Topology optimization of structures subject to self-weight loading under stress constraints

PurposeThe purpose of this paper is to present an approach for structural weight minimization under von Mises stress constraints and self-weight loading based on the topological derivative method. Although self-weight loading topology has been the subject of intense research, mainly compliance minimization has been addressed.Design/methodology/approachThe resulting minimization problem is solved with the help of the topological derivative method, which allows the development of efficient and robust topology optimization algorithms. Then, the derived result is used together with a level-set domain representation method to devise a topology design algorithm.FindingsNumerical examples are presented, showing the effectiveness of the proposed approach in solving a structural topology optimization problem under self-weight loading and stress constraint. When the self-weight loading is dominant, the presence of the regularizing term in the formulation is crucial for the design process.Originality/valueThe novelty of this research work lies in the use of a regularized formulation to deal with the presence of the self-weight loading combined with a penalization function to treat the von Mises stress constraint.

A topology optimization algorithm based on topological derivative and level-set function for designing phononic crystals

PurposeThis 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.Design/methodology/approachThe 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.FindingsThe 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.Originality/valueThe 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.

Brittle fracture on plates governed by topological derivatives

PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.Design/methodology/approachThe Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.FindingsNumerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.Originality/valueThis is the first work concerning brittle fracture modeling of plate structures based on the topological derivative method.

Experimental validation of a topological derivative-based crack growth control method using digital image correlation

PurposeThe purpose of this paper is to experimentally validate the crack growth control based on the topological derivative of the famous Rice's integral.Design/methodology/approachSingle edge notch tensile specimens with two configurations were tested. Displacement fields near notch were experimentally obtained using the digital image correlation method. These displacements were used to verify the minimization of the associated shape functional, which is defined in terms of the Rice's integral, when a set of controls (holes) positioned according to the topological derivative information, is inserted. Based on the Griffth's energy criterion, this minimization represents an improvement in the fracture toughness of cracked bodies.FindingsThe experimental tests confirmed that a decrease around 27% in the value of the associated shape functional can be obtained following this approach. Therefore, the results allow us to conclude that the predictive methodology for crack growth control based on the topological derivative is feasible.Originality/valueThis is the first work concerning experimental validation of crack growth control based on the topological derivative method.

Topological derivatives via one-sided derivative of parametrized minima and minimax

PurposeThe object of the paper is to illustrate how to obtain the topological derivative as a semidifferential in a general and practical mathematical setting for d-dimensional perturbations of a bounded open domain in the n-dimensional Euclidean space.Design/methodology/approachThe underlying methodology uses mathematical notions and powerful tools with ready to check assumptions and ready to use formulas via theorems on the one-sided derivative of parametrized minima and minimax.FindingsThe theory and the examples indicate that the methodology applies to a wide range of problems: (1) compliance and (2) state constrained objective functions where the coupled state/adjoint state equations appear without a posteriori substitution of the adjoint state.Research limitations/implicationsDirect approach that considerably simplifies the analysis and computations.Originality/valueIt was known that the shape derivative was a differential. But the topological derivative is only a semidifferential, that is, a one-sided directional derivative, which is not linear with respect to the direction, and the directions are d-dimensional bounded measures.