Loss Of Convexity Research Articles

Classifying with uncertain data envelopment analysis

Classifications organize entities into categories that identify similarities within a category and discern dissimilarities among categories, and they powerfully classify information in support of analysis. We propose a new classification scheme premised on the reality of imperfect data. Our computational model uses uncertain data envelopment analysis to define a classification's proximity to equitable efficiency, which is an aggregate measure of intra-similarity within a classification's categories. Our classification process has two overriding computational challenges, those being a loss of convexity and a combinatorially explosive search space. We overcome the first challenge by establishing lower and upper bounds on the proximity value, and then by searching this range with a first-order algorithm. We address the second challenge by adapting the p-median problem to initiate our exploration, and by then employing an iterative neighborhood search to finalize a classification. We conclude by classifying the thirty stocks in the Dow Jones Industrial average into performant tiers, by classifying prostate treatments into clinically effectual categories, and dividing airlines into peer groups.

Comparative modelling results between a separable and a non-separable form of principal stretches–based strain energy functions for a variety of isotropic incompressible soft solids: Ogden model compared with a parent model

This paper presents a comparative analysis on the application of two models from the family of principal stretches–based strain energy functions to a wide range of incompressible rubber-like materials, including hydrogels, foams, silicone elastomers, filled rubbers and biomaterials. The two models of concern here are the seminal Ogden model of the separable form, and a more general non-separable form parent model proposed by the author. To this end, the three-term Ogden model, with corresponding six model parameters, and the one-term parent model with four model parameters, are applied to exemplar extant deformation datasets of the aforementioned rubber-like materials, and the ensuing modelling results are then compared. This comparison encompasses the quality of the obtained fits as measured by the resulting relative errors, and the mechanistic validity of the ensuing strain energy function as assessed by the convexity of the iso-energy plots in the principal stretches λ1,λ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left({\\lambda }_{1},{\\lambda }_{2}\\right)$$\\end{document} plane. The studied modes of deformation involve a wide range of deformation types including uniaxial, biaxial, simple shear and inflation. It is shown that for the considered range of materials and deformations, the optimal fits obtained by the Ogden model either (i) possess higher relative errors; or (ii) offer a marginal improvement but with the loss of convexity. This loss of convexity is further exemplified on considering an additional soft tissue dataset. Finally, an important implication of the separable versus non-separable functional forms of the two models in application to the deformation of elastomers will be demonstrated, where a separable function proves to be an inappropriate choice. In all these applications, the proposed parent model, by contrast, will be shown to provide equally good or better fits, with fewer model parameters, while remaining free of the foregoing undesirable modelling outcomes.

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces

An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the H^1 norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.

A three-parameter structurally motivated robust constitutive model for isotropic incompressible unfilled and filled rubber-like materials

A structurally motivated three-parameter strain energy function W is presented in this paper for application to the constitutive modelling of various types of incompressible isotropic rubber-like materials, from unfilled to filled rubbers, polymers and soft tissues. A particular objective of the current work is to achieve another step towards devising a comprehensive model for application to an assortment of incompressible isotropic hyperelastic materials, not limited only to a specific subset type of rubbers or polymers. To this end, the application of the model to extant datasets from various types of rubber-like materials is presented, via simultaneous fitting of the model to the considered deformation modes. The modelling results are then compared with those of a selected set of established three-parameter models in the literature, including a generalised neo-Hookean strain energy function and another two models which include an I2 term. It will be shown that the proposed model most favourably captures the datasets for all unfilled, filled rubber and polymer specimens considered. Finally, the model is applied to the deformation datasets of a soft tissue, namely the human brain, for which the (one-term) Ogden model is currently known to provide the best fit. It will be demonstrated that the application of the Ogden model will lead to loss of convexity, while our proposed model provides a better fit, with lower relative errors, and remains convex over the deformation domain. By way of the presented examples and applications, it is concluded that the proposed model provides a reliable degree of robustness for a suitable application to constitutive modelling of various types of rubber-like materials.

Sacral preauricular extensions and notches as parts of a 'Pelvic Pattern' may provide information on past pregnancies and parturitions.

During the analyses of several hundred prehistoric individuals from Austria, we observed that some women display a "Pelvic Pattern" at the innominate bones and the sacrum, i.e. specific combinations of pronounced expressions of pelvic features. We recorded classic pelvic features (dorsal pubic pitting, preauricular sulcus, extended pubic tubercle) as well as new ones (SPE: sacral preauricular extension, a ventrally pointing flat bone formation at the ventrosuperior margin of the ala ossis sacri; SPN: sacral preauricular notch, a loss of convexity at the same location; CF: corresponding facets at the ilium), and some less well-known features, i.e. the margo auricularis groove, ventral pubic exostoses and lesions. To quantify the assessed features, we developed a specific formula to calculate the 'Pelvic Pattern Index' (PPI). As pregnancies and/or parturitions are suspected to contribute to or be at least partly causative of the occurrence of pelvic features, we analyzed 48 well-preserved female individuals and 15 males from identified skeletal collections with obstetric information in Geneva and London. In these collections, we found a pelvic pattern of at least four out of ten distinctly expressed pelvic features only in multiparous females, but not in nulli- or primiparous females or in males. This pattern was found in 40.6% of the multiparous females and 29.2% of all females from the identified collections, compared to 56.1% of well-preserved prehistoric females with unknown parity status from Austria (n = 41). The mean PPI of the multiparae from the identified collections is 0.25, compared to a mean PPI of 0.19 for all women from the identified collections, and 0.28 for the prehistoric female individuals. We conclude from this that a high PPI (≥ 0.30), especially in cases where SPE or SPN are present, can give insights into past motherhood.

Finite element analysis of the impact of the properties of dental wedge materials on functional features

Purpose: Defect of the interproximal wall of the tooth is filled with use the shaped matrix and wedge which seals bottom margin during filling. Better fit of the wedge and equalization of the pressure forces on the matrix is achieved by the compliance of the wedge structure through cuts and perforations and the use of silicone materials and unidirectionally expanded polytetrafluoroethylene (ePTFE). The work presents a methodology for model studies of the mechanics of dental wedges in order to evaluate and compare the impact of wedge materials on functional features. The hypothesis of the work was that the mechanical properties of ePTFE determine the effectiveness of the dental wedge. Design/methodology/approach: Effect of modulus of elasticity and friction coefficient of wedge and matrix materials on the functional features of the wedge was studied on the way Finite Element Analysis (FEA). Simulation included contact sliding between wedge and matrix what was simulated in nonlinear large displacements regime. The sealing evaluation criterion was the pressure distribution between the wedge and matrix below the lower edge of the defect. Displacement values were the criterion for the loss of convexity as a result of matrix deformation. Findings: The material for the wedge should be characterized by a low coefficient of friction, low elasticity (ensuring high compliance of the wedge) and at the same time the ability to large permanent deformations, which allows for plastic shaping of the matrix from the side of the defect in order to achieve the required wall convexity and the tangent point. Research limitations/implications: Results show tendency of phenomena in limitation to model simplification of the interdental gap and the ideal adhesion of the matrix to the tooth and linear elasticity of materials. Practical implications: The material that best meets the requirements is unidirectionally expanded polytetrafluoroethylene, which has one of the lowest coefficients of friction and very high plasticity necessary to shape the matrix from the inside of the cavity. Originality/value: Methodology of model study and criteria of functional characteristics of dental wedge was presented.

Exploiting Characteristics in Stationary Action Problems

Connections between the principle of least action and optimal control are explored with a view to describing the trajectories of energy conserving systems, subject to temporal boundary conditions, as solutions of corresponding systems of characteristics equations on arbitrary time horizons. Motivated by the relaxation of least action to stationary action for longer time horizons, due to loss of convexity of the action functional, a corresponding relaxation of optimal control problems to stationary control problems is considered. In characterizing the attendant stationary controls, corresponding to generalized velocity trajectories, an auxiliary stationary control problem is posed with respect to the characteristic system of interest. Using this auxiliary problem, it is shown that the controls rendering the action functional stationary on arbitrary time horizons have a state feedback representation, via a verification theorem, that is consistent with the optimal control on short time horizons. An example is provided to illustrate application via a simple mass-spring system.

Further steps on the reconstruction of convex polyominoes from orthogonal projections

A remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

Marrying Stochastic Gradient Descent with Bandits: Learning Algorithms for Inventory Systems with Fixed Costs

We consider a periodic-review single-product inventory system with fixed cost under censored demand. Under full demand distributional information, it is well known that the celebrated (s, S) policy is optimal. In this paper, we assume the firm does not know the demand distribution a priori and makes adaptive inventory ordering decisions in each period based only on the past sales (a.k.a. censored demand). Our performance measure is regret, which is the cost difference between a feasible learning algorithm and the clairvoyant (full-information) benchmark. Compared with prior literature, the key difficulty of this problem lies in the loss of joint convexity of the objective function as a result of the presence of fixed cost. We develop the first learning algorithm, termed the [Formula: see text] policy, that combines the power of stochastic gradient descent, bandit controls, and simulation-based methods in a seamless and nontrivial fashion. We prove that the cumulative regret is [Formula: see text], which is provably tight up to a logarithmic factor. We also develop several technical results that are of independent interest. We believe that the developed framework could be widely applied to learning other important stochastic systems with partial convexity in the objectives. This paper was accepted by Chung Piaw Teo, optimization.

Model reduction by mean-field homogenization in viscoelastic composites. II. Application to rigidly reinforced solids.

The mean-field homogenization scheme proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44, 507-529 (doi:10.1016/j.ijsolstr.2006.04.038)) and revisited in a companion paper (Idiart et al. 2020 Proc. R. Soc. A 20200407 (doi:10.1098/rspa.2020.0407)) is applied to random mixtures of a viscoelastic solid phase and a rigid phase. Two classes of mixtures with different microstructural arrangements are considered. In the first class the rigid phase is dispersed within the continuous viscoelastic phase in such a way that the elastic moduli of the mixture are given exactly by the Hashin-Shtrikman formalism. In the second class, both phases are intertwined in such a way that the elastic moduli of the mixture are given exactly by the Self-Consistent formalism. Results are reported for specimens subject to various complex deformation programmes. The scheme is found to improve on earlier approximations of common use and even recover exact results under several circumstances. However, it can also generate highly inaccurate predictions as a result of the loss of convexity of the free-energy density. An auspicious procedure to partially circumvent this issue is advanced.

A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method

This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton–Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10−12.

A system of ionized gas dynamics

The aim of this paper is to study a system of three equations for ionized gas dynamics at high temperature in one spatial dimension. In addition to the mass density, pressure, and particle velocity, a further quantity is needed, namely, the degree of ionization. The system is supplemented by the first and second law of thermodynamics and by an equation of state; all of them involve the degree of ionization. Finally, under the assumption of local thermodynamic equilibrium, the system is closed by requiring Saha’s ionization equation. The geometric properties of the system are rather complicated: in particular, we prove the loss of convexity (genuine nonlinearity) for both forward and backward characteristic fields and hence the loss of concavity of the physical entropy. This takes place in a small bounded region, which we are able to characterize by numerical estimates on the state functions. The structure of shock waves is also studied by a detailed analysis of the Hugoniot locus.

Reinforced elastomers: Homogenization, macroscopic stability and relaxation

This paper deals with the stability and post-bifurcation response of reinforced hyperelastic composites under general loading conditions. It has long been known that these types of materials can undergo both microscopic and macroscopic instabilities. In the latter case, when the instability does not result in material failure, the behavior of the composite after the onset of the instability is less well understood. Recent work (Avazmohammadi and Ponte Castañeda, 2016) indicates that it is possible for the “principal” solution, i.e. the solution before the onset of any instability, to bifurcate into a lower energy solution via the formation of domains. These domains form on a scale much larger than that of the heterogeneity, but still smaller than that of the macroscopic specimen. This work is concerned with such domain formation in neo-Hookean laminates under general three-dimensional loading. In order to obtain the post-bifurcation behavior, the quasiconvexification or relaxation of the principal solution is computed explicitly. In addition, it is shown that the macroscopic instabilities are triggered not by the loss of strong ellipticity, but rather by the loss of global rank-1 convexity of the principal solution, which, in general, happens first. The calculation also reveals that the relaxation requires, in general, a rank-2 “laminate-within-a-laminate” microstructure and allows for multiple “perfectly soft” modes of deformation.

A new stabilisation approach for level-set based topology optimisation of hyperelastic materials

This paper introduces a novel computational approach for level-set based topology optimisation of hyperelastic materials at large strains. This, to date, is considered an unresolved open problem in topology optimisation due to its extremely challenging nature. Two computational strategies have been proposed to address this problem. The first strategy resorts to an arc-length in the pre-buckling region of intermediate topology optimisation (TO) iterations where numerical difficulties arise (associated with nucleation, disconnected elements, etc.), and is then continued by a novel regularisation technique in the post-buckling region. In the second strategy, the regularisation technique is used for the entire loading process at each TO iteration. The success of both rests on the combination of three distinct key ingredients. First, the nonlinear equilibrium equations of motion are solved in a consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. Third, and with the purpose of avoiding excessive numerical stabilisation, a scalar degradation function is applied on the regularised linearised elasticity tensor, based on a novel regularisation indicator field. The robustness and applicability of this new methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton-Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10− 12.

Regularization by compressibility of the μ(I) model of dense granular flow

The following article deals with the role of compressibility in regularizing the well-known μ(I) model, i.e., eliminating the short-wavelength (Hadamard) instability revealed by Barker et al. [“Well-posed and ill-posed behaviour of the μ(I)-rheology for granular flow,” J. Fluid Mech. 779, 794–818 (2015)]. In particular, we discuss the compressible-flow models proposed in the recent papers by Heyman et al. [“Compressibility regularizes the μ(I)-rheology for dense granular flows,” J. Fluid Mech. 830, 553–568 (2017)] and Barker et al. [“Well-posed continuum equations for granular flow with compressibility and μ(I)-rheology,” Proc. R. Soc. A 473(2201), 20160846 (2017)]. In addition to a critique of certain aspects of their proposed constitutive models, we show that the main effect of their regularizations is to add viscous effects to the shear response in a way that appears unfortunately to eliminate quasi-static yield stress. Another goal of the present work is to show how the development and analysis of visco-plastic constitutive relations are facilitated by dissipation potentials and the dissipative analog of elastic potentials. We illustrate their utility in Sec. IV of this article, where it is shown that a constant non-zero yield stress leads to loss of convexity that can only be restored by substituting viscous effects or else by adding spatial-gradient effects proposed previously by the present authors [Goddard, J. and Lee, J., “On the stability of the μ(I) rheology for granular flow,” J. Fluid Mech. 833, 302–331 (2017)].

Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics

Reformulating constitutive relation in terms of gradient dynamics (being derivative of a dissipation potential) brings additional information on stability, metastability and instability of the dynamics with respect to perturbations of the constitutive relation, called CR-stability. CR-instability is connected to the loss of convexity of the dissipation potential, which makes the Legendre-conjugate dissipation potential multivalued and causes dissipative phase transitions, that are not induced by non-convexity of free energy, but by non-convexity of the dissipation potential. CR-stability of the constitutive relation with respect to perturbations is then manifested by constructing evolution equations for the perturbations in a thermodynamically sound way. As a result, interesting experimental observations of behavior of complex fluids under shear flow and supercritical boiling curve can be explained.

Conservative oral rehabilitation of the bruxism patient with tooth wear

Tooth wear, a progressive but gradual loss of convexity on the cusp of the tooth, does not cause premature deterioration of the dentition. However, it may evolve into pathological wear if this normal wear becomes accelerated due to abnormal endogenous or exogenous factors. The pathological wear can lead to pathological changes of pulp, impaired function, occlusal disharmony and aesthetic problems. When treating patients with pathological wear, the cause of the wear must be analyzed, and each tooth should be diagnosed individually. Additionally, the loss of the vertical occlusal dimension, as well as the appropriateness of the space for restoration, should be evaluated. These procedures can prevent the progression of the wear, limit the range of any possible restoration, and finally, allow a clinician to establish a conservative treatment plan. This case report presents a patient with pathological wear caused by bruxism, and treated with monolithic zirconia fixed dental prostheses, so as to restore the lost guidance and the collapsed tooth shape; treatment was made with an occlusal adjustment of specific teeth, which showed interference during lateral excursive movement. As the result of the treatment, the functional movement of the mandible was freed, and the recurrence of tooth wear was prevented. The pathological wear displayed by this patient was successfully treated by this conservative method, as well as a limited range of restoration, all the while ensuring functionality and aesthetically satisfying results.

On iterative convexity of diffeomorphism

A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. We give several constructions of iteratively convex diffeomorphisms and explain the phenomenon that the non-existence of convex iterative roots is a typical property of convex functions. We show also that a slight perturbation of iteratively convex functions causes loss of iterative convexity. However, every convex function can be approximate by iteratively convex functions.

Coexistence of low and high overlap phases in a supercooled liquid: An integral equation investigation.

The pair structure, free energy, and configurational overlap order parameter Q of an annealed system of two weakly coupled replicas of a supercooled "soft sphere" fluid are determined by solving the hypernetted-chain (HNC) and self-consistent Rogers-Young (RY) integral equations over a wide range of thermodynamic conditions ρ (number-density), T (temperature), and inter-replicas couplings ε12. Analysis of the resulting effective (or Landau) potential W(ρ,T; Q) and of its derivative with respect to Q confirms the existence of a "precursor transition" between weak and strong overlap phases below a critical temperature Tc well above the temperature To of the "ideal glass" transition observed in the limit ε12→0. The precursor transition is signalled by a loss of convexity of the potential W(Q) and by a concomitant discontinuity of the order parameter Q just below Tc, which crosses over to a mean-field-like van der Waals loop at lower temperatures. The HNC and RY equations lead to the same phase transition scenario, with quantitative differences in the predicted temperatures Tc and To.

Nonlocal continuum analysis of a nonlinear uniaxial elastic lattice system under non-uniform axial load

The static behavior of the Fermi-Pasta-Ulam (FPU) axial chain under distributed loading is examined. The FPU system examined in the paper is a nonlinear elastic lattice with linear and quadratic spring interaction. A dimensionless parameter controls the possible loss of convexity of the associated quadratic and cubic energy. Exact analytical solutions based on Hurwitz zeta functions are developed in presence of linear static loading. It is shown that this nonlinear lattice possesses scale effects and possible localization properties in the absence of energy convexity. A continuous approach is then developed to capture the main phenomena observed regarding the discrete axial problem. The associated continuum is built from a continualization procedure that is mainly based on the asymptotic expansion of the difference operators involved in the lattice problem. This associated continuum is an enriched gradient-based or nonlocal axial medium. A Taylor-based and a rational differential method are both considered in the continualization procedures to approximate the FPU lattice response. The Padé approximant used in the continualization procedure fits the response of the discrete system efficiently, even in the vicinity of the limit load when the non-convex FPU energy is examined. It is concluded that the FPU lattice system behaves as a nonlocal axial system in dynamic but also static loading.