Optimal design of aerospace structures using recent meta-heuristic algorithms

Most conventional optimization approaches are deterministic and based on the derivative information of a problem’s function. On the other hand, nature-inspired and evolution-based algorithms have a stochastic method for finding the optimal solution. They have become a more popular design and optimization tool, with a continually growing development of novel algorithms and new applications. Flexibility, easy implementation, and the capability to avoid local optima are significant advantages of these algorithms. In this study, shapes, and shape perturbation limits of a bracket part, which is used in aviation, have been set using the hypermorph tool. The objective function of the optimization problem is minimizing the volume, and the constraint is maximum von Mises stress on the structure. The grey wolf optimizer (GWO) and the moth-flame Optimizer (MFO) have been selected as nature-inspired evolution-based optimizers.

Shape Perturbation of Grushin Eigenvalues

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $${\mathbb {R}}^N$$ R N . We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues.

Structural interpretation of gravity, topography and seismicity

Global geophysical observations constrain all theories of terrestrial dynamics. We jointly interpret EGM2008 gravity, RET2014 topography and the Global Centroid Moment Tensor database from a structural point of view. We hypothesize that lateral variations of gravity and topography reflect the scale-dependent competence of rocks. We compare the spectral and spatial characteristics of the observed fields with structural predictions from the mechanics of differential grade-2 (DG-2) materials. The results indicate that these viscoelastic materials are a powerful tool for exploring dynamic processes in the Earth. We demonstrate that the known spectral range of Earth's gravity and topography can be explained by the folding, shear banding, faulting and differentiation of the crust, lithosphere and mantle. We show that the low-amplitude long-wavelength bias apparent in the disturbance field can be explained by perturbations to Earth's overall ellipsoidal shape, induced by internal slab loading of the mantle. We find by examining the directional isotropy of the data that the zonal energy in Earth's gravity disturbance is maximized about an axis coincident with the shape-perturbation minimum. The symmetry of tectonic features about this axis, extending from eastern Borneo to Brazil, and its coincidence with the equator suggest the coupling of current plate motions to true polar wander.

Refinement of the Maxwell Formula for Fiber-Reinforced Composites

The effective properties of the fiber-reinforced composite materials with fibers of square cross-section are investigated. The novel formula for the effective coefficient of thermal conductivity refining the classical Maxwell formula (MF) is derived. The methods of asymptotic homogenization, boundary shape perturbation and Schwarz alternating process are applied. It is shown that the principal term of the asymptotic expansion of the refined formula in powers of small size of inclusions coincides with the classical MF. The corrections to the MF are obtained for different values of geometrical and physical properties of the constituents of the composite material. The analytical and numerical analyses are carried out and illustrated graphically. In particular, the derived refined formula and the MF are compared for the limiting values of the geometric dimensions and physical properties of the composite. It is shown that the refined formula is applicable for the inclusions with any conductivity in the entire range of the geometric sizes of inclusions, including the limiting cases of inclusions with zero thermal conductivity and maximally large inclusions.

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

This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet–Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method.