The Finite Element method for nonlinear structural problems

Non Linear Computational Mechanics Athens MP06/2013 The Finite Element method for nonlinear structural problems Vincent Chiaruttini vincent.chiaruttini@onera.fr

Outline Generalities on computational strategies for nonlinear problems Examples (contact, crack propagation, non-linear behaviour, geometrical non-linearities) Classical algorithm for nonlinear or time dependant problems Local numerical aspects of plasticity Elastic-plastic behaviour Local integration of non-linear models Global numerical aspects of plasticity Solution process Consistent tangent matrix Examples of solution process Presentation of Z-mat 2

Non-linearities in structural problems Contact Due to the non-penetration condition Crack propagation problem under time dependant loading When crack propagates the solution becomes non-linearly time dependant Geometrical nonlinearities For large deformation, instabilities can also occur (buckling) Nonlinear constitutive relationship Non linear behaviour: elastoplasticity, damage, viscosity 3

Solution process Iterative algorithm Scalar example nding u j f (u) = 0 For any kind of regular function, no direct process exists Iterative algorithm building un! u j f (u) = 0 Stop when a convergence criterion is satisfied rank k j jf (uk )j < "crit Newton method Built on the linear verification of the first order Taylor development nullity f (uk+1 ) ¼ f (uk ) + (uk+1 uk ) f 0 (uk ) = 0 ) uk+1 = uk f (uk )=f 0(uk ) u0 4 u1 u2 f (u) u

Solution process Newton method Built on the linear verification of the first order Taylor development nullity f (uk+1 ) ¼ f (uk ) + (uk+1 uk ) f 0 (uk ) = 0 ) uk+1 = uk f (uk )=f 0(uk ) Convergence depends on u0 When converges Rank k error ek = uk u Recurrence on error relationship ek+1 ek Taylor expansion closely to the exact solution = uk+1 uk = f (uk )=f 0 (uk ) 1 00 f (uk ) = f (u)ek + f (u)e2k + o(e2k ) 2 f 0 (uk ) = f 0 (u) + f 00 (u)ek + f 000 (u)e2k + o(e2k ) 00 2f 0ek + f 00 e2k f (u) 2 2 2 2 ek+1 = ek 0 + o(e ) = e + o(e ) = O(e k k k k) 2 00 000 0 2f + f ek + f ek 2f (u) 0 Quadratic convergence Close enough to the solution each iteration produces twice more significant new digits 5

Solution process Newton method Quadratic convergence Require to update the derivative at each iteration Modified Newton methods Constant direction f 0 (uk ) ¼ K = cst = f 0 (u0 ) uk+1 = uk f (uk )=K 0 Linear convergence ek+1 = (1 f (u)=k)ek + o(jek j) = O(jek j) f (u) u0 6 u1 u2 u

Solution process Newton method Quadratic convergence Require to update the derivative at each iteration Modified Newton methods Constant direction f 0 (uk ) ¼ K = cst = f 0 (u0 ) Linear convergence Secant update uk+1 uk uk 1 = uk + f (uk ) f (uk ) f (uk 1 ) Golden ratio convergence order f (u) u0 7 u1 u2 u ek+1 = O(jek j p 1+ 5 2 )

Solution process Newton method for a set of equations Vectorial function F (U ) = 0 At each iteration a linear system is solved i @F j 0 = F (U k ) + (U ) U k @U j k where the operator constitutes the rigidity matrix at Uk Quadratic convergence when close to solution The neighbourhood where such kind of convergence is observed depends on the vectorial function properties Convergence is insured for convex functions 8

Examples of convergence using a Newton algorithm 9

Examples of convergence using a Newton algorithm 10

ODE integration Time dependant problem are ruled by differential equations Reduce high order differential systems to first order 2 d y dy + g(t) = r(t), 2 dt dt General formulation [Y_ ] = [F (t; [Y ])] ½ dy dt dz dt = z(t) = r(t) g(t) z(t) [Y_ (t = t0 )] = [Y0 ] Euler-type integration schemes Finite difference time discretization Y^ (tn ) = Yn ; tn+1 = tn + t Y n+1 Y n Y (t) = F (t; Y ) forward methods Y (tn+1 ) ¼ t θ-method (method-b) Y n+1 Y n = µf n+1 (Y n+1 ) + (1 µ)f n (Y n ) t Explicit Forward Euler (θ=0) Conditionally stable 1st order accurate 11 Crank-Nicholson (θ=0.5) Unconditionally stable 2nd order accurate Implicit Euler (θ=1) Unconditionally stable 1st order accurate

ODE integration Time dependant problem are ruled by differential equations Runge-Kutta explicit integration Minimal multiple evaluations of F (t; y(t)) on a given time increment to insure a specific 0 + ( t=2)yn00 + o( t2 ) order accuracy, based on Taylor expansion yn+1 = yn + tyn RK1 is the forward explicit Euler scheme RK2 using one mid-point sub calculation at tn+ 1 2 t yn+1=2 = yn + f (tn ; yn ) 2 yn+1 = yn + t f tn+1=2 ; yn+1=2 2nd order method RK4 popular method using 4 sub calculations yn+1 = yn + t=6 (k1 + 2k2 + 2k3 + k4 ) k1 k2 k3 k4 = f (tn ; yn ) = f (tn+1=2 ; yn + ( t=2)k1 ) = f (tn+1=2 ; yn + ( t=2)k2 ) = f (tn+1=2 ; yn + tk3 ) Initial slope on the interval Mid interval slope using k1 Mid interval slope using k2 Final slope using k3 on the interval 4th order method 12

Local numerical aspects of plasticity Assumptions Small strain: linearised kinematic Quasi-static loadings: no dynamic effects Small strain elastoplastic problem Elastic domain f (¾) < 0 A usual elastic behaviour is applied Yield surface f (¾) = 0 Irreversible/dissipative phenomena can occur Example Von Mises criterion q 1 f (¾) = 3=2¾D : ¾D R with ¾ D = ¾ Tr(¾) I 3 14

Local numerical aspects of plasticity Small strain elastoplastic problem Plastic strain " = "e + "in = "e + "p Strain partition Rigidity ¾ = A : "e = A : (" "p ) Yield surface evolution (convex), hardening When f =0 the yield surface evolves Isotropic hardening yield surface increase Kinematic hardening yield surface translation Flow rules Plasticity (normality rule) @f _ (¾) = 0 "_ = ; 0; f (¾) 0; f @¾ p Cumulated plastic strain rate p(t) = Z tr 0 15 ³ 3 p "_ ( ) : "_ p ( ) d = _ for a Von Mises criterion 2

Local numerical aspects of plasticity Common formalism for viscoplasticity/multipotentials/large strain Strain partition Small strain e in "=" +" Elasticity e p = " +" +" vp Large strain F =EP ¾ = A : "e Flow rules Plasticity (state must stay on the yield surface at the end of the increment) p "_ = X _ s ns _ s from f_s = 0 s Viscoplasticity (state expected to be on the relevant equipotential at the end of the increment) s Àn X vp s s f s "_ = v_ m v_ = Hardening rules s K Y_ I = f ct(yi ; "p ; "vp ) 16

Numerical solution process Elastoplastic example Relations t 2 [0; T ] Equilibrium and strain definition r¾ + f = 0 Behaviour ¾ = A : (" "p ) 3 "_ = p_ eq ¾ D 2¾ p 1 T "= ru+r u 2 r 3 eq ¾ = k¾ D k 2 p_ 0 ¾eq R(p) 0 Boundary conditions (initial equilibrium and no plasticity) u = ud on @u ¾ n = F on @f ²p (t = 0) = 0 17 p_ (¾ eq R(p)) = 0

Numerical solution process Elastoplastic example ª p Mechanical state S(x; t) = u(x; t); "(x; t); " (x; t); ¾(x; t) Temporal discretization Incremental temporal approach using a regular grid tn+1 = tn + t The solution is search incrementaly at each time step t n+1 (all previous step states being known) Spatial discretization FE method in displacement 0 8u 2 Uad ; Z ¾ n+1 : " d = Z f n+1 :u d + Z @F Nonlinearity comes from the non linear relation between stress and strain relation Local integration of the mechanical behaviour At each point of the structure process is defined by the following process (un+1 ; Sn )! ¾ n+1 = F(un+1 ; Sn ) Global equilibrium Verified by the solution of the nonlinear variational formulation 0 R(un+1 ; u ; Sn ) = 0; 8u 2 Uad 18 F n+1 :u ds

Local behaviour integration process Generic interface for any constitutive equation Gauss point process (where the element variational formulation is integrated) Definition External parameters (ep) imposed as input Integrated variables (vint) Auxiliary variables (vaux) just for output Coefficients (coef) material parameters Primal and dual variables prescribed variables and associated fluxes Primal: strain increment on the current time interval (input to the local integration process) Dual: stress obtained useful for variational formulation (as output of the local integration process) F(un+1 ; Sn ) Primal variables " 19 Local GP integration ODE integration to get vint, vaux, dual variables p; "p Dual variables ¾

Local behaviour integration process Time integration Normality rule as an ODE 3 p "_ = p_ eq ¾ D, "pn+1 = "pn + 2¾ Z tn+1 "_ p ( ) d tn Explicit integration Runge-Kutta (RK4 usually, beware of stability condition) Implicit integration θ-methods (stable but requires a local Jacobian computation) 20

Local behaviour integration process Elastic prediction and correction algorithm eq Solution depends on the test ¾ R(p) 0 (is the evolution purely elastic on t?) Elastic prediction ¾ en+1 = ¾n + A : "en Convexity of the yield surface gives f (¾ en+1 ; pn ) f (¾ n+1 ; pn + pn ) 8 e < ¾ n+1 = ¾ n+1 f (¾ en+1 ; pn ) 0 purely elastic evolution, prediction is correct "pn+1 = "pn : pn+1 = pn Else a plastic evolution is observed A correction must be applied on the elastic prediction pn > 0; pn f (¾n+1 ; pn + pn ) = 0 ) f (¾n+1 ; pn + pn ) = 0 r 3 p p e ¾ n+1 = ¾n+1 A : "n "n = pn N n+1 pn > 0 2 where 21 N n+1 is the outward unit normal to the final yeld surface

Local behaviour integration process Elastic prediction and correction algorithm eq Solution depends on the test ¾ R(p) 0 (is the evolution purely elastic on t?) Elastic prediction ¾ en+1 = ¾n + A : "en Convexity of the yield surface gives f (¾ en+1 ; pn ) f (¾ n+1 ; pn + pn ) 8 e < ¾ n+1 = ¾ n+1 f (¾ en+1 ; pn ) 0 purely elastic evolution, prediction is correct "pn+1 = "pn : pn+1 = pn Else a plastic evolution is observed A correction must be applied on the elastic prediction pn > 0; pn f (¾n+1 ; pn + pn ) = 0 ) f (¾n+1 ; pn + pn ) = 0 r 3 p p e ¾ n+1 = ¾n+1 A : "n "n = pn N n+1 pn > 0 2 where 22 N n+1 is the outward unit normal to the final yeld surface

Radial return algorithm For von Mises based criteria or isotropic or linear kinematic hardening p The corrective term A : " is oriented by the final normal that can be computed a priori n For von Mises criterion the final normal is collinear to the elastic predictor r deviator N n+1 = ¾ edn+1 k¾edn+1 k eq e;eq ¾n+1 = ¾n+1 3¹ pn ¾ n+1 = A : pn e ¾Dn+1 f (¾n+1 ; pn+1 ) = 0 ¾Dn+1 3¹ pn R(pn + pn ) = 0 that allows to find f (¾n ; pn ) = 0 pn ¾Dn 23 3 N n+1 2 ¹ Lamé coefficient Final state on the yeld surface gives to verify e;eq ¾n+1 ¾en+1

Generalized radial return algorithm Closest point projection technique For a generalized normality rule @f " = p @¾ f (¾n+1 ; pn+1 ) = 0 @f I = p @YI p e ¾Dn+1 ¾Dn+1 f (¾n ; pn ) = 0 Fluxes P ¾ = A : " " ¾Dn YI = MI I = 24 ¾ YI = " A : " 0 # " A 0 0 MI #" @f @¾ @f @YI # p

Generic formulation of the local integration process 25

Global aspects Solving the global discrete problem Global problem 0 with R(un+1 ; u ; Sn ) = R(un+1 ; u ; Sn ) = 0; 8u 2 Uad Z Z Z F (un+1 ; Sn ) : " d f n+1 :u d @F F n+1 :u ds Global Newton algorithm At each time step the solution is searched iteratively starting from the last converged solution leading to the following linear system at iteration i i 0 R(uin+1 ; u ; Sn ) + R0 (uin+1 ; u ; Sn )[ui+1 u ] = 0; 8u 2 U n+1 ad n+1 To obtain the residual evaluation, it is necessary to perform the local integration to get F(uin+1 ; Sn ) To calculate the global consistent tangent operator, it is necessary to obtain a local consistent tangent operator (that can be kept constant Zif a modified Newton algorithm with constant operator is involved) @F R = (un+1 ; Sn ) : " d @u i+1 When the residual is small enough the solution is reached R(un+1 ; u ; Sn ) < N L 0 27 (uin+1 ; u ; Sn )

Local consistent tangent operator In the local integration process 28

Viscoelastic with isotropic hardening 30

Viscoelastic with isotropic hardening Implementation 31

Viscoelastic with isotropic hardening Implementation 32

Viscoelastic with isotropic hardening Implementation 33

Large strain viscoelastic with isotropic hardening 34

System of residuals 35

Algorithm 36

Tangent matrix 37

Presentation of the Z-mat library 39

Inside Z-mat 40

Object oriented design 41

Isotropic and non-linear kinematic model 42

Datafile examples 43

ZebFront interface layer for efficient behaviour development 44

Explicit model implementation - ZebFront 45

Implicit model implementation - ZebFront 46

Multimat capabilities 47

Multimat capabilities 48

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