A popular optimization objective is to maximize the overall stiffness of
a structure, or minimize its compliance under a given amount of mass removal.
Compliance is a measure of the overall flexibility or softness of a structure,
and is the reciprocal of stiffness. The global compliance is equal to the sum of the
element elastic or strain energies. Minimizing the global compliance, C, is equivalent
to maximizing the global stiffness. The optimization algorithm, through an iterative
process, seeks to resolve the element densities (which are the optimization design
variables) that minimize the global compliance of the structure.
[ue] is the nodal displacement vector of element e, [Ke] is the stiffness
of element e, and vector {ρ} contains the elements' relative densities
ρe.
During each optimization iteration, the target mass constraint, the global
force-stiffness equilibrium, and the required functional constraints must be
satisfied:
v
e is the element volume, and M
target is the target mass of the optimization.
[K{
ρ}] is the global stiffness matrix modulated by the vector of
relative densities, {u} is the displacement vector, and {F} is the external force
vector.
The formula above contains design response constraints such as
limits on stresses, displacements, eigenfrequencies, etc.