any order achieved with a simple recursion relationship! • Direct link to CAD: optimisation problems! • Smooth geometries: contact problems • Elaborated enforcement of essential boundary conditions! • Require knowledge on CAD! • More demanding than low order finite elements