Metamaterial for elastostatic cloaking under thermal gradients

Abstract

We introduce the optimization-based method for the design of thermo-mechanical metamaterials and, particularly, for the elastostatic cloaking under thermal loads. It consists of solving a large-scale, nonlinear constrained optimization problem, where the objective function is the error in the cloaking task accomplishment. The design variables define the required metamaterial distribution. In this way, the cloaking task is accomplished, if not exactly, optimally. Further, the design variables dictate how to fabricate the metamaterial, avoiding the uncertainty of simultaneously mimicking several thermal and mechanical effective properties, as required by transformation-based metamaterial design methods.

A Brief Review on Thermal Metamaterials for Cloaking and Heat Flux Manipulation

“…Such an annular device allows the redirection of the electromagnetic field to surround the object, while also remains unperturbed outside the metamaterial region. Important achievements in electromagnetic cloaking were made for both microwave and optical frequencies, and such ideas were successfully extended to other fields, such as thermodynamics, acoustics, and mechanics . In the case of heat‐transfer problems, the key to achieve invisibility is also the heat conduction equation invariance under curvilinear transformations …”

“…Important achievements in electromagnetic cloaking were made for both microwave [26][27][28] and optical frequencies, [29][30][31][32][33][34][35][36][37][38][39] and such ideas were successfully extended to other fields, [40] such as thermodynamics, [1,[13][14][15][41][42][43][44][45][46][47][48][49] acoustics, [50][51][52][53] and mechanics. [54][55][56][57] In the case of heat-transfer problems, the key to achieve invisibility is also the heat conduction equation invariance under curvilinear transformations. [58,59]…”

Design via topology optimisation and experimental assessment of thermal metadevices for conductive heat flux shielding in transient regime

“…Nowadays, the increasing development of metamaterials and metadevices (devices made of metamaterials or that macroscopically behave as such) allows the manipulation of several physical fields in extreme and unprecedented ways. Such an outstanding potential of both metamaterials and metadevices has found noteworthy applications in optics and electromagnetism [1] , sounds and vibrations [2,3] , heat transfer [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] , mass transfer [22][23][24][25] , mechanics [26][27][28] and thermomechanics [29][30][31] . Most of the metamaterials conceived for the aforementioned applications have been designed using the standard coordinates transformation approach originally introduced by Leonhardt [32] and Pendry [33] in the context of electromagnetism, whose particular implementation in the search for invariant forms of the heat conduction equation for several conductive heat flux manipulation purposes has given rise to the concept of transformation thermodynamics (TT) [18,34,35] .…”

“…In addition to the widely known TT and SC methods, numerical optimisation is a technique that is becoming increasingly important in the design of thermal metamaterials [13,15] and metadevices [10,11,30,36,37,[42][43][44][45] . Numerical optimisation has actually been used by Peralta et al [42] to introduce a new paradigm in the design of thermal metadevices, which is the possibility of conceiving heat flux manipulation devices without a previous knowledge of the thermal property distributions dictated by TT or SC procedures.…”

Optimized lattice-based metamaterials for elastostatic cloaking

“…The elliptic equations governing elastostatics are not form-invariant under coordinate transformations [11]; thus, far fewer attempts have been made to achieve elastostatic cloaking [16,17]. Topology optimization has been pursued for elastostatic cloak design [18,19], but a formulation in the discrete setting that is capable of achieving multi-directional elastostatic cloaking devices that mitigate the influence of circular or elliptical holes has not been fully explored.

Figure 1Elastostatic cloak design in 2D lattices: ( a ) reference lattice; ( b ) lattice with a prescribed circular hole surrounded by a region in which a cloak should be designed; ( c ) lattice in which the cloak geometry is defined by a coordinate transformation of the reference lattice nodes.

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