Conformal Coordinate Transformation Research Articles

Design and adaptive finite element simulation of conformal transformation optics devices with isotropic materials

Traditional transform optical devices are designed from anisotropic magnetic materials which are difficult to fabricate. In this paper, we design and simulate a conformal carpet invisible cloak and bending waveguide device with isotropic material. According to the existing conformal coordinate transformation, we deduce the perfect parameters required by the conformal device. And the finite element method and error analysis of the model equation and the basic process of adaptive algorithm are given. We first simulated a carpet cloak and bending waveguide device with perfect parameters; then we used the idea of parameter segmentation to design a carpet cloak that requires 35 isotropic materials and use the idea of layering to design the conformal bending waveguide device, and relevant numerical results are presented; finally, we designed the adaptive finite element algorithm to realize the numerical simulation of Maxwell's equations in isotropic material.

Mapping topological states with compacted dimensions

With conformal coordinate transformations, compacted dimensions might be realized in photonic systems. Here we show, with a one-dimensional (1D) photonic model system, that the essential topological nature of a two-dimensional (2D) nanoribbon is conserved in a mapping without conformal transformations. We take topological edge states as an example. They exist in physical systems of different dimensions, such as Su-Schrieffer-Heeger (SSH) model in 1D with dimerized polyacetylene, or a Haldane model in a 2D hexagonal lattice with spin-orbit coupling. In addition, the 2D graphene nanoribbon model also hosts topological edge states if the edge is of ``zigzag'' type with nontrivial Zak phase. In this Letter, we introduce a mapping formalism which establishes the correspondence between the momentum space of the nanoribbon and the parameter space of a 1D diatomic chain. It explicitly demonstrates that topological origin of the nanoribbon system can be reduced to the SSH model, which reveals the unification of topological nature across different dimensions. As an example, we reconstruct the 2D graphene nanoribbon band structure, including topological edge states, using 1D optical waveguide arrays. The coupling coefficients between waveguides are directly transformed from nanoribbons' electron momenta. We believe that it is a powerful tool to connect physical phenomena in different dimensions and offers a new experimental methodology based on photonic platforms for condensed-matter research.

Space-time wave packets localized in all dimensions

Optical wave packets that are localized in space and time, but nevertheless overcome diffraction and travel rigidly in free space, are a long sought-after field structure with applications ranging from microscopy and remote sensing, to nonlinear and quantum optics. However, synthesizing such wave packets requires introducing non-differentiable angular dispersion with high spectral precision in two transverse dimensions, a capability that has eluded optics to date. Here, we describe an experimental strategy capable of sculpting the spatio-temporal spectrum of a generic pulsed beam by introducing arbitrary radial chirp via two-dimensional conformal coordinate transformations of the spectrally resolved field. This procedure yields propagation-invariant ‘space-time’ wave packets localized in all dimensions, with tunable group velocity in the range from 0.7c to 1.8c in free space, and endowed with prescribed orbital angular momentum. By providing unprecedented flexibility in sculpting the three-dimensional structure of pulsed optical fields, our experimental strategy promises to be a versatile platform for the emerging enterprise of space-time optics.

Conformal Symmetry and Supersymmetry in Rindler Space

This paper addresses the fate of extended space-time symmetries, in particular conformal symmetry and supersymmetry, in two-dimensional Rindler space-time appropriate to a uniformly accelerated non-inertial frame in flat 1+1-dimensional space-time. Generically, in addition to a conformal co-ordinate transformation, the transformation of fields from Minkowski to Rindler space is accompanied by local conformal and Lorentz transformations of the components, which also affect the Bogoliubov transformations between the associated Fock spaces. I construct these transformations for massless scalars and spinors, as well as for the ghost and super-ghost fields necessary in theories with local conformal and supersymmetries, as arising from coupling to two-dimensional (2-D) gravity or supergravity. Cancellation of the anomalies in Minkowski and in Rindler space requires theories with the well-known critical spectrum of particles that arise in string theory in the limit of infinite strings, and it is relevant for the equivalence of Minkowski and Rindler frame theories.

Equivalence properties of 3D conformal transformations and their application to reverse transformations

Seven-parameter conformal coordinate transformations, also known as Helmert transformations, can be constructed in more than one way. Two possible orderings of the rotations are in common use, giving rise to Helmert versions 1 and 2. It is demonstrated how the rotation parameters of either version can be converted into the rotation parameters of the other. This is useful when software is designed for the other version. It also enables computation of the same-formula inverse transformation by changing the sign of the equivalent ‘other version’ parameters. These results were primarily intended for conformal transformations between geodetic datums. They can, however, be extended to coordinate transformations in disciplines such as photogrammetry where rotations sometimes exceed 90 degrees.

Using Weighted Total Least Squares and 3-D Conformal Coordinate Transformation to Improve the Accuracy of Mobile Laser Scanning

With the aid of global position system (GPS), mobile laser scanning (MLS) is able to provide 3-D geo-referenced point cloud that has centimeter-level accuracy. The MLS accuracy, however, degrades significantly due to the trajectory errors of the laser scanner and the residual systematic errors from the geo-referencing transformation process in the GPS-free environments. To solve this problem, this article presents a novel integration algorithm based on the weighted total least squares (WTLS) and the 3-D conformal coordinate transformation (3DCCT). In this new method, the 3-D point measurement model and the error propagation parameter vector in the MLS can be updated in real-time, and they can also adjust the geo-referenced coordinate transformation parameters and eliminate the influences of the residual systematic errors during MLS. In this article, the MLS mathematical model is first established, followed up by a detailed analysis for MLS error budget interpreting the effects of the individual error sources. Second, WTLS is used to correct the 3-D point measurement model of MLS and the error of propagation parameter vector; 3DCCT, WTLS, and ground control target feature constraints are applied to eliminate the residual systematic errors in the geo-referencing transformation process. Finally, several data sets from outdoor scenarios are used to evaluate and validate the proposed method. The experimental results demonstrate that the proposed method can significantly improve the overall accuracy of the MLS system.

A generalized Grad-Shafranov equation with plasma flow under a conformal coordinate transformation

We employ a conformal mapping transformation to solve a generalized Grad-Shafranov equation with incompressible plasma flow of arbitrary direction and construct particular up-down asymmetric D-shaped and diverted tokamak equilibria. The proposed method can also be employed as an alternative quasi-analytic method to solving two dimensional elliptic partial differential equations.

Application of Least Squares Estimation Techniques in 2D Conformal Coordinates Transformation from Local to National

A two-dimensional conformal coordinate transformation is a similarity transformation which is also known as the four parameter transformation since it maintains scale relationships between the two coordinates system. This transformation type uses a mathematical model that establish a geometrical relationship between coordinates of points in different reference frames. It gained wider acceptance because of its capability to retain the shape of an area represented. This paper utilizes three control points; PT02, PT03 and PT04 to compute the transformation parameters and later all validated the derived coordinates of point PT04 known in the local system to national system using manual computation check. The second approach was to derive the transformation parameters using four control points. The result obtained was validated and found to agree with the derived national coordinates system for point PT07. Subsequently, all other points within the areas covered in the scope of work were transformed by writing script and running it in MATLAB software environment. The total area covered was 457.457hectares.

New Software for Simulation of Electrochemical Reaction Mechanisms of Any Complexity

We introduce here our scientific software KISSA designed for automatic simulation of electrochemical reaction mechanisms of any complexity, including those leading to ECL emission at planar, (hemi)spherical and (hemi)cylindrical working electrodes (KISSA-1D) as well as at microelectrodes such as disk or band (KISSA-2D). Simulations are performed by KISSA using a novel approach [1, 2] developed by the authors which involves conformal coordinate transformations to overcome the numerical difficulties arising from diffusional propagation of species and automatic grid adaptation based on the detection of large kinetic terms to obtain accurate results for both concentrations and electrochemical currents even in the presence of steep concentration gradients. This approach allows for efficient resolution of all physicochemical and mathematical issues connected both with edge effects (nano- and microelectrodes) and considerable disparity of time and length scales of the underlying processes arising due to quickly changing electrode potential and/or extremely rapid homogeneous chemical reactions for any complex mechanisms. The software offers a possibility to take into account spontaneous natural convection which limits the extent of the diffusion layer and leads to the establishment of a steady state current even at macroelectrodes. Furthermore, this enables setting correct and thermodynamically consistent initial conditions within the stagnant layer when the system is pre-equilibrated at a given rest potential for a finite period of time [3]. The software also allows simulating complex adsorption dynamics, i.e., kinetics and chemical reactions involving surface-bound species [4, 5]. The programs offer convenient entry of a reaction mechanism and parameters and provide access to the computed dynamic spatial distributions of all important quantities such as currents and current densities, concentration distributions of all reactants, surface coverage of adsorbed species [4, 5] as well as ECL intensity [6] (if applicable) through a graphical user interface [7]. [1] C. Amatore, O.V. Klymenko, I. Svir, Electrochem. Commun. 12 (2010) 1170. [2] O.V. Klymenko, I. Svir, A. Oleinick, C. Amatore, ChemPhysChem 13 (2012) 845. [3] C. Amatore, O.V. Klymenko, I. Svir, Anal. Chem. 84 (2012) 2792. [4] O.V. Klymenko, I. Svir, C. Amatore, J. Electroanal. Chem. 688 (2013) 320. [5] O.V. Klymenko, I. Svir, C. Amatore, Molecular Physics, 112 (2014) 1273. [6] O.V. Klymenko, I. Svir, C. Amatore, ChemPhysChem, 14 (2013) 2237. [7] http://kissagroup.com/index.php/bibliography

Modeling the spatial evolutions of nonlinear unidirectional surface gravity waves with fully nonlinear numerical method

Statistical properties of mechanically generated unidirectional nonlinear wave series are simulated with the fully nonlinear Chalikov-Sheinin model, which is based on a non-stationary conformal surface-following coordinate transformation that reduces the principal equations of potential waves into two simple evolutionary equations for the surface elevation and the velocity potential on the surface. Meanwhile, the numerical simulations performed by the temporal version of modified nonlinear Schrödinger equation (MNLS) are also compared with the laboratory experiments in this study. As a result, except for the wave height distribution, Chalikov-Sheinin or simply called CS model performs a little better in simulating the observed statistics than MNLS equation, particularly in the aspects of fourth-order normalized cumulants and the intermediate probability of wave crest and trough exceedance distributions in the presence of strong nonlinearity.

Implementation of CPCT based BIEs for 3D elasticity and its application in fracture mechanics

The accurate evaluation of singular boundary integrals is a long-debated issue for the successful implementation of BIEs. In the past decades, various methods have been developed, among which the singularity subtraction technique (SST) has been proved to be a unified method for various orders of singularities. However, the accuracy of original SST is sensitive to element shape due to the near singularity caused by element shape distortion. In this paper, a new conformal polar coordinate transformation (CPCT) for quadrilateral elements is proposed to eliminate the shape effect. Besides, an improved sigmoidal transformation is introduced in the angular direction to assign Gauss points more reasonably. By combination of the two strategies with original SST, a more efficient and robust numerical integration scheme for singular integrals can be obtained. Several numerical examples including crack problems are presented to demonstrate the accuracy and feasibility to coarse meshes for CPCT based BIEs and dual BEM.

The CPCT based CBIE and HBIE for potential problems in three dimensions

In this paper, the authors present a more efficient and robust implementation of conventional and hypersingular BIEs for potential problems in three dimensions under the framework of boundary face method (BFM). The focus is laid on the accurate evaluation of singular curved surface integrals, and three aspects related are considered simultaneously: (a) the near singularity caused by distorted element shape; (b) the near singularity derived from the angular direction; (c) the singularity in the radial direction. A conformal polar coordinate transformation (CPCT) is employed to eliminate the shape effect of distorted integration cells, which can retain the shape characteristic. Besides, an improved sigmoidal transformation is introduced to alleviate the near singularity in the angular direction. By combination of the two strategies with previous singularity subtraction method, an efficient numerical integration scheme has been obtained for various orders of singularities. Some numerical examples including parallelogram plate, sphere and hollow cylinder examples with coarse meshes are presented to demonstrate the accuracy and flexibility of the proposed method.

Empirical Research of Interpolation Methods in Distortion Modeling for the Coordinate Transformation between Local and Global Geodetic Datums

Conformal coordinate transformations very often are not accurate enough to be used for the coordinate transformations between local and global geodetic datums because of the inhomogeneity of the spatial data in different geodetic datums. Therefore, distortion modeling is used to improve the accuracy of the coordinate transformations. This paper presents empirical research on distortion modeling methods used to improve the coordinate transformation process from local to global geodetic datum and vice versa. Various models of distortion shifts in different grid resolutions were computed, evaluated, and compared with the official Croatian model of distortion shifts. Tested modeling methods are inverse distance to a power, kriging, minimum curvature, modified Shepard’s method, natural neighbor, nearest neighbor, polynomial regression, radial basis function, triangulation with linear interpolation, moving average, local polynomial, and least-squares collocation. Distortion model performance was evaluated using summary statistics derived from the back-interpolation for the independent control spatial data set. The present paper proposes the most appropriate method(s) of distortion modeling for the Croatian case and gives brief instructions on possible coordinate transformation improvements of the official Croatian coordinate transformation.

Efficient evaluation of integrals with kernel 1/rχ for quadrilateral elements with irregular shape

In this paper, integrals with kernel 1/rχ are concerned with the following three aspects: a). the near singularity caused by distorted element shape; b). the near singularity derived from the angular direction; c). the singularity/near singularity in the radial direction. A conformal polar coordinate transformation (CPCT) is proposed to eliminate the shape effect of elements, which can keep the shape characteristic of distorted elements, and an improved sigmoidal transformation is introduced to alleviate the near singularity in the angular direction. By combination of the two strategies with existing methods, such as singularity subtraction method and distance transformation method utilized in this paper, an efficient and robust numerical integration approach can be obtained for various orders of singular/nearly singular integrals, and a distorted curved quadrilateral element extracted from a cylinder surface is provided to demonstrate the efficiency and robustness of the proposed method.

Standing Gravity Waves in a Horizontal Circular Eccentric Annular Tank

Standing gravity waves in half-full horizontal cylindrical containers with eccentric tube are analyzed using the linear theory of water waves. The problem solution is obtained by the method of conformal coordinate transformation, leading to standard truncated matrix Eigen-value problem from which fluid motion characteristics (Eigen-frequencies and wave modes) are calculated. The effects of tube eccentricity and radius ratio upon the three lowest antisymmetric and symmetric sloshing frequencies and the associated hydrodynamic pressure mode shapes are examined. Also, convergence of the adopted approach with respect to the eccentricity condition, and radius ratio is discussed. Accuracy of the present analysis is checked by comparison with the known results of the limiting cases.

Research on three-dimensional computer-generated holographic algorithm based on conformal geometry theory

A novel method of three-dimensional computer-generated holographic algorithm based on conformal geometry theory was introduced. Firstly, three-dimensional object is transformed to a corresponding two-dimensional image via conformal transformation by making use of conformal geometry theory in mathematics, and then the Fresnel computer-generated hologram of the two-dimensional image is produced. During the reconstruction process, the three-dimensional reconstruction image of the original object is obtained by means of conformal coordinate transformation. The research results show that the algorithm introduced in this paper is characterized by fast running speed and good effects, providing a good reference for the research on three-dimensional computer-generated holographic algorithm.

Wide-angle scannable reflector design using conformal transformation optics

A flat reflector capable of scanning over wide angles is designed using a transformation optics approach. This reflector is derived from its virtual parabolic counterpart using a conformal coordinate transformation that determines the permittivity profile of the flat reflector. By changing the permittivity profile, the flat reflector is then capable of scanning up to 47° away from broadside while maintaining good beam characteristics across a wide frequency range. Moreover, its directivity is comparable to that of the virtual parabolic reflector, even at high scan angles. We use the Schwarz-Christoffel transformation as a versatile tool to produce perfect conformal mapping of coordinates between the virtual and flat reflectors, thereby avoiding the need to monitor the anisotropy of the material that results when employing quasi-conformal methods.

A new strategy for simulation of electrochemical mechanisms involving acute reaction fronts in solution under spherical or cylindrical diffusion

In this paper we present a combined simulation approach for solving complex electrochemical kinetic problems at (hemi)spherical and (hemi)cylindrical electrodes based on the simultaneous application of analytical conformal coordinate transformations to accurately treat the diffusional transport and adaptive grid compression to locally increase the accuracy of the approximation of acute reaction fronts. This strategy is shown to be very efficient and leads to extremely accurate results both for steady state and transient situations and is equally powerful for both pure diffusion and that complicated by severe kinetic distortions.

Closed-form solution for vertical fracture width in anisotropic elastic formations

A closed-form analytical solution for estimating the hydraulic fracture width in anisotropic tight shales is presented, as a function of the anisotropic elastic properties and fluid pressure. The suggested formula is a generalization of the well-known Sneddon solution for fracture width in isotropic formations that is widely used by petroleum engineers for hydraulic fracture applications. The classical isotropic formula becomes inappropriate for applications in unconventional tight shale formations, where the ratio of moduli in orthogonal directions may reach 4.0. This new fracture width closed form solution was derived for infinite, homogeneous, anisotropic elastic medium, and for the particular case of a vertical fracture of given height, aligned with the axes of elastic anisotropy. This solution is less general than the known 2D analytic solutions for the infinite elliptic cavity arbitrarily oriented to the anisotropy axes a medium. These general analytic solutions require multiple complex steps to compute fracture width, including the selection of roots of high-order algebraic equations, and conformal coordinate transformations. In contrast, the presented closed-form solution allows one to quickly and easily understand the influence of rock elastic properties and fluid pressure on fracture width, without extensive numerical simulations. Thus, it is well suited for parametric studies and for handling large amounts of input parameter combinations.

Application of Pareto optimality to linear models with errors-in-all-variables

In some geodetic and geoinformatic parametric modeling, the objectives to be minimized are often expressed in different forms, resulting in different parametric values for the estimated parameters at non-zero residuals. Sometimes, these objectives may compete in a Pareto sense, namely a small change in the parameters results in the increase of one objective and a decrease of the other, as frequently occurs in multiobjective problems. Such is the case with errors-in-all-variables (EIV) models, e.g., in the geodetic and photogrammetric coordinate transformation problems often solved using total least squares solution (TLS) as opposed to ordinary least squares solution (OLS). In this contribution, the application of Pareto optimality to solving parameter estimation for linear models with EIV is presented. The method is tested to solve two well-known geodetic problems of linear regression and linear conformal coordinate transformation. The results are compared with those from OLS, Reduced Major Axis Regression (TLS solution), and the least geometric mean deviation (GMD) approach. It is shown that the TLS and GMD solutions applied to the EIV models are just special cases of the Pareto optimal solution, since both of them belong to the Pareto-set of the problems. The Pareto balanced optimum (PBO) solution as a member of this Pareto optimal solution set has special features and is numerically equal to the GMD solution.