Symmetry in the composite plate problem

In this paper we deal with the composite plate problem, namely the following optimization eigenvalue problem $$\inf_{\rho \in \mathrm{P}} \inf_{u \in \mathcal{W}\setminus\{0\}} \frac{\int_{\Omega}(\Delta u)^2}{\int_{\Omega} \rho u^2},$$ where $\mathrm{P}$ is a class of admissible densities, $\mathcal{W}= H^{2}_{0}(\Omega)$ for Dirichlet boundary conditions and $\mathcal W= H^2(\Omega) \cap H^1_{0}(\Omega)$ for Navier boundary conditions. The associated Euler-Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator $\Delta^2$. In the spirit of [10], we study qualitative properties of the optimal pairs $(u,\rho)$. In particular, we prove existence and regularity and we find the explicit expression of $\rho$. When $\Omega$ is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of $u$ and radial symmetry of both $u$ and $\rho$.Comment: 26 page

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors \BC_1>0 and \BC_2=0 comprising the set U=\{\BC_1,\BC_2\} and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor \BC_2=0 corresponds to a material which is void. (For technical reasons \BC_2 is actually taken to be nonzero and we take the limit \BC_2\to 0). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.Comment: 39 pages, 11 figure

Extremal Spectral Gaps for Periodic Schr\"odinger Operators

The spectrum of a Schr\"odinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the m-th spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the m-th gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices.Comment: 34 pages, 14 figure

On the stability of a nonlinear nonhomogeneous multiply hinged beam

The paper deals with a nonlinear evolution equation describing the dynamics of a nonhomogeneous multiply hinged beam, subject to a nonlocal restoring force of displacement type. First, a spectral analysis for the associated weighted stationary problem is performed, providing a complete system of eigenfunctions. Then, a linear stability analysis for bimodal solutions of the evolution problem is carried out, with the final goal of suggesting optimal choices of the density and of the position of the internal hinged points in order to improve the stability of the beam. The analysis exploits both analytical and numerical methods; the main conclusion of the investigation is that nonhomogeneous density functions improve the stability of the structure

Mini-Workshop: Nonlocal Analysis and the Geometry of Embeddings (hybrid meeting)

Both self-avoidance and self-contact of geometric objects can be modeled using repulsive energies that separate isotopy classes. Giving rise to nonlocal operators, they are interesting objects in their own right. Moreover, their analytical structure allows for devising numerical schemes enjoying robust features such as energy stability. This workshop aimed at discussing recent trends in this matter, including potential applications to modeling

Lightweight Vehicle Structures that Absorb and Direct Destructive Energy Away from the Occupants

One of the main thrusts in current automotive industry is the development of occupant-centric vehicle structures that make the vehicle safe for the occupants. A design philosophy that improves vehicle survivability by absorbing and redirecting destructive energy in underbody blast events should be developed and demonstrated. On the other hand, the size and weight of vehicles are also paramount design factors for the purpose of providing faster transportation, great fuel conservation, higher payload, and higher mobility. Therefore, developing a light weight vehicle structure that provides a balance between survivability and mobility technologies for both vehicle and its occupants becomes a design challenge in this research. One of the new concepts of absorbing blast energy is to utilize the properties of “softer” structural materials in combination with a damping mechanism for absorbing the destructive energy through deformation. These “softer” materials are able to reduce the shock loads by absorbing energy through higher deformation than that of characteristic of normal high strength materials. A generic V-hull structure with five bulkheads developed by the TARDEC is used in the study as the baseline numerical model for investigating this concept. Another new concept is to utilize anisotropic material properties to guide and redirect the destructive energy away from the occupants along pre-designated energy paths. The dynamic performance of multilayer structures is of great interest because they act as a mechanism to absorb and spread the energy from a blast load in the lateral direction instead of permitting it to enter occupant space. A reduced-order modeling (ROM) approach is developed and applied in the preliminary design for studying the dynamic characterization of multilayer structures. The reliability of the ROM is validated by a spectral finite element analysis (SFEA) and a classic finite element analysis by using the commercial code Nastran. A design optimization framework for the multilayer plate is also developed and used to minimize the injury probability, along with a maximum structural weight reduction. Therefore, the goal of designing a lightweight vehicle structure that has high levels of protection in underbody blast events can be achieved in an efficient way.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135895/1/leaduwin_1.pd

Critical adsorption on curved objects

A systematic fieldtheoretic description of critical adsorption on curved objects such as spherical or rodlike colloidal particles immersed in a fluid near criticality is presented. The temperature dependence of the corresponding order parameter profiles and of the excess adsorption are calculated explicitly. Critical adsorption on elongated rods is substantially more pronounced than on spherical particles. It turns out that, within the context of critical phenomena in confined geometries, critical adsorption on a microscopically thin `needle' represents a distinct universality class of its own. Under favorable conditions the results are relevant for the flocculation of colloidal particles.Comment: 52 pages, 10 figure