Introduction to Isogeometric Elements in LS-DYNA T.J.R. Hughes Entwicklerforum October 12th, 2011, Stuttgart, Germany Stefan Hartmann Development in cooperation with: D.J. Benson: Professor of Applied Mechanics, University of California, San Diego, USA Some slides borrowed from: T.J.R. Hughes: Professor of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, USA 1 Outline Isogeometric Analysis - motivation / definition / history From B-splines to NURBS (T.J.R. Hughes) - basis functions / control net / refinements NURBS-based finite elements in LS-DYNA -*ELEMENT_NURBS _PATCH _2D Example 1: Square Tube Buckling - description / deformation Example 2: Underbody Cross Member (Numisheet 2005) - diti/description / compar ison o f resu lt/lts / summary Summary and Outlook Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 2 Isogeometric Analysis – motivation & definition reduce effort of geometry conversion from CAD into a suitable mesh for FEA ISOPARAMETRIC (FE-Analysis) use same approximation for geometry and deformation (normally: low order Lagrange polynomials ---- in LS-DYNA basically only linear elements) GEOMETRY DEFORMATION ISOGEOMETRIC (CAD - FEA) same description of the geometry in the design (CAD) and the analysis (FEA) CAD FEA common geometry descriptions in CAD - NURBS (Non-Uniform Rational B-splines) most commonly used - T-splines enhancement of NURBS - subdivision surfaces mainly used in animation industry -andthd others Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 3 Isogeometric Analysis -history start in 2003 - summer: Austin Cotrell starts as PhD Student of Prof. T.J.R. Hughes at the University of Texas, Austin - autumn: first NURBS-based FE-code for linear, static problems provides promising results, the name „ISOGEOMETRIC“ is used the first time 2004 up to now: many research activities to various topics - non-linear structural mechanics shells with and without rotational DOFs implicit gradient enhanced damage XFEM - shape- und topology-optimization - efficient numerical integration - turbulence and fluid-structure-interaction - acoustics - refinement strategies -… January 2011: first thematic conference on Isogeometric Analysis -“Isogeometric Analysis 2011: Integrating Integrating Design and Analysis “, University of Texas at Austin Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 4 From B-splines to NURBS B-spline basis functions - constructed recursively - ppy(ggpy)ositive everywhere (in contrast to Lagrange polynomials) - shape of basis functions depend on: knot-vector and polynomial degree - knot-vector: non-decreasing set of coordinates in parameter space - normally C(P-1)-continuity e.ggq. lin. / quad. / cub. / quart. La gggrange: C0 / C0 / C0 / C0 e.g. lin. / quad. / cub. / quart. B-spline: C0 / C1 / C2 / C3 Example of a uniform knot -vector: p 0: p=0 1 if ii1 Ni,0 0 otherwise p=1 p 0: ip1 NN i N ip, ip,1 i 1,1 p ip i ip11 i p=2 T.J.R. Hughes Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 5 From B-splines to NURBS B-spline curves - control points Bi / control polygon (control net) - knots linear combination: n CB Nip, i i1 T.J.R. Hughes Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 6 From B-splines to NURBS NURBS – Non-Uniform Rational B-splines - weights at control points leads to more control over the shape of a curve - ppjrojective transformation of a B-spline weight 9=09=19=29=59=10 Nw p ip, i 9 Ri W 8 n with: W Nip, w i i1 n 3 p CB Rii 7 i1 2 4 10 1 5 6 11 13 0 12 Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 7 From B-splines to NURBS Smoothness of Lagrange polynomials vs. NURBS T.J.R. Hughes Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 8 From B-splines to NURBS NURBS – surfaces (tensor-product of univariate basis) p,q NMwijiji,,,p j q ij Rij, , W , nm with:, W Nip,,, M jq w ij ij11 T.J.R. Hughes Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 9 From B-splines to NURBS - summary B-spline basis functions - recursive - dependent on knot -vector and polynomial order - normally C(P-1)-continuity - „partition of unity“ (like Lagrange polynomials) - refinement (h/p and k) without changing the initial geometry adaptivity - control points are normally not a part of the physical geometry (non-interpolatory basis functions) NURBS - B-spline basis functions + control net with weights - all mentioned properties for B-splines apply for NURBS Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 10 NURBS-based finite elements in LS-DYNA A typical NURBS-Patch and the definition of elements - elements are defined through the knot-vectors (interval between different values) - shape functions for each control-point z polynomial order: - quadratic in r-direction (pr=2) y - quadtiidratic in s-direc tion (ps= 2) x s M 3 s 5 M 4 2 r NURBS-Patch ,2,3,3,3] M3 1 (physical space) M2 M1 0 r 0 1 2 ot=[0,0,0,1 CtlControl-PitPoints nn N1 N4 N2 N3 sk Control-Net NURBS-Patch rknot=[0,0,0,1,2,2,2] „Finite Element“ (parameter space) Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 11 NURBS-based finite elements in LS-DYNA A typical NURBS-Patch – Connectivity of elements - Possible „overlaps“ ( higher continuity!) NURBS-Patch - Size of „overlap“ depends on polynomial order (and on knot-vector) (parameter space) s M5 3 M 4 2 M3 1 M2 M 1 0 r 0 1 2 N1 N4 N2 N3 Control-Points NURBS-Patch Control-Net (h(phys ica l space ) „Finite Element“ Connectivity of „FiFinitenit e ElElement“ement“ Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 12 NURBS-based finite elements in LS-DYNA New Keyword: *ELEMENT_NURBS_PATCH_2D - definition of NURBS-surfaces - 4 different shell formulations with/without rotational DOFs Pre- and Postprocessing - work in ppgrogress for LS-PrePost … current status ((pplspp3.2beta) visualization of 2D-NURBS-Patches import IGES-format and construct *ELEMENT_NURBS_PATCH_2D modification of 2D-NURBS geometry … much more to come! Postprocessing and boundary conditions (i.e. contact) currently with - interpp(y)olation nodes (automatically created) - interpolation elements (automatically created) Analysis capabilities - implicit and explicit time integration - eigenvalue analysis - other capabilities (e.g. geometric stiffness for buckling) implemented but not yet tested LS-DYNA material library available (including umats) Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 13 NURBS-based finite elements in LS-DYNA *ELEMENT_NURBS_PATCH_2D $---+--EID----+--PID----+--NPR----+---PR----+--NPS----+---PS----+----7----+----8 11 12 4 2 5 2 $---+--WFL----+-FORM----+--INT----+-NISR----+-NISS----+IMASS----+----7----+----8 0 0 1 2 2 0 $rk-+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 0.0 0.0 0.0 1.0 2.0 2.0 2.0 $sk-+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 0.0 0.0 0.0 1.0 2.0 3.0 3.0 3.0 $net+---N1----+---N2----+---N3----+---N4----+---N5----+---N6----+---N7----+---N8 1 2 3 4 5 6 7 8 91011129 10 11 12 13 14 15 16 18 17 18 19 20 14 19 17 13 15 20 10 16 9 11 Control Points 12 5 6 Control Net s 7 r 2 z 3 8 1 y x 4 Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 14 NURBS-based finite elements in LS-DYNA ... $---+--WFL----+-FORM----+--INT----+-NISR----+-NISS----+IMASS 0 0 1 220 NISR/NISS – Number of Interpolation Elements per Nurbs-Element (r-/s-dir.) important for post-processing, boundary conditions and contact treatment 18 automatically created: 14 19 z 17 Nurbs-Element y 13 15 Interpolation Node x 20 10 16 Interpolation Element 9 11 12 5 6 s 7 r 2 3 8 1 input: Control Points 4 Control Net Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 15 NURBS-based finite elements in LS-DYNA LSPP: Preppgrocessing LSPP: Postppgrocessing - control-net - Interpolation nodes/elements - nurbs surface nisr=niss=2 nisr=niss=10 Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 16 Square Tube Buckling – Description standard benchmark for automobile crashworthiness quarter symmetry to reduce cost perturbation to initiate buckling mode J2 plasticity with linear isotropic hardening mesh: - 640 quartic (P=4) elements - 1156 control points - 3 integration points through thickness D.J. Benson Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 17 Square Tube Buckling D.J. Benson Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 18 Square Tube Buckling D.J. Benson Introduction to Isogeometric Elements in LS-DYNA Entwicklerforum, October 12th, 2011, Stuttgart, Germany 19 Underbody Cross Member – Description Z X Upper die (rigid) Y Binder (rigid) Blank 1.6mm Lower punch (rigid) Tool moving directions Radius of Drawbead - Lower punch: stationary about 4mm - UdiUpper