Sequential integer programming methods for stress-constrained shape and topology optimization

This presentation deals with stress-constrained shape and topology optimization problems of loadcarrying structures. The structure is approximated by a finite element model, where each element is either filled with material or void. The starting point of the optimization is a nonlinear integer programming formulation in which the binary design variable vector (

x

1

, ...,

x

n

) describes completely the shape and topology of the discretized structure:

x

j

= 1 if the j:th element is filled with material, while

x

j

= 0 if it is void.

For nonlinear optimization problems with continuous design variables instead of binary, a fundamental algorithmic approach is to generate and solve a sequence of approximating subproblems. In each subproblem, the original objective- and constraint functions are replaced by explicit, relatively simple, approximating functions which are based on calculated derivatives of the original functions at the current iteration point. This is the framework for several well-known optimization methods like sequential linear programming, method of moving asymptotes, and sequential quadratic programming.

In our considered problems, the design variables are binary and not continuous. Then there are no derivatives, but there are corresponding natural definitions of sensitivities of a function

f

i

with respect to either one binary variable

x

j

or with respect to two binary variables

x

j

and

x

l

. In a recent work we developed efficient methods to calculate these discrete first and second order sensitivities if the function

f

i

stands for e.g. the von Mises stress in the ith element. In the current work we investigate several “sequence of subproblem” approaches, based on these discrete sensitivities, in particular a sequential integer all-quadratic programming method.