Numerical Steepest Descent Path Method Research Articles

The Polar Coordinate Transformation Numerical Steepest Descent Path Method for Calculating Physical Optical Scattering Field of Quadric Concave or Convex Surfaces

In this letter, the polar coordinate transformation numerical steepest descent path (PCT-NSDP) method is proposed to calculate the physical optics (PO) scattered field from concave or convex quadratic surfaces. In the proposed method, only three triangles integral domains are calculated, which avoids the complicated decomposition in the former NSDP methods. The PCT-NSDP method has a closed-form formula of the contribution of PO stationary phase point. Therefore, the calculation time can be much smaller. Moreover, the workload is frequency-independent. The numerical examples of the radar cross section of a quadratic convex surface demonstrate the high accuracy and efficiency of the PCT-NSDP method.

The Efficient Method for Calculating the Physical Optics Scattered Fields from the Concave Surfaces

In this work, the numerical steepest descent path (NSDP) method is proposed to compute the highly oscillatory physical optics (PO) scattered fields from the concave surfaces, including both the monostatic and the bistatic cases. Quadratic variations are adopted to approximate the integrands of the PO type integral into the canonical form. Then, on involving the NSDP method, we deform the integration paths of the integrals into several NSDPs on the complex plain, through which the highly oscillatory integrands are converted to exponentially decay integrands. The RCS results of the PO scattered field are calculated and are compared with the high frequency asymptotic (HFA) method and the brute force (BF) method. The results demonstrate that the proposed NSDP method for calculating PO scattered fields from concave surfaces is frequency-independent and error-controllable. Numerical examples are provided to verify the efficiencies of the NSDP method.

The Two-Dimensional Numerical Steepest Descent Path Method for Calculating the Physical Optics Scattered Fields From Different Quadratic Patches

In this article, the Two-Dimensional numerical steepest descent path (TD-NSDP) method is proposed to calculate the physical optics (PO) scattered fields from the quadratic patches. Compared with the NSDP method, the TD-NSDP method avoids adopting the complementary error function, and hence, the TD-NSDP method could significantly improve the accuracy of the PO scattered fields by around four to five digits. Moreover, the computational workload is also frequency independent. To demonstrate the efficiencies of the TD-NSDP method, numerical examples for both quadratic convex and concave surface patches are conducted. Finally, the radar cross section (RCS) simulation from the realistic and complex scatterer is conducted to verify the practicability of this method in electromagnetic (EM) scattering application.

Frequency-independent approach to calculate physical optics radiations with the quadratic concave phase variations

In this work, we develop the numerical steepest descent path (NSDP) method to calculate the physical optics (PO) radiations with the quadratic concave phase variations. With the surface integral equation method, the physical optics (PO) scattered fields are formulated and further reduced to the surface integrals. The high frequency physical critical points contributions, including the stationary phase points, the boundary resonance points and the vertex points are comprehensively studied via the proposed NSDP method. The key contributions of this work are twofold. One is that together with the PO integrals taking the quadratic parabolic and hyperbolic phase terms, this work makes the NSDP theory be complete for treating the PO integrals with quadratic phase variations. Another is that, in order to illustrate the transition effect of the high frequency physical critical points, in this work, we consider and further extend the NSDP method to calculate the PO integrals with the coalescence of the high frequency critical points. Numerical results for the highly oscillatory PO integral with the coalescence of the critical points are given to verify the efficiency of the proposed NSDP method. The NSDP method could achieve the frequency independent computational workload and error controllable accuracy in all the numerical experiments, especially for the case of the coalescence of the high frequency critical points.

A Frequency-Independent Method for Computing the Physical Optics-Based Electromagnetic Fields Scattered From a Hyperbolic Surface

In this communication, we propose a frequency-independent approach, the numerical steepest descent path (NSDP) method, for computing the physical optics scattered electromagnetic field on the quadratic hyperbolic surface. Due to the highly oscillatory nature of the physical optics integral, the proposed method relies on deforming the integration path of the integral into the NSDP on the complex plane. Numerical results for the PO-based EM fields from the hyperbolic surface illustrate that the proposed NSDP method is frequency independent in computational cost and error controllable in accuracy.

THE MODERN HIGH FREQUENCY METHODS FOR SOLVING ELECTROMAGNETIC SCATTERING PROBLEMS (Invited Paper)

The high frequency scattering problems of electromagnetic fields scattered from electrically large scatterers are important and challenging. On the calculation of the reflected and diffracted wave fields, the high frequency methods could be classified into the current based method and the ray based method. In this paper, first, we give a review on the progress of the modern high frequency methods for solving the electromagnetic scattering problems. Next, due to the highly oscillatory property of the high frequency electromagnetic scattered fields, we propose the numerical steepest descent path method. Finally, we comprehensively address the high frequency wave physics, including the high frequency critical point contributions, the Keller’s cone, the shadow and reflection boundaries and the creeping wave fields.

The Contour Deformation Method for Calculating the High-Frequency Scattered Field by the Fock Current on the Surface of the 3-D Convex Cylinder

In this paper, the high-frequency diffracted waves like the creeping waves are comprehensively analyzed by the Fock currents. On invoking the contour deformation method, the highly oscillatory Fock currents are efficiently calculated. Furthermore, the workload for the calculation of Fock currents is frequency-independent. To capture the high-frequency wave physics phenomenon, the Fock current is separated into the classical physical optics (PO) current and the nonuniform (NU)-Fock current along the shadow boundary and in the deep shadow region. To calculate the highly oscillatory scattered wave fields from the Fock current, quadratic approximations of the phase functions in the integrand are adopted. On invoking the numerical steepest descent path (NSDP) method, the scattered wave fields are efficiently calculated with frequency-independent computational effort and error controllable accuracy in each frequency-independent segment. Meanwhile, the high-frequency creeping wave coming from the NU-Fock current is efficiently captured by the NSDP method. Numerical results for the Fock currents, the high-frequency NU-diffracted and scattered far fields on the convex cylinders are given to validate the efficiency of the proposed method. Furthermore, the contour deformation method for computing the Fock currents offers a clear physical picture for the high-frequency wave fields on the convex scatterer.

The Numerical Steepest Descent Path Method for Calculating Physical Optics Integrals on Smooth Conducting Quadratic Surfaces

In this paper, we use the numerical steepest descent path (NSDP) method to analyze the highly oscillatory physical optics (PO) integral on smooth conducting parabolic surfaces, including both monostatic and bistatic cases. Quadratic variations of the amplitude and phase functions are used to approximate the integrand of PO integral. Then the surface PO integral is reduced into several highly oscillatory line integrals. By invoking the NSDP method, these highly oscillatory PO line integrals are defined on the corresponding NSDPs. Furthermore, the critical point contributions for the PO integral are exactly extracted and represented based on the NSDPs. The proposed NSDP method for calculating the PO integral on the smooth conducting surfaces is frequency-independent and error-controllable. Compared with the traditional asymptotic expansion approach, the NSDP method significantly improves the PO integral accuracy by around two digits when the working wave frequencies are not extremely large. Numerical results are given to validate the NSDP method.

Computing highly oscillatory physical optics integral on the polygonal domain by an efficient numerical steepest descent path method

In this work, the computation of physical optics (PO) type integral with the integrand of quadratic phase and amplitude is studied. First, we apply the numerical steepest descent path (NSDP) method to calculate the highly oscillatory PO integral on the triangular patch. Then, we rigorously extend the proposed NSDP method to analyze the PO integral on polygonal domains. Furthermore, the contributions of critical points on polygonal domains, including the stationary phase point, resonance and vertex points, are comprehensively studied in terms of the NSDP method. Compared to the traditional high frequency asymptotic (HFA) method, when the wave frequency is not very high but in the high frequency regime, the NSDP method has improved the PO integral accuracy by one to two digits. Meanwhile, the computational cost by using the proposed NSDP method is independent of the wave frequency.

Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: the convex case.

We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical results illustrating the high-order convergence of our algorithm as well as its asymptotically bounded computational cost as the frequency increases.