Nonlinear Potential Problems Research Articles

Berezin density and planar orthogonal polynomials

We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density B ( z , ⋅ ) B(z,\cdot ) for the polynomial Bergman space, where the point z ∈ C z\in \mathbb {C} is fixed. When z = ∞ z=\infty , the Berezin density is expressed in terms of the squared modulus of the corresponding normalized orthogonal polynomial P P . We use an approximate version of this characterization to study the asymptotics of the orthogonal polynomials in the context of exponentially varying weights. This builds on earlier works by Its-Takhtajan and by the first author on a soft Riemann-Hilbert problem for planar orthogonal polynomials, where in place of the Laplacian we have the ∂ ¯ \bar \partial -operator. We adapt the soft Riemann-Hilbert approach to the nonlinear potential problem, where the nonlinearity is due to the appearance of | P | 2 |P|^2 in place of P ¯ \overline {P} . Moreover, we suggest how to adapt the potential theory method to the study of the asymptotics of more general Berezin densities B ( z , w ) B(z,w) in the off-spectral regime, that is, when z z is fixed outside the droplet. This is a first installment in a program to obtain an explicit global expansion formula for the polynomial Bergman kernel, and, in particular, of the one-point function of the associated random normal matrix ensemble.

Moving pseudo-boundary method of fundamental solutions for nonlinear potential problems

In the method of fundamental solutions (MFS), the solution of a boundary value problem (BVP) is approximated by a linear combination of fundamental solutions expressed in terms of sources which are located on a pseudo–boundary outside the domain of the problem. A major issue in such applications is the optimal placement of these sources. In this work we try to address this issue by letting some parameters which control the location of the sources free, to be determined as well as the coefficients of the fundamental solutions in the MFS approximations. When this approach is applied to nonlinear BVPs, this results in a system of nonlinear equations which can be solved by means of the MATLAB© routines lsqnonlin and fsolve. We investigate the effect of providing the exact Jacobian of the system to these routines to the overall efficacy of the approach. The results of several numerical experiments are presented and analyzed.

A FIXED POINT APPROACH TO STATIONARY HEAT TRANSFER IN ELECTRIC CABLES

In this paper we present a novel procedure for the determination of temperature in electric conductors. A Helmholtz-to-Poisson estimate is proved, that justifies to restrict the temperature dependence of the electrical resistivity to the conductor boundary. Hence we obtain a nonlinear potential problem for the relevant boundary temperatures, where the temperature dependence of the heat transfer coefficient is fully regarded. Using boundary integral operators, we represent the unknowns as the fixed point of a contraction. Finally a benchmark example is given in the rotationally symmetric case.

2D internal flux compatibility equation of the flux Green element method for transient nonlinear potential problems

AbstractThis article presents the derivation and implementation of the normal directional flux compatibility equation (relationship) at internal nodes when the Green element formulation that consistently provides accurate estimates of the primary variable, and its normal directional derivative (normal flux) is applied in 2D heterogeneous media to steady and transient potential problems. Such a relationship is required to resolve the closure problem due to having fewer integral equations than the number of unknowns at internal nodes. The derivation of the relationship is based on Stokes' theorem, which transforms the contour integral of the normal directional fluxes into a surface integral that is identically zero. The numerical discretization of the compatibility equation is demonstrated with four numerical examples using the six‐node quadratic triangular and the four and eight‐node rectangular elements. The incorporation of triangular elements into the current formulation demonstrates that the internal compatibility equation can be successfully implemented on irregular grids. The direct calculation of the fluxes significantly enhances the accuracy of the formulation, so that high accuracy, exceeding that of the finite element method, is achieved with very coarse spatial discretization. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Coupled Finite and Boundary Element Tearing and Interconnecting solvers for nonlinear potential problems

The present paper deals with the numerical solution of boundary value problems for nonlinear potential equations of the form –∇· [ν(|∇ u|) ∇ u] = f arising e. g. in 2D magnetic field computations. We apply Newton's method to the nonlinear variational formulation and solve the linearized problems with coupled Finite and Boundary Element Tearing and Interconnecting (FETI/BETI) methods. The spectrum of the subdomain Jacobi matrices may show high variation due to the nonlinear behavior of the reluctivity on ferromagnetic subdomains. A special preconditioner is proposed to overcome these problems. In order to get a good initial guess for the Newton iteration, we use the nested iteration strategy based on a hierarchy of nested grids. Finally, we discuss our first numerical results obtained from the numerical solution of some 2D magnetostatic model problems.

Indentation at a bimaterial interface: the line force solution

The problem considered is that of a line force acting normal to the free surface of a bimaterial welded together at the bimaterial interface. The constitutive equations of each constituent of the bimaterial are such that the stress–strain relation is linear in one half but non-linear in the other. The case where both satisfy non-linear stress–strain relations can also be dealt with by the method of this paper. It is shown that in general, there is no separable solution, singular at the application of the line force, which satisfies the field equations of each medium. Instead asymptotic solutions are constructed for the cases where the non-linear medium is incompressible with power law stress–strain relation (i.e. σ≈ ε N , with N>0) distinguishing between the cases where N is less or greater than unity. The characteristics of the asymptotic solutions are first illustrated by modelling non-linear potential problems, which can be viewed as anti-plane strain deformation.

Trefftz indirect method applied to nonlinear potential problems

Boundary type solution methodologies for linear elliptic boundary value problems (BVPs) are now well established. Unfortunately, this class of problems represents very few realistic practical situations due to the presence of nonlinearities in either boundary conditions or material properties. In this paper, we demonstrate an indirect formulation of the Trefftz method for solution of nonlinear BVPs arising in potential problems. While the concept of Trefftz method itself is fairly established, its application to nonlinear BVPs has been quite limited. Past experience indicates that Trefftz complete functions often give rise to polynomial type of nonlinearities and the resulting equations that govern the unknowns are almost always poorly conditioned, for which numerical solutions are difficult to obtain from standard procedures. For the solution of such nonlinear equations, we adopt a highly robust stabilized continuation method with inherent numerical error attenuation properties. Our numerical examples demonstrate this methodology when applied to 2D heat conduction problems with material and boundary condition nonlinearities, and the results indicate good potential for the use of indirect Trefftz method for nonlinear problems.

Application of the multiple reciprocity BEM to nonlinear potential problems

In this paper types of nonlinear potential problems are discussed and some of these problems are solved by the boundary element method. In order to avoid domain discretization the multiple reciprocity method is used. Solution of the problem is expressed in terms of a series which involves the higher order fundamental solutions and Laplacians of source terms. Convergence criteria as well as numerical examples are included.

A generalized Lane‐Emden‐Fowler equation

AbstractThe nonlinear potential problem Δ ϕ = –r2m 0, n = m is studied with r ⩾ 0, n ε (½, ∞), m ε (−1, ∞) and n m > 0.With various additional conditions we have regular and singular solutions which can be studied explicitly by homology transformation. It is proved that if m − n + 1 = 0 the solution with ϕ(0) = 1, ≠ (0) = 0 is radiusstable. Numerical analysis of a number of cases with m − n + 1 = 0 yields the same conclusion.

An algorithm for the calculation of poisson's equation in 2‐d semiconductor problems

AbstractA robust algorithm is described for the solution of the highly nonlinear potential problem in the analysis of semiconductors under a zero current assumption. Some examples are given showing the use of the algorithm.

Perturbation of nonlinear potential problems

Invariant imbedding can be viewed as a nonclassical perturbation technique which converts nonlinear boundary-value problems to quasilinear initial-value problems. The application of these ideas to the nonlinear partial differential equations of the type arising in potential theory is investigated.

On a non-linear theory of thin jets. Part 2. A linear theory for small injection angles

A non-linear potential problem which describes the injection of a high speed jet into a uniform flow of lower total head is linearized when the jet injection angle is small. The resulting linear problem is solved using the Wiener–Hopf technique. Numerical results for the position of the streamline separating the jet from the free stream and the pressure coefficient along the upstream wall are obtained for various injection angles.

On the Interaction of a Two‐Dimensional Jet with a Parallel Flow

Abstract : The injection of a two-dimensional jet of total head H1 from an infinite plate into a uniform stream of lower total head H2 is considered, the fluids being assumed inviscid and incompressible. Steady, irrotational solutions of Euler's equations are found for H1/H2 approaching infinity. The region behind the jet is treated as a stagnant wake with constant pressure equal to that of the undisturbed stream, and the jet injection angle is fixed across the jet opening. When a thin jet approximation is combined with Bernoulli's principle, a non-linear boundary condition is derived along the vortex sheet separating the jet from the external flow. The resulting non-linear potential problem (in the plane of the complex velocity potential) for the external flow is shown by Pal to be equivalent to a variational problem. A numerical procedure based on the variational principle and the Ritz-Galerkin method is used to solve for the case of normal injection with a digital computer. The pressure distribution along the plate upstream and streamline diagrams are given.