  The work explores a specific scenario for structural computational
optimization based on the following elements: (a) a relaxed optimization
setting considering the ersatz (bi-material) approximation, (b) a treatment
based on a nonsmoothed characteristic function field as a topological design
variable, (c) the consistent derivation of a relaxed topological derivative
whose determination is simple, general and efficient, (d) formulation of the
overall increasing cost function topological sensitivity as a suitable
optimality criterion, and (e) consideration of a pseudo-time framework for the
problem solution, ruled by the problem constraint evolution. In this setting,
it is shown that the optimization problem can be analytically solved in a
variational framework, leading to, nonlinear, closed-form algebraic solutions
for the characteristic function, which are then solved, in every time-step, via
fixed point methods based on a pseudo-energy cutting algorithm combined with
the exact fulfillment of the constraint, at every iteration of the non-linear
algorithm, via a bisection method. The issue of the ill-posedness (mesh
dependency) of the topological solution, is then easily solved via a Laplacian
smoothing of that pseudo-energy. In the aforementioned context, a number of
(3D) topological structural optimization benchmarks are solved, and the
solutions obtained with the explored closed-form solution method, are analyzed,
and compared, with their solution through an alternative level set method.
Although the obtained results, in terms of the cost function and topology
designs, are very similar in both methods, the associated computational cost is
about five times smaller in the closedform solution method this possibly being
one of its advantages.
