Packing Argument Research Articles

Combined mean-field and full-field homogenization of porous elasto-plastic materials and composites under arbitrary stress triaxialities

A predictive micromechanical approach is proposed for porous materials where the unreinforced or reinforced matrix phase is elasto-plastic with hardening. The cavities can be spheres, long or short cylinders. The approach is based on an alternative microstructure made of elasto-plastic inhomogeneities embedded in a homogenized porous matrix phase, and the volume fractions are determined from a maximum packing argument. The effective properties of single hollow solids are computed with an energy-based approach coupled with full-field finite element (FE) analyses. Next, the alternative microstructures are homogenized with mean-field (MF) models. For reinforced porous materials, a two-level method is adopted, where the proposed approach is used at the lower level to obtain a fictitious homogenized matrix, in which reinforcements are embedded at the upper level. The present work is restricted to monotonic and proportional loadings and to the secant formulation of isotropic or transversely isotropic elasto-plasticity. However, no constitutive models are supposed or identified. The predictions were verified against reference full-field FE results on the actual microstructures in 3D and 2D plane strain or stress, for arbitrary stress triaxialities, and good agreement was found in all cases.

Relationship of the atomic dynamics and excess volume of a copper rich Cu76Ti24 alloy melt

We investigate the atomic dynamics of a Cu-rich Cu76Ti24 alloy between 1223 K and 1453 K using the quasi-elastic neutron scattering technique. The obtained mean Cu/Ti self-diffusion coefficient D exhibits an Arrhenius-like temperature dependence with an activation energy of 0.46�0.02 eV per atom. Compared with those of the pure elements at a given temperature, D is slower than that for pure Cu, but similar to pure Ti. This slowing down of the liquid dynamics upon alloying has been observed for most binary alloy systems. However, in this study, the packing fraction of the Cu-Ti alloy is lower than that of the pure elements, which can be explained by the positive excess volume of the Cu-Ti system. Therefore, the compositional dependence of the atomic dynamics cannot be linked to a macroscopic packing argument. Only, the atomic dynamics is not correlated with an increase of the average packing fraction of the melt.

Microscopic topology of entangled polymeric liquids

We extensively study the polymer chain packings of various entangled polymer systems. Surprisingly, the number of entanglement strands per cubed tube diameter does not obey the overlap criterion for entanglement. By analyzing microscopic topology of entangled polymeric liquids, an ideal polymer-model-independent method to estimate the entanglement length as a complement of primitive path analyses is proposed in this paper. Using this method, the discrepancy observed in entanglement lengths of a short chain branching entangled system using different approaches is solved, and the results of the system given by different approaches coincide with the packing argument perfectly.

Runlength-Limited Sequences and Shift-Correcting Codes: Asymptotic Analysis

This work is motivated by the problem of error correction in bit-shift channels with the so-called $ (d,k) $ input constraints (where successive $ 1 $'s are required to be separated by at least $ d $ and at most $ k $ zeros, $ 0 \leq d < k \leq \infty $). Bounds on the size of optimal $ (d,k) $-constrained codes correcting a fixed number of bit-shifts are derived, with a focus on their asymptotic behavior in the large block-length limit. The upper bound is obtained by a packing argument, while the lower bound follows from a construction based on a family of integer lattices. Several properties of $ (d, k) $-constrained sequences that may be of independent interest are established as well; in particular, the exponential growth-rate of the number of $ (d, k) $-constrained constant-weight sequences is characterized. The results are relevant for magnetic and optical information storage systems, reader-to-tag RFID channels, and other communication models where bit-shift errors are dominant and where $ (d, k) $-constrained sequences are used for modulation.

Porous plasticity: Predictive second moment homogenization models coupled with Gurson's single cavity stress-strain solution

An original and rather simple micromechanical approach is proposed for the case of a rigid plastic von Mises matrix containing multiple spherical cavities and subjected to linear velocities on the boundary. The approach couples mean-field homogenization with Gurson's single cavity stress-strain solution. The actual porous microstructure is modeled alternatively as spherical rigid plastic inclusions in a homogenized Gurson matrix phase and the volume fractions are determined from a maximum packing argument. The alternative microstructures are homogenized with the Generalized self-consistent (GSC) and Mori-Tanaka (MT) models which are formulated with the secant method and second statistical moments of per phase strain fields. Numerical predictions of effective yield envelopes in the plane of hydrostatic and von Mises stresses are verified against full-field finite element (FE) and analytical results for a large range of porosities. No fitting or empirical parameters are used in GSC or MT. It is found that the agreement between GSC and FE predictions is excellent, except when both the porosity and the stress triaxiality are small. The evolution of porosity as a function of the mean accumulated plastic strain in the solid matrix phase of the original microstructure is computed and illustrated for two effective stress triaxiality ratios.

Packing Curves on Surfaces with Few Intersections

Abstract Przytycki has shown that the size $\mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as $|\chi(S)|^{k^{2}+k+1}$. In this article, we narrow Przytycki’s bounds, obtaining \[ \mathcal{N}_{k}(S) =O \left( \frac{ |\chi|^{3k}}{ ( \log |\chi| )^2 } \right)\!. \] In particular, the size of a maximal 1-system grows sub-cubically in $|\chi(S)|$. The proof uses a circle packing argument of Aougab and Souto and a bound for the number of curves of length at most $L$ on a hyperbolic surface. When the genus $g$ is fixed and the number of punctures $n$ grows, we use a different argument to show \[ \mathcal{N}_{k}(S) \leq O(n^{2k+2}). \] This may be improved when $k=2$, and we obtain the sharp estimate $\mathcal{N}_2(S)=\Theta(n^3)$.

Multiscale Modeling of the Effects of Salt and Perfume Raw Materials on the Rheological Properties of Commercial Threadlike Micellar Solutions

We link micellar structures to their rheological properties for two surfactant body-wash formulations at various concentrations of salts and perfume raw materials (PRMs) using molecular simulations and micellar-scale modeling, as well as traditional surfactant packing arguments. The two body washes, namely, BW-1EO and BW-3EO, are composed of sodium lauryl ethylene glycol ether sulfate (SLEnS, where n is the average number of ethylene glycol repeat units), cocamidopropyl betaine (CAPB), ACCORD (which is a mixture of six PRMs), and NaCl salt. BW-3EO is an SLE3S-based body wash, whereas BW-1EO is an SLE1S-based body wash. Additional PRMs are also added into the body washes. The effects of temperature, salt, and added PRMs on micellar lengths, breakage times, end-cap free energies, and other properties are obtained from fits of the rheological data to predictions of the "Pointer Algorithm" [ Zou , W. ; Larson , R.G. J. Rheol. 2014 , 58 , 1 - 41 ], which is a simulation method based on the Cates model of micellar dynamics. Changes in these micellar properties are interpreted using the Israelachvili surfactant packing argument. From coarse-grained molecular simulations, we infer how salt modifies the micellar properties by changing the packing between the surfactant head groups, with the micellar radius remaining nearly constant. PRMs do so by partitioning to different locations within the micelles according to their octanol/water partition coefficient POW and chemical structures, adjusting the packing of the head and/or tail groups, and by changing the micelle radius, in the case of a large hydrophobic PRM. We find that relatively hydrophilic PRMs with log POW < 2 partition primarily to the head group region and shrink micellar length, decreasing viscosity substantially, whereas more hydrophobic PRMs, with log POW between 2 and 4, mix with the hydrophobic surfactant tails within the micellar core and slightly enhance the viscosity and micelle length, which is consistent with the packing argument. Large and very hydrophobic PRMs, with log POW > 4, are isolated deep inside the micelle, separating from the tails and swelling the radius of the micelle, leading to shorter micelles and much lower viscosities, leading eventually to swollen-droplet micelles.

A Tight Upper Bound on Channel Capacity for Visible Light Communications

Since the optical wireless channel in visible light communication (VLC) systems is subject to the non-negativity of the signal and the average optical power, the classic Shannon channel capacity formula is not applicable to VLC systems. To derive a simple closed-form upper bound on channel capacity, sphere packing argument method has been applied previously. However, there is an obvious gap between the existing sphere-packing upper bounds and the lower bounds at high optical signal-to-noise-ratios (OSNRs), which is mainly caused by the inaccurate mathematical approximation of the intrinsic volumes of the simplex. In this letter, a tight sphere-packing upper bound is derived with a new approximation method. Numerical results demonstrate that compared to the existing sphere-packing upper bounds, our proposed upper bound is tighter at high OSNRs.

An Improvement to Levenshtein's Upper Bound on the Cardinality of Deletion Correcting Codes

We consider deletion correcting codes over a q-ary alphabet. It is well known that any code capable of correcting s deletions can also correct any combination of s total insertions and deletions. To obtain asymptotic upper bounds on code size, we apply a packing argument to channels that perform different mixtures of insertions and deletions. Even though the set of codes is identical for all of these channels, the bounds that we obtain vary. Prior to this paper, only the bounds corresponding to the all-insertion case and the all-deletion case were known. We recover these as special cases. The bound from the all-deletion case, due to Levenshtein, has been the best known for more than 45 years. Our generalized bound is better than Levenshtein's bound whenever the number of deletions to be corrected is larger than the alphabet size.

Tight Bounds on Channel Capacity for Dimmable Visible Light Communications

In this paper, the upper and lower bounds for the channel capacity of dimmable visible light communications(VLC) systems are investigated. Because the information is modulated into the instantaneous optical intensity, the transmitted optical intensity is represented by a nonnegative input that is corrupted by an additive white Gaussian noise. Considering the illumination support in a VLC system, the transmitted optical intensity signal must satisfy the illumination constraint, i.e., the average transmitted optical intensity is constrained by a target illumination intensity, which is determined by the nominal optical intensity of the light source devices and dimming target. An upper bound on the channel capacity is derived based on the signal space geometry via a sphere packing argument. A lower bound on the channel capacity is derived by maximizing the mutual information between the channel input and output. Both the upper and lower bounds are presented in closed forms. Furthermore, a closed-form optimal intensity distribution of the transmitted optical signal is obtained. The numerical results show that the presented bounds are very tight at the application zone of dimmable VLC links.

Capacity Bounds for Wireless Optical Intensity Channels With Gaussian Noise

Lower and upper bounds on the capacity of wireless optical intensity pulse amplitude modulation channels under nonnegativity and average optical power constraints are derived. A lower bound is derived based on source entropy maximization over a family of discrete nonuniform distributions with equally spaced mass points. A closed form for the maxentropic discrete input distribution is provided. Compared to previously reported bounds, the derived lower bound is tight at both low and high signal-to-noise ratios (SNRs). In addition, a closed-form upper bound is derived based on signal space geometry via a sphere packing argument. The proposed bound is tight at low SNRs and incurs a small gap to the channel capacity at high SNRs. The derived bounds asymptotically describe the optical intensity channel capacity at low SNRs, where a majority of such links operate.

Two-Dimensional Array Coloring With Many Colors

Given an mtimesn array and k distinct colors with 2lesmlesn and 2lesk<mn, we consider the problem of marking each location in the array using one of the k given colors such that any two locations in the array marked by the same color are separated as much as possible. This problem is related to two-dimensional (2-D) interleaving schemes for correcting cluster errors where the goal is to rearrange the codeword symbols so that an arbitrarily shaped error cluster of size t can be corrected for the largest possible value of t. In a recent paper, the authors have shown that, for the case 2lesklesmn/2, the maximum coloring distance is given by lfloorradic2krfloor if kleslceilm 2/2rceil, and by m+lfloor(k-lceilm 2/2rceil)/mrfloor if lceilm 2/2rceillesklesmn/2. In this work, we extend these results to the case mn/2<k<mn. We show that in such cases, the maximum coloring distance is given by to m+lfloor(k-lceilm 2/2rceil)/mrfloor if mn/2<k<mn-lfloorm 2/2rfloor, and by m+n-lceilradic2(mn-k)rceil if mn-lfloorm 2/2rfloorlesk<mn. In particular, we generalize the partial sphere packing argument to derive the new bound and consequently propose a new type of construction achieving optimal coloring for the case kgesmn-lceilm 2/2rceil.

Entanglement-Assisted Capacity of Quantum Multiple-Access Channels

We find a regularized formula for the entanglement-assisted (EA) capacity region for quantum multiple-access channels (QMAC). We illustrate the capacity region calculation with the example of the collective phase-flip channel which admits a single-letter characterization. On the way, we provide a first-principles proof of the EA coding theorem based on a packing argument. We observe that the Holevo-Schumacher-Westmoreland theorem may be obtained from a modification of our EA protocol. We remark on the existence of a family hierarchy of protocols for multiparty scenarios with a single receiver, in analogy to the two-party case. In this way, we relate several previous results regarding QMACs.

A note on stability results for scattered data interpolation on Euclidean spheres

The present work considers the interpolation of the scattered data on the d-sphere by spherical polynomials. We prove bounds on the conditioning of the problem which rely only on the separation distance of the sampling nodes and on the degree of polynomials being used. To this end, we establish a packing argument for well separated sampling nodes and construct strongly localized polynomials on spheres. Numerical results illustrate our theoretical findings.

Intermolecular Interactions between Dendrimer Molecules in Solution Studied by Small-Angle Neutron Scattering

Small-angle neutron scattering has been applied for structural investigations of amine-terminated poly(propylene imine) dendrimer (Astramol) solutions in D2O as a function of concentration and acidity. The forward scattering, dΣ/dΩ(Q = 0), decreases progressively by increasing dendrimer concentration due to interdendrimer interference, giving rise to a well-defined interference peak. Upon increasing dendrimer concentration, a shift is observed in the peak position, Qmax, with the empirical scaling relation Qmax ∼ φ0.55±0.05, which is significantly larger than that of a simple packing argument Qmax ∼ φ1/3. At high Q values, all data sets collapse to a single curve, indicating that the internal structure of the dendrimers are unaffected by interactions between neighboring molecules. Upon addition of acid, the dendrimers become charged, giving rise to a pronounced interference peak. These Coulomb interactions can be screened by addition of either excess acid or salts.

Lower Bounds for Transversal Covers

A transversal cover is a set of gk points in k disjoint groups of size g and a collection of b transversal subsets, called blocks, such that any pair of points not contained in the same group appears in at least one block. A central question is to determine, for given g, the minimum possible b for fixed k, or, alternatively, the maximum k for fixed b. The case g=2 was investigated and completely solved by Sperner sperner:28, Renyi renyi:71, Katona katona:73, and Kleitman and Spencer kleitman:73. For arbitrary g, asymptotic results are known but little is understood for small values of k. Constructions exist but these only produce upper bounds on b. The present article is concerned with lower bounds on b. We develop three general lower bounds on b for fixedg and k. The first one is proved using one of the principal constructions brett:97a, the second comes from the study of intersecting set-systems, and the third is shown by a set packing argument. In addition, we investigate upper bounds on k for small fixed b. This proves useful to reduce or eliminate the gap between lower and upper bounds on b for some transversal covers with small k.

A short proof of a classical covering lemma

We give a short proof of a general covering lemma for locally compact groups, using an elementary packing argument.