Users:Structural Optimization/Response Functions/Stress

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'.

This function was originally designed as a continuous alternative for max(σ_i). The larger the value for ρ, the closer max(σ_i) and KS will be. In that case, the value of KS is almost only dependent on the maximal stress value and the effect of the function will be limited to the neighbourhood of this maximal stress. If ρ is small, the KS function will always have a value that is higher than max(σ_i). When reducing ρ, more stresses will influence the value. The effect of the function becomes then more global. A reasonable range of ρ is between 1 and 100.

The value of the exponential function becomes infinite very fast, even for reasonable values of the argument. Therefore, the value of σ_max should not be too small. The best results are obtained if the stresses σ_i are similar to σ_max. If all stresses in the structure are much smaller than the allowed stress, there will be no reasonable value of ρ that is large enough to make KS similar to max(σ_i). As a consequence, KS will be similar for all proposed structures and thus become useless.

So to summarize: 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 be rather low to use the global influence of the 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. Using this constraint is conservative since KS will always be at the safe side. To reduce this, it is necessary to choose a sufficient high ρ to be sure that KS is close to max(σ_i). However, ρ should not be too high, since this will introduce similar numerical problems as using the discontinuous constraint max(σ_i).

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.

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