Shape optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints. In many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable domain.
Topology optimization is, in addition, concerned with the number of connected components/boundaries belonging to the domain. Such methods are needed since typically shape optimization methods work in a subset of allowable shapes which have fixed topological properties, such as having a fixed number of holes in them. Topological optimization techniques can then help work around the limitations of pure shape optimization.
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Famous quotes containing the word shape:
“Strange that so few ever come to the woods to see how the pine lives and grows and spires, lifting its evergreen arms to the light,—to see its perfect success; but most are content to behold it in the shape of many broad boards brought to market, and deem that its true success! But the pine is no more lumber than man is, and to be made into boards and houses is no more its true and highest use than the truest use of a man is to be cut down and made into manure.”
—Henry David Thoreau (1817–1862)