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[[Category: Users:Structural Optimization]]


== General Description ==

=== Short Info ===
In the field of structural optimization it is often necessary to apply a constraining of surface curvature in order to maintain manufacturing constraints. To this purpose, ''Carat++'' provides an estimation tool to approximate the mean curvature at a surface node.



=== Estimation of nodal curvature  ===
The nodal curvature is estimated using the surface normal vector and access vectors to the surrounding nodes. '''The estimation is based on a local spherical approximation of the geometry (see sketch below).''' So the radius of curvature r can be computed within the isosceles triangle.

 [[ File:curv_ks_sketch.png |500px | center |Sketch of curvature estimation in 1D ]]

Having computed this approximated radius of curvature for each surrounding node, the mean curvature of the surface is estimated by a weighted sum over all surrounding nodes ''i'', whereat v''i'' is the access vector to the ith node.

 [[ File:curv_ks_formula.png |250px | center | ]]

=== Kreisselmeier-Steinhauser weighting ===
Instead of constraining each single nodal curvature, ''Carat++'' provides a technique to control the curvature within bigger element patches. To this purpose a Kreisselmeier-Steinhauser weighting is used.

 [[ File:curv_ks_formula2.png |350px | center | KS formulation for one patch ]]

Via the weighting parameter ρ the user can control to which extend exceedings of the maximum allowable curvature are weighted.


== Input Parameters ==

{| border="1" cellpadding="3" cellspacing="0"
|colspan="3" style="background:#efefef;"| Block headline
|-
!Parameter
!Values, Default(*)
!Description
|-
!OPT-RESPONSE_FCT 
|''int'' : CURVATURE_PATCH_KS
|Function ID and response function type.
|-
{{Users:Structural Optimization/Response Functions/Common Parameters}}
|-
|colspan="3" style="background:#efefef;"| Specific parameters
|-
!NODES
|ND-SET ''int''
|Node set that includes all nodes this response function yields onto.
|-
!PATCH_SIZE
|''int''
|Size of patches (number of element rows) into which the ''NODES'' is sub-divided.
|-
!RHO
|''real''
|KS-weighting factor.
|-
{{Users:Structural Optimization/Response Functions/Common Parameters for Constraints}}
|-
|}

=== Example of a Complete Input Block ===

<pre>
OPT-RESPONSE_FCT 1 : EIGENVALUE
 WEIGHT=1.0 ANALYSIS=PC-ANALYSIS 1 ETA=1e-06
 GRAD=ADJOINT SA=SEMI_ANALYTIC FDA=FOREWARD
! -- response function dependant parameters
 NUM_EIGENVALUE = 1
! -- constraint parameters
 REL_LIMIT = 1.1
 REL_TOLERANCE = 0.1
 CONSTRAINT_TYPE = INEQUALITY_LT
 LAMBDA_ABS_MAX = 20
</pre>