12.13 Layout Optimization
LAYOUT_OPTIMIZATION
LAYOUT_OPTIMIZATION step_number = ns , etc...
The procedure computes the distribution of material for structure supported on its boundaries and subjected to a given loading condition, such that its compliance (objective function) is optimized. The amount of material is constrained and the spatial distribution of material is limited to the design domain. The design domain can have holes (void regions) and regions with fixed solid material. The structure can be discretized in a number of different ways (e.g., using truss, beam, plate or solid elements), and can be modeled as two- or three-dimensional.
Stagger_name(s) string [main] Stagger name(s). Name(s) must
be enclosed in quotation marks.
Element_group(s) string [all] Element group(s) for which optimization
is
to be performed. Name(s) must be
enclosed
in quotation marks.
Step_number integer [1] Step number at which optimization is
to
be
initiated.
Frequency integer [0] Optimization frequency
(1) Average_density real [0.0] Target average material density
Minimum_density real [0.01] Minimum density 
Maximum_density real [1.0] Maximum density 
Initial_density list [*] Initial density
uniform uniform
random random
deviates deviates
Seed integer [7654321] Seed for random number generation;
> 100,000 and < 1,000,000
Extreme_size real [0.5] Extreme size £ 0.5
Extreme_prob real [0.5] Extreme probability £ 1.0
(2) Exponent real [1.0] Density exponent p £ 3
Damping real [0.5] Damping 
> 0
cont'd
(cont’d)
Move_limit real [0.2] Move limit
Filter_size real [0.0] Filter size
Max_number_of_iterations integer [100] Maximum number of iterations
EXAMPLE
stagger_name = "main" /
element_group(s) = "group 1 " /
step_number = 1 /
average_density = 0.75 /
minimum_density = 0.001 /
maximum_density = 1.0 /
exponent = 1.0 /
damping = 0.8 /
max_number_of_iterations = 15
Notes /
(1) The
optimization algorithm minimizes the compliance of the structure for a fixed
amount of available material in the design domain. Let 
denote the density of
material in element 
with volume 
, then the total volume of the structure is computed as
where r
is bounded by:
(2) The stiffness of any given element is computed as:
where 
= element stiffness
for solid material, and p = density exponent.
References / Bibliography
1. Bendsoe, M.P., Optimization of Structural Topology, Shape, and Material, Springer, (1995).
Notes . .
Notes . .