Quasi-harmonic Functions Research Articles

On the spatial spread of active control in a van der Pol ring via synchronisation and its stabilisation using parameter mismatch

This work explores spatial propagation of amplitude control in a network of self-excited oscillators having ring topology. We consider a ring of N Van der Pol oscillators, with a first order control model connected to an arbitrary kth oscillator in the network. It is shown that even such a simple one-dimensional control, applied to the kth oscillator, can be made to propagate to other oscillators due to the synchronising property of the network, bringing about the control of their amplitudes also. The slow flow equations of amplitude and phase of the complete model, obtained analytically, are used to study the spatial propagation of control in the network. The oscillator responses are approximated by quasi-harmonic functions whereas the first order control response is approximated by the superposition of relaxation and oscillatory terms. The obtained analytical approximations are validated by comparing them with numerical simulation. The effect of the synchronising property of the network on the spread of amplitude control is investigated and it is shown that the extent and speed of amplitude control increases with increase in coupling strength. It is further shown that the control strategy is not effective when the control gain parameter is greater than the square of linear natural frequency of the oscillator (γ>ωk2). We propose the use of parameter mismatch between oscillators as a tool for overcoming this limitation. Frequency detuning, brought about by parameter mismatch, is shown to stabilise the fixed point at the origin and cause amplitude control even outside the effective range of the control mechanism. This shows that slight differences between the oscillators, unavoidable during manufacturing, can be beneficial for the spatial spread of control in the network.

Green’s functions of steady-state transversely isotropic hygro-thermo-magneto-piezoelectric cones

Green’s functions of transversely isotropic hygro-thermo-magneto-piezoelectric (HTMP) cones are established. The governing equations of transversely isotropic HTMP media are first simplified by introducing two displacement functions. Steady-state general solutions of transversely isotropic HTMP media expressed by seven quasi-harmonic functions are obtained by the operator theory, superposition principle, and generalized Almansi’s theorem. Furthermore, the three-dimensional problems of a semi-infinite HTMP cone subjected to a point moisture source and a point heat source are analyzed by introducing the potential functions which are determined by moisture, thermal, electric, magnetic equilibrium relationships and boundary conditions. Furthermore, the fundamental solutions of the HTMP cones under the action of the point sources are obtained. The numerical examples and the contours of the coupled fields of HTMP cones are presented, and the variation regularity of stress, electric displacement, and magnetic induction are discussed. The effects of different apex angles of cones on the coupled fields are also analyzed.

Classification of metric measure spaces and their ends using p-harmonic functions

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite \(p\)-energy \(p\)-harmonic and \(p\)-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local \(p\)-Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the \(p\)-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant \(p\)-harmonic functions with finite \(p\)-energy as spaces having at least two well-separated \(p\)-hyperbolic sequences of sets towards infinity. We also show that every such space \(X\) has a function \(f \notin L^p(X) + \mathbf{R}\) with finite \(p\)-energy.

Removable singularities for bounded [formula omitted]-(super)harmonic and quasi(super)harmonic functions on weighted [formula omitted

It is well known that sets of p-capacity zero are removable for bounded p-harmonic functions, but on metric spaces there are examples of removable sets of positive capacity. In this paper, we show that this can happen even on unweighted Rn when n>p, although only in very special cases. A complete characterization of removable singularities for bounded A-harmonic functions on weighted Rn, n≥1, is also given, where the weight is p-admissible. The same characterization is also shown to hold for bounded quasiharmonic functions on weighted Rn, n≥2, as well as on unweighted R. For bounded A-superharmonic functions and bounded quasisuperharmonic functions on weighted Rn, n≥2, we show that relatively closed sets are removable if and only if they have zero capacity.

Analysis of 3D planar crack problems in one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part I: Theoretical formulations

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are performed systematically. Part I of this work derives a series of theoretical formulations that are then used to study the 3D planar crack problems in the QCs. The simple layer potential functions with the extended displacement discontinuities (EDDs) as the unknown variables and the general solution based on quasi-harmonic functions for the QCs under consideration are used to deduce the boundary equations that govern 3D planar crack problems. The hypersingular integral equation method is used to analyze the asymptotic singularities of the coupled thermal-electrical-phonon-phason fields near the crack edge. Expressions are then presented for the extended stress intensity factors (ESIFs) of a mixed model crack in terms of EDDs for arbitrarily-shaped cracks in the QCs, and the basic relationships between the energy release rate and the ESIFs are established. Closed-form solutions for some typical cracks, including an elliptical crack that is subjected to coupled electrical-phonon-phason loadings and a penny-shaped crack that is subjected to antisymmetric thermal loading, are determined via Fabrikant's analysis method. Additionally, both the physical quantities on the crack plane and the corresponding variables in the coupled thermal-electrical-phonon-phason field in the full space are given. The theoretical formulations derived in this paper provide a fundamental basis for development of the numerical approach proposed in Part II of our work, and can also serve as benchmarks for numerical solutions.

General solutions for elasticity of transversely isotropic materials with thermal and other effects: A review

The well-known general solution for uncoupled thermoelasticity of isotropic bodies was proposed by Goodier, which has been utilized extensively since its birth in 1937. When the steady-state response is considered, the temperature field satisfies Laplace’s equation, and the corresponding elastic field can be expressed in terms of a harmonic function, giving rise to the Williams solution. However, for anisotropic bodies, there are no steady-state thermoelastic general solutions that are expressed in terms of (quasi)harmonic functions until 2000. This article presents a short review of the harmonic general solutions for uncoupled elasticity of transversely isotropic materials with thermal and other effects. These solutions are obtained by simply but forcedly combining the heat conduction equation with other governing equations (e.g., the Navier equations). We will show that the Williams solution as well as some other solutions all can be deduced as the special cases. Two application scenarios of the general solutions are also highlighted to demonstrate their elegance and versatility.

Three-dimensional steady-state general solution for isotropic hygrothermoelastic media

ABSTRACTThe potential theory method was utilized to derive the steady-state general solution for three-dimensional (3D) hygrothermoelastic media. Two displacement functions are introduced to simplify the governing equations, with which the elastic, moisture, and temperature fields are thus simplified. Using the differential operator theory and superposition principle, all the physical quantities can be expressed in terms of two functions, one of which satisfies a harmonic equation and the other satisfies an eight-order partial differential equation. With the aid of generalized Almansi’s theorem, all the physical quantities like displacements, moisture, and temperature are expressed in terms of five quasi-harmonic functions for various cases of material characteristic roots. The obtained general solutions are in simple form and thus they may bring more convenience to certain boundary problems. As an example, the fundamental solutions for a point moisture source combined with heat source in the interior of infinite hygrothermoelastic body are presented by virtue of the obtained general solution. A planar crack of arbitrary shape in an infinite medium subjected to mechanical, moisture, and temperature loads is investigated to illustrate the application of the solution in boundary value problems. Specifically, for a penny-shaped crack under uniform combined loads, the complete, exact solutions are presented.

On the Uniqueness of the Solution of the Dirichlet Boundary Value Problem for Quasiharmonic Functions in a Non-Unit Disk

In this article the uniqueness of the solution of Dirichlet boundary value problem for quasiharmonic functions in arbitrary disk Tr+ = {z: |z| < r}, where r ≠ 1, is established. Also the non-uniqueness of the solution of this problem in a unit disk is proven.

О НЕУСТОЙЧИВОСТИ РЕШЕНИЙ ОДНОРОДНОЙ КРАЕВОЙ ЗАДАЧИ РИМАНА ДЛЯ КВАЗИГАРМОНИЧЕСКИХ ФУНКЦИЙ В КРУГОВЫХ ОБЛАСТЯХ

We consider the boundary value problem of Riemann type (the conjugation problem) in the classes of piecewise quasiharmonic functions. A homogeneous problem of a Riemann type problem in the classes of piecewise quasiharmonic functions of the second kind in circular domains is studied in details. In particular, in this case a clear solution method is developed for a homogeneous problem of Riemann type, the logical essence of which consists in reducing the solution of the homogeneous problem under consideration to a sequential solution of the common homogeneous Riemann problem for analytic functions and two second-order linear differential Euler equations. Moreover, instability of the solutions of the homogeneous problem is determined with respect to the change in the radius value of the considered circular domain, and a complete picture of its solvability for different values of the index of the problem and the radius of the circular domain is constructed. It is proved that the main reason for the instability of the solutions of a homogeneous problem of Riemann type in classes of piecewise quasiharmonic functions of the second kind in circular domains with respect to the change in the radius value of the considered circular domain is the fact that the number of linear independent analytic solutions of homogeneous differential Euler equations, to which the desired Riemann type problem is reduced, depends essentially on the radius of the considered circular region. In this paper we use the methods of the theory of functions of a complex variable, the theory of integral equations, and the analytic theory of differential equations.

Quantum Metric of Classic Physics

By methods of differential geometry and number theory the following has been established: All fundamental physical constants are the medians of quasi-harmonic functions of relative space and relative time. Basic quantum units are, in fact, the gradients of normal distribution of standing waves between the points of pulsating spherical spiral, which are determined only by functional bonds of transcendental numbers PI and E. Analytically obtained values of rotational speed, translational velocity, vibrational speed, background temperature and molar mass give the possibility to evaluate all basic quantum units with practically unlimited accuracy. Metric of quantum physics really is two-dimensional image of motion of waves in three-dimensional space. Standard physical model is correct, but SI metric system is insufficiently exact at submillimeter distances.

Three-dimensional fundamental thermo-elastic solutions applied to contact problems

This paper aims to develop three-dimensional fundamental thermo-elastic solutions for an infinite/half-infinite space of a two-dimensional hexagonal quasi-crystal, which is subjected to a point heat source. Starting from the newly developed general solution in terms of quasi-harmonic potential functions, the corresponding fundamental solutions are derived by means of the trial-and-error technique. Six appropriate potential functions involved in the general solution are observed. The present fundamental solutions are applied to construct boundary integral equations governing the contact problems. Numerical calculations are performed to show the distributions of the thermo-elastic coupling field variables in a half-space subjected to a point thermal source.

О КАРТИНЕ РАЗРЕШИМОСТИ ОДНОРОДНОЙ КРАЕВОЙ ЗАДАЧИ ГИЛЬБЕРТА ДЛЯ КВАЗИГАРМОНИЧЕСКИХ ФУНКЦИЙ В КРУГОВЫХ ОБЛАСТЯХ

A Hilbert-type boundary value problem in the classes of quasi-harmonic functions is considered. Quasi-harmonic functions are regular solutions of an elliptic differential equation form $\frac{\partial ^2 W}{\partial z\partial \overline z} + \frac{n(n + 1)}{(1 + z \overline z)^2}W = 0$, where $\frac{\partial }{\partial z} = \frac{1}{2} ( \frac{\partial }{\partial x} - i\frac{\partial }{\partial y})$ , $\frac{\partial }{\partial \overline z} = \frac{1}{2} (\frac{\partial }{\partial x} + i\frac{\partial }{\partial y})$ , and n is a given positive integer. Using the fact that a circle is an analytic curve, we have developed an explicit method for finding solutions of the Hilbert homogeneous boundary value problem for quasi-harmonic functions in circular domains. The principal logic of this method consists of two stages. At stage one we are using a representation of quasi-harmonic function via analytic function and its derivatives to reduce the problem to the classical Hilbert problem for some auxiliary analytic function in the circular domain. A solution Φ( z ) for this problem will be used at stage two, when we solve the linear differential Euler equation of order n with the right-hand side Φ( z ). General solution for the problem can be explicitly expressed in terms of the solution of the Euler equation. Moreover, we have established that the solvability for the considered boundary-value problem depends essentially on whether a unit circumference is the carrier of boundary conditions or a non-unit circle

On the Green’s functions for a two-phase soft electroactive medium subjected to biasing fields

We consider the problem of a point force and/or a point charge applied in the interior of an infinite two-phase medium, which consists of two perfectly or smoothly bonded soft electroactive half-spaces. Both two half-spaces are initially subjected to an arbitrary multi-directional stretch and a biasing electric field. A linearized incremental theory is adopted to describe the deformation caused by the point force and/or point charge. A concise general solution to the governing equations of the perturbed infinitesimal elastic and electric fields is employed. By the trial-and-error method, the analytical expressions for the quasi-harmonic functions in the general solution are constructed explicitly, but with several undetermined constants. These constants can be easily determined from the conditions on the perfect or smooth interface as well as some other considerations. As special cases of the present two-phase Green’s functions, the generalized Mindlin solution and Lorentz solution for an electroactive half-space subjected to an initial biasing field are discussed. The more specified generalized Boussinesq solution and Cerruti solution are also given in the appendix.

Fundamental solutions in a half space of two-dimensional hexagonal quasicrystal and their applications

Fundamental phonon-phason field in a half-infinite space of two-dimensional hexagonal quasicrystal is derived, on the basis of general solutions in terms of quasi-harmonic functions, by virtue of the trial-and-error technique. Extended Boussinesq and Cerruti problems are studied. Appropriate potential functions are assumed and corresponding fundamental solutions are explicitly derived in terms of elementary functions. The boundary integral equations governing the contact and crack problems are constructed from the present fundament solutions. The obtained analytical solutions can serve as guidelines for future indentation tests via scanning probe microscopy and atomic force microscopy methods.

General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect

By introducing four potential functions, the governing equations of plane problems in 1D orthorhombic quasicrystals with piezoelectric effect are composed of four second-order partial differential equations, in which the quasi-harmonic functions are the essential unknowns. The general solution of these equations is further established, and all expressions are expressed in terms of the potential functions. As an application of the general solution, the closed-form solutions are obtained for wedge problems or half-plane problems of 1D orthorhombic piezoelectric quasicrystals.

General Solutions for Three-Dimensional Thermoelasticity of Two-Dimensional Hexagonal Quasicrystals and an Application

By introducing four displacement functions, the governing equations of the three-dimensional thermoelasticity of two-dimensional hexagonal quasicrystals are decoupled into two uncorrelated problems. Two higher-order displacement functions are introduced to represent the general solutions, which are eighth-order and fourth-order, respectively, for the two problems. By taking a decomposition and superposition procedure, the general solutions are further simplified in seven cases in terms of six quasi-harmonic displacement functions. To show the application of the general solutions obtained, a closed form solution is obtained for an infinite space containing a penny-shaped crack, subjected to a uniformly distributed temperature at the crack surface.

Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect

In this paper, a set of 3D general solutions to static problems of 1D hexagonal piezoelectric quasicrystals is obtained by introducing two displacement functions and utilizing the rigorous operator theory. All the physical quantities are expressed by five quasi-harmonic functions. Based on the general solutions and with the help of the superposition principle, fundamental solutions for infinite/half-infinite spaces are presented by trial-and-error technique. The general solutions can be conveniently used to solve the boundary value problems regarding dislocations, cracks and inhomogeneities. The fundamental solutions are of primary significance to development of numerical codes such as boundary element method.

On 2D Green's functions for 1D hexagonal quasi-crystals

This paper cokncerns itself with two-dimensional fundamental solutions for infinite and semi-infinite planes, subjected to point heat sources, of one-dimensional hexagonal quasi-crystals. From the basic equations for plane problem in the context of thermo-elasticity of QCs, the rigors operator theory and generalized Almansi's theorem are employed to derive the general solutions in terms of four quasi-harmonic functions. In the framework of the present general solutions, three concreted examples are investigated. Appropriate potentials are set by the trail-and-error technique, and corresponding fundamental thermo-elastic fields are obtained in a complete and exact fashion. Since the fundamental solutions are all in elementary functions, the present solutions not only bear theoretical merits, but also can serve as benchmarks to clarify various approximate methods.

Indentation of a compressible soft electroactive half-space: Some theoretical aspects

We theoretically study the indentation response of a compressible soft electroactive material by a rigid punch. The half-space material is assumed to be initially subjected to a finite deformation and an electric biasing field. By adopting the linearized theory for incremental fields, which is established on the basis of a general nonlinear theory for electroelasticity, the appropriate equations governing the perturbed infinitesimal elastic and electric fields are derived particularly when the material is subjected to a uniform equibiaxial stretch and a uniform electric displacement. A general solution to the governing equations is presented, which is concisely expressed in terms of four quasi-harmonic functions. By adopting the potential theory method, exact contact solutions for three common perfectly conducting rigid indenters of flat-ended circular, conical and spherical geometries can be derived, and some explicit relations that are of practical importance are outlined.

Three-dimensional thermo-elastic general solutions of one-dimensional hexagonal quasi-crystal and fundamental solutions

This Letter presents three-dimensional general solutions for static problems in thermo-elasticity of one-dimensional hexagonal quasi-crystals. Two displacement potentials are introduced to simplify the equilibrium equations in terms of displacement and temperature. Rigorous operator theory and generalized Almansiʼs theorem are applied to derive the general solutions in terms of five quasi-harmonic functions. To show the significance of the general solutions, a semi-infinite space and an infinite space, both of which are subjected to a heat source, are considered. In these two cases, closed form fundamental phonon–phason-elastic fields are expressed by elementary functions, which play an important role in numerical simulations.