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[[Category: Users:Structural Optimization]]


== General Description ==

=== Short Info ===
Strain energy or compliance is a very popular objective function for structural optimization problems. Minimizing the strain energy of a structure yields to designs with optimal load carrying behavior and minimal displacements. At the same time the material strains and stresses are reduced. 

Linear strain energy is commonly formulated by  

<div style="font-size:150%">
:: f = 1/2 '''d'''<sup>T</sup> '''K'''<sub>el</sub> '''d'''
</div>

with the linear elastic stiffness matrix '''K'''<sub>el</sub> and the displacement field '''d'''. The response function for linear strain energy requires the ID of a [[Users:General FEM Analysis/Analyses Reference/Static Linear | linear static analysis]] specified for parameter 'ANALYSIS'.

The formulation of nonlinear strain energy differs significantly from the linear formulation. It is defined by

<div style="font-size:150%">
:: f = 1/2 '''f'''<sub>ext</sub><sup>T</sup> '''d'''
</div>

where '''f'''<sub>ext</sub> describes the external forces. Whenever a model shows significant structural displacements, the optimization should the performed with respect to nonlinear strain energy. In this case the load carrying behavior of the deformed structure is optimized which usually ends up in more efficient designs. The response function for nonlinear strain energy requires the ID of a [[Users:General FEM Analysis/Analyses Reference/Static Nonlinear | nonlinear static analysis]] specified for parameter 'ANALYSIS'.

== Input Parameters ==

{| border="1" cellpadding="3" cellspacing="0"
|colspan="3" style="background:#efefef;"| Block headline
|-
!Parameter
!Values, Default(*)
!Description
|-
!OPT-RESPONSE_FCT 
|''int'' : STRAIN_ENERGY_LIN or STRAIN_ENERGY_NONLIN
|Function ID and type mechanical problem.
|-
{{Users:Structural Optimization/Response Functions/Common Parameters}}
|-
{{Users:Structural Optimization/Response Functions/Common Parameters for Constraints}}
|-
|}

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

<pre>
OPT-RESPONSE_FCT 1 : STRAIN_ENERGY_LIN

  ! -- 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

  ! -- constraint parameters
  ABS_LIMIT = 1-e3
  REL_TOLERANCE = 0.1
  CONSTRAINT_TYPE = INEQUALITY_LT
  LAMBDA_ABS_MAX = 20
</pre>

== Examples ==

=== Cup Structure ===

This example shows the optimal shape of a cup structure with respect to linear strain energy. The structure was motivated by natural shapes of some types of mushrooms or the calyx of flowers. The inner ring of the structure is supported and the cup is subjected to dead load. The initial design is completely flat, whereas the optimal structure has a specified height of 8m. The material is described by the St.Venant-Kirchhoff material model with standard concrete properties. The overall dimensions of the optimized structure are:
* Height = 8m
* Radius of inner ring = 1m
* Radius of outer ring = 10m
* Thickness (const.) = 0.05m   

The optimization problem is solved by the Augmented Lagrangian Multiplier method where the structural height of 8m is implemented as equality constraint.

<gallery caption="" widths="512px" heights="384px" perrow="2" center>
  File:Kelch_design_updates_bottom_view.gif‎ | Bottom view to Design Development
  File:Kelch_design_updates_top_view.gif    | Top view to Design Development
</gallery>


=== Bead optimization ===

Another important application of strain energy minimization is bead optimization of metal sheets. Beads can be considered as local stiffeners that improve the load carrying behavior. The optimal orientation, length, and width of the beads is a challenging job for numerical optimization schemes. The figure below shows an optimized metal sheet with corner support and central loading. The bead height is restricted by proper variable bounds. It can be easily observed that the optimized structure has much better mechanical properties (smaller displacements, smaller stresses) than the flat blank. 

<gallery caption="" widths="400px" heights="210px" perrow="2" center>
  File:Quad_plate_40x40_opt_geom.png‎ | Rendered view of optimal bead geometry
  File:Quad_plate_40x40_opt_geom_color.png    | Colored view to optimal bead geometry
</gallery>