Linear Coordinate Transformation Research Articles

An analytic probability density function for partially premixed flames with detailed chemistry

Laminar premixed flame profiles of methane/air free flames and strained flames at different fuel/air ratios and strain rates are analyzed using detailed chemistry with Lewis numbers equal to one. It is shown that the detailed chemistry flame profiles of progress variables CO2 + CO and H2O + H2 in canonically stretched coordinates can be fitted accurately by a slight generalization of recently proposed analytical presumed flame profiles over a wide range of fuel/air ratios through adaptation of a single model parameter. Strained flame profiles can be reproduced using an additional linear coordinate transformation, emulating the compression of the preheat zone by strain as predicted by premixed flame theory. The model parameter can alternatively be determined using only the laminar flame speeds and the fully burnt temperatures from the laminar flame calculations. The stretch factor of the coordinate transformation is proportional to cp/lambda, which drops by a factor up to 4 across the laminar flame. It is shown how the non-constant cp/lambda modifies the laminar flame probability density function (pdf) and a polynomial fit to cp/lambda as a function of the progress variable allows analytical results for the laminar flame pdf and the mean value of the progress variable and of the reaction source term. An analytic pdf for partially premixed flames is proposed based on Bayes's theorem as a combination of a beta pdf for the mixture fraction and the laminar flame pdf's evaluated at the respective fuel/air ratio.

Non-closed acoustic cloaking devices enabled by sequential-step linear coordinate transformations

Hitherto acoustic cloaking devices, which conceal objects externally, depend on objects' characteristics. Despite previous works, we design cloaking devices placed adjacent to an arbitrary object and make it invisible without the need to make it enclosed. Applying sequential linear coordinate transformations leads to a non-closed acoustic cloak with homogeneous materials, creating an open invisible region. Firstly, we propose to design a non-closed carpet cloak to conceal objects on a reflecting plane. Numerical simulations verify the cloaking effect, which is completely independent of the geometry and material properties of the hidden object. Moreover, we extend this idea to achieve a directional acoustic cloak with homogeneous materials that can render arbitrary objects in free space invisible to incident radiation. To demonstrate the feasibility of the realization, a non-resonant meta-atom is utilized which dramatically facilitated the physical realization of our design. Due to the simple acoustic constitutive parameters of the presented structures, this work paves the way toward realization of non-closed acoustic devices, which could find applications in airborne sound manipulation and underwater demands.

The required acoustic parameters simplification of invisibility cloaks and concentrators using the impedance-tunable coordinate transformation

Transformation acoustics, as an unconventional theory, provides a powerful tool to design various kinds of acoustic devices with excellent functionalities. However, the required ideal parameters, which are prescribed by the method, are both complex and hard to implement-even using acoustic metamaterials. Furthermore, simplified parameter materials are generally favored in transformation-acoustic design due to its easier realization with artificial structures. In this letter, we propose a coordinate transformation methodology for achieving simplified parameters by tuning the impedance distribution in the geometric limit, where the transformation media parameters can be derived by setting tunable impedance functions in the original space and a combination of suitable linear or nonlinear coordinate transformation. Based on this approach, both two-dimensional acoustic cloak and concentrators are designed with different sets of simplified parameters. Numerical simulations indicate good performance of these devices with minimized scattering at higher frequencies. The proposed method provides more opportunities to realize the designed acoustic devices experimentally, and can also be used for other transformation-acoustic designs including 3D cases.

The special Galileon as Goldstone of diffeomorphisms

The special Galileon stands out amongst scalar field theories due to its soft limits, non-linear symmetries and scattering amplitudes. This prompts the question what the origin of its underlying symmetry is. We show that it is intimately connected to general relativity: the special Galileon is the Goldstone mode of the affine group, consisting of linear coordinate transformations, analogous to the dilaton for conformal symmetries. We construct the corresponding metric, and discuss various relations to gravity, Yang-Mills and the non-linear sigma-model.

Bases for Structures and Theories I

Sometimes structures or theories are formulated with different sets of primitives and yet are definitionally equivalent. In a sense, the transformations between such equivalent formulations are rather like basis transformations in linear algebra or co-ordinate transformations in geometry. Here an analogous idea is investigated. Let a relational signature P = {P_i}_{i in I_P} be given. For a set Phi = {phi _i}_{i in I_{Phi }} of L_P-formulas, we introduce a corresponding set Q = {Q_i}_{i in I_{Phi }} of new relation symbols and a set of explicit definitions of the Q_i in terms of the phi _i. This is called a definition system, denoted d_{Phi }. A definition system d_{Phi } determines a translation functiontau _{Phi } : L_Q rightarrow L_P. Any L_P-structure A can be uniquely definitionally expanded to a model A^{+} models d_{Phi }, called A + d_{Phi }. The reduct A + d_{Phi } to the Q-symbols is called the definitional imageD_{Phi }A of A. Likewise, a theory T in L_P may be extended a definitional extension T + d_{Phi }; the restriction of this extension T + d_{Phi } to L_Q is called the definitional imageD_{Phi }T of T. If T_1 and T_2 are in disjoint signatures and T_1 + d_{Phi } equiv T_2 + d_{Theta }, we say that T_1 and T_2 are definitionally equivalent (wrt the definition systems d_{Phi } and d_{Theta }). Some results relating these notions are given, culminating in two characterization theorems for the definitional equivalence of structures and theories.

General design of 3D piecewise homogeneous illusion devices with arbitrary shapes

In this work, a general method is presented for the design of arbitrarily shaped 3D illusion devices with piecewise homogeneous parameters based on geometric divisions and linear coordinate transformations. Three illusion devices that can reshape the sizes or positions of the wrapped objects are demonstrated, namely, shrinking, amplifying, and shifting devices. The shrinking device can shrink a larger object into a smaller one with different material parameters, whereas the amplifying device can enlarge a smaller object into a larger one, and a shifting device can generate a new image with an identical size but located at a different position. In addition, based on the presented shrinking device, a perfect 3D invisibility cloak is achieved by shrinking the wrapped object to sufficiently small dimensions as compared to the operating frequency. An electromagnetic concentrator is also obtained by replacing the coated object of the amplifying device with a compression medium. The presented design approach can be easily extended to the design of other electromagnetic devices and even to other physical fields. It is believed that the presented piecewise homogeneous devices are more practicable in reality and can accelerate the potential applications of illusion devices in both military and commercial fields.

Käyräviivaiset koordinaatistot kontinuumimekaniikassa

Artikkelissa tarkastellaan suoraviivaisista avaruuskoordinaatistoista yleistettyjä käyräviivaisia koordinaatistoja ja niiden välisiä muunnoksia. Teoria esitetään suoraviivaisten koordinaatistojen teorian pohjalta. Yleistämisen keskeisenä ominaisuutena pidetään suoraviivaisten koordinaatistojen kantajärjestelmän yksikäsitteistä määrittelyä. Sovelluksina tarkastellaan kontinuumimekaniikan tärkeitä koordinaatistoja ja niiden välisiä muunnoksia, kannan vaihtoa, vaihdetun kannan derivaattoja sekä niissä tarvittavia niin kutsuttuja Christoffelin symboleja. Pohditaan myös mitä hyötyä käyräviivaisista koordinaatistoista ja niiden muunnoksista on.

Handling Gaussian blur without deconvolution

The paper presents a new theory of invariants to Gaussian blur. Unlike earlier methods, the blur kernel may be arbitrary oriented, scaled and elongated. Such blurring is a semi-group action in the image space, where the orbits are classes of blur-equivalent images. We propose a non-linear projection operator which extracts blur-insensitive component of the image. The invariants are then formally defined as moments of this component but can be computed directly from the blurred image without an explicit construction of the projections. Image description by the new invariants does not require any prior knowledge of the blur kernel parameters and does not include any deconvolution. The invariance property could be extended also to linear transformation of the image coordinates and combined affine-blur invariants can be constructed. Experimental comparison to three other blur-invariant methods is given. Potential applications of the new invariants are in blur/position invariant image recognition and in robust template matching.

Dynamic output feedback sliding mode control for non-minimum phase systems with application to an inverted pendulum

In this paper, a class of nonlinear systems is considered, where the nominal system representation is allowed to be non-minimum phase. A sliding surface is proposed which is a function of the measured system output and an estimated state. A linear coordinate transformation is introduced so that the stability analysis of the reduced order sliding mode dynamics can be conveniently performed. A robust output feedback sliding mode control (OFSMC) is then designed to drive the system states to the sliding surface in finite time and maintain a sliding motion thereafter. A simulation example is used to demonstrate the effectiveness of the proposed method and the method is successfully applied to an inverted pendulum.

Design of elastic metasurfaces for controlling shear vertical waves using uniaxial scaling transformation method

Controlling wave propagation is a popular research field; however, progress with regard to the study of elastic waves is slow because of the coupling of different types of waves. In this paper, we report a novel approach to design elastic metasurfaces for controlling elastic shear vertical (SV) waves in solids. The metasurface is composed of separate parallel thin plates connected at both ends to half-space solids. The elastic SV-waves in solids were controlled by manipulating the induced flexural waves in the thin plates of the metasurface. Based on the form invariance of the governing equation for flexural waves in a thin plate under linear coordinate transformations, the uniaxial scaling transformation method (USTM) has been put forward to alter the material to constitute the thin plates of the metasurface. The transformed material is transversely isotropic, which greatly reduces implementation difficulties. To tune the traveling times of the flexural waves in the thin plates, different values of the scaling parameter α and plate thickness h were assigned for different thin plates based on the Kirchhoff plate theory. By combining the plate thickness variation approach with the USTM, the plate length of the metasurface could be effectively controlled to be smaller than the wavelength at the working frequency. To demonstrate the performance of the proposed approach, two designs of elastic metasurfaces for wave focusing and abnormal refraction were described in detail, and full numerical simulations were conducted. The present approach opens a promising avenue toward the realization of metasurfaces for controlling elastic waves.

Electromagnetic time-harmonic and static field polygonal rotator with homogeneous materials

We propose a scheme of designing polygonal rotator with homogenous materials by using linear coordinate transformation. Our strategy is available for both time-harmonic electromagnetic field case and static field case. In particular, we found that only one anisotropic material is needed in static field case, and the density of field in the central region can be altered to be denser or sparser, or stay the same. The magnetostatic field rotator can be realized by multilayered structure composed of ferromagnetic materials and superconductor, and the direct current rotator can be realized by metals with different conductivity. Numerical results verify the effectiveness of our strategy in both time-harmonic field case and static field case.

Design of an acoustic illusion device based on a shifting medium and multi-folded transformation.

An acoustic illusion device that can act as an invisible cloak or a shifting medium depending on the value of shift distance, which is about twice the circum-radius of the outer polygon, is proposed and designed based on linear coordinate transformation. A multi-folded transformation approach is used to design an illusion device with a circular opening window that allows for information interaction with the outside world. The results show that the proposed device can hide objects with arbitrary shapes or positions. Furthermore, in order to remove the material anisotropy of the proposed illusion device, a layered structure composed of homogenous and isotropic material is used based on the effective medium theory. The combination of the layered structure and the circular opening provide a flexible and feasible approach to achieve the partial implementation of the illusion device. It is hoped that these results may open an avenue for designing and implementing invisibility cloaks or illusion devices, and speed up potential applications for noise shielding, target camouflage, or target protection from active sonar signals.

Data Registration with Ground Points for Roadside LiDAR Sensors

The Light Detection and Ranging (LiDAR) sensors are being considered as new traffic infrastructure sensors to detect road users’ trajectories for connected/autonomous vehicles and other traffic engineering applications. A LiDAR-enhanced traffic infrastructure system requires multiple LiDAR sensors around intersections, along with road segments, which can provide a seamless detection range at intersections or along arterials. Each LiDAR sensor generates cloud points of surrounding objects in a local coordinate system with the sensor at the origin, so it is necessary to integrate multiple roadside LiDAR sensors’ data into the same coordinate system. None of existing methods can integrate the data from roadside LiDAR sensors, because the extensive detection range of roadside sensors generates low-density cloud points and the alignment of roadside sensors is different from mapping scans or autonomous sensing systems. This paper presents a method to register datasets from multiple roadside LiDAR sensors. This approach innovatively integrates LiDAR datasets with 3D cloud points of road surface and 2D reference point features, so the method is abbreviated as RGP (Registration with Ground and Points). The RGP method applies optimization algorithms to identify the optimized linear coordinate transformation. This research considered the genetic algorithm (global optimization) and the hill climbing algorithm (local optimization). The performance of the RGP method and the different optimization algorithms was evaluated with field LiDAR sensors data. When the developed process can integrate data from roadside sensors, it can also register LiDAR sensors’ data on an autonomous vehicle or a robot.

Exploiting transformation optics for arbitrary manipulation of antenna radiation pattern

Using transformation optics, a general method for tailoring the radiation pattern of a monopole antenna encompassed by a coating layer is proposed. Unlike previous studies, the propounded approach is not restricted to special patterns and can produce arbitrary radiation pattern with customisable beam parameters such as number, direction, and directivity in both azimuthal and elevation planes. A linear coordinate transformation is established to simplify the coating layer realisation via offering homogeneous materials. As proof-of-principle, two different antennas capable of generating multiple beams and a single-directive beam, are elaborately acquired by a meta-structure consisting of a split-ring resonator-wire array composite. It was observed that the experimental results corroborate numerical simulations. The proposed approach is believed to have potential applications in antenna technologies, satellite communication, and multiple-input multiple-output systems.

Kink scattering in a hybrid model

In this work we consider a model where the potential has two topological sectors connecting three adjacent minima, as occurs with the ϕ6 model. In each topological sector, the potential is symmetric around the local maximum. For ϕ>0 there is a linear map between the model and the λϕ4 model. For ϕ<0 the potential is reflected. Linear stability analysis of kink and antikink lead to discrete and continuum modes related by a linear coordinate transformation to those known analytically for the λϕ4 model. Fixing one topological sector, the structure of antikink-kink scattering is related to the observed in the λϕ4 model. For kink-antikink collisions a new structure of bounce windows appear. Depending on the initial velocity, one can have oscillations of the scalar field at the center of mass even for one bounce, or a change of topological sector. We also found a structure of one-bounce, with secondary windows corresponding to the changing of the topological sector accumulating close to each one-bounce windows. The kink-kink collisions are characterized by a repulsive interaction and there is no possibility of forming a bound state.

A bi-functional illusion device based on transformation optics

An illusion device based on a shifting-distance concept has been investigated using a linear coordinate transformation method. Distinguishing it from previously designed illusion devices, we construct a shift-illusion device featuring resizing and rotating for both active and passive scatterers, in case the shifting region and physical space shared the same origin point. Similarly, when the shifting region is foreign to the designed device, the proposed illusion design performs as a rotation and relocation-shifting device to change the location of the scatterer. In addition, these proposed devices behave like the stealth effect in the absence of any scatterer as the outer-boundary field distribution is not disturbed. Meanwhile, the proposed concept is further extended for different mapping styles to validate the shifting-illusion concept. The constitutive parameters of the proposed devices are homogeneous, which makes them more appreciable for real implementation. It is expected that the proposed design will find potential applications in illusion devices, such as object protection, camouflage, etc.

Quantum mappings acting by coordinate transformations on Wigner distributions

We prove two results about Wigner distributions. Firstly, that the Wigner transform is the only sesquilinear map $\mathcal{S}(\mathbb{R}^n) \times \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^{2n})$ which is bounded and covariant under phase-space translations and linear symplectomorphisms. Consequently, the Wigner distributions form the only set of quasidistributions which is invariant under linear symplectic transformations. Secondly, we prove that the maximal group of (linear or non-linear) coordinate transformations that preserves the set of (pure or mixed) Wigner distributions consists of the translations and the linear symplectic and antisymplectic transformations.

Multidimensional Data Visualization Based on the Minimum Distance Between Convex Hulls of Classes

The problem of data visualization in the analysis of two classes in a multidimensional feature space is considered. The two orthogonal axes by which the classes are maximally separated from each other are found in the mapping of classes as a result of linear transformation of coordinates. The proximity of the classes is estimated based on the minimum-distance criterion between their convex hulls. This criterion makes it possible to show cases of full class separability and random outliers. A support vector machine is used to obtain orthogonal vectors of the reduced space. This method ensures the obtaining of the weight vector that determines the minimum distance between the convex hulls of classes for linearly separable classes. Algorithms with reduction, contraction, and offset of convex hulls are used for intersecting classes. Experimental studies are devoted to the application of the considered visualization methods to biomedical data analysis.

New Integrated Guidance and Control of Homing Missiles with an Impact Angle against a Ground Target

A new integrated guidance and control (IGC) law is investigated for a homing missile with an impact angle against a ground target. Firstly, a control-oriented model with impact angle error of the IGC system in the pitch plane is formulated by linear coordinate transformation according to the motion kinematics and missile dynamics model. Secondly, an IGC law is proposed to satisfy the impact angle constraint and to improve the rapidity of the guidance and control system by combining the sliding mode control method and nonlinear extended disturbance observer technique. Thirdly, stability of the closed-loop guidance and control system is proven based on the Lyapunov stability theory, and the relationship between the accuracy of the impact angle and the estimate errors of nonlinear disturbances is derived from stability of the sliding mode. Finally, simulation results confirm that the proposed IGC law can improve the performance of the missile guidance and control system against a ground target.

Underwater acoustic ground cloak development and demonstration

Acoustic ground cloaks, which conceal an object on a rigid surface, utilize a linear coordinate transformation to map the flat surface to a triangular void by compressing space into two triangular cloaking regions consisting of a homogeneous anisotropic acoustic metamaterial. Transformation acoustics allows for the realization of a coordinate transformation through a reinterpretation of the scale factors as a new material in the original coordinate system. An underwater acoustic ground cloak was constructed from perforated steel plates and experimentally tested to conceal an object on a pressure release surface. The perforated plate acoustic ground cloak successfully cloaked the scattered object. There was excellent agreement between the phase of the surface reflection and cloak reflection with a small amplitude difference. Above 15 [kHz], the cloaking performance decreased as the effective material parameters of the perforated plate metamaterial deviated from the required material parameters.