Convex Inclusions Research Articles

Asymptotic analysis of the stress concentration between two adjacent stiff inclusions in all dimensions

In the region between two closely located stiff inclusions, the stress, which is the gradient of a solution to the Lamé system with partially infinite coefficients, may become arbitrarily large as the distance between interfacial boundaries of inclusions tends to zero. The primary aim of this paper is to give a sharp description in terms of the asymptotic behavior of the stress concentration, as the distance between interfacial boundaries of inclusions goes to zero. For that purpose, we capture all the blow‐up factor matrices, whose elements comprise of some certain integrals of the solutions in the case when two inclusions are touching. Then, we are able to establish the asymptotic formulas of the stress concentration in the presence of two close‐to‐touching ‐convex inclusions in all dimensions. Furthermore, an example of curvilinear squares with rounded‐off angles is also presented for future application in numerical computations and simulations.

Gradient estimates for the insulated conductivity problem: The non-umbilical case

We study the insulated conductivity problem with inclusions embedded in a bounded domain in Rn, for n≥3. The gradient of solutions may blow up as ε, the distance between the inclusions, approaches to 0. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on Sn−2.

Estimates for stress concentration between two adjacent rigid inclusions in Stokes flow

In this paper, we establish the estimates for the gradient and the second-order partial derivatives for the Stokes flow in the presence of two closely located strictly convex inclusions in dimension three. Moreover, the blow-up rate of the gradient is showed to be optimal by a pointwise upper bound and a lower bound in the narrowest region. We also show the optimal blow-up rate of Cauchy stress tensor. In dimensions greater than three, the upper bounds of the gradient are established. These results answer the questions raised in [25].

A versatile and highly efficient algorithm to generate representative microstructures for heterogeneous materials

In this study, a novel approach based on the maximum penetration-biased (MPB) algorithm is proposed to rapidly generate representative volume elements (RVEs) for advanced composites. This method is capable of handling various microstructural features such as nonuniform distribution of high-volume fractions, concave or convex inclusions and the control of inter-inclusion distances. Statistical functions from short-range to long-range demonstrate that the resulting microstructures are complete spatial random, and the microstructures observed in realistic composites can be reproduced by the proposed algorithm. Accurate prediction for the elastic properties and transverse isotropy of the generated microstructures with varying inclusion shapes using homogenization method further verify the validity of the developed method. A nonlinear damage study compared to the circular inclusion reveals that the capsule shape leads to a reduction of strength of unidirectional fiber reinforced composites under transverse tension or compression while the lobular inclusion presents a deterioration of strength under tension but an enhancement of strength in compression. The MPB algorithm provides an effective tool for micromechanical assessment, quantitative research and data-driven studies such as machine learning of composites.

Optimal estimates for transmission problems including relative conductivities with different signs

We study the gradient and higher order derivative estimates for the transmission problem in the presence of closely located inclusions. We show that in two dimensions, when relative conductivities of circular inclusions have different signs, the gradient and higher order derivatives are bounded independent of ε, the distance between the inclusions. We also show that for general smooth strictly convex inclusions, when one inclusion is an insulator and the other one is a perfect conductor, the derivatives of any order is bounded independent of ε in any dimensions n≥2.

Gradient estimates for the insulated conductivity problem: The case of m-convex inclusions

We consider an insulated conductivity model with two neighboring inclusions of m-convex shapes in Rd when m ≥ 2 and d ≥ 3. We establish pointwise gradient estimates for the insulated conductivity problem and capture the gradient blow-up rate of order ɛ−1/m+β with β=[−(d+m−3)+(d+m−3)2+4(d−2)]/(2m)∈(0,1/m) as the distance ɛ between these two insulators tends to zero. In particular, the optimality of the blow-up rate is also demonstrated for a class of axisymmetric m-convex inclusions.

Optimal estimates for the conductivity problem close to the boundary in dimension greater than two

<p style='text-indent:20px;'>In high-contrast composite materials, the electric field (or stress field) may blow up in the narrow region when a convex inclusion is closely to the boundary of the matrix domain. The gradient of solutions depends on the parameters–the distance between the inclusions and boundaries. An optimal upper bound was obtained for convex inclusions of arbitrary shape in dimension greater than two. Our results extend that in [<xref ref-type="bibr" rid="b3">3</xref>], where the disk inclusion case in 2D is considered.</p>

An extended Flaherty-Keller formula for an elastic composite with densely packed convex inclusions

In this paper, we are concerned with the effective elastic property of a two-phase high-contrast periodic composite with densely packed inclusions. The equations of linear elasticity are assumed. We first give a novel proof of the Flaherty-Keller formula for elliptic inclusions, which improves a recent result of Kang and Yu (Calc Var Partial Differ Equ 59(1):3, 2020). We construct a family of auxiliary functions consisting of the Keller-type functions and additional correctors which depend on the coefficients of Lamé system and the geometry of inclusions, to capture the full singular term of the gradient. On the other hand, this method allows us to deal with the inclusions of arbitrary shape, even with zero curvature. An extended Flaherty-Keller formula is proved for m-convex inclusions, $$m>2$$ , curvilinear squares with round off angles, which minimize the elastic energy under the same volume fraction of hard inclusions.

An Extreme Problem for a Volterra Type Integral Inclusion

In the work, we have studied the dependencies of the solutions to integral in-clusions from perturbation and investigated an extremal problem for integral inclusions. We obtained necessary and sufficient minimum conditions for ex-tremal problems of Volterra type convex inclusions. We also studied a non-convex extremal problem for the Volterra type inclusion. We obtained a high order necessary condition in the extremal problem for the Volterra type inclu-sion.

Closed Fejer cycles for inconsistent systems of convex inequalities

In this paper we consider certain questions, concerning the construction of generalized discrete solutions to inconsistent systems of convex inequalities based on the notion of a closed cycle. This idea is close to the procedure of the successive (cyclic) projection used in the analysis of an inconsistent system of convex inclusions. Let us adduce the necessary conceptual apparatus, as well as denotations and terms:

Weak sharp minima revisited, Part III: error bounds for differentiable convex inclusions

The notion of weak sharp minima unifies a number of important ideas in optimization. Part I of this work provides the foundation for the theory of weak sharp minima in the infinite-dimensional setting. Part II discusses applications of these results to linear regularity and error bounds for nondifferentiable convex inequalities. This work applies the results of Part I to error bounds for differentiable convex inclusions. A number of standard constraint qualifications for such inclusions are also examined.

Error bounds for set inclusions

A variant of Robinson-Ursescu Theorem is given in normed spaces. Several error bound theorems for convex inclusions are proved and in particular a positive answer to Li and Singer's conjecture is given under weaker assumption than the assumption required in their conjecture. Perturbation error bounds are also studied. As applications, we study error bounds for convex inequality systems.

On stability in local controllability problems for discrete nonlinear systems*

Sufficient conditions for various types of local controllability for a general class of discrete nonlinear inclusions have been given in Ref.15. In this paper we prove that these conditions also imply local controllability of perturbed inclusion as well. The stability of controllability for discrete convex Inclusions is also considered

Second order necessary and sufficient conditions for convex composite NDO

In convex composite NDO one studies the problem of minimizing functions of the formF:=h ○f whereh:ℝ m → ℝ is a finite valued convex function andf:ℝ n → ℝ m is continuously differentiable. This problem model has a wide range of application in mathematical programming since many important problem classes can be cast within its framework, e.g. convex inclusions, minimax problems, and penalty methods for constrained optimization. In the present work we extend the second order theory developed by A.D. Ioffe in [11, 12, 13] for the case in whichh is sublinear, to arbitrary finite valued convex functionsh. Moreover, a discussion of the second order regularity conditions is given that illuminates their essentially geometric nature.

Series in locally convex spaces and inclusions between FK spaces

Singer [ 10 ] defined a series Σ x k in a Banach space X to be weakly p -unconditionally Cauchy if and only if Σλ k x k converges in X for all λ∊l p , where 1 p 0 Singer characterized such series as those for which where 1/ p + 1/ q = 1 and X ′ is the dual space of X .