Minimum Energy Solution Research Articles

Three-dimensional buckling analysis of stiffened plates with complex geometries using energy element method

A novel numerical method, energy element method (EEM), is proposed for the three-dimensional (3D) buckling analysis of stiffened plates with complex geometries. The problem is formulated in a cuboidal domain, and any complex geometric stiffened plate is modeled by assigning cutouts within the cuboidal domain. The stiffened plate is considered as an energy body and is discretized using Gauss points with variable stiffness properties to simulate its energy distribution. Incorporating the extended interval integration, Gauss quadrature, variable stiffness properties, and Chebyshev polynomials, the strain energy of stiffened plates with complex geometries can be numerically simulated by putting the stiffness and thickness of Gauss points in the cutouts to zero in the cuboidal domain. Using the principle of minimum potential energy and Ritz solution procedure, the deformation and buckling behaviors of stiffened plates with complex geometries can be captured. As a result of the new formulations in EEM, new standard energy functionals and solving procedures have been developed. In addition, Gauss points are generated within the energy elements accounting for the geometric boundaries of the stiffened plate, which are characterized by level set functions. EEM is employed to investigate complex-shaped stiffened plates with straight or curvilinear stiffeners, and the results are compared to those obtained using FEM or mesh-free method. The precision, generalization, and stability of EEM are demonstrated.

A New Solution of the Pulsar Equation

We present the first new type of solution of the pulsar equation since 1999. In it, the whole magnetosphere is confined inside the light cylinder and an electrically charged layer wraps around it and holds it together. The reason this new solution has never been obtained before is that all current time-dependent simulations are initialized with a vacuum dipole configuration that extends to infinity; thus, their final steady-state solution also extends to infinity. Under special conditions, such a confined configuration may be attained when the neutron star first forms in the interior of a collapsing star during a supernova explosion, or when it accretes from an external wind or disk from a donor star. It is shown that this new maximally closed non-decelerating solution is the limit of a continuous sequence of standard magnetospheres with open and closed field lines when the amount of open field lines gradually drops to zero. The minimum energy solution in this sequence is a standard magnetosphere in which the closed field line region extends up to about 80% of the light cylinder. We estimate that the released energy when the new solution transitions to the minimum energy one is enough to power a fast radio burst.

Energy-optimal trajectories for skid-steer rovers

This paper presents the energy-optimal trajectories for skid-steer rovers on hard ground, without obstacles. We obtain 29 trajectory structures that are sufficient to describe minimum-energy motion, which are enumerated and described geometrically; 28 of these structures are composed of sequences of circular arcs and straight lines; there is also a special structure called whirls consisting of different circular arcs. Our analysis identifies that the turns in the trajectory structures (aside from whirls) are all circular arcs of a particular turning radius, R′, the turning radius at which the inner wheels of a skid-steer rover are not commanded to turn. This work demonstrates its paramount importance in energy-optimal path planning. There has been a lack of analytical energy-optimal trajectory generation for skid-steer rovers, and we address this problem by a novel approach. The equivalency theorem presented in this work shows that all minimum-energy solutions follow the same path irrespective of velocity constraints that may or may not be imposed. This non-intuitive result stems from the fact that with this model of the system the total energy is fully parameterized by the geometry of the path alone. With this equivalency in mind, one can choose velocity constraints to enforce constant power consumption, thus transforming the energy-optimal problem into an equivalent time-optimal problem. Pontryagin’s Minimum Principle can then be used to solve the problem. Accordingly, the extremal paths are obtained and enumerated to find the minimum-energy path. Furthermore, our experimental results by using Husky UGV provide the experimental support for the equivalency theorem.

Global minimizers of coexistence for strongly competing systems involving the square root of the Laplacian

This paper is concerned with the spatial behavior of the strongly competing systems involving the square root of the Laplacian. The coexistence of minimal energy solutions is discussed and a mechanism to ensure coexistence is given. Moreover, in the case of two densities the global minimizer converges, as the interspecies interaction tends to infinity, to a spatially segregated distribution where the two densities coexist and solve a non-coupled variational problem.

Design of ultra-lightweight and energy-efficient civil structures through shape morphing

This paper gives a new formulation to synthesize load-responsive structures through shape morphing. The design method is based on the minimization of the whole-life energy, which comprises an embodied share in the material and an operational share for adaptation. Target shapes that counteract the effect of the design loads are obtained through geometry and sizing optimization. Adaptation through shape morphing enables effective stress redistribution so that the design is not dominated by peak loads with long return periods. Material utilization is maximized, and therefore embodied energy is significantly reduced. The geometry is controlled through length changes of optimally-placed linear actuators. The actuator commands are computed through minimization of the operational energy to satisfy safety and serviceability criteria. Minimum energy solutions are obtained through a univariate optimization process in which embodied and operational energy minimization are nested. Numerical benchmarks between adaptive and mass-optimized passive solutions are provided on a truss bridge and multi-story frame configurations. Parametric studies show that mass and energy savings are significant for slender structures and when the live load is dominant over the permanent load. Generally, larger shape bounds result in a greater reduction of mass and embodied energy at the cost of a larger operational energy.

Generalized equipartition method from an arbitrary viewing angle

ABSTRACT The equipartition analysis yields estimates of the radius and energy of synchrotron self-absorbed radio sources. Here we generalize this method to relativistic off-axis viewed emitters. We find that the Lorentz factor Γ and the viewing angle θ cannot be determined independently but become degenerate along a trajectory of minimal energy solutions. The solutions are divided into on-axis and off-axis branches, with the former reproducing the classical analysis. A relativistic source viewed off-axis can be disguised as an apparent Newtonian one. Applying this method to radio observations of several tidal disruption events, we find that the radio flare of AT 2018hyz, which was observed a few years after the optical discovery, could have been produced by a relativistic off-axis jet with a kinetic energy of $\sim 10^{53}\, \rm erg$ that was launched around the time of discovery.

Spherically Constrained Relative Motion Trajectories in Low Earth Orbit

This paper considers a satellite inspection mission where the deputy satellite is required to remain a fixed distance away from the chief satellite. This constraint on the deputy’s state is modeled as the surface of a sphere. Finding minimum energy solutions to travel from one point to another point on the sphere requires solving a nonconvex optimal control problem. Various novel suboptimal algorithms are proposed that can guarantee a feasible solution. These algorithms are benchmarked by optimal values obtained via a global continuation method, and their computational performance is analyzed in a statistical setting.

Major 3 Satisfiability logic in Discrete Hopfield Neural Network integrated with multi-objective Election Algorithm

<abstract> <p>Discrete Hopfield Neural Network is widely used in solving various optimization problems and logic mining. Boolean algebras are used to govern the Discrete Hopfield Neural Network to produce final neuron states that possess a global minimum energy solution. Non-systematic satisfiability logic is popular due to the flexibility that it provides to the logical structure compared to systematic satisfiability. Hence, this study proposed a non-systematic majority logic named Major 3 Satisfiability logic that will be embedded in the Discrete Hopfield Neural Network. The model will be integrated with an evolutionary algorithm which is the multi-objective Election Algorithm in the training phase to increase the optimality of the learning process of the model. Higher content addressable memory is proposed rather than one to extend the measure of this work capability. The model will be compared with different order logical combinations $ k = \mathrm{3, 2} $, $ k = \mathrm{3, 2}, 1 $ and $ k = \mathrm{3, 1} $. The performance of those logical combinations will be measured by Mean Absolute Error, Global Minimum Energy, Total Neuron Variation, Jaccard Similarity Index and Gower and Legendre Similarity Index. The results show that $ k = \mathrm{3, 2} $ has the best overall performance due to its advantage of having the highest chances for the clauses to be satisfied and the absence of the first-order logic. Since it is also a non-systematic logical structure, it gains the highest diversity value during the learning phase.</p> </abstract>

Electrically charged multi-field configurations

In this work we investigate the presence of electrically charged structures that are localized in two and three spatial dimensions. We use the Maxwell-scalar Lagrangian to describe several systems with distinct interactions for the scalar fields. The procedure relies on finding first order differential equations that solve the equations of motion and ensure stability of the corresponding minimum energy solutions. We illustrate the many possibilities in two and in three spatial dimensions, examining different examples of electrically charged solutions that engender internal structure.

Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.

Positive mass theorem and the CR Yamabe equation on 5-dimensional contact spin manifolds

We consider the CR Yamabe equation with critical Sobolev exponent on a closed contact manifold M of dimension 2n+1. The problem of finding solutions with minimum energy has been resolved for all dimensions except for dimension 5 (n=2). In this paper we prove the existence of minimum energy solutions in the 5-dimensional case when M is spin. The proof is based on a positive mass theorem built up through a spinorial approach.

A Variational Approach to Morphogenesis : Recovering Spatial Phenotypic Features from Epigenetic Landscapes.

We model the process of cell fate determination of the flower Arabidopsis-thaliana employing a system of reaction-diffusion equations governed by a potential field. This potential field mimics the flower's epigenetic landscape as defined by Waddington. It is derived from the underlying genetic regulatory network (GRN), which is based on detailed experimental data obtained during cell fate determination in the early stages of development of the flower. The system of equations has a variational structure, and we use minimax techniques (in particular the Mountain Pass Lemma) to show that the minimal energy solution of our functional is, in fact, the one that traverses the epigenetic landscape (the potential field) in the spatial order that corresponds to the correct architecture of the flower, that is, following the observed geometrical features of the meristem. This approach can generally be applied to systems with similar structures to establish a genotype to phenotype correspondence. From a broader perspective, this problem is related to phase transition models with a multiwell vector potential, and the results and methods presented here can potentially be applied in this case.

Thermal buckling behavior of power and sigmoid functionally graded material sandwich plates using nonpolynomial shear deformation theories

PurposeThe purpose of this article is to carry out the thermal buckling analysis of power and sigmoid functionally graded material Sandwich plate (P-FGM and S-FGM) under uniform, linear, nonlinear and sinusoidal temperature rise.Design/methodology/approachThermal buckling of FGM Sandwich plates namely, FGM face with ceramic core (Type-A) and homogeneous face layers with FGM core (Type-B), incorporated with nonpolynomial shear deformation theories are considered for an analytical solution in this investigation. Effective material properties and thermal expansion coefficients of FGM Sandwich plates are evaluated based on Voigt's micromechanical model considering power and sigmoid law. The governing equilibrium and stability equations for the thermal buckling analysis are derived based on sinusoidal shear deformation theory (SSDT) and inverse trigonometric shear deformation theory (ITSDT) along with Von Karman nonlinearity. Analytical solutions for thermal buckling are carried out using the principle of minimum potential energy and Navier's solution technique.FindingsCritical buckling temperature of P-FGM and S-FGM Sandwich plates Type-A and B under uniform, linear, non-linear, and sinusoidal temperature rise are obtained and analyzed based on SSDT and ITSDT. Influence of power law, sigmoid law, span to thickness ratio, aspect ratio, volume fraction index, different types of thermal loadings and Sandwich plate types over critical buckling temperature are investigated. An analytical method of solution for thermal buckling of power and sigmoid FGM Sandwich plates with efficient shear deformation theories has been successfully analyzed and validated.Originality/valueThe temperature distribution across FGM plate under a high thermal environment may be uniform, linear, nonlinear, etc. In practice, temperature variation is an unpredictable phenomenon; therefore, it is essential to have a temperature distribution model which can address a sinusoidal temperature variation too. In the present work, a new sinusoidal temperature rise is proposed to describe the effect of sinusoidal temperature variation over critical buckling temperature for P-FGM and S-FGM Sandwich plates. For the first time, the FGM Sandwich plate is modeled using the sigmoid function to investigate the thermal buckling behavior under the uniform, linear, nonlinear and sinusoidal temperature rise. Nonpolynomial shear deformation theories are utilized to obtain the equilibrium and stability equations for thermal buckling analysis of P-FGM and S-FGM Sandwich plates.

On minimal energy solutions to certain classes of integral equations related to soliton gases for integrable systems* *The work of the first author is supported by long term structural funding-Methusalem grant of the Flemish Government, by the Fonds Wetenschappelijk Onderzoek-Vlaanderen (FWO) and the Fonds de la Recherche Scientifique-FNRS under EOS Project No. 30889451, and by FWO Flanders projects G.0864.16 and

We prove existence, uniqueness and non-negativity of solutions of certain integral equations describing the density of states u(z) in the spectral theory of soliton gases for the one dimensional integrable focusing nonlinear Schrödinger equation (fNLS) and for the Korteweg–de Vries (KdV) equation. Our proofs are based on ideas and methods of potential theory. In particular, we show that the minimising (positive) measure for a certain energy functional is absolutely continuous and its density u(z) ⩾ 0 solves the required integral equation. In a similar fashion we show that v(z), the temporal analog of u(z), is the difference of densities of two absolutely continuous measures. Together, the integral equations for u, v represent nonlinear dispersion relation for the fNLS soliton gas. We also discuss smoothness and other properties of the obtained solutions. Finally, we obtain exact solutions of the above integral equations in the case of a KdV condensate and a bound state fNLS condensate. Our results is a step towards a mathematical foundation for the spectral theory of soliton and breather gases, which appeared in work of El and Tovbis (2020 Phys. Rev. E 101 052207). It is expected that the presented ideas and methods will be useful for studying similar classes of integral equation describing, for example, breather gases for the fNLS, as well as soliton gases of various integrable systems.

Nonlinear Schrödinger systems with mixed interactions: locally minimal energy vector solutions

This paper is concerned with asymptotic behavior of positive solutions for coupled Schrödinger equations with mixed interactions between components. We construct locally minimal energy solutions that show distinctively different limiting profile for simultaneously large attractive and repulsive couplings. The components of the solutions constructed exhibit partial synchronization and segregation.

Vortex precession and exchange in a Bose-Einstein condensate

Vortices in a Bose-Einstein condensate are modelled as spontaneously symmetry breaking minimum energy solutions of the time dependent Gross-Pitaevskii equation, using the method of constrained optimization. In a non-rotating axially symmetric trap, the core of a single vortex precesses around the trap center and, at the same time, the phase of its wave function shifts at a constant rate. The precession velocity, the speed of phase shift, and the distance between the vortex core and the trap center, depend continuously on the value of the conserved angular momentum that is carried by the entire condensate. In the case of a symmetric pair of identical vortices, the precession engages an emergent gauge field in their relative coordinate, with a flux that is equal to the ratio between the precession and shift velocities.

Minimum Energy Control of a Unicycle Model Robot

Abstract Point-to-point path planning for a kinematic model of a differential-drive wheeled mobile robot (WMR) with the goal of minimizing input energy is the focus of this work. An optimal control problem is formulated to determine the necessary conditions for optimality and the resulting two point boundary value problem is solved in closed form using Jacobi elliptic functions. The resulting nonlinear programming problem is solved for two variables and the results are compared to the traditional shooting method to illustrate that the Jacobi elliptic functions parameterize the exact profile of the optimal trajectory. A set of terminal constraints which lie on a circle in the first quadrant are used to generate a set of optimal solutions. It is noted that for maneuvers where the angle of the vector connecting the initial and terminal point is greater than a threshold, which is a function of the radius of the terminal constraint circle, the robot initially moves into the third quadrant before terminating in the first quadrant. The minimum energy solution is compared to two other optimal control formulations: (1) an extension of the Dubins vehicle model where the constant linear velocity of the robot is optimized for and (2) a simple turn and move solution, both of whose optimal paths lie entirely in the first quadrant. Experimental results are used to validate the optimal trajectories of the differential-drive robot.

Electrically charged localized structures

This work deals with an Abelian gauge field in the presence of an electric charge immersed in a medium controlled by neutral scalar fields, which interact with the gauge field through a generalized dielectric function. We develop an interesting procedure to solve the equations of motion, which is based on the minimization of the energy, leading us to a first order framework where minimum energy solutions of first order differential equations solve the equations of motion. We investigate two distinct models in two and three spatial dimensions and illustrate the general results with some examples of current interest, implementing a simple way to solve the problem with analytical solutions that engender internal structure.

A new general third-order zigzag model for asymmetric and symmetric laminated composite beams

This paper proposes a new third-order zigzag model for asymmetric and symmetric laminated composite beams. In the framework of a general displacement form, a new zigzag shear strain shape function with layerwise coefficients corresponding to constant, linear, quadratic and cubic terms is proposed. By comparing transverse shear stress fields obtained by constitutive and equilibrium equations respectively, layerwise constants of quadratic and cubic terms are determined. With further consideration of interlaminar continuity and free stress conditions on traction free surfaces, other coefficients are achieved. As interlaminar continuity and dissimilar characteristics between layers are fully considered, the present model which has only three unknowns can accurately describe zigzag effects and provide Cz0 quadratic distribution results of transverse shear stress field by constitutive equations directly. Characteristics and advantages of the present model are discussed by comparing with other classical theoretical models. Moreover, governing equations are formulated using the principle of minimum potential energy, and analytical and finite element solutions for static analysis of laminated beams are presented. Lastly, numerical validations are presented, and comparisons with exact solutions, other theoretical models and high-fidelity finite element models well demonstrate accuracy and effectiveness of the present model for laminated beams with different features.

Fast decaying ground states for elliptic equations with exponential nonlinearity

We show the existence of positive minimal energy solutions for the equation −div(K(x)∇u)=K(x)f(u) in R2, where K(x)=exp(|x|2∕4) and f:R→R is a superlinear continuous function with exponential subcritical or exponential critical growth. We use as a main tool the Nehari manifold method.