Harmonic Polynomials Research Articles

An adaptive harmonic polynomial cell method with immersed boundaries: Accuracy, stability, and applications

AbstractWe present a 2D high‐order and easily accessible immersed‐boundary adaptive harmonic polynomial cell (IB‐AHPC) method to solve fully nonlinear wave‐structure interaction problems in marine hydrodynamics using potential‐flow theory. To reduce the total number of cells without losing accuracy, adaptive quad‐tree cell refinements are employed close to the free‐surface and structure boundaries. The present method is simpler to implement than the existing IB‐HPC alternatives, in that it uses standard square cells both in the fluid domain and at the boundaries, thus without having to use the more complex and expensive overlapping grids or irregular cells. The spurious force oscillations on moving structures, which is a known issue for immersed boundary methods (IBMs), are eliminated in this study by solving a separate boundary value problem (BVP) for a Lagrangian acceleration potential. We also demonstrate that solving a similar BVP for the corresponding Eulerian acceleration potential is far less satisfactory due to the involved second derivatives of the velocity potential in the body‐boundary condition, which are very difficult to calculate accurately in an IBM‐based approach. In addition, we present, perhaps for the first time since the HPC method was developed, a linear matrix‐based stability analysis for the time‐domain IB‐AHPC method. The stability analysis is also used in this study as a general guide to design robust and stable numerical algorithms, in particular related to the treatment of boundary conditions close to the intersection between a Dirichlet and a Neumann boundary, which is essential in time‐domain wave‐structure interaction analyses using IBMs. We confirm theoretically through the stability analysis that square cells have the best stability properties. The present method has been verified and validated satisfactorily by various cases in marine hydrodynamics, including a moving structure in an infinite fluid, fully nonlinear wave generation and propagation, and fully nonlinear diffraction and radiation of a ship cross section.

Further results on modified harmonic functions in three dimensions

The Weinstein equation , with , considered in , is a modification of the classical Laplace equation . Its solutions are called k‐modified harmonic functions. Whereas for positive integers k the Weinstein equation is relatively well understood, little is known if the parameter k is negative.The main result of this article is the statement that in case the negative integers are even, i.e., , we still have a Fischer‐type decomposition. For , the classical harmonic functions, this decomposition is well known. But also in case , a Fischer‐type decomposition holds true, a Fischer‐type decomposition holds true. Surprisingly in case or and probably in all higher negative odd cases, the decomposition doesn't hold.In case , we give a complete description of the vector space of homogeneous k‐modified harmonic polynomials of degree n in . Such a result is also at hand in case . Finally, in case of the classical harmonic functions, we give a description of the vector space .

Approximation of Weak Solutions of the Laplace Equation by Harmonic Polynomials

A new proof based on F. Browder’s ideology is given for the theorem on the approximation of weak solutions of the Laplace equation in a bounded domain $$\Omega \subset {{\mathbb{R}}^{n}}$$ , $$n \geqslant 2$$ , with a connected Lipschitz boundary by harmonic polynomials in the Lebesgue space $${{L}_{p}}(\Omega )$$ and the Sobolev space $$W_{p}^{1}(\Omega )$$ .

Computational formulas and identities for new classes of Hermite‐based Milne–Thomson type polynomials: Analysis of generating functions with Euler's formula

The aim of this paper is to construct generating functions for a new family of polynomials, which are called parametric Hermite‐based Milne–Thomson type polynomials. Many properties of these polynomials with their generating functions are investigated. These generating functions give us generalization of some well‐known generating functions for special polynomials such as Hermite type polynomials, Milne–Thomson type polynomials, and Apostol type polynomials. Using the Euler formula, functional equation method for generating function, and differential operator technique, we give relations among parametric Hermite‐based Milne–Thomson type polynomials, the Bernoulli numbers, the Euler numbers, the Chebyshev polynomials, the Bernstein basis functions, homogeneous harmonic polynomials, and parametric kinds of Apostol type polynomials. Moreover, some computational formulas for these polynomials are derived. Finally, using Wolfram Mathematica version 12.0, some of these polynomials and their generating functions are illustrated by their plots under the special conditions. Potential relationships and connections of this paper's results with the results of previous and future research are pointed out.

Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic form

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions are constructed in all dimensions and solutions are classified in dimension at most $4$. Techniques are given for determining when two solutions are linearly conformally inequivalent.

On a subclass of close-to-convex harmonic mappings

For $\alpha > -1$ and $\beta >0, $ let $\mathcal{B}_{\mathcal{H}}^0(\alpha, \beta)$ denote the class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk $\mathbb{D}$ satisfying $|zh''(z)+\alpha(h'(z)-1)|\leq \beta-|zg''(z)+\alpha g'(z)|.$ First, we establish that each function belonging to this class is close-to-convex in the open unit disk if $\beta \in (0, 1+\alpha]$. Next, we obtain coefficient bounds, growth estimates and convolution properties. We end the paper with applications and will construct harmonic univalent polynomials belonging to this class.

The Electrostatic Potential of Periodic Crystals

The electrostatic potentials $\phi$ associated with neutral periodic crystals are defined by lattice sums that are never absolutely convergent. The sum depends on the order of summation. The mean zero periodic solution of Poisson's equation, denoted ${\underline \phi}$, is a natural potential. So is the potential obtained by the Mellin transform algorithm of [Borwein, Borwein, and Taylor, J. Math. Phys., 26 (1985), pp. 2999--3009]. We prove that these two are equal and are both equal to the potential obtained by Abel summation. The sum defining $\partial^\alpha \phi$ converges absolutely for $|\alpha|\ge 3$ to $\partial^\alpha{\underline \phi}$. The indeterminacy in the potential is at most a harmonic polynomial of degree 2.

Cartesian harmonic polynomials for a problem of deformation of a long, current-carrying elastic cylindrical conductor of nearly-circular normal cross-section

Harmonic Cartesian polynomial and rational functions are used to obtain a semi-analytical solution to the uncoupled, two-dimensional problem of thermo-magnetoelasticity for a long, transversely isotropic elastic cylinder of nearly-circular normal cross-section carrying an axial, steady electric current. Numerical results and plots are provided and discussed. Comparison with the case of a circular cross-section allows to assess the in uence of imperfection of the cross-sectional shape on the quantities of practical interest. The effect of dependence of the magnetic permeability on strain is investigated. We use the magnetic vector potential which is parallel to the axis of the cylinder. The mathematical problem reduces to the determination of ve harmonic functions, interrelated by a set of boundary conditions. Two of these harmonic functions combine to yield Airy's stress function in the cross-sectional domain in case the mechanical problem is solved in stresses. A third harmonic function enters in the formulation of the thermal problem for temperature distribution. The two remaining harmonic functions enter in the description of the vector potential distribution inside the conductor and in the surrounding region. The results nd application in calculating the deformation of straight central sections of electric conductors in various instruments, in particular busbars in electric power stations, and the stresses induced in them due to the electric current.

Concurrent Tri-band CMOS Power Amplifier Linearized by 3D Improved Memory Polynomial Digital Predistorter

This paper presents the design and post-layout results of a 5G power amplifier (PA) in 130 nm CMOS technology. The circuit incorporates a pre-amplification stage that uses a current reuse technique followed by a power stage. The PA operates in the 2–5 GHz range, with a power gain of 16–20.9 dB, output saturation power of 19.1–21.7 dBm, an output compression point of 15.8–20.4 dBm and maximum power-added efficiency of 9.28–22.4%. A three-dimensional improved memory polynomial (3DIMP) and a three-dimensional harmonic improved memory polynomial (3DHIMP) are introduced to the behavioral modeling and digital baseband predistortion of the designed PA for the cases without and with multiplicity among carriers, respectively. Moreover, an ascendant factor algorithm is presented for selecting the truncation factors. Simulation results adopting IEEE 802.11n at channel 1, LTE at channel 2 and IEEE 802.11ac at channel 3 show that the proposed 3DIMP and 3DHIMP, in comparison with previous three-dimensional multi-band (3DMB) and three-dimensional harmonic memory polynomial (3DHMP) having the same amount of parameters, can improve the normalized mean square error by up to 8.3 dB and 16.6 dB, respectively, when applied to the PA inverse behavioral modeling. Additional reductions in error vector magnitude up to 0.39 p.p. and 1.23 p.p. are also achieved when the 3DIMP and 3DHIMP are responsible for the PA linearization instead of previous 3DMB and 3DHMP, respectively.

Sensitivity of the Euler–Poinsot Tensor Values to the Choice of the Body Surface Triangulation Mesh

The inertial characteristics of celestial bodies can be calculated using their triangle partitions based on photometric observations. Such partitions can be refined along with the accumulation of necessary information. In this regard, the question arises to what extent the approximations of the inertial characteristics of celestial bodies, in particular, the approximations of the components of the Euler–Poinsot tensor of different orders, are susceptible to the choice of such partitions. Such components enter into the expansion of the gravitational potential in harmonic polynomials. In this paper, for some small celestial bodies, a comparison of such coefficients is carried out as coarse partitions are replaced with finer ones.

Generalized Eshelby Problem in the Gradient Theory of Elasticity

A generalized Eshelby problem of arbitrary order in the gradient elasticity for a multilayer inclusions of spherical shape with a polynomial strain field at infinity is considered. For this problem we propose a constructive method of solution in a closed finite form, using generalized Papkovich–Neuber representation and the system of canonical potentials based on harmonic polynomials. We use also the Gauss theorem on the decomposition of an arbitrary homogeneous polynomials. The solutions of the generalized Eshelby problem are applied in the method of asymptotic homogenization of the gradient elasticity to accurately calculation of the effective characteristics of composite materials with scale effects.

A structure theorem for almost low-degree functions on the slice

The Fourier-Walsh expansion of a Boolean function f: {0, 1}n → {0, 1} is its unique representation as a multilinear polynomial. The Kindler-Safra theorem (2002) asserts that if in the expansion of f, the total weight on coefficients beyond degree k is very small, then f can be approximated by a Boolean-valued function depending on at most O(2k) variables. In this paper we prove a similar theorem for Boolean functions whose domain is the ‘slice’ $$\left({\matrix{{\left[n \right]} \cr {pn} \cr}} \right) = {\rm{\{}}x \in {{\rm{\{}}0,1{\rm{\}}}^n}:\sum\nolimits_i {{x_i} = pn} {\rm{\}}}$$ , where 0 ≪ p ≪ 1, with respect to their unique representation as harmonic multilinear polynomials. We show that if in the representation of $$f:\left({\matrix{{\left[n \right]} \cr {pn} \cr}} \right) \to {\rm{\{}}0,1{\rm{\}}}$$ , the total weight beyond degree k is at most ϵ, where ϵ =min(p, 1 − p)O(k), then f can be O(ϵ)-approximated by a degree-k Boolean function on the slice, which in turn depends on O(2k) coordinates. This proves a conjecture of Filmus, Kindler, Mossel, and Wimmer (2015). Our proof relies on hypercontractivity, along with a novel kind of a shifting procedure. In addition, we show that the approximation rate in the Kindler-Safra theorem can be improved from ϵ + exp(O(k))ϵ5/4 to ϵ + ϵ2(2 ln(1/ϵ))k/k!, which is tight in terms of the dependence on ϵ and misses at most a factor of 2O(k) in the lower-order term.

A mimetic method for polygons

A new method for mimetic interpolation on polygonal meshes is described. This new method is based on harmonic function interpolation. Explicit formulas for harmonic functions on general polygons do not exist, so truncated harmonic polynomial expansions are used for computational efficiency. We show that the naive harmonic polynomial expansion, is not stable for arbitrary polygons. However, higher level truncations of harmonic interpolations are stable and accurate. This new method is shown to be a direct extension of the lowest order Raviart-Thomas finite elements to polygons. This method is also a direct extension of the finite volume MAC method to polygons. The accuracy of the interpolation is shown to be first-order irrespective of the polynomial truncation level. However the accuracy of vector Laplace equation solutions using this inner product is shown to be second-order accurate, in keeping with other lowest order Mimetic methods. The versatility of this numerical method is demonstrated on a multiphase incompressible flow problem with a density jump of 1000, on a moving polygonal mesh.

Factorization of special harmonic polynomials of three variables

We consider harmonic polynomials of real variables $x,y,z$ that are eigenfunctions of the rotations about the axis $z$. They have the form $(x\pm yi)^{n}p(x,y,z)$, where $p$ is a rotation invariant polynomial. Let ${\mathfrak R}_{m}$ be the family of the polynomials $p$ of degree $m$ which are reducible over the rationals. We describe ${\mathfrak R}_{m}$ for $m\leq5$ and prove that ${\mathfrak R}_{6}$ and ${\mathfrak R}_{7}$ are finite.

Neumann Type Problems for the Polyharmonic Equation in Ball

For Neumann type problems for the homogeneous polyharmonic equation in the unit ball we obtain necessary solvability conditions in the form of orthogonality of homogeneous harmonic polynomials to linear combinations of boundary functions with coefficients taken from the integer Neumann triangle. Bibliography: 18 titles.

On a subclass of close-to-convex harmonic mappings

In this paper, we introduce a new class of sense preserving harmonic mappings [Formula: see text] in the open unit disk and prove that functions in this class are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the sections of functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.

On -reflection of harmonic functions and -approximation by harmonic polynomials

We obtain several new sharp -continuity conditions, both necessary and sufficient, for operators of harmonic reflection of functions over boundaries of simple Carathéodory domains in . These results are based on a new criterion (also obtained in this paper) for -continuity of the Poisson operator in the aforesaid domains. As corollaries, we give new sufficient conditions for -approximability of functions by harmonic polynomials on boundaries of simple Carathéodory domains in . Bibliography: 17 titles.

Gradient systems of harmonic polynomials

We characterize all local phase-portraits of the finite and infinite singular points of the gradient systems defined by the real harmonic polynomials in two variables.We classify all the non-equivalent topological phase portraits of the gradient systems in the Poincaré disc defined by harmonic polynomials of degree less than five.

A high‐order harmonic polynomial method for solving the Laplace equation with complex boundaries and its application to free‐surface flows. Part I: Two‐dimensional cases

SummaryA high‐order harmonic polynomial method (HPM) is developed for solving the Laplace equation with complex boundaries. The “irregular cell” is proposed for the accurate discretization of the Laplace equation, where it is difficult to construct a high‐quality stencil. An advanced discretization scheme is also developed for the accurate evaluation of the normal derivative of potential functions on complex boundaries. Thanks to the irregular cell and the discretization scheme for the normal derivative of the potential functions, the present method can avoid the drawback of distorted stencils, that is, the possible numerical inaccuracy/instability. Furthermore, it can involve stationary or moving bodies on the Cartesian grid in an accurate and simple way. With the proper free‐surface tracking methods, the HPM has been successfully applied to the accurate and stable modeling of highly nonlinear free‐surface potential flows with and without moving bodies, that is, sloshing, water entry, and plunging breaker.

Orientation at singularities of harmonic functions

We find a simple expression in complex terms for homogeneous harmonic polynomials, which we use to express the Laurent series of a harmonic function around an isolated singularity. Also, we show a residue theorem and study the orientation at isolated singularities through the use of complex dilatation, focusing on those points where orientation is not preserved nor reversed, making essential the concept of exceptional set and extending it to isolated singularities.