The role of structural mechanics in material design, control and optimization

1. The role of structural mechanics in material design, control and optimization Adnan Ibrahimbegovic Ecole Normale Superieure / LMT-Cachan, France e-mail : ai@lmt.ens-cachan.fr, fax : +33147402240 TU Gdansk, Poland, - December 8, 2003

2. LMT-Cachan : Laboratory of Mechanics and Technology (Ecole Normale Sup / Univ. Pierre & Marie Curie / CNRS) LMT-Cachan Is located in Paris  Distance cca 3km : 30 min on foot 5 min by metro RER 2 – 60 min by car !

3. Introduction and motivation LMT-Cachan : Laboratory of Mechanics and Technology (Ecole Normale Sup / Univ. Pierre & Marie Curie / CNRS • LMT-Cachan is: • - cca 150 researchers (with 45 Profs/Lecturers) • Center for experiments and measurements& Computer center • 3 Divisions at LMT: • • Div. « Material Science »(UPMC / J. Lemaitre) • • Div. « Mechanical Eng.  » (GM-ENS / P. Ladeveze) • • Div. « Civil & Env. Eng. »(GC-ENS /A. Ibrahimbegovic) • privileged environement for scientific research with: • at least a couple of national or international meetings each year, • seminar IDF / LMT, invited professors (for 1 month) … http://www.lmt.ens-cachan.fr

4. The role of structural mechanics in material design, control and optimization Outline : Part I: Micro-macro material modeling Part II: Structural optimization and control Acknowledgements : D. Brancherie, D. Markovic, A. Delaplace, F. Gatuingt C.Knopf-Lenoir, P. Villon, UTC, France A. Kucerova, EU – Erasmus, French Ministry of Research – ACI 2159

5. Micro-macro material modeling • Outline – part I • Introduction and motivation • Micro-macro modeling of inelastic behavior of structures • Discrete model ingredients • Time integration schemes for dynamic fracture analysis • Crack pattern analysis • Conclusions

6. Introduction and motivation Final (worthy) goal Reliable description of limit state behaviour of structures in their environment (multi-physics problems) Available technologies Macroscopic models (damage, plasticity, etc.) FE implementation (limit load, softening, etc.) Goals of the present study Reliable interpretation of damage mechanisms Representing local behaviour in heavily damaged areas (crack pattern, spacing, opening) Flow comp. & crack propagation, dissipation cyclic loading, spalling Proposed Approach Micro-macro modeling with or without separation of scales Discrete models at micro- (and/or meso-) scale accounting for randomness of material microstructure

7. Multi-scale modeling Objectives : dialogue mecanics - materials rupture criterion for: 1.Metalic materials 2.Concrete or RC 3. Global (vs. detailed) analysis of damaged zone with a good estimate of total inelastic dissipation (no crack profiles) (difficulties w/r to multi-physics, coupling e.g. durability?), A Ibrahimbegovic, D. Brancherie [2003] Comp. Mech. (M. Crisfield issue)

8. Multiscale modeling : porous materials = 2phase • Multi-scale coupling : • 3pt. - bending test • i) FEM2 (Suquet et al. [1990]) • ii) Strong coupling of scales • Each element“macro” repres. • witha meshof “micro”elements • AIbrahimbegovic,D. Markovic [2003] CMAME (guest ed. P. Ladevèze) contours show : macro-trace of stress tensor effectivemicro-plastic def. Remark : at “macro” scale we get a (novel) smeared model of coupled damage - plasticity A Ibrahimbegovic, D. Markovic, F.Gatuingt [2003] REEF Gurson/

9. FEM representation of microstructure: bi-phase material Exact vs. structured representations Simple tension test Incompatible modes Integration point filtering Exact

10. Structured representation of a bi-phase material Simple tension test Force-displacement diagram Integration point filtering Incompatible modes Exact Smeared

11. Structured representation of a bi-phase material Three-point bending test Integration point filtering Exact Smeared Deviatoric stress Force-displacement diagram Hydrostatic stress Integration point filtering Incompatible modes Effective plastic strain Exact Smeared

12. Discrete model ingredients • Structure (material) is described as an assembly of ”rigid” particles • Particles constructed as Voronoi cells (dual of Delauney triangularization) • Behavior characterized by cohesive links - elastic/fracture (isotropy) • Heterogeneity by mesh grading & random distribution of fracture thresholds. Pj Pi tf Link behaviour Φ Random variables

13. Cohesive force – beam link • 2 types of beam links employed: • Euler-Bernoulli: only small strain & small rotation ! • Reissner (Ibrahimbegovic A., Frey F. [1993]): • large motion (displ. & rot.) of any beam link t n q g2 v g1 Position vector in deformed configuration: u a x l where z is the coordinate along beam cross-section Idea : new position of cell interface defined by vector t

14. Cohesive force – beam link Deformation gradient: whereS, K, G generalised strain measures (Reissner [1972], A.I., F. Frey [1993]). Compute (material) strains in matrix notation S = (S G K)T, where Spatial strain measures: Stress resultants (Biot) : with E and G - Young modulus and shear modulus A and I - cross section area and moment of inertia

15. Contact forces Contact forces: two particles not linked with cohesive forces overlap at later stage. Penalization: contact force proportional to overlapping area (aij). cohesion ( U , F ) contact

16. Numerical examples : microscale F F Tension test –softening-like stress-strain diag. Biaxial traction – crack pattern

17. Numerical examples: mesoscale Computation on structures in large relative motion Mesoscale model: integration multi-layer, deterministic (A. Ibrahimbegovic, A. Delaplace [2003] CS) 0,13F F Anisotropic cont. damage model ------ Present discrete model ------ 2295 particles

18. Numerical examples: mesoscale Remark: -model robust (contrary to anisotropic damage) -crack pattern computed 3 pt, bending: numerical simulation

19. f fc u Time integration schemes for dynamic fracture Goal: damp undesirable high frequencies Application: high rate loading

20. Explicit time integration scheme: central difference • Second order accurate • Time step: smaller than a (non-constant) critical CFL value • Rayleigh viscous damping : all frequencies affected

21. Newmark time integration scheme: implicit • Second order accurate for g=0.5, β=0.25 (trapezoidal rule) • Unconditional stable for g≥0.5 (for linear analysis) • Numerical damping for g>0.5, but no longer of second order accuracy g=0.7

22. HHT time integration scheme (Hilbert et al., 77) High frequency damping only for linear analysis ! a = -0.2

23. Energy decaying time integr. scheme (Ibrahimbegovic, et al 99. 02) a1 : dissipation in internal . forces a2 : dissipation in inertia . . terms

24. Numerical examples: dynamic fracture Impact loading – shock propagation , Compute spalling (large displacement!) Dynamic fracture: numerical simulation

25. Crack pattern: anisotropic damage ds (Nooru-Mohamed, 92) ds = 10 µm/min

26. Crack pattern: study of the crack roughness Topographic description of the surface area, geometrical analysis (Delaplace et al., 99)

27. Crack pattern: study of the crack roughness 32400 particles Self-affine properties z ~ 0,6 ± 0,2

28. Current developments: 3D analysis

29. Conclusions – part I • Multi-scale strongly coupled model of inelastic behavior with • structured FE representation of microstructure • Discrete models – structural mechanics based modelling of • fracture behaviour of heterogeneous materials . • with brittle fracture at low or high rate loading. • Representation of local behaviour & Inelastic dissip. in damaged zone • . (crack spacing, crack opening and crack propagation) • + Post-processing - high accuracy description of cracked area • . (Delaplace et al., 01) • Capabilities beyond traditional cont. mech. models : e.g. spalling • Essential role played by large diplacement/rotation structural theory!

30. Optimal control and optimal design of structures undergoing large rotations: Outline – part II : -Introduction and motivation -Problem model in nonlinear structural mechanics : 3D beam - Optimization : coupled mechanics-optimization problem - Solution procedures for coupled problem: -Conclusions Acknowledgements : C.Knopf-Lenoir, P. Villon, UTC, France A. Kucerova, EU – Erasmus, French Ministry of Research,

31. -Optimization : unavoidable “constraint” of modern times … -Optimal design : choose mechanical and/or geometric properties of a structural system to achieve a goal (cost or objective function) design variables : thickness, shape, …, Young’s modulus, … applications : shape design, material design, etc. -Optimal control : choose loading on a struct. system to achieve a goal control variables : forces, temperatures, … applications : construction sequence, structure testing procedure, etc. Introduction and motivation

32. Introduction and motivation -Traditional approach to design and control : sequential procedure separate design and control from mechanics (even different comp.codes) -advantage : simple computer program architecture / each specialist contrib. -disadvantage : inefficient for non-linear mechanics problems (many “useless” iterations for non-converged value of opt.) -Proposed approach to design and control - simultaneous procedure: bring mechanics eqmb. eqs. on the same level as design and control mechanics state variables independent from design and control variables -advantage : iterate simultaneously on mechanics (eqmb.) and optimal design and/or control A. Ibrahimbegovic, C. Knopf-Lenoir, CMES [2002] A. Ibrahimbegovc, A. Kucerova, C. Knopf-Lenoir, P. Villon, IJNME [2003]

33. Model problem in mechanics : geom. exact 3D beam g2 t n q g1 v u a x l -Configuration space: Displacement update : additive Rotation updates : multiplicative φt = φ + t δφΛt = Λexp [t δΘ] ; δΘv = δθ×v

34. Model problem in mechanics : geom. exact 3D beam - Strain measures δε = ΛtTδφ’ + εt× δθ δω = δθ’ + ωt× δθ - Stress resultants

35. Model problem in mechanics : geom. exact 3D beam • Variational formulation : min. potential energy principle • (problem of minimization without constraints) -where: weak form of equilibrium equations G (φt, Λt) := ∫l [ δφ’ • Λtnt + δθ • (Et nt + Ωt mt) + δθ’ • mt ] ds - Gext -exception : non-conservative load (e.g. follower force, moment)

36. -Optimal design : i) Traditional approach : constrained minimization of cost function Optimization : coupled mechanics-optimization problem • -where : d – design variables (e.g. cross-section diameter) ii) Proposed approach : simultaneous solution procedure idea: use Lagrange multipliers to ‘lift’ mechanics to the level of design -where : mechanics state variables and design variables independent !

37. Optimization : coupled mechanics-optimization problem - Kuhn-Tucker optimality conditions – optimal design

38. Optimization : coupled mechanics-optimization problem -Example 1: Cost function – thickness optimization -Example 2: Cost function – shape optimization

39. Optimal design : Finite element approximations -mechanics state variables : isoparametric interpolations Optimization : coupled mechanics-optimization problem -where : Na Lagrange polynomials -Lagrange multipliers: (isoparametric interpolations) -optimal design variables: design element (Bezier interpolation)

40. -System of coupled mechanics-optimization discretized equations: -Optimal design - cost fcn. J(.) = V Optimization : coupled mechanics-optimization problem

41. -Optimal control: i) Traditional approach : constrained minimization of cost function Optimization : coupled mechanics-optimization problem -where: v control variables, f(t) = F0v(t) , F0 - fixed external load pattern ii) Proposed approach - simultaneous solution procedure idea: use Lagrange multipliers to ‘lift’ mechanics to the level of control -where : state variables and control variables independent

42. Optimization : coupled mechanics-optimization problem - Kuhn-Tucker optimality conditions – optimal control

43. Optimization : coupled mechanics-optimization problem -Example 1: cost function

44. Optimization : coupled mechanics-optimization problem -System of coupled mechanics-control discretized equations: - Optimal control: -isoparametric interpolation or “control” elem. -eliminate Lagrange multipliers (if α≠0) -Remark: similar to arc-length procedure (tangent plane projection)

45. Solution procedure for coupled problem • - 1. Diffuse approximation based response surface • two-stage sequential solution procedure • r(φ, Λ, d) = 0 φ(d), Λ(d)  d • approximate solution computation, but efficient • 2. Genetic algorithm • simultaneous solution procedure : • r(•) = 0  min { || r||2 = rTr } • exact solution computed (optimization and mechanics ) • goal: computation should be robust, later efficient …

46. Response Surface Optimization Cost function evaluation Large comp. cost with FEanalysis Industrial Software « black box» type Impossible to compute analytic grad. Possible improvements regular / irregular grid adaptivity

47. Diffuse approximation based optimization method Moving least square – best fit written as minimization problem First order optimality condition(x – fixed): -which implies

48. Updating the evaluation point Computing the cost function in the neighboring points Computating coefficients a Computing increment Diffuse approximation based optimization method

49. cost function , constant volume discretization with 10 elements : large displacements, large rotations, small strains, elasticity shape parameters: 2 quadratic splines design variables h(x=0), h(x=l) Shape optimization of cantileverbeam

50. Solve: r(x) = 0  min { || r||2 = rTr } Choose a population of ‘chromosomes’ (size: 10n , n – nb. of eqs.) i-th chromosome of g-th generation : xi(g) = [ xi1(g), xi2(g),…, xin(g) ] Producing a new generation: i)-chromosome mutation: xi(g+1) = xi(g) + MR ( RP - xi(g) ) where: RP = xr(g) – random chromosome; MR – algo. const. ½ ii)-chromosome cross operator: xi(g+1) = xp(g) + CR ( xq(g)- xr(g) ) where: xp(g), xq(g),xr(g) – randomly chosen chromosomes, CR – radioact. 0.3 iia)-gradient-like modification of cross operator xi(g+1) = max(xq(g), xr(g)) + SGCR ( xq(g)- xr(g) ) where: SG – sign selection w/r to gradient iii) tournament of chromosome pairs from two populations  reduce population size of generation g+1 again to 10n Genetic algorithm – solution procedure