Isogeometric Analysis Application to Car Crash Simulation

Isogeometric Analysis Application to Car Crash Simulation S. Bouabdallah 2, C. Adam 2,3, M. Zarroug 2, H. Maitournam 3 De Vinci Engineering Lab, École Supérieure d Ingénieurs Léonard de Vinci 2 Direction Scientifique et Technologies Futures, PSA Peugeot Citroën 3 Laboratoire de Mécanique des Solides, École Polytechnique Forum Teratec 24 - July 2, 24

Outline Introduction Industrial context Numerical locking 2 Timoshenko beams Pure bending problem Membrane and transverse shear locking D B-Spline reduced quadrature rules 3 Reissner-Mindlin plates Pure bending problem 2D B-Spline reduced quadrature rules 4 Reissner-Mindlin shells Shell model Shell obstacle course 5 Non-linear solid shells Benchmark problems 6 Conclusions S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation / 26

Industrial context Good predictive capability expected. Complete mesh size: 3 million elements. = Explicit simulations performed on the whole model. Stable time step: µs. hours to simulate 3ms on 96 CPUs. = Excessive computational cost. 9% of the geometry meshed with linear Reissner-Mindlin shell elements. Figure: Model size evolution S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 2 / 26

Industrial context Geometry has proved to be very significant. Need to ine some components. = Stable time step greatly penalized. Accurate and smooth geometry perred for contact problems. Large time used for preparing and meshing parts. 32 4 2 4 6 6 8 4 5 Figure: Estimation of relative time costs for model generation and analysis process S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 3 / 26

Industrial context = IsoGeometric Analysis for automobile crashworthiness. Parametrized CAD geometry for optimisation process. Analysis and Design share the same model. Higher regularity between elements within a patch. Figure: Numerical simulation process in isogeometric analysis. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 4 / 26

Numerical locking Fully integrated low order elements, based on Timoshenko/Mindlin hypothesis, present poor performances with thin beams, plates and shells. Transverse shear and membrane locking appear. Classical Lagrange-based reduced and selective integration rules do not remove these pathologies. = Need to define B-Spline/NURBS-based integration schemes. Framework: Multi-patch problems with uniform regularity within each patch. Quadratic and cubic polynomials, regularity r from to p. C r C S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 5 / 26

Introduction Industrial context Numerical locking 2 Timoshenko beams Pure bending problem Membrane and transverse shear locking D B-Spline reduced quadrature rules 3 Reissner-Mindlin plates Pure bending problem 2D B-Spline reduced quadrature rules 4 Reissner-Mindlin shells Shell model Shell obstacle course 5 Non-linear solid shells Benchmark problems 6 Conclusions S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 6 / 26

Pure bending problem Thick curved beam of length L and radius R clamped at one end. Moment of inertia for a rectangular section: I = bh3 2. Distributed loading moment proportional to the bending stiffness: m(s) = EI ( π 2L) 2 sin ( π 2L s ). = Exact solution known, depends only of L and R: u n (s) = 2L ( ( π s ) ( π )) ( 2L cos cos πr )2 R 2L s. Exact membrane and shear strains are zero. Severe test for membrane and shear locking. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 7 / 26

Membrane and transverse shear locking One element with B-Splines of degree p from to 5. Shear locking for slenderness ratio L/h and membrane locking for L/h R/L. Both shear and membrane locking appear for all degrees p. Rotation relative error with R/L = 3 Rotation relative error with R/L = 3 2 θ b ex θb h L 2 / θ b ex L 2 8 6 4 2 SHEAR LOCKING MEMBRANE LOCKING Linear Quad. Cubic Quartic Quintic 2 R/L 3 4 5 6 L/h θ b ex θb h L 2 / θ b ex L 2 SHEAR LOCKING MEMBRANE LOCKING Linear 2 Quad. Cubic 3 Quartic 4 Quintic 2 R/L 3 4 5 6 L/h (a) Linear scale (b) Logarithmic scale Figure: One element curved thick beam: rotation relative error in L 2 norm. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 8 / 26

S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam 2 Teratec 24 - IGA Application to Car Crash Simulation 9 / 26 Transverse shear locking Different effects on convergence according to the couple (p,r). Degree : no pre-asymptotic convergence. Degree 2: slow pre-asymptotic convergence when C and no pre-asymptotic convergence with few elements when C. Spurious constraints strengthened by the high regularity. = Excessive bending stiffeness of the structure. θ b ex θb h L 2 / θ b ex L 2 2 4 6 Rotation relative error with L/h = 3 Linear Quad. C Quad C Cubic C Cubic C 3 5 2 Number of elements / l (a) Pre-asymptotic convergence with L/h = 3 2 4 θ b ex θb h L 2 / θ b ex L 2 2 4 6 8 3 Rotation relative error with L/h = 2 2 2 4 Linear Quad. C Quad C Cubic C Cubic C 5 2 3 Number of elements / l (b) Pre-asymptotic and asymptotic convergence with L/h = 2 4 3

Mathematical framework Based on the work in FEA of [Prathap,93]. Field-consistency paradigm in thick structural elements. Transverse shear strain in straight beam: γ s (s) = w,s (s) θ(s). In the limit of an infinitely thin beam, the shear strain energy must vanish. = Different physical and spurious constraints are obtained according to the couple (p,r). Mathematical induction is performed adding elements one-by-one. New unknowns and conditions are numbered. An over-constrained linear system leads to numerical locking whereas an under-constrained system produces zero energy modes. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation / 26

D B-Spline reduced quadrature rules Optimal selective reduced integration schemes adapted to high regularity basis functions. The higher the regularity is, the fewer Gauss points are needed. Shear and membrane locking are completely removed. Improved accuracy of curved beam B-Spline elements. Order C Order 2 C Order 2 C Order 3 C Order 3 C Order 3 C2 Reduced integration for transverse shear: Gauss points ξ = s / L θ ex θ h L 2 / θ ex L 2 2 4 6 C full C selec. C full C selec. Quadratic B spline 3 2 Number of elements 2 4 θ ex θ h L 2 / θ ex L 2 2 4 6 Cubic B spline C full C selec. C full C selec. C 2 full C 2 selec. 2 Number of elements Figure: Rotation relative error in L 2 norm with full and selective reduced integrations. 4 3 5 S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation / 26

Pure bending problem Same behaviours as with beams. High regularity wanted for more accurate results with fewer control points. Need to unlock B-Splines especially when C p regularity. Possibility to extend integration rules from beams to plates and shells. Vertical displacement of point C: concentrated load 8 w c / w c 6 4 2 Linear C L/h = 3 Linear C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 5 5 Number of elements in each direction Figure: Central deflection of a simply supported plate (L/h = 3, 5 ). S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 3 / 26

Pure bending problem Same behaviours as with beams. High regularity wanted for more accurate results with fewer control points. Need to unlock B-Splines especially when C p regularity. Possibility to extend integration rules from beams to plates and shells. Vertical displacement of point C: uniform pressure w c / w c 8 6 4 2 Linear C L/h = 3 Linear C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 5 5 Number of elements in each direction Figure: Central deflection of a simply supported plate (L/h = 3, 5 ). S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 3 / 26

2D B-Spline reduced quadrature rules Uni-dimensional B-Spline-based reduced quadrature rules extended to bi-dimensional rules by tensor product. The high regularity lowers the required number of Gauss points. Hourglass modes are suppressed using additional quadrature points in boundary elements, where the control points accumulate. B spline selective integration: Gauss points B spline selective integration: Gauss points B spline selective integration: Gauss points.75.75.75 η.5 η.5 η.5.25.25.25.25.5.75 ξ (a) Quadratic C.25.5.75 ξ (b) Cubic C.25.5.75 ξ (c) Figure: B-Spline selective integration quadrature rules and central deflection. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 4 / 26

2D B-Spline reduced quadrature rules Resulting under-integrated elements are free from transverse shear locking. Numerical solutions independent of the thickness. Improved accuracy of plate isogeometric elements. No spurious zero energy modes. Vertical displacement of point C: concentrated load Vertical displacement of point C: concentrated load 8 8 w c / w c 6 4 2 Linear C L/h = 3 Linear C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 5 5 Number of elements in each direction w c / w c 6 4 2 Linear C L/h = 3 Linear C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 5 5 Number of elements in each direction (a) Full integration (b) Reduced integration Figure: Central deflection of a simply supported plate (L/h = 3, 5 ). S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 5 / 26

2D B-Spline reduced quadrature rules Resulting under-integrated elements are free from transverse shear locking. Numerical solutions independent of the thickness. Improved accuracy of plate isogeometric elements. No spurious zero energy modes. Vertical displacement of point C: uniform pressure Vertical displacement of point C: uniform pressure w c / w c 8 6 4 2 Linear C L/h = 3 Linear C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 5 5 Number of elements in each direction w c / w c 8 6 4 2 Linear C L/h = 3 Linear C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 Quad. C L/h = 3 Quad. C L/h = 5 5 5 Number of elements in each direction (c) Full integration (d) Reduced integration Figure: Central deflection of a simply supported plate (L/h = 3, 5 ). S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 5 / 26

Shell model Reissner-Mindlin shell model obtained from a degenerated three-dimensional model. First-order kinematic description through the thickness with transverse shear. Exact geometry of the shell defined by x(ξ) = nm A= R A (ξ, η)x A + h ζn(ξ, η). 2 Interpolated displacements described by u(ξ) = nm A= R A (U A + h ) 2 ζθ A n. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 7 / 26

Characterization of the normal Several possibilities: Exact normal: n = x,ξ x,η x,ξ x,η 2. 2 Collocated normal: n = n A A [, nm ]. a. n A defined by the orthogonal projection of the associated control point X A onto the mid-surface of the shell, b. n A defined by a uniform distribution of the shell normals in the parametric space, c. n A defined at the Greville abscissae in the parametric space. Exact normal formulation is incompatible with reduced integration. Collocation at Greville abscissae results in good compromise between accuracy and computation time. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 8 / 26

Shell obstacle course problems Evaluate time efficiency and accuracy of the proposed under-integrated elements. Linear elasticity benchmark problems. Bending dominated problems: transverse shear and membrane locking. (a) Scordelis-Lo roof (b) Pinched cylinder (c) Pinched hemisphere Figure: The shell obstacle course: problem descriptions and data. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 9 / 26

Accuracy Vertical displacement of point B : full integration Vertical displacement of point C: full integration Horizontal displacement of point A: full integration w B / w B 8 6 4 2 Linear C Quad. C Quad. C Cubic C Cubic C w C / w C 8 6 4 2 Linear C Quad. C Quad. C Cubic C Cubic C u A / u A 8 6 4 2 Linear C Quad. C Quad. C Cubic C Cubic C 5 5 2 25 3 35 Number of elements per side 5 5 2 25 3 35 Number of elements per side 5 5 2 25 3 35 Number of elements per side (a) Roof: full (b) Cylinder: full (c) Hemisphere: full Vertical displacement of point B : reduced integration Vertical displacement of point C: reduced integration Horizontal displacement of point A: reduced integration 8 8 8 w B / w B 6 4 2 Linear C Quad. C Quad. C Cubic C Cubic C w C / w C 6 4 2 Linear C Quad. C Quad. C Cubic C Cubic C u A / u A 6 4 2 Linear C Quad. C Quad. C Cubic C Cubic C 5 5 2 25 3 35 Number of elements per side 5 5 2 25 3 35 Number of elements per side 5 5 2 25 3 35 Number of elements per side (d) Roof: reduced (e) Cylinder: reduced (f) Hemisphere: reduced Figure: Shell obstacle course: displacement convergence S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 2 / 26

Time efficiency Vertical displacement of point B : full integration Vertical displacement of point C: full integratione Horizontal displacement of point A: full integration w B / w B 8 6 4 2 Linear C Quad. C Quad. C Cubic C Cubic C CubicC 2 w C / w C 8 6 4 2 Linear C Quad. C Quad. C Cubic C Cubic C u A / u A 8 6 4 2 Linear C Quad. C Quad. C Cubic C Cubic C 2 Computational time (s) 2 Computational time (s) 2 Computational time (s) (a) Roof: full (b) Cylinder: full (c) Hemisphere: full Vertical displacement of point B : reduced integration Vertical displacement of point C: reduced integration Horizontal displacement of point A: reduced integration 8 8 8 6 6 6 w B / w B 4 2 Linear C Quad. C Quad. C Cubic C Cubic C w C / w C 4 2 Linear C Quad. C Quad. C Cubic C Cubic C u A / u A 4 2 Linear C Quad. C Quad. C Cubic C Cubic C 2 Computational time (s) 2 Computational time (s) 2 Computational time (s) (d) Roof: reduced (e) Cylinder: reduced (f) Hemisphere: reduced Figure: Shell obstacle course: displacement convergence S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 2 / 26

Total number of Gauss points Lagrange-based quadrature: n bending GP = ( e ( )) 2p+ 2 2 B-Spline-based quadrature: and n shear GP = ( e ( )) 2p+ 2 2. n bending GP = ngp shear = ( e ( ) ) 2p+ 2 2 r + 2r. Computational effort depends on the difference p r when performing B-Spline-based reduced integration. Speed up factor greater than 5 (resp. 6) is reached with the under-integrated quadratic (resp. cubic) C p elements. Computation time: full integration Computation time: selective integration Computation time (s) 2 5 5 Linear C Quad. C Quad. C Cubic C Cubic C Computation time (s) 2 5 5 Linear C Quad. C Quad. C Cubic C Cubic C 5 5 2 25 3 35 Number of elements per side (a) Full integration 5 5 2 25 3 35 Number of elements per side (b) Selective integration Figure: Computation time for the pinched cylinder problem. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 22 / 26

Benchmark problems Solid-shell elements with one quadratic element in the thickness. Geometric non-linear elasticity benchmark problems. Bi-variate B-Spline-based quadrature rules applied. (a) Pinched hemisphere (b) Stretched cylinder Figure: Geometric non-linear analysis of shells: problem descriptions and data. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 24 / 26

Accuracy and time efficiency Bi-quadratic C shape functions in the in-plane directions. Improved accuracy and time efficiency. Speed up factor of approximately 3. Load P in % of P max (N) 8 6 4 Load displacement curve 2 2x2 full 2x2 reduced Shell erence S4R 2 4 6 8 Displacement (m) (a) Pinched hemisphere Load P in % of P max (N) 8 6 4 2 Load displacement curve 2x2 full 2x2 reduced Shell erence S4R 2 3 4 5 Displacement (m) (b) Stretched cylinder Figure: Displacement convergence with Total Lagrangian Formulation S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 25 / 26

Conclusions Severe numerical locking appear in quadratic and cubic C p elements. = Reduced integration to improve the performances of IG elements. Classical reduced/selective quadrature rules remove numerical locking in C elements only. Mathematical framework to define B-Spline-based reduced D quadrature rules. Extension to 2D quadrature rules by tensor product. = Efficient thick shell elements obtained as in FEA. Improved accuracy and time efficiency. Computational cost depends on the difference p r. S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 24 - IGA Application to Car Crash Simulation 26 / 26