Formulation Of Boundary Value Problem Research Articles

A variational principle for problems with a hint of convexity

A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As a result, we study several super-critical semilinear Elliptic problems.

MATHEMATICAL MODELS AND METHODS OF SOLVING SPATIAL GENERALIZED BOUNDARY VALUE PROBLEM HEAT ROTATING HOLLOW PIECEWISE HOMOGENEOUS CYLINDER

Context. The phenomenological theory of heat conduction speed of heat propagation is assumed infinitely large. However, the high intensity of the observed transient processes such as explosions, supersonic flow, high speeds of rotation of the use of this assumption leads to errors, so it is necessary to take into account that the distribution of heat takes place at a finite rate.Objective. Development of a new generalized mathematical model of the temperature distribution in the hollow piecewise uniform cylinder in the form of a boundary value problem of mathematical physics for the heat equation and the solution of the resulting boundary value problem.Method. The use of known integral Laplace transforms, Fourier series, and developed a new integral transformation for piecewise homogeneous space.Results. Found Polga transient temperature field of a circular cylinder in a cylindrical coordinate system, a piecewise homogeneous polar radius direction, which rotates at a constant angular velocity about the axis OZ, with the ultimate heat propagation speed. Thermal properties of each layer does not depend on the temperature at an ideal thermal contact between the layers, and there are no internal sources of heat.Conclusions. For the first time developed a mathematical model of the temperature distribution in the empty piecewise uniform cylinder, which rotates at a constant angular velocity about the axis OZ, taking into account the finite speed of propagation of heat in the form of mathematical physics boundary value problem for hyperbolic partial differential equations of heat conduction with boundary conditions of the first kind. Thermal properties are in each layer does not depend on the temperature at an ideal thermal contact between the layers, and there are no internal sources of heat.Created a new integral transform of a piecewise-homogeneous space, with which found the temperature field of the hollow piecewise homogeneous circular cylinder in the form of convergent orthogonal series of Bessel functions and of Fourier. The obtained analytical solution of a generalized boundary value problem of heat transfer cylinder, which rotates, given a finite amount of heat propagation velocity can be used for modeling of temperature fields, which occur in many technical systems (satellites, forming rolls, turbines, etc.).

Modeling of magnetic hystereses in soft MREs filled with NdFeB particles

Herein, we investigate the structure-property relationships of soft magnetorheological elastomers (MREs) filled with remanently magnetizable particles. The study is motivated from experimental results which indicate a large difference between the magnetization loops of soft MREs filled with NdFeB particles and the loops of such particles embedded in a comparatively stiff matrix, e.g. an epoxy resin. We present a microscale model for MREs based on a general continuum formulation of the magnetomechanical boundary value problem which is valid for finite strains. In particular, we develop an energetically consistent constitutive model for the hysteretic magnetization behavior of the magnetically hard particles. The microstructure is discretized and the problem is solved numerically in terms of a coupled nonlinear finite element approach. Since the local magnetic and mechanical fields are resolved explicitly inside the heterogeneous microstructure of the MRE, our model also accounts for interactions of particles close to each other. In order to connect the microscopic fields to effective macroscopic quantities of the MRE, a suitable computational homogenization scheme is used. Based on this modeling approach, it is demonstrated that the observable macroscopic behavior of the considered MREs results from the rotation of the embedded particles. Furthermore, the performed numerical simulations indicate that the reversion of the sample’s magnetization occurs due to a combination of particle rotations and internal domain conversion processes. All of our simulation results obtained for such materials are in a good qualitative agreement with the experiments.

Adjoint-Based Optimization Procedure for Active Vibration Control of Nonlinear Mechanical Systems

In this paper, a new computational algorithm for the numerical solution of the adjoint equations for the nonlinear optimal control problem is introduced. To this end, the main features of the optimal control theory are briefly reviewed and effectively employed to derive the adjoint equations for the active control of a mechanical system forced by external excitations. A general nonlinear formulation of the cost functional is assumed, and a feedforward (open-loop) control scheme is considered in the analytical structure of the control architecture. By doing so, the adjoint equations resulting from the optimal control theory enter into the formulation of a nonlinear differential-algebraic two-point boundary value problem, which mathematically describes the solution of the motion control problem under consideration. For the numerical solution of the problem at hand, an adjoint-based control optimization computational procedure is developed in this work to effectively and efficiently compute a nonlinear optimal control policy. A numerical example is provided in the paper to show the principal analytical aspects of the adjoint method. In particular, the feasibility and the effectiveness of the proposed adjoint-based numerical procedure are demonstrated for the reduction of the mechanical vibrations of a nonlinear two degrees-of-freedom dynamical system.

Analysis of meshless weak and strong formulations for boundary value problems

This paper introduces a weak meshless procedure combined with a multi-resolution numerical integration and its comparison with a strong local meshless formulation for approximating displacement and strain modeled in the form of Elliptic Boundary Value Problems (EBVPs) in one- and two-dimensional spaces. Assets and losses of both strong and weak meshless approaches are considered in detail. The meshless weak formulation considered in the current paper is the well-known Element Free Galerkin (EFG) method whereas the Local Radial Basis Functions Collocation Method (LRBFCM) is taken as a strong formulation. First aspect of the current work is implementation of the new numerical integration techniques introduced in Siraj-ul-Islam et al. (2010) and Aziz et al. (2011) [1,2] in the EFG method and its comparison with numerical integration based on standard Gaussian quadrature, adaptive integration and stabilized nodal integration techniques used in the context of EFG and other allied weak meshless formulations. Second aspect of the current work is analysis of comparative performance of the localized versions of strong and weak meshless formulations. Standard numerical tests are conducted to validate performance of both the approaches.

Analysis of stable model inversion methods for constrained underactuated mechanical systems

Inverse dynamics methods are presented and analyzed in detail for the feedforward control of underactuated manipulators. Thereby, two different exact strategies are discussed: the standard stable inversion method, which relies on the formulation of a two-point boundary value problem, and an alternative optimization problem formulation, which does not require any boundary conditions. This paper proves that the solution of the optimization problem converges to the solution of the boundary value problem in the limit case where the lengths of the pre- and post-actuation phases tend to infinity. It is also shown that both strategies can be directly based on the DAE form of the equations of motion, so that there is no need to explicitly derive the input-output normal form, and standard multibody modeling codes can be used. The developments are illustrated using examples of manipulators with passive and compliant joints.

Numerical simulation of heat and mass transfer in a closed two – phase geothermal thermosyphon

Mathematical model of the heat and mass transfer process in a two-phase thermosyphon with allowance for the coolant phase transformations is presented in this study. The formulated boundary-value problem was solved by the finite differences method. The numerical experiment results are presented and the temperature fields are obtained for different operation times of the geothermal thermosyphon.

Operators of Minimal Norm via Modified Green’s Function for the Elastic Two-Dimensional Case Using the Simple Multipole Coefficients

The problem of non-uniqueness arising in the integral formulation of an exterior boundary value problem for the elastic two-dimensional case can be faced using the fundamental solution technique. In this work a criterion based on the minimization of the norm of the modified integral operator is established using simple multipole coefficients. As an example of the proposed procedure the case of the circle and perturbations of circle are examined.

Microscale modeling and simulation of magnetorheological elastomers at finite strains: A study on the influence of mechanical preloads

Herein, we present a numerical study on the deformation dependent behavior of magnetorheological elastomers with structured and unstructured particle distributions. To this end, finite element simulations are performed in order to calculate the effective magnetization and macroscopic actuation stresses for different specimens with realistic microstructures and varying mechanical preloads. Since the proposed microscale model is based on a continuum formulation of the magnetomechanical boundary value problem, the local magnetic and mechanical fields are resolved explicitly within the microstructures. The consideration of finite strains results in a finite element implementation of the coupled field problem for which a consistent linearization scheme is presented. In order to provide a better understanding of the deformation dependent behavior in real specimens, a study on chain-like structures is performed. It reveals that the interaction of the constituents in chain-like structures yields different material responses depending on their position. These findings are used to explain the influence of mechanical preloads on the behavior of samples with structured and unstructured arrangements of the particles. All of our results are in good agreement with experimental investigations which have been carried out for magnetorheological elastomers comprising a structured particle distribution.

Generalized compatibility equations for tensors of high ranks in multidimensional continuum mechanics

Compatibility equations are derived for the components of generalized strains of rank m associated with generalized displacements of rank m − 1 by analogs of Cauchy kinematic relations in n-dimensional space (multi-dimensional continuous medium) (m ≥ 1, n ≥ 2). These relations can be written in the form of equating to zero all components of the incompatibility tensor of rank m(n − 2) or its dual generalized Riemann–Christoffel tensor of rank 2m. The number of independent components of these tensors is found; this number coincides with that of compatibility equations in terms of generalized strains or stresses. The inequivalence of the full system of compatibility equations to any of its weakened subsystems is discussed, together with diverse formulations of boundary value problems in generalized stresses in which the number of equations in a domain can exceed the number of unknowns.

Mixed finite-element formulations in piezoelectricity and flexoelectricity.

Flexoelectricity, the linear coupling of strain gradient and electric polarization, is inherently a size-dependent phenomenon. The energy storage function for a flexoelectric material depends not only on polarization and strain, but also strain-gradient. Thus, conventional finite-element methods formulated solely on displacement are inadequate to treat flexoelectric solids since gradients raise the order of the governing differential equations. Here, we introduce a computational framework based on a mixed formulation developed previously by one of the present authors and a colleague. This formulation uses displacement and displacement-gradient as separate variables which are constrained in a 'weighted integral sense' to enforce their known relation. We derive a variational formulation for boundary-value problems for piezo- and/or flexoelectric solids. We validate this computational framework against available exact solutions. Our new computational method is applied to more complex problems, including a plate with an elliptical hole, stationary cracks, as well as tension and shear of solids with a repeating unit cell. Our results address several issues of theoretical interest, generate predictions of experimental merit and reveal interesting flexoelectric phenomena with potential for application.

Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems

The fourth-order boundary value problems of one parameter gradient-elastic bar and plane strain/stress models are formulated in a variational form within an H2 Sobolev space setting. For both problems, the existence and uniqueness of the solution is established by proving the continuity and coercivity of the associated symmetric bilinear form. For completeness, the full sets of boundary conditions of the problems are derived and, in particular, the new types of boundary conditions featured by the gradient-elastic models are given the additional attributes singly and doubly. By utilizing the continuity and coercivity of the continuous problems, corresponding error estimates are formulated for conforming Galerkin formulations. Finally, numerical results, with isogeometric Cp−1-continuous discretizations for NURBS basis functions of order p≥2, confirm the theoretical results and illustrate the essentials of both static and vibration problems.

HYPERBOLIC MODEL OF NON -STATIONARY THERMAL CONDUCTIVITY

The article presents fundamentally new results on the analytical theory of thermal conductivity for hyperbolic transport models. The questions of the correct formulation of boundary value problems are considered. A technique for finding exact analytical solutions of a rather complex class of boundary value problems based on the method of Green's functions and operational calculus is developed.

Unidirectional flows of binary mixtures within the framework of the Oberbeck–Boussinesq model

Existence conditions are investigated for the unidirectional flows of binary mixtures describable by the equations of motion in the Oberbeck–Boussinesq approximation with account for the thermodiffusion effect. Possible solutions are classified, known and novel classes of exact solutions being distinguished. For the solutions obtained different formulations of boundary-value problems are proposed. Flows between two rigid walls with given heat distribution laws are described.

Dissipative formulation of initial boundary value problems for Friedrichs’ systems

ABSTRACTIn this article we present a dissipative definition of a solution for initial boundary value problems for Friedrichs’ systems posed in the space . We study the information contained in this definition and prove an existence and uniqueness theorem in the non-characteristic case and with constant coefficients. Finally, we compare our choice of boundary condition to previous works, especially on the wave equation and show how to model additional constrained problems in view of initial boundary value problems for viscoplastic equations.

Four-wave mixing and transverse–longitudinal oscillatory modes in plasmonic nanoparticles: nonlinear theory from variational principles and mathematical simulation

In this study we propose a theory of nonlinear plasmonic oscillations in metal nanoparticles subjected to optical irradiation which is further reduced to solvable mathematical model by applying variational procedures and numerical methods. One of the goals pursuing under construction of the theory has been to minimize its axiomatics. A governing equation of motion is deduced on the basis of the least action principle and supplemented by a continuity equation that results from variational and differential formulations of charge conservation. The presence of irreversible processes is taken into account by leveraging of a polynomial dissipation term embedded into the action functional. An initial boundary value problem formulation for nonlinear integrodifferential equations constituting the model of electron gas dynamics has been given. On the ground of a finite-difference approximation an iterative method for treating the problem as well as the algorithm and solver code have been worked out. By utilizing these tools we investigate the third order field effects such as third harmonic generation and four-wave mixing by a group of metal nanoparticles illuminated by a multispectral source of light. An excitation of transverse–longitudinal mode of plasmonic oscillations has been observed under plane wave illumination of two spherical particles. The charge and current distributions intrinsic to the mode have been investigated numerically together with an electromagnetic field scattered by the particles into far zone. The estimate of the density function parameter providing regular dynamics has been analytically obtained by the contraction principle.

Optical Short-Arc Association Hypothesis Gating via Angle-Rate Information

Given a short-arc optical observation with estimated angle rates, the admissible region is a compact region in the range/range-rate space defined such that all likely and relevant orbits are contained within it. An alternative boundary value problem formulation has recently been proposed where range/range hypotheses are generated with two angle measurements from two tracks as input. In this paper, angle-rate information is reintroduced as a means to eliminate hypotheses by bounding their constants of motion before a more computationally costly Lambert solver or differential correction algorithm is run.

A coupled finite element/equivalent source formulation for transient structural-acoustic problems

Recent research suggests that instability problems that occur in the time domain boundary element formulations are manifestations of nonuniqueness difficulties, and that remedies similar to those used in frequency domain formulations can address the difficulties and improve solution stability. For frequency domain equivalent source formulations, the difficulties can be addressed by combining simple and dipole sources together to form “tripole sources”. The basic goals of the current research are to develop an analogous equivalent source formulation for transient acoustic boundary value problems and to combine it with structural finite element analyses to solve transient coupled structural-acoustic problems. A brief description of the equivalent source formulation is given along with several validation cases for acoustic boundary value problems, including one with a closed boundary surface where nonexistence difficulties should occur. The formulation requires convolution summations, which are independent of each other and a brief discussion is given of how the computations can be parallelized. A brief description is also given of the coupled finite element/equivalent source formulation for transient structural-acoustic boundary value problems. Several examples are given to validate the formulation and to demonstrate that the tripole source formulation for the equivalent sources is more stable than the simple source version.

Periodic Solutions of Mathematical Models of Micromechanics Under Periodic Impulse Action

We consider mathematical models of two microelectromechanical systems describing the motion of a movable electrode in a micro-gap under the action of the repetitive pulse electrostatic field between movable and fixed electrodes. We formulate boundary value problems with periodicity conditions and found the range of parameters corresponding to the existence of two solutions such that one of them is stable, whereas the other is unstable. Bibliography: 11 titles. Illustrations: 9 figures.

Axisymmetric problem on the indentation of a hot circular punch into an arbitrarily nonhomogeneous half-space

This paper addresses an axisymmetric quasi-static contact problem on both thermal and mechanical interactions of a circular punch and a nonhomogeneous elastic half-space whose mechanical and thermophysical properties are assumed to be arbitrary functions of the depth coordinate. To solve the formulated boundary value problem, the technique of Hankel’s integral transformation is used. Using the bilateral asymptotic method, an approximate analytical–numerical solution is constructed for the heat flux, boundary displacements of the half-space, and contact stress under the punch. Numerical implementation is performed for different dependences of the material properties on the depth of the half-space.