Results In Potential Theory Research Articles

Polyharmonic potential theory on the Poincaré disk

We consider the open unit disk D equipped with the hyperbolic metric and the associated hyperbolic Laplacian L. For λ∈C and n∈N, a λ-polyharmonic function of order n is a function f:D→C such that (L−λI)nf=0. If n=1, one gets λ-harmonic functions. Based on a Theorem of Helgason on the latter functions, we prove a boundary integral representation theorem for λ-polyharmonic functions. For this purpose, we first determine nth-order λ-Poisson kernels. Subsequently, we introduce the λ-polyspherical functions and determine their asymptotics at the boundary ∂D, i.e., the unit circle. In particular, this proves that, for eigenvalues not in the interior of the L2-spectrum, the zeroes of these functions do not accumulate at the boundary circle. Hence the polyspherical functions can be used to normalise the nth-order Poisson kernels. By this tool, we extend to this setting several classical results of potential theory: namely, we study the boundary behaviour of λ-polyharmonic functions, starting with Dirichlet and Riquier type problems and then proceeding to Fatou type admissible boundary limits.

Numerical Evaluation of Hydrodynamic Coefficients in Roll Motion

The determination of ship motions in a seaway heavily depends on hydrodynamic coefficients. However, the coefficients related to roll-motion exhibit nonlinear behavior due to viscous effects and relatively small wave damping, which pose significant challenges. To overcome these challenges, several studies have employed computational fluid dynamics (CFD) to analyze hydrodynamic performance, leveraging the advancements in computational performance. Nevertheless, limited research has investigated the error and uncertainty of these numerical solutions. In this study, we conducted a forced oscillation test on a ship-like section that induces lateral vortex shedding using the OpenFOAM with overset grid method. We validated the numerical results of the roll hydrodynamic coefficients with experiments and the results of potential theory. We used the procedure proposed by the ITTC to perform uncertainty analysis on the grid size and time step size, which presented the overall uncertainty of the numerical analysis results. Our findings indicate that the grid uncertainty of the added moment of inertia was significant in the low frequency region, while the time step uncertainty was significant in the high frequency region.

Potential theory with multivariate kernels

In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions between triples or, more generally, $n$-tuples of particles. Such objects, which arise naturally in various fields, present subtle differences and complications when compared to the classical two-input case. We introduce appropriate analogues of conditionally positive definite kernels, establish a series of relevant results in potential theory, explore rotationally invariant energies on the sphere, and present a variety of interesting examples, in particular, some optimization problems in probabilistic geometry which are related to multivariate versions of the Riesz energies.

Experimental Investigation of the Flow Field in the Vicinity of an Oscillating Wave Surge Converter

The main objective of this paper is to characterize the flow field on the front face of an oscillating wave surge converter (OWSC) under a regular wave. For this purpose, the longitudinal and vertical velocity components were measured using an Ultrasonic Velocity Profiler (UVP). In order to explain the main trends of the OWSC’s dynamics, the experimental data were firstly compared with the analytical results of potential theory. A large discrepancy was observed between experimental and analytical results, caused by the nonlinear behavior of wave-OWSC interaction that determine the turbulent field and the boundary layer. The experimental velocity field shows a strong ascendant flow generated by the mass transfer over the flap (overtopping) and flow rotation generated by the beginning of the flap deceleration and acceleration. These features (overtopping and flow rotation) have an important role on the power capture of OWSC and, therefore, analytical results are not accurate to describe the complex hydrodynamics of OWSC.

Recurrence of Markov chain traces

Nous montrons que les graphes pour lesquels la marche aléatoire simple est transiente n’admettent pas de chaîne de Markov transiente aux plus proches voisins (même non réversible) visitant toutes les arêtes avec probabilité positive, tandis qu’il en existe une pour le réseau carré $\mathbb{Z}^{2}$. En particulier, le réseau $d$-dimensionnel $\mathbb{Z}^{d}$ admet une telle chaîne de Markov seulement lorsque $d=2$. Lorsque $d=2$ nous présentons un exemple de Gady Kozma, et le résultat général est obtenu en prouvant que la trace de toute chaîne de Markov sur un espace d’état dénombrable qui admet une mesure stationnaire est presque sûrement récurrente pour la marche simple. Le cas où la chaîne est réversible a été traité par Gurel-Gurevich, Lyons et le premier auteur. Nous exploitons des résultats récents de théorie du potentiel sur les chaînes de Markov non réversibles pour étendre leur résultat au cas non réversible.

Effect of the angle of attack on the transient lift during the interaction of a vortex with a flat plate. Potential theory and experimental results

The dynamics of a two-dimensional vortex interacting with a flat plate at different angles of attack α is analysed using potential flow theory based on conformal mapping varying the nondimensional separation distance h∕c of the upstream incoming vortex to the plate (c is the chord length of the plate) and the vortex intensity Γl. Transient lift forces measured in a wind tunnel are also compared with the potential theory results for a given Γl and several values of h∕c and α. For the Reynolds number considered in the experiments (about 25 000) it is found that the potential theory predicts reasonably well the transient fluctuation in the lift force provided that the separation distance is not too close to the critical one h∗∕c at which the vortex trajectory given by the potential theory bifurcates. We find that the separation distance generating the highest induced lift is around this critical value h∗∕c, which, according to the potential theory, is displaced about −2.3(1−0.07|Γl|1∕2)α in relation to the zero angle of attack for the same Γl. Potential theory also predicts that the maximum peak of the lift fluctuation depends on α only through the relative separation |h−h∗|∕c, and that the maximum lift is substantially larger when a clockwise vortex passes below the plate than when it passes above the plate, for the same vortex intensity Γl and relative separation distance. The opposite happens for a counter-clockwise vortex. This asymmetry in the maximum lift fluctuation increases slightly with |Γl|, approaching a ratio of almost two for large |Γl|.

Gravity Anomaly of Polyhedral Bodies Having a Polynomial Density Contrast

We analytically evaluate the gravity anomaly associated with a polyhedral body having an arbitrary geometrical shape and a polynomial density contrast in both the horizontal and vertical directions. The gravity anomaly is evaluated at an arbitrary point that does not necessarily coincide with the origin of the reference frame in which the density function is assigned. Density contrast is assumed to be a third-order polynomial as a maximum but the general approach exploited in the paper can be easily extended to higher-order polynomial functions. Invoking recent results of potential theory, the solution derived in the paper is shown to be singularity-free and is expressed as a sum of algebraic quantities that only depend upon the 3D coordinates of the polyhedron vertices and upon the polynomial density function. The accuracy, robustness and effectiveness of the proposed approach are illustrated by numerical comparisons with examples derived from the existing literature.

Analytical Solution of the Cerruti Problem Under Linearly Distributed Horizontal Loads over Polygonal Domains

The classical Cerruti problem of an isotropic homogeneous half-space subject to a concentrated load tangential to its surface is extended to cope with linear distributions of loads acting over a polygonal domain. The approach is based upon a generalized version of the Gauss theorem and recent results of potential theory which consistently take into account the singularities affecting the expressions of the fields of interest. This issue, which has been recently dealt within the literature by exploiting generalized constitutive theories, is successfully addressed in the paper within classical elasticity theory by proving that uneliminable singularities can be experienced only at the vertices of the loading region and only for a single component of stress. Analytical expressions of displacements, strains and stresses are derived at an arbitrary point of the half-space as a function of the loading function, assumed to be linear, and of the position vectors which define the boundary of the loaded region. The proposed approach is validated by numerical examples.

A General Approach to the Solution of Boussinesq’s Problem for Polynomial Pressures Acting over Polygonal Domains

We outline a general approach for extending the classical Boussinesq’s solution to the case of pressures distributed according to a polynomial law of arbitrary order over a polygonal domain. To this end we exploit a generalized version of the Gauss theorem and recent results of potential theory which consistently take into account the singularities affecting the expressions of the fields of interest, an issue which seems to have been overlooked in the existing literature. For linearly varying pressures we derive analytical expressions of displacements, strains and stresses at an arbitrary point of the half-space as a function of the loading function and of the position vectors which define the boundary of the loaded region. We briefly discuss how bilinear and more general pressure distributions can be accommodated in our formulation since the paper is mainly motivated by the interest in developing efficient computational tools for solving 3D problems in foundation engineering and contact mechanics. Finally, comparisons with existing solutions and numerical examples are discussed.

The Gravity Anomaly of a 2D Polygonal Body Having Density Contrast Given by Polynomial Functions

An analytical solution is presented for the gravity anomaly produced by a 2D body whose geometrical shape is arbitrary and where the density contrast is a polynomial function in both the horizontal and vertical directions. Approximating the real shape of the body by a polygon, the solution is expressed as sum of algebraic quantities that depend only upon the coordinates of the vertices of the polygon and upon the polynomial density function. The solution presented in the paper, which refers to a third-order polynomial function as a maximum, exhibits an intrinsic symmetry that naturally suggests its extension to the case of higher-order polynomials describing the density contrast. Furthermore, the gravity anomaly is evaluated at an arbitrary point that does not necessarily coincide with the origin of the reference frame in which the density function is assigned. Invoking recent results of potential theory, the solution derived in the paper is shown to be singularity-free and numerically robust. The accuracy and effectiveness of the proposed approach is witnessed by the numerical comparisons with examples derived from the existing literature.

Contact problems for a finitely deformed incompressible elastic halfspace

This paper examines the class of problems related to the interaction between a finitely deformed incompressible elastic halfspace and contacting elements that include smooth, flat rigid indenters with elliptical and circular shapes and a thick plate of infinite extent. The contact between the finitely deformed elastic halfspace and the contacting elements is assumed to be bilateral. The interaction between both the rigid circular indenter and the finitely deformed halfspace is induced by a Mindlin force that acts at the interior of the halfspace regions and by exterior loads. Similar considerations apply for the contact between the flexible plate of infinite extent and the finitely deformed elastic halfspace. The theory of small deformations superposed on large deformations proposed by Green et al. (Proc R Soc Ser A 211:128–155, 1952) is used as the basis for the formulation of the problem, and results of potential theory and integral transform techniques are used to develop the analytical results. In particular, explicit results are presented for the displacement of the rigid elliptical indenter and the maximum deflection of the flexible plate induced by the Mindlin forces, when the finitely deformed halfspace region has a strain energy function of the Mooney–Rivlin form.

On a generalized Love's problem

We present explicit expressions for computing the displacements induced in a homogeneous, linearly elastic half-space by uniform vertical pressure applied over an arbitrary polygonal region of the horizontal surface. By suitably applying Gauss theorem and recent results of potential theory we derive formulas which allow one to evaluate the displacements at an arbitrary point of the half-space solely as a function of the position vectors of the boundary of the loaded region assumed to be polygonal. Representative numerical examples referred to geodetically observed elastic displacements of the Earth surface due to water loads show the effectiveness and the flexibility of the proposed approach. Actually, it allows for a more realistic evaluation of displacements distribution and to achieve a considerable simplification in data handling since it is now possible to avoid tiling of complex regions by the simple load shapes, such as circles or rectangles, for which analytical solutions are currently available in the literature.

BIBEE: a Rigorous and Computationally Efficient Approximation to Continuum Electrostatics

The computational costs associated with modeling biomolecular electrostatics using continuum theory have motivated numerous approximations, such as Generalized-Born (GB) models, that can be computed in much less time. Unfortunately, most of these approximate models abandon physics in favor of computational efficiency. On the other hand, a new approximation method for molecular electrostatics, called BIBEE (boundary-integral-based electrostatics estimation), retains the underlying physics of continuum theory, but is nearly as efficient as Generalized-Born models. The BIBEE approach derives from well-known results in potential theory and the theory of boundary-integral equations. Three main results demonstrate the value BIBEE may hold for biomolecular analysis and design. First, the integral-equation theory clarifies the origin of accuracy of the Coulomb-field approximation (CFA). Second, BIBEE models offer significantly better accuracy for individual pairwise interactions, relative to GB methods. Third, BIBEE readily provides provable upper and lower bounds to the electrostatic solvation free energy of the original (exact) continuum-theory problem.

A FEM-BEM approach for electro-magnetostatics and time-harmonic eddy-current problems

In this paper we propose a method for solving the electro-magnetostatics problem and the eddy current problem in terms of suitable potentials. A new variational formulation is devised, in which standard results of potential theory are used to reduce the problem in the external domain to an integral equation on the boundary of a computational domain containing the conductor. The existence and uniqueness of the solution is proved, by showing that the associated sesquilinear form is coercive. A numerical approximation scheme, based on nodal finite elements in the computational domain and boundary elements on its boundary, is devised and proved to be convergent. It is also shown that the solution of the time-harmonic eddy current problem tends to the solution of the electro-magnetostatics problem as the frequency tends to 0. The same convergence holds, uniformly with respect to the mesh size, for the finite element solutions.

A global Riesz decomposition theorem on trees without positive potentials

We study the potential theory of trees with nearest-neighbor transition probability that yields a recurrent random walk and show that, although such trees have no positive potentials, many of the standard results of potential theory can be transferred to this setting. We accomplish this by defining a non-negative function H, harmonic outside the root e and vanishing only at e, and a substitute notion of potential which we call H-potential. We define the flux of a superharmonic function outside a finite set of vertices, give some simple formulas for calculating the flux and derive a global Riesz decomposition theorem for superharmonic functions with a harmonic minorant outside a finite set. We discuss the connection of the H-potentials with other notions of potentials for recurrent Markov chains in the literature.

The potential theory method for a half-plane crack and contact problems of piezoelectric materials

The half-plane crack and contact problems for transversely isotropic piezoelectric materials are exactly analyzed. The potential theory method is employed with the resulting integro-differential (for crack problem) and integral (for contact problem) equations having identical structures with those reported earlier in the literature. Existing results in potential theory are thus utilized to obtain complete solutions of the problems under consideration. In particular, for the half-plane crack, both the permeable and impermeable electric conditions at the crack surfaces are considered. The solutions for the permeable crack and half-plane contact are entirely new to the literature.

Nonlinear transient wave–body interactions in steady uniform currents

Fully nonlinear monochromatic wave three-dimensional body interactions are studied with and without the presence of steady uniform currents. The mixed initial boundary value problem is set-up in a numerical wave tank (NWT) and solved by an indirect desingularized boundary integral equation method (DBIEM). The Laplace equation is solved at each time-step and time-dependent nonlinear free surface boundary conditions are simultaneously integrated by a mixed Eulerian–Lagrangian (MEL) method. A piston-type wave maker is used for generating the incident waves and a steady current is imposed everywhere in the computational domain at t=0 +. A spatially varying sponge layer on the free surface is devised to dissipate the outgoing waves. The resulting influence coefficient matrix is divided into sub-matrices by domain decomposition technique and solved simultaneously by block line Jacobi method (BJM). A specially devised preconditioning matrix is used to accelerate the convergence rate of the matrix equation by obtaining the minimized condition number of the influence coefficient matrix. Computations are performed for the nonlinear diffractions of steep monochromatic waves by a truncated vertical cylinder both without currents and in the presence of uniform coplanar or adverse currents. A φ n -type adaptive beach is developed and its performance compared with some beaches in the literature. Results of NWT simulations are compared with the first-order potential theory (Buchmann et al., Proceedings of the Seventh International Offshore and Polar Engineering Conference, Honolulu, Hawaii, USA, 1997), second-order diffraction theory [Kim and Yue, J. Fluid Mech. 200 (1989) 235–264], modified marker and cell (MAC) method (Park et al., International Journal for Numerical Methods in Fluids 29 (1999) 685–703) and the experimental and second-order potential theory results of Mercier and Niedzwecki [Proceedings of the Seventh International Conference on Behavior of Offshore Structures, vol. 2, TX, USA, 1994, pp. 265–287].

Alternative Methods in Spectral Factorization. A Modeling and Design Tool

Spectral factorization can be used to recover the complex transfer function of a linear, causal, stable, minimum-phase system from merely its amplitude information. Two different approaches are presented, resulting in two consistent expressions for the complex transfer function. Firstly, an approach using Fourier theory is followed (Papoulis, 1977; Priestley, 1981). Secondly, a new approach using potential theory results is presented. Spectral factorization can be successfully used as a modeling tool. Moreover, its capability to serve as a design tool is emphasized. These fields of application are illustrated by means of examples.

Complete and exact solutions of a penny-shaped crack in a piezoelectric solid: antisymmetric shear loadings

This paper presents an exact analysis of the problem of a penny-shaped crack in a transversely isotropic piezoelectric medium subjected to arbitrary shear loading that is antisymmetric with respect to the crack plane. The analysis is based on the general solution of three-dimensional piezoelasticity, which is represented by four quasi-harmonic displacement functions. It is shown that these harmonics can be represented by one complex potential. By using the previous results in potential theory, an exact solution is obtained. In particular, for uniform shear and point shear loadings, complete expressions for the elastoelectric field are derived in terms of elementary functions.

A penny-shaped crack in piezoelectrics: resolved

This paper reconsiders the problem of a penny-shaped crack in a piezoelectric medium under uniform mechanical as well as electric loadings applied at infinity (mode I). The analysis is based on the general solution of three-dimensional piezoelasticity, which is represented by four quasi-harmonic displacement functions. It is shown that these harmonics can be expressed in terms of two potentials of a simple layer. By using the previous results in potential theory, exact solution of the mode I crack is obtained. In particular, the solution is complete and in terms of elementary functions.