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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

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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 !

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Introduction and motivation 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) … LMT-Cachan : Laboratory of Mechanics and Technology (Ecole Normale Sup / Univ. Pierre & Marie Curie / CNRS http://www.lmt.ens-cachan.fr

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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

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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

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Introduction and motivation Reliable description of limit state behaviour of structures in their environment (multi-physics problems) Final (worthy) goal 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

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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)

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

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

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

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

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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. PiPi PjPj tftf Link behaviour Random variables Φ

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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 Cohesive force – beam link l x v u Position vector in deformed configuration: g1g1 g2g2 nt where is the coordinate along beam cross-section Idea : new position of cell interface defined by vector t

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

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

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

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

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

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

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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

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

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

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Energy decaying time integr. scheme (Ibrahimbegovic, et al 99. 02) 2 : dissipation in inertia.. terms 1 : dissipation in internal. forces

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

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Crack pattern: anisotropic damage d s = 10 µm/min dsds (Nooru-Mohamed, 92)

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

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

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Current developments: 3D analysis

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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!

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Optimal control and optimal design of structures undergoing large rotations: Acknowledgements : C. Knopf-Lenoir, P. Villon, UTC, France A. Kucerova, EU – Erasmus, French Ministry of Research, 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

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Introduction and motivation -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.

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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]

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Model problem in mechanics : geom. exact 3D beam -Configuration space: l x v u g1g1 nt g2g2 Displacement update : additive Rotation updates : multiplicative φ t = φ + t δφ Λ t = Λ exp [t δΘ] ; δΘ v = δθ × v

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Model problem in mechanics : geom. exact 3D beam - Strain measures - Stress resultants δε = Λ t T δφ’ + ε t × δθ δω = δθ’ + ω t × δθ

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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 (φ t, Λ t ) := ∫ l [ δφ’ Λ t n t + δθ (E t n t + Ω t m t ) + δθ’ m t ] ds - G ext -exception : non-conservative load (e.g. follower force, moment)

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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 ! -Optimal design : i) Traditional approach : constrained minimization of cost function

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

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

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

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

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Optimization : coupled mechanics-optimization problem -Optimal control: i) Traditional approach : constrained minimization of cost function -where: v control variables, f(t) = F 0 v(t), F 0 - 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

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

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

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

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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 = r T r } exact solution computed (optimization and mechanics ) goal: computation should be robust, later efficient …

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

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

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

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Shape optimization of cantilever beam 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)

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Genetic algorithm – solution procedure Solve: r(x) = 0 min { || r || 2 = r T r } Choose a population of ‘chromosomes’ (size: 10n, n – nb. of eqs.) i-th chromosome of g-th generation : x i (g) = [ x i1 (g), x i2 (g),…, x in (g) ] Producing a new generation: i)-chromosome mutation: x i (g+1) = x i (g) + MR ( RP - x i (g) ) where: RP = x r (g) – random chromosome; MR – algo. const. ½ ii)-chromosome cross operator: x i (g+1) = x p (g) + CR ( x q (g) - x r (g) ) where: x p (g), x q (g), x r (g) – randomly chosen chromosomes, CR – radioact. 0.3 ii a)-gradient-like modification of cross operator x i (g+1) = max(x q (g), x r (g)) + SG CR ( x q (g) - x r (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

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Optimal control of cantilever in form of letter T T letter cantilever : Initial configuration (-----) Final configuration (-----) Intermediate configs. Cost function : J(v) = ½|| u(v) – u d || 2 Control variables: v = (F,M) Remarks: -exact solution: F=40, M=205 -solution sensitivity w/r M, not F (system conditioning)

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Optimal control of cantilever in form of letter T Diffuse appr. (fixed grid) solution : Grid 5x5 F=60 M=205.26 Grid 20x20 F=47.444 M=204.970 Genetic algorithm solution: F=40, M=205

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Optimal control of cantilever in form of letter I I letter cantilever Initial configuration (-----) Final configuration (-----) Intermediate configs. (-----) Cost function :J(v) = ½ || u(v) – u d || 2 Control variables : v = (M,F) (F-follower force !) Solution non-unique : F = 102.63 – 0.49989 M Regularized cost fcn. : sol. unique J(v) = ½ || u(v) – u d || 2 + ½ α || v || 2 α =10 -9 F = 68.466, M = 68.526

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Optimal control of deployment of multibody system Control variables : (M 1, M 2, M 3, H, V) Remark: solution sensitivity w/r Mi, not H, nor V Multibody system Initial configuration (-----) Final configuration (-----) Intermediate configs. (-----)

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Optimal control of deployment of multibody system H V M 1 M 2 M 3

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Optimal control of deployment of multibody system 1 st generation 4 th generation Genetic algorithm based solution procedure – more difficult with larger number of variables

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Optimal control of deployment of multibody system 26 th generation 43 rd generation -Solution : H =0.04, V =-0.05, M 1 =0.78, M 2 =-0.79, M 3 =0.79 -how to accelerate convergence at the last stage …

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Conclusions Presented unified approach for optimal design and/or control in nonlinear structural mechanics coupled problem opt.-mech. Model problem : 3D beam – also shells with drills and 3D solids Solution procedures for solving this optimization problem Response surface algorithm based on diffuse approximation –generality : without gradients –accuracy : adaptive size of searching pattern –efficiency : progressive construct. of a response surface, reuse of data points Genetic algorithm simultaneous solution procedure -robustness : solution computed even if non-unique, isolated peak values … -efficiency: to develop further combining with response surface and/or gradient method To contact me : e-mail : ai@lmt.ens-cachan.fr, fax : +33147402240 : :. :.

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