Invisibility problem in acoustics, electromagnetism and heat transfer. Inverse design method

Abstract

Abstract. Two approaches (direct design and inverse design methods) for solving problems of designing devices providing invisibility of material bodies of detection using different physical fields -electromagnetic, acoustic and static are discussed. The second method is applied for solving problems of designing cloaking devices for the 3D stationary thermal scattering model. Based on this method the design problems under study are reduced to respective control problems. The material parameters (radial and tangential heat conductivities) of the inhomogeneous anisotropic medium filling the thermal cloak and the density of auxiliary heat sources play the role of controls. A unique solvability of direct thermal scattering problem in the Sobolev space is proved and the new estimates of solutions are established. Using these results, the solvability of control problem is proved and the optimality system is derived. Based on analysis of optimality system, the stability estimates of optimal solutions are established and numerical algorithms for solving particular thermal cloaking problem are proposed.

Optimization method in 2D magnetic cloaking problems

“…Приведенное выше предположение (i) позволяет существенно упростить исследование сформулированной выше задачи магнитного рассеяния, записав уравнение для Φ 0 в (2) в полярных координатах r, ϕ. Действительно, напомним сначала, что в полярных координатах gradΦ 0 и произведение диагонального (в полярных координатах) тензора µ = diag(µ r , µ ϕ ) на вектор gradΦ 0 выражаются формулами (см, например, [9, c.101]) (9) gradΦ 0 = ∂Φ 0 ∂r e r + 1 r…”

“…Сущность этого подхода заключается в замене исходной обратной задачи задачей минимизации определенного функционала качества, адекватно отвечающего поставленной обратной задаче. Данный метод, на который в задачах маскировки ссылаются как на метод обратного дизайна [9,16], применялся в работах [17,18,19], посвященных численному анализу 2D задач дизайна слоистых маскировочных оболочек, и в [20,21,22,23,25,24] при теоретическом исследовании задач электромагнитной либо акустической маскировки. Работы [26,27] посвящены изучению проблемы невидимости в томографии.…”

On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

“…We remark that Assumption (D2) on the dual cone stated in the (19) form is advantageous in comparison to the condition on Feasible Set 2:…”

“…This means that the divergence operator in (63) is surjective; thus, Assumptions (G1), (G2) and Assertions (i)-(iii) of Proposition 2 hold true. In the case of equality constraint, the positive cone turns into the whole space K (Ω t ) = L 2 (Ω t ) according to (19). OS (20) and (21) implies the solution pair…”

“…The constraint operator (see (3)) may become: the trace operator under contact conditions [1][2][3], the jump operator for cracks and anticracks [4][5][6], the gradient operator in plasticity [7], the divergence operator under incompressibility conditions [8][9][10], a state-constraint in mathematical programs with equilibrium constraints [11,12], and the like. The constraint problems are related to parameter identification problems (see the theory in References [13][14][15] and application to biological systems in Reference [16]), to inverse problems by the mean of observation data used in mathematical physics [17,18] and in acoustics [19][20][21], to overdetermined and free-boundary problems [22,23]. As an application, in the current paper we focus on the incompressible Brinkman flow problem under a divergence-free condition (see the related modeling of porous medium in References [24,25], well-posedness analysis in Reference [26], and fluid-porous coupling with numerics in References [27,28]).…”