Showing papers on "Elliptic coordinate system published in 2010"
TL;DR: In this paper, a semi-analytical solution methodology for the linear hydrodynamic diffraction induced by arrays of elliptical cylinders subjected to incident waves is presented, where the solution of the Laplace equation in elliptic coordinates for both the incident and the diffracted waves is formulated analytically in terms of the even and odd periodic and radial Mathieu functions.
43 citations
TL;DR: In this paper, a numerical method was developed for computing the shape of an infinite three-dimensional hydrostatic meniscus originating from interior contact lines whose projection in the horizontal plane has a specified shape.
Abstract: A numerical method is developed for computing the shape of an infinite three-dimensional hydrostatic meniscus originating from interior contact lines whose projection in the horizontal plane has a specified shape. The Laplace-Young equation determining the meniscus shape is solved in orthogonal curvilinear coordinates generated by conformal mapping using a finite-difference method. The elevation of the contact lines is either prescribed or computed as part of the solution to ensure a specified contact angle. The method is applied to study the hydrostatic meniscus developing around an elliptical vertical cylinder using elliptic coordinates and between two parallel vertical circular cylinders using bipolar coordinates. The results illustrate variations in the contact angle or contact line distribution due to the boundary geometry and furnish numerical estimates for the capillary force and torque.
21 citations
TL;DR: To extend the applicability to nonflat boundaries, this paper formulates one-way wave equations and corresponding reciprocity theorems in terms of curvilinear coordinates of the semiorthogonal (SO) type.
Abstract: One-way wave equations conveniently describe wave propagation in media with discontinuous and/or rapid variations in one direction, but with smooth and slow variations in the complementary transverse directions. In the past, reciprocity theorems have been developed in terms of one-way wave fields. The boundaries of the integration volumes and the variations of the medium parameters must adhere to strict conditions. The variations must have the smoothness required by pseudodifferential operators, while the boundaries have to be flat. To extend the applicability to nonflat boundaries, this paper formulates one-way wave equations and corresponding reciprocity theorems in terms of curvilinear coordinates of the semiorthogonal (SO) type. In SO coordinate systems, one of the covariant basis vectors is orthogonal to the others, which can be nonorthogonal among each other. The same applies to the contravariant basis vectors. Furthermore, the orthogonal directions coincide; that is, the orthogonal co- and contravariant basis vectors coincide. SO coordinates are characterized by a local property of the basis vectors. An extra specification is necessary to make them conform in any way to nonflat boundaries. This can be done in terms of so-called lateral Cartesian (LC) coordinates. Cartesian coordinates are mapped to LC coordinates by applying an invertible transformation to one coordinate while keeping the others the same. LC coordinates are a straightforward means to describe or conform to nonflat boundaries. Applications of the extended reciprocity theorems include removal of multiple reflections, removal of complex propagation effects, wave field extrapolation, and synthesis of unrecorded data.
17 citations
TL;DR: In this paper, the authors give a complete proof of the existence of an infinite set of eigenmodes for a vibrating elliptic membrane in one to one correspondence with the well-known eigen mappings for a circular membrane.
Abstract: We give a complete proof of the existence of an infinite set of eigenmodes for a vibrating elliptic membrane in one to one correspondence with the well-known eigenmodes for a circular membrane. More exactly, we show that for each pair $(m,n) \in \{0,1,2, \cdots\}^2$ there exists a unique even eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves and, similarly, for each $(m,n) \in \{0,1,2, \cdots\}\times \{1,2, \cdots\}$ there exists a unique odd eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves. Our result is based on directly using the separation of variables method for the Helmholtz equation in elliptic coordinates and in proving that certain pairs of curves in the plane of parameters $a$ and $q$ cross each other at a single point. As side effects of our proof, a new and precise method for numerically calculating the eigenfrequencies of these modes is presented and also approximate formulae which explain rather well the qualitative asymptotic behavior of the eigenfrequencies for large eccentricities.
16 citations
TL;DR: In this paper, the relation between elliptic coordinates and potentials is established, from which the semi-discrete Chen-Lee-Liu equations are decomposed into solvable ordinary differential equations.
16 citations
TL;DR: It is shown that aligning the methylene hydrogens to preserve the sign of a local dihedral angle, rather than minimizing a distance metric, provides significant improvements with respect to connection times and failures.
Abstract: A variety of methods (in total seven) comprising different combinations of internal and Cartesian coordinates are tested for interpolation and alignment in connection attempts for polypeptide rearrangements. We consider Cartesian coordinates, the internal coordinates used in CHARMM, and natural internal coordinates, each of which has been interfaced to the OPTIM code and compared with the corresponding results for united-atom force fields. We show that aligning the methylene hydrogens to preserve the sign of a local dihedral angle, rather than minimizing a distance metric, provides significant improvements with respect to connection times and failures. We also demonstrate the superiority of natural coordinate methods in conjunction with internal alignment. Checking the potential energy of the interpolated structures can act as a criterion for the choice of the interpolation coordinate system, which reduces failures and connection times significantly.
12 citations
TL;DR: The general-purpose partial differential equation (PDE) solver PDE2D uses a Galerkin finite element method, with standard triangular elements of up to fourth degree, to solve PDEs in general 2D regions, and once the global coordinate system is defined, the rest of the input is as simple as if the region were a rectangle.
10 citations
TL;DR: In this paper, a method of adaptive artificial viscosity (AAV2D-3D) for the solution of two-and three-dimensional equations of gas dynamics for Euler variables in the Cartesian coordinates system is considered.
Abstract: A method of adaptive artificial viscosity (AAV2D-3D) for the solution of two-and three-dimensional equations of gas dynamics for Euler variables in the Cartesian coordinates system is considered. This paper continues works [1, 2]. The computational scheme is described in detail and the results of the test case are given.
9 citations
TL;DR: In this article, the second order local approximate DtN boundary condition, suggested by the authors in a previous work, is investigated analytically when employed for solving high-frequency exterior Helmholtz problems.
5 citations
TL;DR: In this article, the material constants of perfectly matched layers (PMLs) in the cylindrical and spherical coordinates in the frequency domain were derived using the coordinate transformation laws of tensors on manifolds, the quotient rule, and complex coordinate stretching.
Abstract: The material constants of perfectly matched layers (PMLs) in the cylindrical and spherical coordinates in the frequency domain are presented. Using the coordinate transformation laws of tensors on manifolds, the quotient rule, and complex coordinate stretching, we obtain the material parameters of PMLs in the real coordinate. Our results show that PML parameters for elastic waves may be determined by the same procedure in the Cartesian coordinates. However, this rule has been determined for PML material constants derived from the analytic continuation in the cylindrical and spherical coordinates by Zheng and Huang in 2002. Our derivation based on differential forms shows that this rule holds for PML parameters in any orthogonal coordinate system.
5 citations
TL;DR: In this paper, the authors investigated the coordinate dependence of non-commutative theory by studying the solutions of noncommutive U(1, 1)× U( 1, 1)-Chern-Simons theory on AdS3 in polar and rectangular coordinates.
Abstract: We investigate the coordinate dependence of noncommutative theory by studying the solutions of noncommutative U(1, 1)× U(1, 1) Chern-Simons theory on AdS3 in polar and rectangular coordinates. We assume that only the space coordinates are noncommuting. When the two coordinate systems are equivalent only up to first order in the noncommutativity parameter θ, we investigate the effect of this non-exact equivalence between the two coordinate systems for two cases, a conical solution and a BTZ black hole solution in noncommutative AdS3, by using the Seiberg-Witten map. In each case, the noncommutative solutions in the two coordinate systems obtained from the same corresponding commutative solution turn out to be different even to the first order in θ.
01 Jan 2010
TL;DR: In this paper, the null-field boundary integral formulation in conjunction with the degenerate kernel and the Mathieu function in the elliptic coordinates is proposed to solve the hydrodynamic scattering problem by multiple circular and elliptical cylinders.
Abstract: In this paper, the null-field boundary integral formulation in conjunction with the degenerate kernel and the Mathieu function in the elliptic coordinates is proposed to solve the hydrodynamic scattering problem by multiple circular and elliptical cylinders. Based on the adaptive observer system, the present method can solve the water wave problem containing circular and elliptical cylinders at the same time in a semi-analytical manner. The closed-form fundamental solution is expressed in terms of the degenerate kernel in the polar and elliptic coordinates for circular and elliptical cylinders, respectively. Several examples are demonstrated to see the validity of the semi-analytical approach. Keyword: null-field boundary integral equation, degenerate kernel, Mathieu function, hydrodynamic scattering, elliptical cylinders.
TL;DR: In this article, an elliptic plate horizontally submerged in waves is investigated within the scope of linear wave theory, where an elliptical coordinate system is adopted to represent the solution in an analytical form, i.e., an expansion of eigen functions.
Abstract: With potential applications as a breakwater, an elliptic plate horizontally submerged in waves is investigated within the scope of linear wave theory. An elliptical coordinate system is adopted, which has an advantage to represent the solution in an analytical form, i.e. an expansion of eigen functions. By means of separation of variables, it turns out that the eigen functions in the elliptical coordinates consist of the Mathieu functions and the modified Mathieu functions. The interaction of the elliptic plate with the waves is studied. The wave loads, as well as the scattered wave field, are evaluated.
TL;DR: In this article, the Coulombian three-body system is studied using hyper-spherical elliptic coordinate system and the corresponding eigenvalue problem has been solved and the adiabatic potential and the binding energy of this system are calculated.
Abstract: Study of Coulombian three-body system is a basic phenomenon in muon catalyzed fusion (μCF). In this investigation, separation of variables in the base of adiabatic expansion, have been applied to the mesic three-body molecule, 3Heμd using hyper-spherical elliptic coordinate system. The corresponding eigenvalue problem has been solved and the adiabatic potential and the binding energy of this system are calculated. The obtained results agreed with the expected values of various theoretical methods.
11 Jul 2010
TL;DR: In this article, the ground plane carpet cloak was designed to control the traveling route of the electromagnetic waves. But, instead of the global coordinate transformation, a strategy of discrete coordinate transformation was proposed to design the ground-plane carpet cloak.
Abstract: The method of coordinate transformation provides us a practical approach to control the travelling route of the electromagnetic waves [1,2]. Coordinate transformation is applied to set up a distorted space from the free space by changing the permittivity or permeability distributions of the space background. The route of the wave is decided by how the space is distorted. Such scheme is firstly used for the design of the 'invisible cloak' [3]. Unfortunately, the general coordinate transformation often results in some singular points where extreme values of permittivities or permeabilities are required. Instead of the global coordinate transformation, a strategy of discrete coordinate transformation is proposed to design the ground-plane carpet cloak [4]. Discrete coordinate transformation is operated by transforming local coordinates to their peers in the other space. It has a good prospect of application and extension for designing flat devices in antenna systems.