Single Scalar Function Research Articles

Displacement-potential analysis of ply-stresses in a partially guided non-uniform short-column of hybrid laminated composites

Reliable and accurate analysis of local stresses in structural problems of fiber-reinforced laminated composites involving nonuniform geometry is of utmost practical importance, especially when they are subjected to local loads and supports. In this research, a new study of ply stresses in a partially guided, nonuniform, short composite column of hybrid laminates, subjected to an eccentrically distributed loading is conducted using an efficient computational scheme. The analytical modeling of the present mixed-boundary-value problem of laminated structure is realized in terms of a single scalar function of space variables (called as displacement-potential function), which is defined by the displacement components of plane elasticity, and is governed by a single fourth-order partial-differential equation of equilibrium. In this modeling approach, the original three-dimensional symmetric laminated column is reduced to a plane-stress scalar-field problem of an equivalent imaginary lamina in an attempt to obtain the corresponding strain field (global laminate strain), which is then used to determine the stress fields of individual plies through the respective transformed reduced stiffness matrices. A finite-difference based computational scheme is developed to solve the representative nonuniform equivalent lamina in terms of the displacement-potential function, which is however well capable of dealing with different ply materials as well as fiber orientations, together with the associated mixed and changeable physical conditions along the boundaries of the column. A 28-ply balanced hybrid laminate composed of boron-epoxy and glass-epoxy plies with various fiber orientation is considered as the structural material for the present column. The stress fields obtained for individual plies as well as the overall laminate are analyzed in the perspective of a number of practical issues of structural design, which include material hybridization, eccentricity of loading and attachment of partial guides to the column. The design stresses of the nonuniform laminated column are identified to be affected significantly by the material and orientation of fibers as well as the laminate hybridization. An attempt is also made to verify the soundness and accuracy of the present computational scheme through the comparison of the present potential-function solutions with those obtained by the conventional computational method.

A new cutting plane method for lexicographic multi-objective integer linear programming

This work presents a new cutting plane method for lexicographic multi-objective integer linear programming (LMOILP). The method uses Grossone Methodology to reformulate a LMOILP problem into one having a single scalar objective function involving Grossone, as done in Cococcioni et al. (2018) (but in that case in the absence of the integer constraints). The problem, without the integer constraints, is solved using the GrossSimplex algorithm presented in Cococcioni et al. (2018) to find a candidate optimal solution. Here a novel cutting plane is introduced, named Gross-based Objective Function Cutting Plane whenever the optimal value of the Gross-scalar objective function is Gross-fractional. When it happens to be not Gross-fractional, cutting planes are generated using the Fractional Cutting Plane, derived from fractional components of the optimal solution. Moreover, by combining them, we proposed an algorithm that we called Gross-based Cutting Plane (GCP) method. It has been proved that it finds the optimal solution of a LMOILP problem and terminates after a finite number of iterations. To speed-up the GCP, at each iteration subsequent the first one, we re-use the optimal basis of the last continuous relaxation, by running the GrossDualSimplex algorithm. This is the well-known warm-start technique, which, however, needs specific attention due to the need to solve a linear problem having a right-hand side involving Grossone-based numbers. In the experimental part, we show the efficacy of the proposed approach.

The average conformation tensor of inter-atomic bonds as an alternative state variable to the strain tensor: definition and first application — the case of nanoelasticity

Most of the mechanical models for solid state materials are in a methodological framework where a strain tensor, whatever it is, is considered as a thermodynamic state variable. As a consequence, the Cauchy stress tensor is expressed as a function of a strain tensor—and, in many cases, of one or more other state variables, such as the temperature. Such a choice for the kinematic state variable is clearly relevant in the case of infinitesimal or finite elasticity. However, one can ask whether an alternative state variable could not be considered. In the case of finite elastoplasticity, the choice of a strain tensor as the basic, kinematic state variable is not totally without issue, in particular in relation to the physical meaning of the internal state variable describing the permanent strains. In any case, this paper proposes an alternative to the strain tensor as a state variable, which is not based on the deformation (Lagrangian) gradient: the average conformation tensor of inter-atomic bonds. The purpose, however, is restricted to (1) a particular type of materials, namely the pure substances (copper or aluminum, for instance), (2) the nanoscale, and (3) the case of elasticity. The very simple case of two atoms of a pure substance in the solid state is first considered. It is shown that the kinematics of the inter-atomic bond can be characterized by a so called ``conformation'' tensor, and that the tensorial internal force acting on it can be immediately deduced from a single scalar function, depending only on the conformation tensor: the state potential of free energy (or interaction potential). Using an averaging procedure, these notions are then extended to a finite set of atoms, namely an atom and its first neighbours, which can be seen as the ``unit cell'' of a pure substance in the solid state considered as a discrete medium. They are also transposed to the Continuum case, where an expression of the Cauchy stress tensor is proposed as the first derivative of a state potential of density (per unit mass) of average free energy of inter-atomic bonds, which is an explicit function of the average conformation tensor of inter-atomic bonds. By applying a standard procedure in Continuum Thermodynamics, it is then shown that the objective part of the material derivative of this new state variable, at least in the case when the pure substance can be considered as an elastic medium, is equal to the symmetric part of the Eulerian velocity gradient, that is the rate of deformation tensor. In the case of uniaxial tension, a simple relationship is eventually set out between the average conformation tensor and a strain tensor, which is correctly approximated by the usual infinitesimal strain tensor as long as the conformation variations (from an initial state of conformation) are ``small''. From this latter result, and assuming an elastic behavior, a simple expression for the state potential of density of average free energy is inferred, showing great similarities with—but not equivalent to—the classical model of isotropic, linear elasticity (Hooke's law).

Global existence of a nonlinear wave equation arising from Nordström’s theory of gravitation

We show global existence of classical solutions for the nonlinear Nordström theory with a source term and a cosmological constant under the assumption that the source term is small in an appropriate norm, while in some cases no smallness assumption on the initial data is required. In this theory, the gravitational field is described by a single scalar function that satisfies a certain semi-linear wave equation. We consider spatial periodic deviation from the background metric, that is why we study the semi-linear wave equation on the three-dimensional torus {mathord {mathbb T}}^3 in the Sobolev spaces H^m({mathord {mathbb T}}^3). We apply two methods to achieve the existence of global solutions, the first one is by Fourier series, and in the second one, we write the semi-linear wave equation in a non-conventional way as a symmetric hyperbolic system. We also provide results concerning the asymptotic behavior of these solutions and, finally, a blow-up result if the conditions of our global existence theorems are not met.

Displacement-function analysis of local stresses in thick beams on simple supports

A new boundary modeling scheme is introduced for the solution of thick beams on simple supports using the displacement-function elasticity approach. The performance of two sets of mixed boundary conditions at the opposing lateral ends of a quasi-infinitely long beam is investigated for the analytical solution of the thick beam. In this approach, a cropped solution strategy is implemented to obtain the elastic field from the central region of the long beam, in which the solution of the beam is obtained in terms of a single scalar function in the form of infinite trigonometric series. The appropriateness and accuracy of the solutions is verified through the comparison with corresponding solutions obtained by FEM and classical beam theory. The convergence characteristics of the series solution are analyzed in the perspective of required computational effort with varying load distribution factor, beam aspect ratio and far-field span ratio. Finally, the method is applied to analyze the local stress and displacement fields, especially in the immediate neighborhood of the discretely applied loads and supports of thick beams. Significant concentrations of bending and shear stresses are observed, especially at the terminal regions of the discrete loading, which are captured in details and verified to be highly sensitive to the loading span and beam aspect-ratio.

Time-asymptotic stability for first-order symmetric hyperbolic systems of balance laws in dissipative compressible fluid dynamics

This paper identifies a non-(or /iso-)thermal variant of Ruggeri’s 1983 formulation of viscous heat-conductive fluid dynamics as a hyperbolic system of balance laws and shows that both the original model and this variant have (a) time-asymptotically stable equilibria and (b) principal parts deriving from a protopotential: a single scalar function that induces the temporospatial flux as an appropriate part of its Hessian.

Displacement-function modeling of thermo-mechanical behavior of fiber-reinforced composite structures

A new modeling scheme has been developed for the analysis of thermo-mechanical stresses in structural problems of fiber-reinforced composite materials with mixed mode of physical conditions. In contrast with conventional approaches, the coupled thermal and mechanical problems of structural mechanics are integrated here using a single scalar function of space variables, which is defined in terms of the displacement components of plane elasticity. The displacement-function is governed by a single partial-differential equation of equilibrium that takes into account the two orthogonal gradients of the resulting non-uniform temperature field through two different versions of the formulation. The temperature-dependent elastic field is determined using a two-step solution scheme that utilizes the two versions of displacement-function formulations in two sequential steps. The scheme is developed in such a flexible fashion that even the solution of the corresponding isotropic problems can readily be obtained with thermal and/or mechanical loading as a limiting case of the formulation. The application of the proposed modeling scheme is demonstrated through the solution of a number of problems of structural mechanics, namely plates and beams subjected to thermal as well as thermo-mechanical loadings with both uniform and mixed mode of physical conditions. An attempt is also made to verify the soundness and accuracy of the resulting thermoelastic fields by comparing them with the corresponding analytical as well as computational solutions available in the literature. Moreover, the correspondence between the versions of the displacement-function formulation is also verified by solving identical problems using different versions or different combinations of the versions of the formulation.

Optical equations for null strings

An optical equation for null strings is derived. The equation is similar to Sachs' optical equations for null geodesic congruences. The string optical equation is given in terms of a single complex scalar function $Z$, which is a combination of spin coefficients at the string trajectory. Real and imaginary parts of $Z$ determine expansion and rotation of strings. Trajectories of strings can be represented by diagrams in a complex $Z$-plane. Such diagrams allow one to draw some universal features of null strings in different backgrounds. For example, in asymptotically flat space-times $Z$ vanishes as future null infinity is approached, that is, gradually shapes of strings are "freezing out". Outgoing gravitational radiation and flows of matter leave ripples on the strings. These effects are encoded in subleading terms of $Z$. String diagrams are demonstrated for rotating and expanding strings in a flat space-time and in cosmological models.

On the thermodynamic consistency of Quasi-linear viscoelastic models for soft solids

Originating in the field of biomechanics, Fung’s model of quasi-linear viscoelasticity (QLV) is one of the most popular constitutive theories employed to compute the time-dependent relationship between stress and deformation in soft solids. It is one of the simplest models of nonlinear viscoelasticity, based on a time-domain integral formulation. In the present study, we consider the QLV model incorporating a single scalar relaxation function. We provide natural internal variables of state, as well as a consistent expression of the free energy to illustrate the thermodynamic consistency of this version of the QLV model. The thermodynamic formulation highlights striking similarities between QLV and the internal-variable models introduced by Holzapfel and Simo. Finally, the dissipative features of compressible QLV materials are illustrated in simple tension.

Measure of complexity in self-gravitating systems using structure scalars

The aim of this paper is to present the definition of complexity for static self-gravitating anisotropic matter proposed in f(G,T)theory, where Gis the Gauss-Bonnet term and Tis the trace of energy momentum tensor. We evaluate field equations, Tolman-Oppenheimer-Volkoff equation, mass functions and structure scalars. Among the calculated modified scalar variables that are obtained from the orthogonal splitting of Riemann tensor, a single scalar function has been identified as the complexity factor. After exploring the corresponding Tolmann mass function, it is seen that the complexity factor along with the f(G,T)terms have greatly influenced its formulation and its role in the subsequent radial phases of the spherical system. We have also used couple of ansatz in order to discuss possible solutions of equations of motion in the study of the structure of compact object.

Complete dimensional collapse in the continuum limit of a delayed SEIQR network model with separable distributed infectivity.

We take up a recently proposed compartmental SEIQR model with delays, ignore loss of immunity in the context of a fast pandemic, extend the model to a network structured on infectivity and consider the continuum limit of the same with a simple separable interaction model for the infectivities beta . Numerical simulations show that the evolving dynamics of the network is effectively captured by a single scalar function of time, regardless of the distribution of beta in the population. The continuum limit of the network model allows a simple derivation of the simpler model, which is a single scalar delay differential equation (DDE), wherein the variation in beta appears through an integral closely related to the moment generating function of u=sqrt{beta }. If the first few moments of u exist, the governing DDE can be expanded in a series that shows a direct correspondence with the original compartmental DDE with a single beta . Even otherwise, the new scalar DDE can be solved using either numerical integration over u at each time step, or with the analytical integral if available in some useful form. Our work provides a new academic example of complete dimensional collapse, ties up an underlying continuum model for a pandemic with a simpler-seeming compartmental model and will hopefully lead to new analysis of continuum models for epidemics.

Designing stress for optimizing and toughening truss-like structures

Optimization of materials and structures is a crucial step in the design of man-made mechanical components for a wide field of engineering applications. It also plays a key role in mechanobiology of living systems, being involved by nature across the scales, from single-cell to tissues and organs, as a strategy to minimize metabolic cost and maximize biomechanical performances. The synergy between the continuously increasing development of high-resolution 3D printing technologies and the possibility to predict chemical and physical properties through molecular dynamics-based numerical analyses has recently contributed to boost the use of both design and topology optimization procedures. They are employed in ab initio simulations as key strategies for deciding microstructures to improve mechanical performances and, concretely, to achieve prototypes of new material components. With this in mind, we here propose to abandon the classical approach of using a single scalar objective function employed in the classical design and topology optimization strategies, to introduce multiple quantities to be minimized, identified as the differences between material yield stress and the maximum von Mises stress. After mathematically justifying the well-posedness of this unconventional choice for the case at hand, it is highlighted that the proposed strategy is based on the concept of "equalizing" a proper stress measure at any point of the body and, for this reason, it is baptized as Galilei’s optimization, in honor of the Italian scholar who somehow first wondered about the possibility of changing sizes of beams to have uniform internal forces and, in turn, minimum weight. By exploiting analytical solutions and ad hoc implementing a parametric finite element algorithm to be applied to a wide variety of solids with arbitrary complex structural geometries, including nested or hierarchically organized architectures, it is first demonstrated that the proposed optimization strategy roughly retraces principles invoked by nature to guide growth, remodeling and shaping of biomaterials. More importantly, by means of several benchmark examples, we finally show the proposed procedure might be also helpfully employed to conceive a new class of micro-structured, eventually 3D-printed materials exhibiting surprising post-elastic properties, such as high overall resilience and toughness, in particular obtaining a decrease of stress concentration and a slowing down of crack propagation as direct effects of the optimization, which de facto minimizes stress gradients wherever in the solid domain.

Conformal properties of soft operators: Use of null states

Soft-operators, loosely speaking, are operators which create or annihilate zero energy massless particles on the celestial sphere in Minkowski space. The Lorentz group acts on the celestial sphere by conformal transformation and the soft-operators transform as conformal primary operators of various dimension and spin. Working in space-time dimensions $D=4$ and $6$, we study some properties of the conformal representations with the (leading) soft photon and graviton as the highest weight vectors. Typically these representations contain null-vectors. We argue, from the $S$-matrix point of view, that infinite dimensional asymptotic symmetries and conformal invariance require us to set some of these null-vectors to zero. As a result, the corresponding soft-operator satisfies linear PDE on the celestial sphere. Curiously, these PDEs are equations of motion of Euclidean gauge theories on the celestial sphere with scalar gauge-invariance, i.e, the gauge parameter is a scalar field on the sphere. These are probably related to large $U(1)$ and supertranslation transformations at infinity. Now, the PDE satisfied by the soft-operator can be converted into PDE for the $S$-matrix elements with the insertion of the soft-operator. These equations can then be solved subject to appropriate boundary conditions on the celestial sphere, provided by conformal invariance. The solutions determine the soft $S$-matrix elements, for different helicities of the soft-particle, in terms of a single scalar function. This makes the Ward-identity for the asymptotic symmetry almost integrable. The result of the integration, which we are not able to perform completely, should of course be Weinberg's soft-theorem. Finally, we comment on the similarity between the roles played by null-states in the context of asymptotic symmetry and in string theory in relation to space-time gauge symmetry.

A method of effective potentials for calculating the frequency spectrum of eccentrically layered spherical cavity resonators

ABSTRACTA novel method for the calculation of eigenfrequencies of non-uniformly filled spherical cavity resonators is developed. The spectrum is determined through the introduction of effective “dynamic” potentials which encode the infill inhomogeneity. For angularly symmetric cavity, regardless of its radial non-uniformity, the set of modes is shown to be a superposition of TE and TM oscillations which can be described in terms of a single scalar function independently of each other. The violation of polar symmetry in the infill dielectric properties, the azimuthal symmetry being preserved, suppresses all azimuthally non-uniform modes of TM oscillations. In the absence of angular symmetry of both electric and magnetic properties of the infill, only azimuthally uniform distribution of both TM and TE fields is expected to occur in the resonator. The method is quite efficient for computational complex algorithms for solving different spectral problems, including those for studying the chaotic properties of systems' spectra.

Earthquake source arrays: optimal configuration and applications in crustal structure studies

SUMMARYA collection of earthquake sources recorded at a single station, under specific conditions, are considered as a source array (SA), that is interpreted as if earthquake sources originate at the station location and are recorded at the source location. Then, array processing methods, that is array beamforming, are applicable to analyse the recorded signals. A possible application is to use source array multiple event techniques to locate and characterize near-source scatterers and structural interfaces. In this work the aim is to facilitate the use of earthquake source arrays by presenting an automatic search algorithm to configure the source array elements. We developed a procedure to search for an optimal source array element distribution given an earthquake catalogue including accurate origin time and hypocentre locations. The objective function of the optimization process can be flexibly defined for each application to ensure the prerequisites (criteria) of making a source array. We formulated four quantitative criteria as subfunctions and used the weighted sum technique to combine them in one single scalar function. The criteria are: (1) to control the accuracy of the slowness vector estimation using the time domain beamforming method, (2) to measure the waveform coherency of the array elements, (3) to select events with lower location error and (4) to select traces with high energy of specific phases, that is, sp- or ps-phases. The proposed procedure is verified using synthetic data as well as real examples for the Vogtland region in Northwest Bohemia. We discussed the possible application of the optimized source arrays to identify the location of scatterers in the velocity model by presenting a synthetic test and an example using real waveforms.

Using both qualitative and quantitative data in parameter identification for systems biology models

In systems biology, qualitative data are often generated, but rarely used to parameterize models. We demonstrate an approach in which qualitative and quantitative data can be combined for parameter identification. In this approach, qualitative data are converted into inequality constraints imposed on the outputs of the model. These inequalities are used along with quantitative data points to construct a single scalar objective function that accounts for both datasets. To illustrate the approach, we estimate parameters for a simple model describing Raf activation. We then apply the technique to a more elaborate model characterizing cell cycle regulation in yeast. We incorporate both quantitative time courses (561 data points) and qualitative phenotypes of 119 mutant yeast strains (1647 inequalities) to perform automated identification of 153 model parameters. We quantify parameter uncertainty using a profile likelihood approach. Our results indicate the value of combining qualitative and quantitative data to parameterize systems biology models.

Multi-particle amplitudes from the four-point correlator in planar mathcal{N} = 4 SYM

A non-trivial consequence of the super-correlator/super-amplitude duality is that the integrand of the four-point correlation function of stress-tensor multiplets in planar mathcal{N} = 4 super Yang-Mills contains a certain combination of n-point amplitude integrands for any n. This combination is the sum of products of all helicity super-amplitudes with their corresponding helicity conjugates. The four-point correlator itself is described by a single scalar function whose loop level integrands possess a hidden permutation symmetry facilitating its computation up to ten loops. We discover that assuming Yangian symmetry and an appropriate basis of planar dual conformal integrands it is possible to disentangle the contributions from the individual amplitudes from this combination. We test this up to seven points and up to two loops. This suggests that any scattering amplitude for any n, with any helicity structure and at any loop order may be extractable from the four-point correlator.

Ultimate swelling described by limiting chain extensibility of swollen elastomers

In this study, we study ultimate swelling characterized by limiting chain extensibility of swollen elastomers. Limiting chain extensibility is introduced into the Flory–Rehner theory using the Arruda–Boyce eight chain model and the Gent phenomenological model. The difference between these models is unified by defining a single scalar function. The inequality derived from this function allows for analysis to provide an ultimate value of swelling ratio. This ultimate value is not exceeded at equilibrium swelling regardless of the set of material constants. Under uniaxial loading at equilibrium swelling, deswelling can occur even in tension. Further, the very large swelling behavior of pH sensitive hydrogels is found to originate from the resistance generated by approaching the ultimate value of swelling ratio.

Hyperbolic theory of relativistic conformal dissipative fluids

We develop a complete description of the class of conformal relativistic dissipative fluids of divergence form, following the formalism carried out by Geroch, Lindblom and Pennisi. This type of theories is fully described in terms of evolution variables whose dynamics is governed by total divergence-type conservation laws. Specifically, we give a characterization of the whole family of conformal fluids in terms of a single master scalar function defined up to second order corrections in dissipative effects, which we explicitly find in general form. This allows us to identify the equilibrium states of the theory, as well as to derive constitutive relations and a Fourier-like law for the corresponding first-order theory heat flux. Finally, we show that among this class of theories-- and near equilibrium configurations-- there exist symmetric hyperbolic ones, implying that for them one can define well posed initial value problems.

Multi-cluster dynamics in coupled phase oscillator networks

In this paper, we examine robust clustering behaviour with multiple nontrivial clusters for identically and globally coupled phase oscillators. These systems are such that the dynamics is completely determined by the number of oscillators N and a single scalar function g(ϕ) (the coupling function). Previous work has shown that (a) any clustering can stably appear via choice of a suitable coupling function and (b) open sets of coupling functions can generate heteroclinic network attractors between cluster states of saddle type, though there seem to be no examples where saddles with more than two nontrivial clusters are involved. In this work, we clarify the relationship between the coupling function and the dynamics. We focus on cases where the clusters are inequivalent in the sense of not being related by a temporal symmetry, and demonstrate that there are coupling functions that give robust heteroclinic networks between periodic states involving three or more nontrivial clusters. We consider an example for N = 6 oscillators where the clustering is into three inequivalent clusters. We also discuss some aspects of the bifurcation structure for periodic multi-cluster states and show that the transverse stability of inequivalent clusters can, to a large extent, be varied independently of the tangential stability.