Shape sensitivities in a Navier‐Stokes flow with convective and grey bodies radiative thermal transfer

Abstract

SUMMARYWe study a shape optimal design problem for a forced convection flow: the steady-state Navier-Stokes equations coupled with an integro-differential thermal model. The thermal transfers are convective, diffusive and radiative with multiple reflections (model of grey bodies, radiosity equation). The inverse problem consists in minimizing a smooth cost function which depends on the solution, with respect to the domain of the equations. We prove the differentiability of the solution with respect to the domain. It follows the cost function differentiability. We introduce the adjoint state equation and obtain the exact differential of the cost function. The computational method of shape sensitivities and the optimization process are presented too.

Numerical Modeling of Electrowetting by a Shape Inverse Approach

“…As mentioned above, we need to compute the shape derivative of the cost function, dj dω (Ω), and the shape derivative of the constraint, dc dω (Ω). This is done using the optimal shape design method, see [15,6,11]; definitions of [7,12] are used. Three approaches are possible: i) we differentiate the equations then we discretize them, thus obtaining the discretized continuous gradient; ii) we discretize the equations then we differentiate them, thus obtaining the discrete gradient; iii) we differentiate directly the direct code (typically, using automatic differentiation).…”

“…The shape derivative of a real valued function is the derivative of the transported function with respect to the transformation. We refer to [15,6], and we follow the definitions and properties presented in [7,12].…”