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This work presents a topology optimization methodology for designing microarchitectures of phononic crystals. The objective is to get microstructures having, as a consequence of wave propagation phenomena in these media, bandgaps between two specified bands. An additional target is to enlarge the range of frequencies of these bandgaps.

The resulting optimization problem is solved employing an augmented Lagrangian technique based on the proximal point methods. The main primal variable of the Lagrangian function is the characteristic function determining the spatial geometrical arrangement of different phases within the unit cell of the phononic crystal. This characteristic function is defined in terms of a level-set function. Descent directions of the Lagrangian function are evaluated by using the topological derivatives of the eigenvalues obtained through the dispersion relation of the phononic crystal.

The description of the optimization algorithm is emphasized, and its intrinsic properties to attain adequate phononic crystal topologies are discussed. Particular attention is addressed to validate the analytical expressions of the topological derivative. Application examples for several cases are presented, and the numerical performance of the optimization algorithm for attaining the corresponding solutions is discussed.

The original contribution results in the description and numerical assessment of a topology optimization algorithm using the joint concepts of the level-set function and topological derivative to design phononic crystals.

Presents an implementation of continuation methods in the context of a code for flexible multibody systems analysis. These systems are characterized by the simultaneous presence of elastic deformation terms and rigid constraints. In our formulation, the latter terms are introduced by an augmented Lagrangian technique, resulting in the presence of Lagrange multipliers in the set of unknowns, together with displacement and rotation associated terms. Essential aspects for a successful implementation are discussed: e.g. the selection of an appropriate metric for computing the path following constraint, a flexible description of control parameters which accounts for conservative and nonconservative loads, imposed displacements and imposed temperatures (dilatation effects), and the inclusion of second order derivatives of rigid constraints in the Jacobian. A large set of examples is presented, with the objective of evaluating the numerical effectiveness of the implemented schemes.