Variational Solution Research Articles

Fundamental Analysis of Singular and Resonance Phenomena in Vibrational Polyads of the Difluorosilylene Molecule

The singular structure of lower vibrational states of the difluorosilylene molecule (up to four total excitation quanta) is studied by expansion of energies of each state into series of the high-order Rayleigh–Schrodinger perturbation theory and analysis of their implicit properties of multiple-valuedness using Pade–Hermite fourth-order approximants. The quartic surface of potential energy in dimensionless normal coordinates is calculated quantum-mechanically at the MP2/cc-pVTZ level. It is shown that one of values of multiple-valued approximants reproduces the variational solution with a high accuracy, while other values (beginning with the fourth polyad) in many cases coincide with energies of other states of the same polyad. The Fermi and Darling–Dennison resonances are analyzed based on facts of coincidence of singular complex branch points of the approximants for mutually interacting states inside a unit circle or near it on the complex plane. It is found that a pair of states can have several coinciding branch points of the solutions (in particular, inside the unit circle). A conclusion is made that this approach is an effective method for the determination of the polyad structure of vibrational states. Calculation parameters necessary for the reproducibility of key results are selected. The calculations are carried out by means of a Fortran-based software package using an arithmetic calculation package with a long mantissa of real numbers (200 digits).

Variational Principle and Approximate Solution for the Fractal Vibration Equation in a Microgravity Space

Under the microgravity space, many theories derived from the earth's surface become untenable, so a modified vibration equation with fractal derivative is presented in this work. With the help of the semi-inverse method, we successfully develop the fractal variational principle, which not only provides conservation laws in an energy form but also provides physical insight into the nature structures of the solutions. Finally, the variational iteration method, together with the two-scale transform, is applied to find the solution of the fractal vibration equation. The obtained results show that the method is powerful and accurate.

A mixed variational formulation of dynamic viscoelastic problems with adhesion and friction

The aim of this work is to study a class of dynamic contact problems coupling adhesion and friction between two viscoelastic bodies with nonlinear viscosity operators. The boundary conditions concern relaxed unilateral contact, pointwise friction and adhesion including possible recoverable behavior. A mixed variational formulation of these problems is given as a five-field evolution implicit equation coupled with a differential inclusion describing the evolution of the intensity of adhesion. Based on several estimates and a classical fixed point theorem for multivalued functions, the existence of a strong variational solution is proved.

Mesoscopic structure of mixed type domain walls in multiaxial ferroelectrics

The structure of 180-degree uncharged rotational domain wall in a multiaxial ferroelectric film is studied within the framework of analytical Landau-Ginzburg-Devonshire (LGD) approach. The Finite Element Modelling (FEM) is used to solve numerically the system of the coupled nonlinear Euler-Lagrange (EL) differential equations of the second order for two components of polarization. We show that the structure of the domain wall and corresponding (meta)stable phase of the film are controlled by a single master parameter, dimensionless ferroelectric anisotropy {\mu}. We fitted the static profile of a solitary domain wall, calculated by FEM, with hyperbolic functions for polarization components, and extracted the five {\mu}-dependent parameters from the fitting to FEM curves. The high accuracy of the fitting results allows us to conclude that the analytical functions can be treated as the high-accuracy variational solution of the static EL equations with cubic nonlinearity. We further derive the two-component analytical solutions of the static EL equations for a polydomain 180-degree domain structure in a multiaxial ferroelectric film. The analysis of the free energy dependence on the film thickness and boundary conditions at its surfaces allows to select the domain states corresponding to the minimal energy. The single-domain state is ground for zero polarization derivative at the surfaces, while the poly-domain states minimize the system energy for zero polarization at the surfaces. Counterintuitively, the energy of the polydomain states split into two levels 0 and 1 for zero polarization at the surfaces, and each of the levels is the infinite set of the close-energy sub-levels, which morphology is characterized by different structure of the two-component polarization nodes.

The obstacle problem for singular doubly nonlinear equations of porous medium type

In this paper we prove the existence of variational solutions to the obstacle problem associated with doubly nonlinear equations \partial_t (|u|^{m-1}u) - \mathrm {div}(D_\xi f(Du)) = 0 with m > 1 and a convex function f satisfying a standard p -growth condition for an exponent p \in (1,\infty) in a bounded space-time cylinder \Omega_T := \Omega \times (0,T) . The obstacle function \psi and the boundary values g are time dependent. The proof relies on a nonlinear version of the method of minimizing movements.

Scattering of The Vector Soliton in Coupled Nonlinear Schrödinger Equation with Gaussian Potential

Nonlinear Schrodinger equation (NLSE) is the fundamental equation which describes the wave field envelope dynamics in a nonlinear and dispersive medium. However, if the fields have many components, one should consider the Coupled Nonlinear Schrodinger equation (CNLSE). We considered the interactions of orthogonally polarized and equal-amplitude vector solitons with two polarization directions. In this paper, we focused on the effect of Gaussian potential on the scattering of the vector soliton in CNLSE. The scattering process was investigated by the variational approximation method and direct numerical solution of CNLSE. Analytically, we analyzed the dynamics of the width and center of mass position of a soliton by the variational approximation method. Soliton may be reflected from each other or transmitted through or trapped. Initially, uncoupled solitons may form the coupled state if the kinetic energy of solitons less than the potential of attractive interaction between solitons but when its’ velocity above the critical velocity, the soliton will pass through each other easily. Meanwhile, a direct numerical simulation of CNLSE had been run to check the accuracy of the approximation. The result of the variational model gives a slightly similar pattern with direct numerical simulation of CNLSE by fixing the parameters for both solutions with the same value. The interaction of the vector soliton with Gaussian potential depends on the initial velocity and amplitude of the soliton and the strength of the external potential.

Assessment of variational fundamental solution based elements in plate bending dynamics

In this paper, an assessment of the accuracy performance of a new fundamental solution based element is carried out. Dynamic analysis of shear-deformable plate bending problems is considered. The element stiffness matrix is obtained by modified hybrid displacement variational statement. The mass matrix is computed using the relevant element shape functions. Free and forced vibrations are considered. It has demonstrated that the element is suitable for modelling both thin and thick plates. The obtained results are very promising as the accuracy level is excellent even with coarse discretization. The present element could be also considered as a locking-free element.

Local Sharp Vector Variational Type Inequality and Optimization Problems

In this paper, our goal was to establish the relationship between solutions of local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, also Minty local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, under generalized approximate η-convexity conditions for locally Lipschitzian functions.

Radial Limits of Nonparametric PMC Surfaces with Intermediate Boundary Curvature

The influence of the geometry of the domain on the behavior of generalized solutions of Dirichlet problems for elliptic partial differential equations has been an important subject for over a century. We investigate the boundary behavior of variational solutions $f$ of Dirichlet problems for prescribed mean curvature equations in a domain $\Omega \subset \mathbb{R}^{2}$ near a point $\mathcal{O} \in \partial \Omega$ under different assumptions about the curvature of $\partial \Omega$ on each side of $\mathcal{O}$. We prove that the radial limits at $\mathcal{O}$ of $f$ exist under different assumptions about the Dirichlet boundary data $\phi$, depending on the curvature properties of $\partial \Omega$ near $\mathcal{O}$.

Existence and concentration behavior of solutions to 1-Laplace equations on [formula omitted

In this paper, the following 1-Laplace equation−εDiv(Du/|Du|)+a(x)u/|u|=h(u)(x∈RN) in BV space is considered, where ε>0 is a small parameter, N≥2; and a:RN→R, h:R→R are suitable functions. The existence of bounded variation solutions are obtained by employing the nonsmooth critical point theorem. Furthermore, the concentration behavior of the solutions are studied and a single spike shape of the solutions is proved when the parameter ε is small enough.

An improved Jacobian-Torsor model for statistical variation solution in aero-engine rotors assembly

Rotor assembly is one of the core components of aero-engine, which basically consists of multistage revolving components. With the influence of parts’ manufacturing errors and practical assembly technology, assembly variations are unavoidable which will cause insecurity and unreliable of the whole engine. Statistical variation solution is a feasible means to analyze assembly precision. When using the three-dimensional variation analysis in rotor assembly, two key issues cannot be well solved, which involve the variation expression (the over-positioning problem of multiple datums) and the variation propagation (revolving characteristic of the rotors). To overcome the deficiency, extended Jacobian matrix and updated torsor equation were derived and unified, which eventually resulted in the improved Jacobian-Torsor model. This model can both provide rotation regulating mechanism by introducing the revolution joint, and characterize the interaction between essential mating features. Multistage rotational optimization of four-stage aero-engine rotors assembly has been performed to demonstrate this solution in statistical way. Results showed that the proposed model was applicable and conducive to precision prediction and analysis in design preliminary stage.

Transition between nonlinear and linear eigenvalue problems

We study convergence of variational solutions of the nonlinear eigenvalue problem−Δu=λ|u|p−2u,u∈H01(Ω), as p↓2 or as p↑2, where Ω is a bounded domain in RN with smooth boundary. It turns out that if λ is not an eigenvalue of −Δ then the solutions either blow up or vanish according to p↓2 or p↑2, while if λ is an eigenvalue of −Δ then the solutions converge to the associated eigenspace.

An Analytical Method to Calculate Groundwater Released From an Aquitard Undergoing Nonlinear Consolidation

AbstractThe estimation of groundwater released from aquitards is important for the groundwater resource management. In this study, a governing equation of drawdown variation within the aquitard, which undergoes nonlinear consolidation, was obtained based on the consolidation theory of Gibson et al. (1967, https://doi.org/10.1680/geot.1967.17.3.261). Then, analytical solutions of drawdown variations within an aquitard were derived for a single aquifer underlying an aquitard, as well as for the case when there are aquifers above and below an aquitard. Drawdown solutions were obtained for both the cases when there was a sudden hydraulic head decline in the confined aquifer(s). The characteristics of the groundwater dynamics and water balance in the aquitard have been analyzed with the analytical solution. A laboratory experiment was conducted to verify the analytical solution, and a type‐curve fitting method was presented to calculate the aquitard's hydraulic parameters. Our study revealed that the cumulative water released from the aquitard predicted by Gibson's theory (1967) is less than that obtained through Terzaghi's theory (1943). Groundwater released from the aquitard increases with increasing compression index, aquitard thickness, and drawdown of the adjacent aquifer, and with decreasing initial void ratio and initial effective stress. The time for the drawdown to reach steady state increases with increasing compression index, initial void ratio, and aquitard thickness, and with decreasing initial effective stress and hydraulic conductivity.

Numerical variational solution of hydrogen molecule and ions using one-dimensional hydrogen as basis functions

The ground state solution of hydrogen molecule and ions are numerically obtained as an application of our scheme to solve many-electron multi-center potential Schrödinger equation by using one-dimensional hydrogen wavefunctions as basis functions. The all-electron sparse Hamiltonian matrix for the given system is generated with the standard order finite-difference method, then the electronic trial wavefunction to describe the ground state is constructed based on the molecular orbital treatment, and finally an effective and accurate iteration process is implemented to systematically improve the result. Many problems associated with the evaluation of the matrix elements of the Hamiltonian in more general basis and potential are circumvented. Compared with the standard results, the variationally obtained energy of H2+ is within 0.1 mhartree accuracy, while that of H2 and H3+ include the electron correlation effect. The equilibrium bond length is highly consistent with the accurate results and the virial theorem is satisfied to an accuracy of −V/T = 2.0.

Uniqueness of Bounded Variation Solutions for Measure Functional Differential Equations with Infinite Delay

Uniqueness of Bounded Variation Solutions for Measure Functional Differential Equations with Infinite Delay

Positive solutions of superlinear indefinite prescribed mean curvature problems

This paper analyzes the superlinear indefinite prescribed mean curvature problem [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with a regular boundary [Formula: see text], [Formula: see text] satisfies [Formula: see text], as [Formula: see text], [Formula: see text] being an exponent with [Formula: see text] if [Formula: see text], [Formula: see text] represents a parameter, and [Formula: see text] is a sign-changing function. The main result establishes the existence of positive regular solutions when [Formula: see text] is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for [Formula: see text] small is further discussed assuming that [Formula: see text] satisfies [Formula: see text] as [Formula: see text], [Formula: see text] being such that [Formula: see text] if [Formula: see text]; thus, in dimension [Formula: see text], the function [Formula: see text] is not superlinear at [Formula: see text], although its potential [Formula: see text] is. Imposing such different degrees of homogeneity of [Formula: see text] at [Formula: see text] and at [Formula: see text] is dictated by the specific features of the mean curvature operator.

Lightweight optimization of passenger car seat frame based on grey relational analysis and optimized coefficient of variation

At present, the safety performance and lightweight of passenger car seat has become more and more important. The multi-objective lightweight optimization of the passenger car seat frame is carried out in this study. The novelty of this study is that we propose a detailed optimization design method and a design process in lightweight optimization for passenger car seat. In more detail, firstly, according to the ratio of energy absorption to mass method, the lightweight design components are selected. Secondly, different seat safety tests are conducted to optimize the corresponding components by considering both continuous thickness variables and discrete material variables. Thirdly, the strain index, the displacement of key points, the material cost, and the total mass of the components which need to be optimized (opti-components) are considered objectives. After that, the grey relational analysis (GRA) is adopted to optimize the material thickness scheme of the lightweight design components. Besides, the optimized coefficient of variation (OCV) method is applied to evaluate the corresponding weighting values of the objectives. Meanwhile, the availability of the grey relational analysis and optimized coefficient of variation (GRA&OCV) is assessed through comparing advantages among the method of GRA&OCV, the grey relational analysis and coefficient of variation (GRA&CV), the technique for order preference by similarity to ideal solution and coefficient of variation (TOPSIS&CV), as well as the technique for order preference by similarity to ideal solution and optimized coefficient of variation (TOPSIS&OCV) in the multi-objective lightweight optimization of passenger car seat frame. As a result, the total material cost and mass of the passenger car seat frame are reduced by 17.00% and by 2.08 kg (12.46%), respectively, with guaranteed vibrational and safety performance. Therefore, GRA&OCV can be effectively applied in multi-objective lightweight optimization of the passenger car seat frame.

Boundary Behavior of Rotationally Symmetric Prescribed Mean Curvature Hypersurfaces in $$\pmb {\varvec{{\mathbb {R}}}}^{4}$$

In a domain $$\Omega \subset {\mathbb {R}}^{n}$$ whose boundary is smooth except on a set $${\mathcal {C}}$$ of codimension (in $$\partial \Omega $$ ) k, the behavior of a nonparametric prescribed mean curvature hypersurface $$z=f\left( \mathbf{x}\right) $$ in the vertical cylinder $$\Omega \times {\mathbb {R}}$$ at a point $${\mathcal {O}}\in {\mathcal {C}}$$ can be largely unknown when $$n\ge 3,$$ depending on the type of boundary condition f satisfies. In a previous note, the authors considered an example in which $$n=3$$ and $$k=2;$$ that is, when $${\mathcal {C}}$$ is a (nonconvex) conical point on $$\partial \Omega .$$ Here, we consider prescribed mean curvature boundary value problems which are rotationally symmetric and investigate the behavior of variational solutions near a point $$P\in {\mathcal {C}}$$ when $$n=3$$ and $$k=1.$$

Vehicle trajectory reconstruction for signalized intersections: A hybrid approach integrating Kalman Filtering and variational theory

Trajectory data provide a rich source of information on spatial and temporal speed fluctuations of individual vehicles. To assemble a complete picture of traffic operations, obtaining vehicle trajectories of the entire traffic flow is essential. The current techniques cannot fulfill this need sufficiently. To this end, we developed a hybrid approach for reconstructing vehicle trajectories at signalized intersections that accounts for queue dynamics. Specifically, an extended variational solution network was established and Kalman Filtering was applied to capture and reproduce the stochastic properties of queue boundary curves. The proposed method was demonstrated using both NGSIM data and simulation data. Results showed that the hybrid approach reproduces accurately the shockwave boundaries and trajectory reconstruction. By fusing different sources of traffic data, the proposed method does not rely on the common assumption of uniform arrival between adjacent probe vehicles, which is the main drive for estimation improvement.

Variational nonlinear elliptic equations in nonreflexive Musielak spaces

We investigate the existence and uniqueness of variational solutions for the Dirichlet problem associated with variational nonlinear elliptic equations in Musielak spaces. We assume neither Δ2 nor ∇2-condition on the Φ-functions but suppose that they satisfy some structural assumptions that allow to define the variational framework and to obtain Poincaré-type inequalities. Within the approach we use, the Galerkin method is coupled with an abstract existence result by F.E. Browder, 1966.