Admissible Functions Research Articles

Identification of the diffusion coefficient in a time fractional diffusion equation

Abstract In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to D t α C ⁢ u - ∇ ⋅ ( k ⁢ ( x ) ⁢ ∇ ⁡ u ) = f {{}^{C}D_{t}^{\alpha}u-\nabla\cdot(k(x)\nabla u)=f} , where the function k = k ⁢ ( x ) {k=k(x)} is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function k. At the first step of the methodology, we give a stability result corresponding to connectivity of k and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given.

On a class of fully anisotropic elliptic equations

We consider a class of nonlinear elliptic equations driven by a fully anisotropic elliptic operator, and with a Carathéodory reaction. The elliptic operator depends on x and on the gradient of u, via the differential (with respect to u) of a general function belonging to a class of admissible functions. We present also several examples showing how our theorem incorporates the non-radial case, as well as that of functions with no polynomial growth.

On the Morse index of least energy nodal solutions for quasilinear elliptic problems

In this paper we study the quasilinear equation $$- \varepsilon ^2 \varDelta u-\varDelta _p u=f(u)$$ in a smooth bounded domain $$\varOmega \subset {\mathbb {R}}^N$$ with Dirichlet boundary condition, where $$p>2$$ and f is a suitable subcritical and p-superlinear function at $$\infty $$. First, for $$\epsilon \ne 0$$ we prove that Morse index is two for every least energy nodal solution. This result is inspired and motivated by previous results by A. Castro, J. Cossio and J. M. Neuberger, and T. Bartsch and T. Weth; and it is connected with a result by S. Cingolani and G. Vannella. Then, for the limit case $$\varepsilon = 0$$ we prove (a) the existence of a least energy nodal solution whose Morse index is two, and (b) Morse index is two for every nodal solution which strictly and locally minimizes the energy functional on the set of sign-changing admissible functions.

Vibrations of cylindrical shells with rectangular cross-section

The free vibrations of cylindrical shells with rectangular cross-section are studied in the work. The precision of Rayleigh method for various admissible functions and different boundary conditions is evaluated. Using finite element method, the natural frequencies in the case of deviation of the cross-section shape from square to rectangular are examined, and the vibration frequencies and modes for the shell joined with the plate are studied for various plate thicknesses. Additionally, fundamental frequency behavior is analyzed in the case of fixed support, as one of the cross-section sides approaches zero.

Research of the Inspection Optimization Strategy for Distribution Network Based on Comprehensive Information Model

To improve the quality and efficiency of inspection tasks, this paper studies the path planning optimization strategy for distribution equipment inspection of distribution networks. The comprehensive information model of each type of distribution equipment to be inspected is established by integrating the real-time monitoring information, geographic information and historic information effectively. The inspection path optimization model with an admissible cost function is established and solved by using the heuristic search A* algorithm. Comparison of the results of several inspection path planning strategies are provided to verify the effectiveness of the proposed strategy.

Saturated Attitude Control for Rigid Spacecraft Under Attitude Constraints

This paper addresses the attitude-maneuver control problem for a rigid-body spacecraft in the presence of attitude-constrained zones as well as input saturation. More specifically, attitude-constrained zones are properly encoded, and an admissible artificial potential function (APF) is developed under the unit-quaternion representation. The elaborately designed APF ensures a unique minimum without requiring convexity of constrained zones, which yields a less stringent condition from previous approaches. Benefiting from the bounded property of the gradient of the proposed APF, a saturated controller is presented to render the almost uniformly ultimate boundedness of maneuver errors while complying with underlying maneuver constraints. The associated stability analysis of the closed-loop system is ensured by using the primal Lyapunov and the dual Lyapunov techniques. Numerical simulation tests are presented to assess the efficiency and demonstrate the advantages of the proposed control scheme.

Free Vibration Analysis of Curved Laminated Composite Beams with Different Shapes, Lamination Schemes, and Boundary Conditions.

A general formulation is considered for the free vibration of curved laminated composite beams (CLCBs) with alterable curvatures and diverse boundary restraints. In accordance with higher-order shear deformation theory (HSDT), an improved variational approach is introduced for the numerical modeling. Besides, the multi-segment partitioning strategy is exploited for the derivation of motion equations, where the CLCBs are separated into several segments. Penalty parameters are considered to handle the arbitrary boundary conditions. The admissible functions of each separated beam segment are expanded in terms of Jacobi polynomials. The solutions are achieved through the variational approach. The proposed methodology can deal with arbitrary boundary restraints in a unified way by conveniently changing correlated parameters without interfering with the solution procedure.

Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

Abstract We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

Existence of solution of a three-point boundary value problem via variational approach

By the use of a constructing adequate real functional and choosing an appropriate admissible function space, the existence and multiplicity of solutions to a three-point second-order differential system is proved via Mountain Pass Lemma. The interesting point is the boundary value conditions are imposed on admissible space rather than the functionals.

A minimum entropy principle in the compressible multicomponent Euler equations

In this work, the space of admissible entropy functions for the compressible multicomponent Euler equations is explored, following up on Harten (J. Comput. Phys. 49 (1983) 151–164). This effort allows us to prove a minimum entropy principle on entropy solutions, whether smooth or discrete, in the same way it was originally demonstrated for the compressible Euler equations by Tadmor (Appl. Numer. Math. 49 (1986) 211–219).

Low Hemoglobin Levels at Admission Are Independently Associated with Cognitive Impairment after Ischemic Stroke: a Multicenter, Population-Based Study.

Data on the association between hemoglobin (Hb) levels and poststroke cognitive function are limited. We investigated the relationship between Hb concentrations at admission and poststroke cognitive function using a multicenter database. In total, 1081 patients were recruited from seven Chinese medical centers within 6months after experiencing ischemic stroke. Cognitive status was evaluated with a series of brief neuropsychological tests. A subgroup of 439 patients from a single center was followed up for 4-6years and was eventually reassessed with a cognitive test. The association between Hb and cognitive impairment was analyzed by multivariable Tobit regression and logistic regression. The mean age of the 920 eligible participants at study entry was 42.5years; 311 (34%) were women, and all participants were Chinese nationals who lived locally. After adjustment for multiple covariables, Hb levels at admission remained positively associated with poststroke Mini-Mental State Examination (MMSE) scores, with a 0.37-point increase in the MMSE score for every 1-standard-deviation increase in the Hb level. Moreover, an optimal Hb level above 15.0g/dl was proposed for preventing or alleviating the development of poststroke cognitive impairment in men. After 4-6years of rehabilitation, the baseline Hb still correlated with MMSE scores. A significant interaction was found between baseline Hb and change in MMSE scores over time, with higher baseline Hb levels predicting faster recovery of global cognitive performance (β, 0.21; 95% confidence interval, 0.03-0.39).These findings warrant further study of anemia as a risk factor for poststroke cognitive impairment.

Average spectrum method for analytic continuation: Efficient blocked-mode sampling and dependence on the discretization grid

The average spectrum method is a promising approach for the analytic continuation of imaginary time or frequency data to the real axis. It determines the analytic continuation of noisy data from a functional average over all admissible spectral functions, weighted by how well they fit the data. Its main advantage is the apparent lack of adjustable parameters and smoothness constraints, using instead the information on the statistical noise in the data. Its main disadvantage is the enormous computational cost of performing the functional integral. Here we introduce an efficient implementation, based on the singular value decomposition of the integral kernel, eliminating this problem. It allows us to analyze the behavior of the average spectrum method in detail. We find that the discretization of the real-frequency grid, on which the spectral function is represented, biases the results. The distribution of the grid points plays the role of a default model while the number of grid points acts as a regularization parameter. We give a quantitative explanation for this behavior, point out the crucial role of the default model and provide a practical method for choosing it, making the average spectrum method a reliable and efficient technique for analytic continuation.

Dynamic analysis of stepped functionally graded piezoelectric plate with general boundary conditions

A stepped functionally graded piezoelectric (FGPM) plate model is proposed for the first time, and its free and forced vibration are studied by using the domain energy decomposition method. The segmentation technique is used to discretize the structure along the length direction. At the structural boundary and piecewise interface, the weight parameters are introduced to satisfy the boundary conditions and the coordination conditions between the piecewise interfaces. On this basis, the boundary conditions of subdomains can be regarded as free boundary a constraint, which reduces the difficulty in constructing the displacement admissible function. Because all the structures of subdomains are the same, the displacement admissible functions of them are uniformly obtained by the two-dimensional Jacobian orthogonal polynomial expansion. The potential energy function of the plate is based on the first-order shear deformation theory. The displacement admissible function is substituted into the potential energy function, then the standard variational operation is used to obtain the solution equation of the dynamic characteristics of the FGPM plate. Through the numerical calculation, the superior calculation performance of the method is proved, and it is not limited to the boundary conditions. On this basis, the effects of geometric and material parameters on free and forced vibration of FGPM plate are also discussed.

A unified modeling method for the rotary enclosed acoustic cavity

The rotary enclosed acoustic cavity has three common forms, which are conical, cylindrical and spherical cavity. They are widely used in practical production and belong to the category of rotary room acoustics. Therefore, the study of their acoustic characteristics is of great engineering significance. In this paper, a unified analytical model for acoustic characteristics of rotary enclosed cavity with various impedance walls is first established. A three-dimensional (3D) modified Fourier series method is proposed to construct the admissible function of sound pressure. Specifically, the sound pressure function is invariably expressed as a 3D trigonometric series superposition, which includes the multiplication of three cosine functions and six complementary polynomials. The introduction of complementary polynomials can effectively solve the impedance acoustic wall. Based on Rayleigh-Ritz energy method, the acoustic characteristics of the unified analysis model can be obtained. The sound pressure response of the cavity under the influence of different impedance walls is further studied by placing a point sound source inside the acoustic cavity. The accuracy of the unified model is verified by comparing the present results with those obtained by finite element method (FEM) and experiment, and the effect of important parameters on the acoustic characteristics is systematically studied.

Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

This research predicts theoretically post-critical axial buckling behavior of truncated conical carbon nanotubes (CCNTs) with several boundary conditions by assuming a nonlinear Winkler matrix. The post-buckling of CCNTs has been studied based on the Euler–Bernoulli beam model, Hamilton’s principle, Lagrangian strains, and nonlocal strain gradient theory. Both stiffness-hardening and stiffness-softening properties of the nanostructure are considered by exerting the second stress-gradient and second strain-gradient in the stress and strain fields. Besides small-scale influences, the surface effect is also taken into consideration. The effect of the Winkler foundation is nonlinearly taken into account based on the Taylor expansion. A new admissible function is used in the Rayleigh-Ritz solution technique applicable for buckling and post-buckling of nanotubes and nanobeams. Numerical results and related discussions are compared and reported with those obtained by the literature. The significant results proved that the surface effect and the nonlinear term of the substrate affect the CCNT considerably.

Application of semi-analytical method to vibration analysis of multi-edge crack laminated composite beams with elastic constraint

In this paper, the semi-analytical method is first extended to study the vibration behavior of multi-edge crack laminated composite beams with rectangular cross-section. The establishment of theoretical analysis model is on the basis of the Timoshenko theory for laminated beam. The laminated composite beam is segmented along the edge crack position, and the flexibility coefficient of fracture mechanics theory is used to ensure the essential continuity condition of the segmented joint interface. The general elastic boundary can be realized by introducing the virtual boundary technique at both ends of the beam. The admissible displacement functions for each beam segment are expressed by Jacobi orthogonal polynomials in a more general form. The convergence and correctness of the proposed method are verified by numerical tests. On this basis, the parametric study for vibration characteristics of multi-edge crack laminated beam is carried out to provide reference data for subsequent researchers.

Exploration of the encased nanocomposites functionally graded porous arches: Nonlinear analysis and stability behavior

This paper explores the nonlinear stability mechanism of the functionally graded porous (FGP) arch reinforced by graphene nanocomposites. Both the pores and the nanocomposites are distributed symmetrically to the mid-surface of the arch but not uniformly in the cross-section so that the bending stiffness can be best improved. The arch is confined in an elastic medium with a radially-pointed concentrated load at the crown position. The confinement of the medium results in a symmetrical deformed shape of the arch, which can be described by an admissible displacement function. Associated with the thin-walled arch theory and the principle of minimum potential energy, analytical predictions are obtained to express the critical buckling load, as well as the hoop force and bending moment. Subsequently, a numerical model is developed to simulate the medium and the arch in ABAQUS software. By introducing the modified arc-length method, the equilibrium paths of the encased arch are traced. After comparison in terms of the critical buckling load and the equilibrium paths, it is found the numerical results are in good accordance with the analytical solutions. Finally, particular attention is paid to the parameters that may impact the buckling load, such as the porosity coefficient, the weight fraction, the central angle, the geometry of the Graphene platelets (GPLs) et al.

On the plastic buckling of curved carbon nanotubes

This research, for the first time, predicts theoretically static stability response of a curved carbon nanotube (CCNT) under an elastoplastic behavior with several boundary conditions. The CCNT is exposed to axial compressive loads. The equilibrium equations are extracted regarding the Euler–Bernoulli displacement field by means of the principle of minimizing total potential energy. The elastoplastic stress-strain is concerned with Ramberg–Osgood law on the basis of deformation and flow theories of plasticity. To seize the nano-mechanical behavior of the CCNT, the nonlocal strain gradient elasticity theory is taken into account. The obtained differential equations are solved using the Rayleigh–Ritz method based on a new admissible shape function which is able to analyze stability problems. To authorize the solution, some comparisons are illustrated which show a very good agreement with the published works. Conclusively, the best findings confirm that a plastic analysis is crucial in predicting the mechanical strength of CCNTs.

Simple Utility Design for Welfare Games under Global Information

Welfare game is a game-theoretic model for resource allocation problem which is to find an allocation to maximize the welfare function. In order to determine it in a distributed way, each agent is assigned to an admissible utility function such that the resulting game possesses desirable properties, for example, scalability, existence and efficiency of pure Nash equilibria, and budget balance. In this paper, supposing that each agent can access the global information, marginal contribution based utility design is proposed. It is shown that utility functions based on the above design have scalability and existence of pure Nash equilibria. Furthermore, efficiency is the same as that of the conventional utility design via Shapley value.

Black's Inverse Investment Problem and Forward Criteria with Consumption

We study an inverse investment problem proposed by Black and provide necessary and sufficient conditions for a given function to be an admissible indirect utility function in a log-normal market; w...