Metrics of curves in shape optimization and analysis - Andrea Carlo ...

2. Calculate the directional derivative:∫DE(c; h) = ∇φ(c) · h + φ(c)(D s h · D s c) dsc∫= ∇φ(c) · h − ( ∇φ(c) · D s c ) (h · D s c) − φ(c)(h · D ss c) dsc∫= h · ((∇φ(c)· N)N − φ(c)κN ) ds . (2.3)3. Deduce the “gradient”:4. Write the gradient flowc∇E = −φκN + (∇φ · N)N .∂c= φκN − (∇φ · N)N .∂t(Note that the flow of geometric energies moves only in orthogonal direction w.r.tthe curve — this phenomenon will be explained in in Section 11.10.)2.3.3 Implicit assumption of H 0 inner productWe have made a critical assumption in going from the directional derivative∫DE(c; h) = h(s) · v(s) dscto deducing that the gradient of E is ∇E(c) = v. Namely, the definition of thegradient 4 is based on the following equality∫〈h(s), ∇E〉 = h(s) · v(s) ds,c }{{} }{{}h 1=h h 2=∇Ethat needs an inner-product structure.This implies that we have been presuming all along that curves are equippedwith a H 0 -type inner-product defined as follows∫〈 〉h1 , h 2 := hH 0 1 (s) · h 2 (s) ds . (2.4)2.4 Problems & goalscWe will now briefly list some examples that show some limits of the usual activecontour method.4 The precise definition of what the gradient is is in section 3.7.14