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The Adjoint State

As mentioned before, we are going to evaluate the shape gradient of functional (35). For further simplifications, we introduce an adjoint system of the form

divcr(u^a) divS^a(w^a, q^a) div4>^a\w^a. q^a)

=    0 in

:    o} ^{in fiP}'    ⁽³⁶⁾

where <r(tv^a), S^a(w^a, q^a) and (j)^a{w^a. q^a), respectively are the adjoint mechanical stress tensor, electromechanical stress tensor and electrical displacement, given by

a(u^a) = CV^su^a ,

S^a{w^a.q^a) = AV^sw^a — PVq^a ,    (37)

(j)^a{w^a.q^a) = -P^TV^sw^a - KVq^a ,

where u^a, w^a and q^a are the adjoint mechanical displacement, electromechanical displacement and electric pofentj^l, respectively.

J. Sokołowski (IECL) Topological Derivatives October 6, 2015 43 / 110