scholarly journals New S-Type Bounds of M-Eigenvalues for Elasticity Tensors with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Zhuanzhou Zhang ◽  
Jun He ◽  
Yanmin Liu ◽  
Zerong Ren
Keyword(s):  
Sufficient Conditions ◽  
Upper Bounds ◽  
Elasticity Tensor ◽  
Strong Ellipticity ◽  
Extreme Eigenvalues ◽  
Elasticity Tensors ◽  
The Given

In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, S-type upper bounds for the M-eigenvalues of elasticity tensors are established. Finally, S-type sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the S-type M-eigenvalue inclusion sets.

Which Elasticity Tensors are Realizable?

1995 ◽  
Vol 117 (4) ◽  
pp. 483-493 ◽  
Author(s):  
Graeme W. Milton ◽  
Andrej V. Cherkaev
Keyword(s):  
Building Blocks ◽  
Elasticity Tensor ◽  
Isotropic Phase ◽  
Order Tensor ◽  
Two Phase ◽  
Effective Elasticity ◽  
Elasticity Tensors ◽  
Two Phases ◽  
The Given ◽  
Fourth Order Tensor

It is shown that any given positive definite fourth order tensor satisfying the usual symmetries of elasticity tensors can be realized as the effective elasticity tensor of a two-phase composite comprised of a sufficiently compliant isotropic phase and a sufficiently rigid isotropic phase configured in an suitable microstructure. The building blocks for constructing this composite are what we call extremal materials. These are composites of the two phases which are extremely stiff to a set of arbitrary given stresses and, at the same time, are extremely compliant to any orthogonal stress. An appropriately chosen subset of the extremal materials are layered together to form the composite with elasticity tensor matching the given tensor.

scholarly journals Sharp Z-eigenvalue inclusion set-based method for testing the positive definiteness of multivariate homogeneous forms

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3131-3139
Author(s):  
Gang Wang ◽  
Linxuan Sun ◽  
Yiju Wang
Keyword(s):  
Convergence Rate ◽  
Numerical Experiments ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Upper Bounds ◽  
Rank One ◽  
Inclusion Set ◽  
Even Order ◽  
Weakly Symmetric ◽  
The Given

In this paper, we establish a sharp Z-eigenvalue inclusion set for even-order real tensors by Z-identity tensor and prove that new Z-eigenvalue inclusion set is sharper than existing results. We propose some sufficient conditions for testing the positive definiteness of multivariate homogeneous forms via new Z-eigenvalue inclusion set. Further, we establish upper bounds on the Z-spectral radius of weakly symmetric nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms. The given numerical experiments show the validity of our results.

Approximating the stationary distribution of an infinite stochastic matrix

1991 ◽  
Vol 28 (1) ◽  
pp. 96-103 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Stationary Distribution ◽  
Numerical Approximation ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Finite System ◽  
Balance Equations ◽  
The Given ◽  
Steady State Balance

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

scholarly journals Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

2021 ◽  
Author(s):  
V. Calisti ◽  
A. Lebée ◽  
A. A. Novotny ◽  
J. Sokolowski
Keyword(s):  
Circular Inclusion ◽  
Elasticity Tensor ◽  
Second Order ◽  
Topological Derivative ◽  
Microstructural Changes ◽  
Topological Derivatives ◽  
Spatial Dimensions ◽  
Elasticity Tensors ◽  
Derivatives Of ◽  
Optimization Of Microstructures

AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

scholarly journals Positivity and Monotonicity Preserving Biquartic Rational Interpolation Spline Surface

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Xinru Liu ◽  
Yuanpeng Zhu ◽  
Shengjun Liu
Keyword(s):  
Sufficient Conditions ◽  
Rational Interpolation ◽  
Rectangular Domain ◽  
Interpolation Spline ◽  
Spline Surface ◽  
Surface Data ◽  
Shape Preserving Interpolation ◽  
The Given ◽  
Surface Construction ◽  
Special Case

A biquartic rational interpolation spline surface over rectangular domain is constructed in this paper, which includes the classical bicubic Coons surface as a special case. Sufficient conditions for generating shape preserving interpolation splines for positive or monotonic surface data are deduced. The given numeric experiments show our method can deal with surface construction from positive or monotonic data effectively.

Creative Design Methodology and the SIT Method

1997 ◽  
Author(s):  
Roni Horowitz ◽  
Oded Maimon
Keyword(s):  
Design Methodology ◽  
Search Strategy ◽  
Sufficient Conditions ◽  
Strategy Selection ◽  
Step Procedure ◽  
Creative Solutions ◽  
General Search ◽  
Problem Reformulation ◽  
Creative Problem ◽  
The Given

Abstract The paper presents SIT (Structured Inventive Thinking) — a structured method for enhancing creative problem solving in engineering design. The method is a three step procedure: problem reformulation, general search strategy selection, and an application of idea provoking techniques. The most innovative part of the method is the problem reformulation stage. The given problem is modified through the application of objectively defined and empirically tested set of sufficient conditions for creative solutions. The paper describes the sufficient conditions and the empirical study that demonstrates their appropriateness. Then the whole SIT mechanism is presented with illustrative examples.

scholarly journals Seismic wave propagation in anisotropic ice – Part 1: Elasticity tensor and derived quantities from ice-core properties

2015 ◽  
Vol 9 (1) ◽  
pp. 367-384 ◽  
Author(s):  
A. Diez ◽  
O. Eisen
Keyword(s):  
Seismic Waves ◽  
Dynamic Behaviour ◽  
Ice Core ◽  
Ice Cores ◽  
Seismic Wave Propagation ◽  
Elasticity Tensor ◽  
Reflection Coefficients ◽  
Seismic Velocities ◽  
Crystal Anisotropy ◽  
Elasticity Tensors

Abstract. A preferred orientation of the anisotropic ice crystals influences the viscosity of the ice bulk and the dynamic behaviour of glaciers and ice sheets. Knowledge about the distribution of crystal anisotropy is mainly provided by crystal orientation fabric (COF) data from ice cores. However, the developed anisotropic fabric influences not only the flow behaviour of ice but also the propagation of seismic waves. Two effects are important: (i) sudden changes in COF lead to englacial reflections, and (ii) the anisotropic fabric induces an angle dependency on the seismic velocities and, thus, recorded travel times. A framework is presented here to connect COF data from ice cores with the elasticity tensor to determine seismic velocities and reflection coefficients for cone and girdle fabrics. We connect the microscopic anisotropy of the crystals with the macroscopic anisotropy of the ice mass, observable with seismic methods. Elasticity tensors for different fabrics are calculated and used to investigate the influence of the anisotropic ice fabric on seismic velocities and reflection coefficients, englacially as well as for the ice–bed contact. Hence, it is possible to remotely determine the bulk ice anisotropy.

scholarly journals Acyclicity Notions for Existential Rules and Their Application to Query Answering in Ontologies

2013 ◽  
Vol 47 ◽  
pp. 741-808 ◽  
Author(s):  
B. Cuenca Grau ◽  
I. Horrocks ◽  
M. Krötzsch ◽  
C. Kupke ◽  
D. Magka ◽  
...  
Keyword(s):  
Knowledge Representation ◽  
Sufficient Conditions ◽  
Query Answering ◽  
Conjunctive Queries ◽  
Practical Solution ◽  
Ontology Language ◽  
Lower Complexity ◽  
The Given ◽  
Theoretical Side

Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a prominent problem in knowledge representation and databases. This problem can be solved using the chase algorithm, which extends the given set of facts with fresh facts in order to satisfy the rules. If the chase terminates, then CQs can be evaluated directly in the resulting set of facts. The chase, however, does not terminate necessarily, and checking whether the chase terminates on a given set of rules and facts is undecidable. Numerous acyclicity notions were proposed as sufficient conditions for chase termination. In this paper, we present two new acyclicity notions called model-faithful acyclicity (MFA) and model-summarising acyclicity (MSA). Furthermore, we investigate the landscape of the known acyclicity notions and establish a complete taxonomy of all notions known to us. Finally, we show that MFA and MSA generalise most of these notions. Existential rules are closely related to the Horn fragments of the OWL 2 ontology language; furthermore, several prominent OWL 2 reasoners implement CQ answering by using the chase to materialise all relevant facts. In order to avoid termination problems, many of these systems handle only the OWL 2 RL profile of OWL 2; furthermore, some systems go beyond OWL 2 RL, but without any termination guarantees. In this paper we also investigate whether various acyclicity notions can provide a principled and practical solution to these problems. On the theoretical side, we show that query answering for acyclic ontologies is of lower complexity than for general ontologies. On the practical side, we show that many of the commonly used OWL 2 ontologies are MSA, and that the number of facts obtained by materialisation is not too large. Our results thus suggest that principled development of materialisation-based OWL 2 reasoners is practically feasible.

On two pursuit differential game problems with state and geometric constraints in a Hilbert space

2021 ◽  
Vol 65 (3) ◽  
pp. 5-16
Author(s):  
Abbas Ja’afaru Badakaya ◽  
Keyword(s):  
Differential Game ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Nonempty Closed Convex Subset ◽  
Geometric Constraints ◽  
First Order ◽  
Control Functions ◽  
Closed Convex Set ◽  
The Given ◽  
First Order Differential Equations

This paper concerns with the study of two pursuit differential game problems of many pursuers and many evaders on a nonempty closed convex subset of R^n. Throughout the period of the games, players must stay within the given closed convex set. Players’ laws of motion are defined by certain first order differential equations. Control functions of the pursuers and evaders are subject to geometric constraints. Pursuit is said to be completed if the geometric position of each of the evader coincides with that of a pursuer. We proved two theorems each of which is solution to a problem. Sufficient conditions for the completion of pursuit are provided in each of the theorems. Moreover, we constructed strategies of the pursuers that ensure completion of pursuit.

scholarly journals Two-Machine Job-Shop Scheduling Problem to Minimize the Makespan with Uncertain Job Durations

Algorithms ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 4 ◽  
Author(s):  
Yuri N. Sotskov ◽  
Natalja M. Matsveichuk ◽  
Vadzim D. Hatsura
Keyword(s):  
Relative Error ◽  
Job Shop ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Maximum Relative Error ◽  
Scheduling Problems ◽  
Lower And Upper Bounds ◽  
Shop Scheduling ◽  
Job Duration ◽  
On Line

We study two-machine shop-scheduling problems provided that lower and upper bounds on durations of n jobs are given before scheduling. An exact value of the job duration remains unknown until completing the job. The objective is to minimize the makespan (schedule length). We address the issue of how to best execute a schedule if the job duration may take any real value from the given segment. Scheduling decisions may consist of two phases: an off-line phase and an on-line phase. Using information on the lower and upper bounds for each job duration available at the off-line phase, a scheduler can determine a minimal dominant set of schedules (DS) based on sufficient conditions for schedule domination. The DS optimally covers all possible realizations (scenarios) of the uncertain job durations in the sense that, for each possible scenario, there exists at least one schedule in the DS which is optimal. The DS enables a scheduler to quickly make an on-line scheduling decision whenever additional information on completing jobs is available. A scheduler can choose a schedule which is optimal for the most possible scenarios. We developed algorithms for testing a set of conditions for a schedule dominance. These algorithms are polynomial in the number of jobs. Their time complexity does not exceed O ( n 2 ) . Computational experiments have shown the effectiveness of the developed algorithms. If there were no more than 600 jobs, then all 1000 instances in each tested series were solved in one second at most. An instance with 10,000 jobs was solved in 0.4 s on average. The most instances from nine tested classes were optimally solved. If the maximum relative error of the job duration was not greater than 20 % , then more than 80 % of the tested instances were optimally solved. If the maximum relative error was equal to 50 % , then 45 % of the tested instances from the nine classes were optimally solved.

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