Users:Structural Optimization/Response Functions/Stress

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	LAMBDA_ABS_MAX = 20		LAMBDA_ABS_MAX = 20
	</pre>		</pre>
		+
		+
		+	== Examples ==
		+
		+	=== Shell cantilever ===
		+
		+	This example shows the optimal shape of a cantilever beam subjected to an end load. The structure is modelled by shell elements. During the optimization, the Kreisselmeier-Steinhauser function is minimized while the structural weight was not allowed to increase more than 5%. To avoid singularity at the free end, a bound constraints the movement of the end nodes. The optimization problem is solved by the Augmented Lagrangian Multiplier method for two values of ρ. Comparing the evolution of the shape, it is clear that the design updates are much more local when using a higher value of ρ.
		+
		+	<gallery caption="" widths="512px" heights="384px" perrow="2" center>
		+	File:Kelch_design_updates_bottom_view.gif‎ | ρ=1
		+	File:Kelch_design_updates_top_view.gif    | ρ=10
		+	</gallery>

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Revision as of 14:43, 13 October 2010

Contents 1 General Description 1.1 Short Info 2 Input Parameters 2.1 Example of a Complete Input Block 3 Examples 3.1 Shell cantilever

General Description

Short Info

The Kreisselmeier-Steinhauser function for the stresses is a global measure of stress in a structure. Stress results are generally very local results but using this responce function, an overall stress indicator is obtained. It can be used as an objective to reduce the overall stress in the structure or as a constraint to limit the stress in the structure to a maximum allowed value.

The Kreisselmeier-Steinhauser function for the stresses is formulated by

with ∑_i a summation over all Gausspoints in the domain and σ_i the stress in the i-th Gausspoint from a linear or nonlinear analysis. σ_max is the maximum allowed stress. ρ is a parameter that determines the importance of the largest stresses amongst all σ_i. The response function requires the ID of a linear static analysis or a nonlinear static analysis specified for parameter 'ANALYSIS'.

(influence of sigma_max) The value of the exponential function becomes infinite very fast, even for reasonable values of the argument. Therefore, the choice of σ_max is very important. The best results are obtained if the stresses σ_i are similar to σ_max. If the stresses in the structure are much smaller than the allowed stress, the value of this response function becomes useless. Since it is in that case a summation of all small values, it will be similar for all structures.

(influence of rho)

(so how to choose the values) If this response function is used as an objective function, the value of σ_max should be chosen such that it is close to the real stresses in the structure. Additionally, the value for ρ should not be high to use the global stress results. If this response function is used as a constraint function, the value of σ_max should be chosen as the real maximum allowed value. Additionally, the value for ρ should be high enough to localize the largest stress.

Input Parameters

Block headline
Parameter	Values, Default(*)	Description
OPT-RESPONSE_FCT	int : STRESS_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
ELEMENTS	El-SET or PART	Determines the elements of which the stress is taken into account
STRESS_TYPE	VMISES_TOP, VMISES_BOTTOM, VMISES, PRINCIPLE_1_TOP, PRINCIPLE_1_BOTTOM, PRINCIPLE_1	Determines which stress type is taken into account
RHO	real	Parameter that determines how local the effect of this response function is
STRESS_LIMIT	integer	Maximum value for the real value of the stress (cannot be combined with ABS_STRESS_LIMIT)
ABS_STRESS_LIMIT	integer	Maximum value for the absolute value of the stress (cannot be combined with STRESS_LIMIT)
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 : STRESS_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

  ! -- specific
  ELEMENTS = EL-SET 1
  STRESS_TYPE = PRINCIPLE_1_TOP
  RHO = 10.0
  ABS_STRESS_LIMIT = 200.0E06

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

Examples

Shell cantilever

This example shows the optimal shape of a cantilever beam subjected to an end load. The structure is modelled by shell elements. During the optimization, the Kreisselmeier-Steinhauser function is minimized while the structural weight was not allowed to increase more than 5%. To avoid singularity at the free end, a bound constraints the movement of the end nodes. The optimization problem is solved by the Augmented Lagrangian Multiplier method for two values of ρ. Comparing the evolution of the shape, it is clear that the design updates are much more local when using a higher value of ρ.

• 

  ρ=1

• 

  ρ=10

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