Orthogonal Diagonalization Research Articles

A recursive trapezoid-based algorithm designed to compute the strength- and stiffness-related geometric properties of beams with polygonal cross-sections

A new recursive trapezoid-based algorithm is proposed for calculating the strength- and stiffness-related geometric characteristics of polygonal beam sections. By taking as input the coordinate vertices of the polygon, the approach may effectively be applied to determine the cross-section area and the position of its centroid along with the axial area moments of inertia and product of inertia relative to the cross-sections’ centroidal axes. As the method combines the theory of planar moments with the well-known trapezoid numerical method, it provides a straightforward tabular procedure that can be easily implemented using any spreadsheet software. In addition, aiming to determine the principal area moments of inertia, a computation matrix formalism based on the theory of orthogonal diagonalization of symmetric matrices is put forward as an alternative to the classical method of vanishing the first derivative and solving for the principal directions. Some benchmark examples are worked out to prove the relevance and accuracy of the proposed approach for several beam sections ranging from the most common triangular and quadrilateral ones to very complex open and closed polygonal profiles. The solution presented here appears to be virtually foolproof compared to the conventional one, as it does not necessitate the decomposition of complex polygonal cross-section area into elementary surfaces of different shapes, then use the parallel axes theorem, and so forth; rather, it requires only insertion of the coordinates of the vertices of the polygon, turn by turn in the clockwise order, relative to an arbitrary cartesian coordinate system, and no further steps.

Numerical solutions of Schrödinger–Boussinesq system by orthogonal spline collocation method

This paper is concerned with the numerical solutions of Schrödinger–Boussinesq (SBq) system by an orthogonal spline collocation (OSC) discretization in space and Crank–Nicolson (CN) type approximation in time. By using the mathematical induction argument and standard energy method, the proposed CN+OSC scheme is proved to be unconditionally convergent at the order O(τ2+h4) with mesh-size h and time step τ in the discrete L2-norm. We devise a new computation method based on the orthogonal diagonalization techniques (ODT) to realize the proposed CN+OSC scheme. In order to compare the performance of ODT, we devise an alternating direction implicit (ADI) method to compute the CN+OSC scheme for high spatial dimension SBq system. As an alternative implementation, the new method ODT not only exhibits more accurate numerical results, but also demonstrates stronger invariance preserving ability. Numerical results are reported to verify the error estimates and the discrete conservation laws.

Enhancing Salp Swarm Optimization with Orthogonal Diagonalization Transformation for Damage Detection in Truss Bridge

Enhancing Salp Swarm Optimization with Orthogonal Diagonalization Transformation for Damage Detection in Truss Bridge

Two energy-preserving Fourier pseudo-spectral methods and error estimate for the Klein–Gordon–Dirac system

We devote the present paper to the convergency of conservative Fourier pseudo-spectral methods for three-dimensional Klein–Gordon–Dirac (KGD) system. By adopting the mathematical induction argument, standard energy method and inverse inequality, the error bound is established under the condition τ<h with time step τ and mesh size h. More specifically, the scalar ϕ and 4-spinor ψ are proved to be convergent in H2 and H1-norm, respectively. In addition, we establish a frame to compute the numerical solutions of three-dimensional KGD system with periodic conditions, and a faster solver is designed to accelerate the computation by means of orthogonal diagonalization technique. Numerical results are reported to verify the error estimate and discrete conservation laws.

Spontaneous parametric down-conversion: Revisiting the parameters of transverse entanglement outside the near zone

The reduced density matrix ${\ensuremath{\rho}}_{r}(x,{x}^{\ensuremath{'}})$ of the transverse biphoton state beyond the near zone is found to be real on and only on diagonals in the plane of the photon's transverse coordinates $(x,{x}^{\ensuremath{'}})$. This remarkable feature is found to occur at any distances of photon propagation with diffraction spreading completely taken into account. The functions of the reduced density matrix at two orthogonal diagonals, ${\ensuremath{\rho}}_{r}(x,x)$ and ${\ensuremath{\rho}}_{r}(x,\ensuremath{-}x)$, are considered as the only two complementary single-particle distributions generated by the reduced density matrix. The ratio of widths of these two distributions is interpreted as the new entanglement parameter ${R}_{\mathrm{diag}}$ which is found to be very close to and completely compatible with the Schmidt entanglement parameter $K$.

Robust topology optimization of multi-material structures under load uncertainty using the alternating active-phase method

The synergy between different constituent materials can drastically improve the performance of composite structures. The optimal design of such structures for practical applications is complicated by the often-encountered non-deterministic loading conditions. This paper proposes an efficient method for robust multi-material topology optimization problems of continuum structures under load uncertainty. Specifically, the weighted sum of the mean and standard deviation of structural compliance is minimized under volume constraints for each material phase. Based on the theory of linear elasticity and using the orthogonal diagonalization of real symmetric matrices, the Monte Carlo sampling is separated from the topology optimization procedure and an efficient procedure for sensitivity analysis is established. By employing an alternating active-phase algorithm of the Gauss-Seidel version, the multi-material topology optimization problem is split into a series of binary topology optimization sub-problems, which can be easily solved using the modified SIMP model. Several 2D examples are presented to demonstrate the effectiveness of the proposed method.

The Dirichlet problem for orthodiagonal maps

We prove that the discrete harmonic function corresponding to smooth Dirichlet boundary conditions on orthodiagonal maps, that is, plane graphs having quadrilateral faces with orthogonal diagonals, converges to its continuous counterpart as the mesh size goes to 0. This provides a convergence statement for discrete holomorphic functions, similar to the one obtained by Chelkak and Smirnov [4] for isoradial graphs. We observe that by the double circle packing theorem [3], any finite, simple, 3-connected planar map admits an orthodiagonal representation.Our result improves the work of Skopenkov [41] and Werness [47] by dropping all regularity assumptions required in their work and providing effective bounds. In particular, no bound on the vertex degrees is required. Thus, the result can be applied to models of random planar maps that with high probability admit orthodiagonal representation with mesh size tending to 0. In a companion paper [21], we show that this can be done for the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield [15,25].

An efficient approach to model updating for a multispan railway bridge using orthogonal diagonalization combined with improved particle swarm optimization

In this paper, a novel approach to model updating for a large-scale railway bridge using orthogonal diagonalization (OD) coupled with an improved particle swarm optimization (IPSO) is proposed. Particle swarm optimization (PSO) is a well-known and widely applied evolutionary algorithm. However, as other evolutionary algorithms (EAs), PSO has two main drawbacks that may reduce its capability to tackle optimization issues. A fundamental shortcoming of PSO is premature convergence. On the other hand, since PSO employs all populations to seek the best solution through iterations, it is very time-consuming. This makes PSO as well as EAs difficult to apply for optimization problems of large-scale structural models. In order to overcome those drawbacks, we propose coupling OD with IPSO (ODIPSO). OD is applied to arrange the position of particles and to select only particles with the best solution for next iterations, which helps to reduce the computational cost dramatically. There are several significant features of ODIPSO: (1) IPSO is employed to tackle the problem of premature convergence of PSO; (2) only one guide is used to update the velocity of particles instead of utilizing both guides, consisting of the local best and the global best; and (3) in each iteration, only the velocity and the position of the best particles are updated. In order to assess the effectiveness of the proposed approach, a large-scale railway bridge calibrated on the field is employed. This paper also introduces the use of wireless triaxial sensors (replacing classical wired systems) to obtain structural dynamic characteristics. The appearance of wireless triaxial transducers increases significantly the freedom in designing an ambient vibration test. The results show that ODIPSO not only outperforms PSO, IPSO and OD combined with PSO (ODPSO) in terms of accuracy, but also dramatically reduces the computational time compared to PSO and IPSO.

Checkerboard patterns with black rectangles

Checkerboard patterns with black rectangles can be derived from quad meshes with orthogonal diagonals. First, we present an initial theoretical analysis of these quad meshes. The analysis reveals many possible applications in geometry processing and also motivates the numerical optimization for aesthetic and functional checkerboard pattern design. Second, we describe an optimization algorithm that transforms initial 2D and 3D quad meshes into quad meshes with orthogonal diagonals. Third, we present a 2D checkerboard pattern design framework based on integer programming inspired by the logo design of the 2020 Olympic games. Our results show a variety of 2D and 3D checkerboard patterns that can be derived from 2D or 3D quad meshes with orthogonal diagonals.

Computational properties of pentadiagonal and anti-pentadiagonal block band matrices with perturbed corners

In this paper, we derive an orthogonal block diagonalization and a number of formulas for pentadiagonal block band symmetric Toeplitz matrices and anti-pentadiagonal block band persymmetric Hankel matrices, both with perturbed corners. Namely, our formulas include block diagonalization, inverse, eigenvalues of these matrices. Our approach uses a suitable modification technique for pentadiagonal block band symmetric Toeplitz matrices and anti-pentadiagonal block band persymmetric Hankel matrices. Some numerical experiments are also presented to demonstrate the performance and effectiveness of the proposed theorems.

On approximate diagonalization of third order symmetric tensors by orthogonal transformations

In this paper, we study the approximate orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to the stationary points of this problem. We study the relationships between these classes, and other well-known objects, such as tensor Z-eigenvalue and Z-eigenvector. We also prove results on convergence of the cyclic Jacobi (or Jacobi CoM2) algorithm.

Globally Convergent Jacobi-Type Algorithms for Simultaneous Orthogonal Symmetric Tensor Diagonalization

In this paper, we consider a family of Jacobi-type algorithms for a simultaneous orthogonal diagonalization problem of symmetric tensors. For the Jacobi-based algorithm of [M. Ishteva, P.-A. Absil, and P. Van Dooren, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 651--672], we prove its global convergence for simultaneous orthogonal diagonalization of symmetric matrices and 3rd-order tensors. We also propose a new Jacobi-based algorithm in the general setting and prove its global convergence for sufficiently smooth functions.

A novel orthogonal PSO algorithm based on orthogonal diagonalization

Abstract One of the major drawbacks of the global particle swarm optimization (GPSO) algorithm is zigzagging of the direction of search that leads to premature convergence by falling into local minima. In this paper, a new algorithm named orthogonal PSO (OPSO) algorithm is proposed that not only alleviates the associated problems in GPSO algorithm but also achieves better performance. In OPSO algorithm, the m particles of the swarm are divided into two groups: one active group of best personal experience of d particles and a passive group of personal experience of remaining (m ‒ d) particles. The purpose of creating two groups is to enhance the diversity in the swarm's population. In each iteration, the d active group particles undergo an orthogonal diagonalization process and are updated in such way that their position vectors are orthogonally diagonalized. The passive group particles are not updated as their contribution in finding correct direction is not significant. In the proposed algorithm, the particles are updated using only one guide, thus avoiding the conflict between the two guides that occurs in the GPSO algorithm. We tested the OPSO algorithm with thirty unimodal and multimodal high-dimensional benchmark functions and compared its performance with GPSO and several competing evolutionary techniques. With extensive simulated experiments, we have shown superiority of the proposed algorithm in terms of convergence, accuracy, consistency, robustness and reliability over other algorithms. The proposed algorithm is found to be successful in achieving optimal solution in all the thirty benchmark functions.

Orthogonal PSO algorithm for economic dispatch of thermal generating units under various power constraints in smart power grid

Particle swarm optimization (PSO) algorithm has been successfully applied to solve various optimization problems in science and engineering. One such popular one is called global PSO (GPSO) algorithm. One of major drawback of GPSO algorithm is the phenomenon of “zigzagging”, that leads to premature convergence by falling into local minima. In addition, the performance of GPSO algorithm deteriorates for high-dimensional problems, especially in presence of nonlinear constraints. In this paper we propose a novel algorithm called, orthogonal PSO (OPSO) that alleviates the shortcomings of the GPSO algorithm. In OPSO algorithm, the m particles of the swarm are divided into two groups: active group and passive group. The d particles of the active group undergo an orthogonal diagonalization process and are updated in such way that their position vectors become orthogonally diagonalized. In the OPSO algorithm, the particles are updated using only one guide, thus avoiding the conflict between the two guides that occurs in the GPSO algorithm. We applied the OPSO algorithm for solving economic dispatch (ED) problem by taking three power systems under several power constraints imposed by thermal generating units (TGUs) and smart power grid (SPG), for example, ramp rate limits, and prohibited operating zones. In addition, the OPSO algorithm is also applied for ten selected shifted and rotated CEC 2015 benchmark functions. With extensive simulation studies, we have shown superior performance of OPSO algorithm over GPSO algorithm and several existing evolutional computation techniques in terms of several performance measures, e.g., minimum cost, convergence rate, consistency, and stability. In addition, using unpaired t-Test, we have shown the statistical significance of the OPSO algorithm against several contending algorithms including top-ranked CEC 2015 algorithms.

Discrete Riemann surfaces based on quadrilateral cellular decompositions

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian differentials. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann–Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann–Roch Theorem, and a discrete Abel–Jacobi map.

Elliptic visualizing optical resolution and kinetic energy

Diffraction limited resolution as introduced by Abbe is well established, but interference limited resolution was not well known until holographic interferometry was introduced. The holodiagram is used to simplify holography and in a new way visualize the distribution, ratio, and relation among resolutions of different optical techniques, including relativistic phenomena. Resolution, when measured by optical methods based on the number of wavelengths of light, is defined in the following as the minimum distance between resolvable points, or the largest object needed to be resolved. Everywhere in the diagram this resolution is represented by two orthogonal diagonals of rhombs.

Vision based persistent localization of a humanoid robot for locomotion tasks

Abstract Typical monocular localization schemes involve a search for matches between reprojected 3D world points and 2D image features in order to estimate the absolute scale transformation between the camera and the world. Successfully calculating such transformation implies the existence of a good number of 3D points uniformly distributed as reprojected pixels around the image plane. This paper presents a method to control the march of a humanoid robot towards directions that are favorable for visual based localization. To this end, orthogonal diagonalization is performed on the covariance matrices of both sets of 3D world points and their 2D image reprojections. Experiments with the NAO humanoid platform show that our method provides persistence of localization, as the robot tends to walk towards directions that are desirable for successful localization. Additional tests demonstrate how the proposed approach can be incorporated into a control scheme that considers reaching a target position.

Gain-Enhanced Patch Antennas With Loading of Shorting Pins

A new gain-enhanced patch antenna with loading of shorting pins is proposed in this paper. Four metallic pins are symmetrically placed in the two diagonals of a square patch resonator to electrically short the patch and ground plane. These shorting pins tremendously perturb the field distribution beneath the patch due to their shunt inductive effect. As these four pins are moved outward along the two orthogonal diagonals away from the center, their influence on the field distribution over the patch is strengthened to gradually raise its dominant mode, i.e., TM010 mode, resonant frequency as the pin-to-pin spacing is enlarged. At a fixed resonant frequency, the overall area of this proposed patch antenna with four pins results to be increased. As such, its radiation directivity or gain gets to be enhanced due to the enlarged antenna area. After extensive analysis is executed, two square patch antennas with and without loaded pins are designed and fabricated. The simulated and measured results agree with each other, and they have evidently demonstrated that the radiation directivity can be enhanced up to 11.0 dBi, or about 2.9 dB increment, by virtue of the proposed approach.

High-Gain Circularly Polarized Microstrip Patch Antenna With Loading of Shorting Pins

A single-fed microstrip patch antenna (MPA) with loading of shorting pins for high-gain circularly polarized (CP) radiation is proposed in this paper. Two sets of metallic pins are symmetrically placed along the two orthogonal diagonals of a square patch radiator at first. Due to the shunt inductive effect brought by these shorting pins, the resonant frequency of the dominant mode in this MPA is progressively tuned up so as to enlarge the electrical size of this pin-loaded patch resonator and to enhance its radiation directivity. After the optimal loading position is investigated for maximum directivity of linear polarization, one pair of the inner pins is slightly shifted in an offset to properly separate the two degenerate modes, so that the CP radiation can be excited. Moreover, upon request, either left-handed or right-handed circular polarization can be obtained by means of different position-offset scheme of the inner pins along the diagonals. After extensive analysis is executed, two equal-size CP MPAs with and without shorting pins are fabricated and tested. Simulated and measured results show good agreement and demonstrate that the CP directivity is enhanced from 8.0 (conventional MPA) to 10.8 dBic, indicating a 2.8-dB increment by means of the proposed approach.

On anti-pentadiagonal persymmetric Hankel matrices with perturbed corners

In this paper we express the eigenvalues of real anti-pentadiagonal persymmetric Hankel matrices with perturbed corners as the zeros of explicit rational functions. From these prescribed eigenvalues we give an orthogonal diagonalization for these matrices and a formula to compute its integer powers. In particular, an explicit expression not depending on any unknown parameter for the determinant and the inverse of complex anti-pentadiagonal persymmetric Hankel matrices with perturbed corners is provided.