• Документ: 2013 The Finite Element method for nonlinear structural problems Vincent Chiaruttini Outline Generalities on computational strategies
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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 FE analysis for non linear mechanics – Athens MP06 - V.Chiaruttini 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 FE analysis for non linear mechanics – Athens MP06 - V.Chiaruttini 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 ) f (u) u0 u1 u2 u 4 FE analysis for non linear mechanics – Athens MP06 - V.Chiaruttini 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 = uk+1 ¡ uk = ¡f (uk )=f 0 (uk ) Taylor expansion closely to the exact solution 0 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 ) 2f 0ek + f 00 e2k 2 f 00 (u) 2 2 2 ek+1 = ek ¡ 0 00 000 2 + o(e k ) = 0 e k + o(e k ) = O(e k) 2f + f ek + f ek 2f (u) Quadratic convergence Close enough to the solution each iteration produces twice more significant new digits 5 FE analysis for non linear mechanics – Athens MP06 - V.Chiaruttini 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 u1 u2 u 6 FE analysis for non linear mechanics – Athens MP06 - V.Chiaruttini 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 p uk ¡ uk¡1 1+ 5 uk+1 = uk + f (uk ) ek+1 = O(jek j 2 ) f (uk ) ¡ f (uk¡1 ) Golden ratio convergence order f (u) u0 u1 u2 u 7 FE analysis for non linear mechanics – Athens MP06 - V.Chiaruttini 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 @U j k k where th

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