By Grove K., Madsen I.H., Pedersen E.K. (eds.)

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Def This yields a transformation X → Tt (X) = x(t; X) : R2 → R2 of the plane, and it is natural to introduce the following notion of semiderivative: def dJ(Ω; V ) = lim t 0 J(Tt (Ω)) − J(Ω) . t For t ≥ 0 small, the velocity ﬁeld must be chosen in such a way that triangles are moved onto triangles and the point Mi is moved in the direction e : Mi → Mit = Mi + t e . This is achieved by choosing the following velocity ﬁeld: Vi (t, x) = bMit (x) e , where bMit is the piecewise P 1 basis function associated with node Mit : bMit (Mj ) = δij for all i, j.

5) 0 over all length L > 0 and shape function R subject to the constraint T (x, y, z) ≤ Tf (typ. 50◦ C), ∀(x, y, z) ∈ Σ1 . In this analysis we shall drop the requirement (ii) in the introduction. 2 Chapter 1. 8): x → ξ1 = x , R0 y → ξ2 = ˜= L, L R0 y(ξ1 , ξ2 , ζ) = def y , R0 z→ζ= z , L 0 ≤ ζ ≤ 1, R(Lζ) ˜ R(ζ) = , R0 σεR0 k 1/3 T (R0 ξ1 , R0 ξ2 , Lζ), ˜ 2 . 7) on S2 , ∂y 3 ˜ ˜ + Ly|y| =L ∂νA σεR0 k 1/3 qin qs R0 k qin on S3 , where ν denotes the outward normal to the boundary surface S and ∂y/∂νA is the conormal derivative to the boundary surface Σ, ∂y ˜ 2 ν1 ∂y + ν2 ∂y =L ∂νA ∂ξ1 ∂ξ2 ζ ∂y .

Following V. Caselles, R. Kimmel, and G. Sapiro [1], it is important to choose objective functionals that are intrinsically deﬁned and do not depend on an arbitrary parametrization of the boundary. 20) where the integrand is the normal derivative of I. Using the velocity method the Eulerian shape directional semiderivative is given by the expression (cf. 21) where H = ∆bΩ is the mean curvature and n = ∇bΩ is the outward unit normal. Proceeding in a formal way a necessary condition would be H ∂ ∂I + ∂n ∂n ∂I ∂n = 0 on Γ ⇒ ∆bΩ ∇I · ∇bΩ + ∇(∇I · ∇bΩ ) · ∇bΩ = 0 ⇒ ∆bΩ ∇I · ∇bΩ + D2 I∇bΩ · ∇bΩ + D2 bΩ ∇I · ∇bΩ = 0 ⇒ D2 I n · n + H ∂I = 0 on Γ.