Optimal design
Optimal design is a remarkable application of scientific computing. It is aimed at building a systematic tool for the design of devices, structures, or systems with optimal properties. The main difference with most applications of scientific computing is that topology optimization does not attempt to replicate some physical experiments on a pre-existing design, but to find the design that will perform optimally for that experiment.
The idea is not new. Already Newton in the
Principia mentioned an interest in computing the
shape of a rotationally invariant object with minimum drag
in a viscous fluid. More recently, it has gained a lot of
attention from the engineering and mathematical communities
since the 70’s. Indeed, soon after it became possible to
simulate simple experiments, engineers tried to compute
sensitivities with respect to variation of the
design, by studying the dependence of the performance of
designs on small variations of their shape. The methods
originating from this idea are usually called shape
optimization methods. Optimal Design (often also
called topology optimization) is a broader version of this
problem, where one does not only consider variations of the
boundaries of designs, but instead tries to solve an
optimization problem among all admissible designs.
Despite its apparent simplicity, the theoretical study and
numerical implementation of optimal design problem raise
very challenging issues. Indeed, most optimal designs are
by nature ill-posed. Their solutions can not be described
by classical designs, but require infinitely small patterns
like micro-laminations or micro-perforations. An intuitive
explanation of that mathematical fact can be found,
observing many steel structures.
The overall design of the Eiffel tower in Paris consists
of four main pillars meeting at the top, and two main
floors. In order to improve the stiffness to weight ratio,
Gustave Eiffel did not use massive steel columns, but opted
for a truss structure. Of course, the same logic applies to
each element of the truss structure, and the overall
stiffness to weight ratio will increases when they are
replaced with arrays of beams. The process is endless!
For practical reasons, Eiffel stopped after two steps, but
in the mathematical world, nothing prevents us from
iterating this process ad vitam eternam... The
resulting structure would not be possible to describe as a
classical design, but would require the concepts of
microstructures and all the related mathematical framework.
Literature on this subject is abundant, including several
recently published or revised monographies.
References
[All02] G. Allaire, Shape optimization by the homogenization method, Springer-Verlag, New York, 2002.[BS03] M.P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications, Springer, 2003.
[Che00] A. Cherkaev, Variational methods for structural optimization, Springer-Verlag, New York, 2000.
Links
TopOpt group at the Technical University of Denmark.Shape and Topology Optimization Group at CMAP (École Polytechnique, France)
The International Society for Structural and Multidisciplinary Optimization
The journal of Structural and Multidisciplinary Optimization