Symmetries and bifurcations in optimal design

by Elena Cherkaev

The talk discusses a problem of optimal design of a structure that 
minimizes the elastic energy under the most unfavorable loading. 
The external forces on the boundary of the domain (loadings) are 
unknown; only an integral constraint for the class of admissible 
loadings is given. The extreme, most unfavorable loading is defined 
as the one that maximizes the structure's energy. This formulation leads 
to a min-max problem. If the min-max problem has a saddle point, then 
the necessary conditions of optimality determine the extreme loading 
and the optimal design. Otherwise, the problem is characterized by a 
multiplicity of extreme, equally unfavorable loadings. Continuous 
change of the constraint can change the number of extreme loadings; 
this leads to bifurcation of the solution. Symmetric solutions 
always occur if the set of constraints is invariant to a particular 
type of symmetry. The multiplicity of the most dangerous loadings 
corresponds to a symmetry of an optimal design which results in a 
structure that minimizes the energy for all of the extreme loadings. 
The paper is our joint work with Andrej Cherkaev.