A novel lattice structure topology optimization method with extreme anisotropic lattice properties

This research presents a lattice structure topology optimization (LSTO) method that significantly expands the design space by creating a novel candidate lattice that assesses an extremely large range of effective material properties. About the details, topology optimization is employed to design lattices with extreme directional tensile or shear properties subject to different volume fraction limits and the optimized lattices are categorized into groups according to their dominating properties. The novel candidate lattice is developed by combining the optimized elementary lattices, by picking up one from each group, and then parametrized with the elementary lattice relative densities. In this way, the LSTO design space is greatly expanded for the ever increased accessible material property range. Moreover, the effective material constitutive model of the candidate lattice subject to different elementary lattice combinations is pre-established so as to eliminate the tedious in-process repetitive homogenization. Finally, a few numerical examples and experiments are explored to validate the effectiveness of the proposed method. The superiority of the proposed method is proved through comparing with a few existing LSTO methods. The options of concurrent structural topology and lattice optimization are also explored for further enhancement of the mechanical performance.

ZIRCONIUM AND HAFNIUM DIOXIDES DOPED BY OXIDES OF YTTRIUM, SCANDIUM AND ERBIUM: NEW METHODS OF SYNTHESIS AND PROPERTIES

The results of elaborating a method for the synthesis of zirconia and hafnia doped by rare earths (yttrium, erbium and scandium) by using low-hydrated hydroxides of zirconium and hafnium as precursors are reported. The low-hydrated zirconium and hafnium hydroxides were prepared using the heterophase reaction. The physicochemical properties (including sorption properties) of low-hydrated zirconium and hafnium hydroxides ZrxHf1-x(OH)3÷1O0.5÷1.5·0.9÷2.9H2Owere studied. The scheme of thermal decomposition of low-hydrated hydroxides in air was determined. The sorption properties of the low-hydrated hafnium hydroxide are less pronounced owing to the lower amount of sorption centers, in this case, hydroxo and aqua groups. The sequence of stages of thermal decomposition of rare earth acetates was elucidated. Single-phase zirconia and hafnia doped by rare earths (yttrium, erbium and scandium) were obtained. The parameters of the elementary lattice were calculated for each phase. It was established that the stabilization of zirconium dioxide with yttria leads to the formation of interstitial solid solutions based on tetragonal zirconia (in the case of the composition Y2O3×4ZrO2 - cubic modification), with erbium oxide - interstitial solid solutions based on cubic zirconia; with scandium oxide - solid solutions based on tetragonal zirconia. The article presents the results of measuring electrical conductivity.

A Note on Palindromic $\delta$-Vectors for Certain Rational Polytopes

Let $P$ be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if $P$ is also a lattice polytope then the Ehrhart $\delta$-vector of $P$ is palindromic. Perhaps less well-known is that a similar result holds when $P$ is rational. We present an elementary lattice-point proof of this fact.

Diagonals and -maximal sets

In analogy to the definition of the halting problem K, an r.e. set A is called a diagonal iff there is a computable numbering ψ of the class of all partial recursive functions such that A = {i ∈ ω: ψi(i)↓} (in that case we say that A is the diagonal of ψ). This notion has been introduced in [10]. It captures all r.e. sets that can be constructed by diagonalization.It was shown that any nonrecursive r.e. T-degree contains a diagonal, that for any diagonal A there is an r.e. nonrecursive nondiagonal B ≤TA, and that there are r.e. degrees a such that any r.e. set from a is a diagonal.In §2 of the present paper we show that the property “A is a diagonal” is elementary lattice theoretic (e.l.t.). This result complements and generalizes previous results of Harrington and Lachlan, respectively. Harrington (see [16, XV. 1]) proved that the property of being the diagonal of some Gödelnumbering (i.e., of being creative) is e.l.t., and Lachlan [12] proved that the property of being a simple diagonal (i.e., of being simple and not hh-simple [10]) is e.l.t.In §§3 and 4 we study the position of diagonals and nondiagonals inside the lattice of r.e. sets. We concentrate on an important class of nondiagonals, generalizing the maximal and hemimaximal sets: the -maximal sets. Using the results from [6] we are able to classify the -maximal sets that can be obtained as halfs of splittings of hh-simple sets.

Diagonals and semihyperhypersimple sets

The most basic construction of an r.e. nonrecursive set—e.g. of the halting problem—proceeds by taking the diagonal of a recursive enumeration of all r.e. sets. We will answer the question of which r.e. sets can be constructed in this manner.If ψ is a computable numbering of some class of partial recursive functions, we define the diagonal of ψ to be the set Kψ ≔ {i ∈ ω ∣ ψi(i)↓}- It is well known that Kφ is creative if φ is a Gödelnumbering, and that for each creative set K there exists a Gödelnumbering φ such that K = Kφ. That is to say, the class of diagonals of Gödelnumberings is characterized as the class of creative sets. This class was shown to be elementary lattice theoretic (e.l.t.) by Harrington (see [So87, XV. 1.1]).We give a characterization of diagonals of arbitrary computable numberings of the class P1 of all partial recursive functions. To this end we introduce the notion of a semihyperhypersimple (shhs) set, which generalizes the notion of hyperhypersimplicity to nonsimple sets. It is shown that the diagonals of numberings of P1 are exactly the non-shhs sets. Then, properties of shhs sets are discussed. For example, for each nonrecursive r.e. set A there exists a nonrecursive shhs set B ≤TA, but not every r.e. T-degree contains a shhs set. These results build upon previous work by Downey and Stob [DSta].The question whether the property “shhs” is (elementary) lattice theoretic remains open. A positive answer would give both an analog of Harrington's result mentioned above, and a generalization of the well-known fact, due to Lachlan [La68], that hyperhypersimplicity is e.l.t. Therefore, we suspect that shhs sets turn out to be useful in the study of the lattice of r.e. sets.Previously, for several constructions from recursion theory the role of the underlying numbering of P1 was investigated; see Martin ([Ma66a] or [So87, V.4.1]) and Lachlan ([La75] or [Od89, III.9.2]) for Post's simple set, and Jockusch and Soare ([JS73]; cf. also [So87, XII.3.6, 3.7]) for Post's hypersimple set. However, only Gödelnumberings were considered. An explanation for the greater variety which arises when arbitrary numberings of P1 are admitted is provided by the fact that the index set of Gödelnumberings is less complex than the index set of all numberings of P1. The former is Σ1-complete; the latter is Π4-complete.