Users:Structural Optimization/Response Functions/Linear Buckling

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General Description

Short Info

This response function works similar to the EIGENFREQUENCY_KS response function. The difference is that this response function treat buckling load factors in the Kreisselmeier-Steinhauser function. The provided analysis has to be a linear buckling analysis. By minimizing the Kreisselmeier-Steinhauser function the buckling load factors will be increased.

For this response function the following Kreisselmeier-Steinhauser function is implemented:

ψ = 1/ρ * ln ∑i e-ρ*λi

with the scaling parameter ρ and the i-th eigenvalue λi of the linear buckling problem

(Kel - λ*Kgeo) φ = 0.

Input Parameters

Block headline
Parameter Values, Default(*) Description
OPT-RESPONSE_FCT int : LIN_BUCKLING_KS Function ID and type mechanical problem.
Common compulsory parameters
ETA real Finite difference disturbance for sensitivity analysis
GRAD DIRECT, ADJOINT Method of gradient computation
SA GLOBAL_FD, SEMI_ANALYTIC, EXACT_SEMI_ANALYTIC, ANALYTIC Method of derivative computations inside sensitivity analysis
FDA FOREWARD, CENTRAL, BACKWARD Method of finite difference approximation (if neccessary for the chosen sensitivity analysis method)
DESVAR OPT-VAR vector of integers Design variables that are considered in the sensitivity analysis of this response function
Common optional parameters
WEIGHT real, 1.0* The weighting factor for this response function in multi-objective optimization
ANALYSIS PC-ANALYSIS int ID of the underlying analysis
Specific parameters
RHO float Rho parameter of Kreisselmeier-Steinhauser function. Refer to the page Users:Structural Optimization/Response Functions/Eigenfrequency for more information.
Common Compulsory Parameters for Constraints
Parameter Values, Default(*) Description
REL_LIMIT real Relative limit for constraint, depending on the actual value.
ABS_LIMIT real Absolute limit for constraint. Only one limit can be defined for a constraint.
CONSTRAINT_TYPE INEQUALITY_LT, INEQUALITY_GT, EQUALITY Type of constraint
Common Optional Parameters for Constraints
REL_TOLERANCE real, 0* Upper relative limit until which an inactive constraint is concidered as an active one
LAMBDA_ABS_MAX real, 1/cepsilon* Upper limit for lagrangian multiplier

Example of a Complete Input Block

OPT-RESPONSE_FCT 1 : LIN_BUCKLING_KS

  ! -- basic stuff
  WEIGHT=1.0 ANALYSIS=PC-ANALYSIS 1 ETA=1e-06
  GRAD=ADJOINT SA=SEMI_ANALYTIC FDA=FOREWARD
  DESVAR=OPT-VAR 1,2,3,4,5,6

  ! -- response function dependant parameters
  RHO = 2.0

  ! -- constraint parameters
  ABS_LIMIT = 1-e3
  REL_TOLERANCE = 0.1
  CONSTRAINT_TYPE = INEQUALITY_LT
  LAMBDA_ABS_MAX = 20


Example

This example considers a flat plate subjected to in-plane compression loading. After reaching a specified load level, the structure starts to buckle. The four smallest buckling modes are presented in the figure below. They result from a linear buckling analysis.

The goal of the optimization problem is to increase the buckling load factors. For this aim the four smallest buckling load factors are formulated in a Kreisselmeier-Steinhauser function. This function serves as objective and is decreased during optimization. All normal nodal coordinates are defined as design variables. This results in 441 design variables. The problem is solved by the Conjugate Gradient Method using a 3-Point Line Search.

The figure below shows the design updates of the first five optimization steps. It can be clearly observed that the optimized structure has a much higher buckling load than the initial structure.

Design updates of flat blank buckling optimization problem

This is also shown by the curves of the buckling load factors.

Design updates of flat blank buckling optimization problem




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