Interior-Point Method for the Computation of Shakedown Loads for Engineering Systems

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1 Institute of General Mechanics RWTH Aachen University Interior-Point Method for the Computation of Shakedown Loads for Engineering Systems J.W. Simon, D. Weichert ESDA 2010, 13. July 2010, Istanbul, Turkey

2 INTRODUCTION PHENOMEN NOLOGY Schematic illustration of different material behaviors under varying thermo-mechanical loading purely elastic instantaneous collapse

3 INTRODUCTION PHENOMEN NOLOGY Schematic illustration of different material behaviors under varying thermo-mechanical loading ratcheting alternating plasticity shakedown

4 CONTENTS INTRODUCTION LOWER-BOUND SHAKEDOWN ANALYSIS SOLUTION BY INTERIOR-POINT T METHOD NUMERICAL EXAMPLES CONCLUSIONS

5 LOWER-BOUND SHAKEDOWN ANALYSIS Statical shakedown theorem by Melan*: If there exists a loading factor α > 1 and a time-independent residual stress field ρ such that the yield condition F 0 is satisfied for all loads contained within the loading domain Ω at any time t and at all points x V in the volume V of the considered structure, then the system will shake down. E F ( ασ ( x, t) + ρ( x), σ Y ( x) ) 0 where: σ Y ( x) : yield stresss E σ ( x, t) : elastic reference stress * Melan, E.: Sitzber. Akad. Wiss., Abt. IIA 145, (1936 )

6 LOWER-BOUND SHAKEDOWN ANALYSIS Mathematical formulation as an optimization problem: maxα NG C r= 1 r ρ = 0 r [ 1, ], [ 1, ] r NG j NC E, j ( ασ r + ρr, σ Y, r ) F ασ +, 0 : linear objective function (convex) affine linear equality constraints nonlinear, convex inequality constraints where: r: considered Gaussian point NG: total number of Gaussian points j: considered corner of the loading domain Ω NC: total number of corners of the loading domain C r : equilibrium matrixes which guarantee that ρr is self-equilibrated

7 SOLUTION BY INTERIOR-POIN NT METHOD min f ( x) = α A x = 0 c( x) 0 x R n introduce slack variables and split variables y and w z improved formulation: v. Mises yield criterion: c ( x) = 2σ u αa 0 2 j r, j Y, r r r solution vector: ( α ) x = u v R r n = 6* NG + 1 T n 2 2 min f ( x) A x = 0 c( x) w = 0 x y + z = 0 w 0, y 0, z 0 introduce barrier parameter µ min ( x, y, z, w) f µ A x = 0 c( x) w = 0 x y + z = 0 w > 0, y > 0, z > 0 where: fµ ( x) = f ( x) µ log w + j log y + i log z i

SOLUTION BY INTERIOR-POIN NT METHOD Karush-Kuhn-Tucker (KKT) conditions: solution is optimal if the Lagrangian L of the problem possesses a saddle point (necessary and sufficient condition for convex

8 SOLUTION BY INTERIOR-POIN NT METHOD Karush-Kuhn-Tucker (KKT) conditions: solution is optimal if the Lagrangian L of the problem possesses a saddle point (necessary and sufficient condition for convex problems) Lagrangian: ( ) ( ) ( ) L = f µ ( x, y, z, w) λe A x λi c( x) w s x y + z with Lagrange multipliers: λ E R m E, λ I mi R, s R n + + Saddle point condition: L = 0 in each direction System of nonlinear equations approximately solved by Newton s method

9 SOLUTION BY INTERIOR-POIN NT METHOD Newton s method: New variables Π k +1 of the subsequent iteration step k+1 are computed from the variables of the previous one k: Π k Π k + 1 = Π + ϒ Π k k k Step values Π k are the solution of the linearized system: J ( Π ) Π = L( Π ) k k k where: J ( Π k ): Jacobian of L( Π) ϒ k : diagonal matrix of damping factors

10 SOLUTION BY INTERIOR-POIN NT METHOD Reduced KKT-system: ( Q ) T T + E1 A C x x f ( x) 0 A 0 λe = C 0 E λ 2 I T T A λe C λi s + E 1 b1 A x 1 c( x) + µ Λ I e 2 2 where: = L = ( k ( ) ) m I Q c x λ x x I, k k = ( ) E = S Y + R Z E = W Λ C = c( x) 1 I x ( ) ( ) 1 b = x + z + µ R S e + R S z T e = 1 1 in proper dimension problem dimension: (6 NG + 1) + me + mi due to zero-block on diagonal regularization necessary!

11 SOLUTION BY INTERIOR-POIN NT METHOD Improvement by condensation of KKT-system: Improved formulation leads to specific structure: Hu 0 H = Q + E + C E C = 0 H T h T 0 v h 0 H α and ( ) A = A A a u v α Condensation of KKT-system: T Hu h A u u T T H α α α h a = 1 T Au aα Av Hv Av λe rhs problem dimension: (5 NG + 1) + me no regularization necessary!

12 SOLUTION BY INTERIOR-POIN NT METHOD Initialization: elastic stresses C-matrix transformation Solve KKT update barrier Damp with Non-negativity Done YES NO Break cond. outer iter. update KKT NO Damp with Linesearch update variables YES Break cond. inner iter.

13 NUMERICAL EXAMPLES Square plate with a circular hole: P x and P y vary independently Length L in [mm] 100 Thickness t in [mm] 2 Diameter D in [mm] 20 Young s modulus in [MPa] 2.1x10 5 Poisson s ratio 0.3 Yield stress in [MPa] 20

14 NUMERICAL EXAMPLES FE-mesh and relevant numbers of optimization problem: Elements NE 400 Gaussian points NG Corners NC 4 Variables Equality constraints m E Inequality constraints m I

15 NUMERICAL EXAMPLES Results: Reference value: Present result: Relative error: 4 % Reduction of CPU: 15 %

16 NUMERICAL EXAMPLES Square plate with a circular hole: Additional temperature load All three loads P x,p y and T vary independently: 0 P x µ x P 0 0 P µ P y y 0 T µ T T 0 0 All material parameters are considered as temperature-independent.

17 NUMERICAL EXAMPLES Result in three-dimensional loading space:

18 CONCLUSIONS Concluding remarks: use of interior-point methods leads to efficient algorithms in shakedown analysis improved formulation has been presented for v.mises yield criterion condensation of the KKT-system provides reduction of CPU-time (much higher reduction for more complex structures) method allows calculation of systems with multidimensional loading Perspectives: extension to larger classes of materials: kinematical hardening, SMA numerical advancements: selective algorithm industrial application

HONGJUN LI Department of Mechanical Engineering University of Strathclyde Glasgow, Scotland, UK

HONGJUN LI Department of Mechanical Engineering University of Strathclyde Glasgow, Scotland, UK Introduction FEA is an established analysis method used in Pressure Vessel Design Most DBA is still based