Convexity Requirement Research Articles

Convexity of Yield Surface with Directional Distortional Hardening Rules

The present paper examines the convexity of the yield surface in the directional distortional hardening models by Feigenbaum and Dafalias. In these models anisotropy develops through kinematic and directional distortional hardening, supplemented by the classical isotropic hardening, and the associative flow rule is used. However, the issue of convexity, which naturally arises due to the distortion of the yield surface, was not fully addressed. The present paper derives the necessary and sufficient conditions to ensure convexity of the yield surface for the simpler Feigenbaum and Dafalias models, but it is not as straightforward to derive corresponding conditions for convexity of the Feigenbaum and Dafalias model version which contains an evolving fourth-order tensor. In this case convexity will be addressed first in general and then at the limit state for which simple restrictions on the material constants to ensure convexity are derived. Numerical examples will show that some of the yield surfaces simulated in the original Feigenbaum and Dafalias publication will not stay convex if loaded beyond what was done in these publications. Therefore the material constants for these cases are recalibrated based on the derived relations for satisfaction of the convexity requirement, and the fitting of the yield surfaces is repeated with the new set of constants and compared with the previous case.

On the superiority of pulsing under a concave advertising market potential function

The authors study the superiority of advertising pulsing policy (turning advertising on and off in a cyclic fashion) over its uniform (constant spending) counterpart that costs the same under the assumption that sales dynamics follow a modified Vidale–Wolfe aggregate advertising model. The authors show that pulsing can be superior if the product of the concave market potential function and the linear or concave advertising response function is convex in advertising. Similar to previous studies in the literature, the average undiscounted profit over the infinite planning horizon is considered as a performance measure according to which alternative advertising pulsation policies are compared. Employing a well-known data set relating advertising to sales, the above convexity requirement is empirically supported and the superiority of pulsing is established numerically. The performance of the proposed model is found to be superior to two rival models using a one-step-ahead forecasting procedure. The analytical findings of the study are documented into six theoretical results for which proofs are relegated to a separate appendix. Managerial implications of the study and directions for future research are also discussed.

Data Envelopment Analysis - Basic Models and their Utilization

Data Envelopment Analysis - Basic Models and their Utilization Data Envelopment Analysis (DEA) is a decision making tool based on linear programming for measuring the relative efficiency of a set of comparable units. Besides the identification of relatively efficient and inefficient units, DEA identifies the sources and level of inefficiency for each of the inputs and outputs. This paper is a survey of the basic DEA models. A comparison of DEA models is given. The effect of model orientation (input or output) on the efficiency frontier and the effect of the convexity requirements on returns to scale are examined. The paper also explains how DEA models can be used to assess efficiency.

Representation of preference relations on [formula omitted]-algebras of nonatomic measure spaces: Convexity and continuity

The purpose of this paper is to show how preference relations on σ -algebras can be represented by means of nonadditive set functions that satisfy appropriate requirements of convexity and continuity on σ -algebras. To this end, we introduce convex combinations of measurable sets, and quasiconcave and concave functions on σ -algebras, which conform with the standard results in convex analysis. We formulate the convexity and the continuity axioms for preference relations on σ -algebras with a metric topology and show the existence of utility functions for convex continuous preference relations. We also show that monotone continuous preference relations are representable by fuzzy measures.

Parallelohedra and topological transitions in cellular structures

In 1891, Fedorov showed that tilings of three-dimensional space by congruent convex polyhedra, in which all the tiles are in the same orientation, belong to just five topological classes. In 1953, Cyril Stanley Smith generalized Fedorov's result by dispensing with the convexity requirement. In this and the later work of Ferro and Fortes, solutions were obtained by applying topological transitions, of the kind that occur during grain growth, to the well-known space-filling by Kelvin 14-hedra. We demonstrate an anomalous solution to the generalized Fedorov problem that is not derivable by this method, and which provides a counter example to some conjectures suggested by O’Keeffe. Finally, a further generalization is proposed, that has relevance in the study of periodic networks. We conclude with a few examples to indicate some interesting directions for possible future developments of the idea that Fedorov and Smith had initiated.

An exact multiplicity result for a class of symmetric problems

We consider positive solutions of a class of semilinear problems u'' + λa(x)f(u) = 0, \qquad −1 < x < 1,\qquad u(−1) = u(1) = 0, with even and positive a(x) , depending on a positive parameter λ . In case f(u) is convex, an exact multiplicity result was given in P. Korman, Y. Li and T. Ouyang [6]; see also P. Korman [4] for the details. It was observed by P. Korman and J. Shi [7] that convexity requirement can be relaxed for large u (see also [5]). We show that convexity requirement can also be relaxed for small u .

Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect

A class of nonsingular yield conditions depending on three parameters is analyzed for isotropic materials exhibiting strength differential effect and pressure insensitivity. The yield condition can then be expressed in terms of the second and third stress deviator invariants. The convexity requirement is considered and the constraints imposed on the material parameters are discussed in detail. The dual dissipation function is derived in the analytical form. The condition can be applied in the analysis of high strength alloys (such as Inconnel 718) or of shape memory alloys (such as NiTi, NiAl, CuZnGa, or CuAlNi) in order to specify the onset of yield, or of martensitic or austenitic transformation. The conditions can easily be generalized to account for mixed hardening and back stress anisotropy. Some experimental data are provided to verify the proposed conditions.

Covers for Angleworms

A set in the plane is a cover for a family of planar arcs if it is convex and if it contains a congruent copy of each arc of the family. For a given family of arcs, one is typically interested in the cover of least area. Although many such problems, commonly called problems, have been considered over the years, few have actually been solved. In this note we investigate what is perhaps the simplest case, the family of all two-segment unit arcs, and we discover to our surprise that the cover of least area is not unique. The prototype of these problems, L. Moser's 1966 question of determin ing the smallest cover for the family of all unit arcs?the so-called classical worm problem?remains unsolved, even unapproachable. In the final section we forego the convexity requirement and show that (a) for each s > 0 there is a non-convex cover for the family of two-segment unit arcs that is a connected set with simply connected interior whose area is less than e, and (b) for each s > 0 there is a non-convex Jordan domain that is a cover for this family and has area less than s.

A modified conservation principles theory leading to an optimal Galerkin CFD algorithm

AbstractA modified conservation principles theory in one, then multi‐dimensions, admits the prediction of an optimally accurate algorithm construction for the unsteady incompressible Navier–Stokes (INS) equations. Via a time Taylor series (TS) operation, followed by a pseudo‐limit process, the theory generates a modified, but still analytical, INS system parameterized by a set of coefficients constrained only by a convexity requirement. A spatially discretized finite element implementation of a Galerkin weak statement on this modified INS system, termed the ‘Taylor weak statement (TWS), ’ generates a parameterized CFD algorithm for analysis. TWS algorithm phase velocity and amplification factor error functions are derived for linear and bilinear basis implementations assembled at the generic node. A subsequent TS expansion in wave number space admits analytical identification of parameter set options affecting lowest order error terms. The results of definitive verification‐ and validation‐class computational experiments for a range of published CFD algorithms belonging to the TWS class, reported herein, clearly confirm theoretical prediction of the optimal TWS algorithm for INS thermal/fluid transport applications. Copyright © 2007 John Wiley & Sons, Ltd.

Infinitesimal local operations and differential conditions for entanglement monotones

Much of the theory of entanglement concerns the transformations that are possible to a state under local operations with classical communication (LOCC); however, this set of operations is complicated and difficult to describe mathematically. An idea which has proven very useful is that of the entanglement monotone: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. In this paper we look on LOCC as the set of operations generated by infinitesimal local operations, operations which can be performed locally and which leave the state little changed. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone. We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requirement of convexity. We then demonstrate that a number of known entanglement monotones satisfy these differential criteria. Finally, as an application, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states. It is our hope that this approach will avoid some of the difficulties in the theory of multipartite and mixed-state entanglement.

Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for d>~3 dimensions

Renormalization-group (RG) flow equations have been derived for the generalized sine-Gordon model (GSGM) and the Coulomb gas (CG) in d >= 3 of dimensions by means of Wegner's and Houghton's, and by way of the real-space RG approaches. The UV scaling laws determined by the leading-order terms of the flow equations are in qualitative agreement for all dimensions d >= 3, independent of the dimensionality, and in sharp contrast to the special case d = 2. For the 4-dimensional GSGM it is demonstrated explicitly (by numerical calculations), that the blocked potential tends to a constant effective potential in the infrared (IR) limit, satisfying the requirements of periodicity and convexity. The comparison of the RG flows for the three-dimensional GSGM, the CG, and the vortex-loop gas reveals a significant dependence on the renormalization schemes and the approximations used.

Combined Preisach–Mayergoyz-neural-network vector hysteresis model for electrical steel sheets

Three versions of a vector hysteresis model for electrical steel sheets are presented, based on the function approximation capabilities of feed-forward neural networks and the memory mechanism of vector hysteresis proposed by Mayergoyz. The first model handles arbitrary vector magnetization patterns, but requires a very extended data set for the training of the neural network. The second model is suitable for convex induction loci and allows a reduction of the required training set. The third model handles the features of the considered magnetization pattern in an alternative way and relaxes the convexity requirement. The choice of the specific model, its parameters, and the network training set depends on the types of magnetization patterns concerned. Arbitrary high accuracy can be reached by extending the complexity of the model and/or the size of the training set. Experimental results for the third model are presented and show the good accuracy of the approach. Standard neural network algorithms are used.

Plastic Resistance of Pipe Sections: Upper Bound Solution

In a previous paper, interaction relations based on the lower bound theorem of plasticity were derived for moderately thick pipes subjected to biaxial shear and bending, axial force, twisting moments, and internal or external pressure. The developments provided a lower bound solution based on assumed simplified stress fields. In this paper it is demonstrated that the assumptions made in the previous solution were indeed equilibrium requirements. Pipe interaction relations are obtained using the upper bound theorem of plasticity. Because (1) the assumed strain fields are based on the general solution of the compatibility equations; (2) the stress fields meet the equilibrium conditions; and (3) the yield condition is met over the whole pipe cross section, the interaction relations obtained are exact within the limitations of the formulation. The yield surface obtained meets the Drucker convexity requirement and is suitable as a potential surface characterizing the behavior of generalized plastic hinges in the elastoplastic analysis of pipe elements. The yield surface is mathematically described by two intrinsic functions. A simple mathematical procedure is devised in order to express the normality condition to the yield surface.

Right and wrong near critical endpoints

The implications of convexity (or thermodynamic stability) for isothermal density (or composition) phase diagrams near critical endpoints are investigated. In particular, Schreinemakers’ rules for the geometry of three coexisting phases in a space of densities are refined and extended physically to apply at and in the vicinity of a critical endpoint. Seemingly plausible phase diagrams are presented which violate the extended rules (and, in most cases, the Second Law of Thermodynamics). The requirements of convexity, supplemented for Ising-related systems by the Griffiths–Kelly–Sherman inequalities, also restrict the signs of important expansion coefficients of thermodynamic scaling functions for criticality and provide bounds on universal critical amplitude ratios.

Closed form line spring yield surfaces for deep and shallow cracks: formulation and numerical performance

The line-spring finite element is a versatile numerical tool for performing engineering fracture mechanics analysis of surface cracked shells. The accuracy of the line-spring finite element solution for deep/medium-sized cracks has shown to be higher than the accuracy for the shallow-sized ones. Proper treatment of shallow cracks is important because they are the ones most frequently encountered in engineering practice. Accurate yield surfaces of plane-strain single edge-cracked specimens having shallow, as well as deep, cracks are developed here in order to improve the overall performance of the line-spring element. The yield surface is represented by equations that automatically satisfy the convexity requirement and that fit the result of limit load analyses. The present study addresses, furthermore, the performance of the backward Euler return algorithm for one of the accurate yield surface formulated here, by means of iso-error maps.

Hoops: 3D Curves as Conservative Occluders for Cell-Visibility

Most visibility culling algorithms require convexity of occluders. Occluder synthesis algorithms attempt to construct large convex occluders inside bulky non-convex sets. Occluder fusion algorithms generate convex occluders that are contained in the umbra cast by a group of objects given an area light. In this paper we prove that convexity requirements can be shifted from the occluders to their umbra with no loss of efficiency, and use this property to show how some special non-planar, non-convex closed polylines that we call hoops can be used to compute occlusion efficiently for objects that have no large interior convex sets and were thus rejected by previous approaches.

Thermodynamically self-consistent approximations to the liquidus and solidus of Hg1−xCdxTe

Based on liquidus and solidus temperatures and associated confidence limits reported by Brice, Capper, and Jones [J. Cryst. Growth, 75, 395 (1986)], we develop polynomial approximations to the liquidus and solidus curves for the pseudobinary material Hg1−xCdxTe. These approximations are “thermodynamically consistent” in that they satisfy the requirement that the liquidus and solidus temperatures must agree at x=0 and 1. A linear programming approach is used to find the lowest-degree polynomial approximations to the liquidus and solidus temperatures falling within the confidence limits of Brice et al. at each of their 21 compositions, and also satisfying the convexity requirement imposed by Brice et al. on their own manually-drawn curves. Finally, we present a twoparameter rational approximation that fits the tabulated segregation coefficients for Hg1−xCdxTe better than an earlier five-parameter polynomial approximation.

Generalized dual to the monotone nonlinear complementarity problem

It was recently shown that duality between the nonlinear programming problem corresponding to the nonlinear complementarity problem and its Wolfe dual holds under a monotonicity assumption without the usual convexity requirements. Here it is shown that duality also hold for a number of Mond-Weir duals under the monotonicity assumption and without the usual convexity conditions.

Optimality Conditions for Multiobjective Programming with Generalized ([formula omitted], ρ, θ)-Convex Set Functions

Necessary conditions for Pareto optimality in multiobjective programming with subdifferentiable set functions are established in Theorem 12 of H. C. Lai and L. J. Lin (J. Math. Anal. Appl.132,1988, 558–571). In this paper, we establish some sufficient conditions under which a feasible solution of such a problem will be Pareto optimal provided that a weaker convexity requirement is satisfied; for instance (I,ρ,θ)-convexity is assumed for both objective and constraint set functions. Some duality models are also discussed. Wolfe-type and Mond–Weir-type duality theorems are proved.

Tensor Polynomial Failure Criterion: Coefficient Limits Based on Convexity Requirements

The tensor polynomial failure criterion is a widely used tool for predicting failure of unidirectional composite lamina. Typically, the tensor polynomial failure criterion is used in the quadratic form, commonly referred to as the Tsai-Wu failure criterion. Some recent work has presented the cubic form as an alternative to the widely used Tsai-Wu failure criterion. This paper presents limits on the coefficients of the tensor polynomial failure criterion for the quadratic, cubic, and quartic forms based on convexity of the failure surface in the stress domain. The results for the quadratic form of the polynomial include those commonly found in the literature and a further limit that is not normally found due to assumptions on the affect of shear. Further, the limits on the quadratic coefficients are used to prove that the quadratic tensor polynomial failure function is invariant with respect to coordinate system transformation. Results for the cubic tensor polynomial show that, when the entire stress space is considered, the cubic tensor polynomial cannot generate a convex failure surface, and is therefore not an acceptable failure criterion. However, using constraints identified in this paper, it is possible to determine regions in the stress domain where the failure criterion is convex and gives valid results. A further conclusion is that the "hybrid method" of Tennyson, MacDonald, and Nanyaro, used to determine the cubic interaction coefficients, yields results that violate convexity requirements and therefore should not be considered as a valid method. Results for the quartic form of the tensor polynomial failure criterion indicate the cubic terms may be included in the polynomial provided the appropriate quartic terms are present. The derivation of the limits on the coefficients is presented in an appendix and the limits are presented in the body. Illustratory examples of quartic tensor polynomial are presented.