A finite elastic body with a curved crack loaded in anti-plane shear - ScienceDirect
Export JavaScript is disabled on your browser. Please enable JavaScript to use all the features on this page., 1993, Pages Author links open overlay panelShow moreAbstractThis paper presents a Boundary Integral Equation Method (BIEM) for an arbitrarily shaped, linearly elastic, homogeneous and isotropic body with a curved crack loaded in anti-plane shear. The crack must be modeled as an arc of a circle and wholly inside the solid—otherwise its position and orientation with respect to the boundary of the body is arbitrary.The effect of the crack on the stress field is incorporated in an augmented kernel developed for the mode III crack problem such that discretization of the cutout boundary is no longer necessary. This modification of the kernel of the integral equation leads to solutions on and near the cutout with great accuracy. An asymptotic analysis is conducted in order to derive the Stress Intensity Factor (SIF) KIII, at each crack tip, in closed form. In this formulation, a straight crack can be viewed as a particular case of the more general curved crack. In particular, attention is paid to the influence of crack curvature and edge effect on the stress intensity factors at the right and left crack tips. A rigorous mathematical formulation is developed, the main aspects of the numerical implementation are discussed and several representative numerical examples are presented in this paper.Check if you have access through your login credentials or your institution.ororRecommended articlesCiting articles (0)Efficient analysis of electromagnetic scattering and radiation from patches on finite, arbitrarily curved, grounded substrates - Yuan - 2004 - Radio Science - Wiley Online Library
Advertisement
[1]&A precorrected-fast Fourier transform (FFT) accelerated surface integral equation approach formulated using the homogeneous medium Green's function is presented for the analysis of patch arrays on finite, arbitrarily shaped, grounded substrate. The integral equation is solved by the method of moments, and the precorrected-FFT method is applied to reduce the memory requirement and computational complexity of the solution procedure. The memory required for this algorithm is O(N1.5), and the computational complexity is NiterN1.5log N, where N is the number of unknowns and Niter is the iteration number. Numerical results are presented to demonstrate the accuracy and capability of the method.1.&Introduction[2]&The prediction of radiation and scattering from realistic microstrip antennas is essential in the design of many communication systems. Among the various methods developed for the analysis of microstrip antennas, the most accurate and widely used method utilizes the familiar Green's function/Sommerfeld approach [; ]. Several fast algorithms, such as the conjugate gradient-fast Fourier transform [], the adaptive integral method (AIM) [], etc., have been applied to expedite the solution of the method of moments (MOM) matrix equation that resulted from the mixed potential integral equation. However, these methods work only with the assumptions that both the ground plane and dielectric substrate are of infinite extent in the transverse direction, and thus the associated multilayer media Green's functions are available. These assumptions are not practical because the diffraction due to the presence of the edges of substrate and ground plane would affect various characteristics of the antennas. Furthermore, when the substrate and the ground plane form part of a curved surface, the &classical approach& fails miserably.[3]&To overcome this difficulty, several methods have been proposed in the literature, such as the method of lines [], the combined mixed potential integral equation-weak form conjugate gradient-fast Fourier transform method [], the ray-based asymptotic methods (e.g., geometrical theory of diffraction) [], and the hybrid integral equation approaches formulated using the free space Green's function [; ]. Unfortunately, the former methods are restricted to patch antennas on finite-sized planar substrate and certain simplifying assumptions. On the other hand, the hybrid integral equation approach was found to be capable of providing complete and rigorous solution. There are at least two such approaches available. One is the hybrid volume-surface integral equation (VSIE) formulation [], which uses the surface integral equation on the conducting surface and the volume integral equation in the dielectric region. Another approach is the surface integral equation (SIE) formulation [], which uses the surface integral equations on both conducting and dielectric surfaces. For either approach, since all the microstrip patches, penetrable substrates, and ground planes must be accurately modeled, the MOM-based assessment is a computationally challenging task because of the resultant large number of unknowns. To alleviate this problem, ,
have applied the multilevel fast multipole algorithm to reduce the CPU time and memory storage required to solve the VSIE. In this paper, a precorrected-FFT-accelerated SIE approach for the accurate and rapid analysis of microstrip structures with finite substrate and ground plane truncation effects is presented. The approach is also an efficient tool to simulate conformal antennas fabricated on arbitrarily curved substrate and ground plane. The SIE formulation has the advantage of yielding fewer unknowns in comparison with the VSIE, although it is limited to piecewise homogeneous media. The precorrected-FFT method achieves its fast processing and memory reduction by mapping the original moment method discretization onto a uniform grid and applying the FFTs to carry out the matrix-vector multiplications. The memory requirement and computational complexity are reduced to O(N1.5) and NiterN1.5 log N, respectively, where N is the number of unknowns and Niter is the iteration number.[4]&The paper is organized as follows. In
we present the surface integral equation formulation for the problem. Next, the MOM solution and the application of the precorrected-FFT method are described in
and , respectively. Finally, in , numerical results for patch arrays on various supporting structures are presented to demonstrate the validity and capability of the proposed method.2.&Integral Equation Formulation[5]&For clarity of illustration we present the surface integral equation formulation for a single conducting body and a single dielectric region immersed in free space. It is straightforward to extend the formulation to multiple conductors and dielectric regions.[6]&Consider a perfectly conducting body Sc and a dielectric body Sd placed in an infinite homogeneous medium (ɛ1, &1), as shown in . The dielectric body is characterized by relative permittivity and permeability (ɛ2, &2), which can be complex if the body is lossy. Suppose this composite structure is illuminated by an incident plane wave (Einc, Hinc). By invoking the equivalence principle, two equivalent problems are formulated, each valid for regions exterior and interior to the dielectric material. For the exterior equivalent problem, as shown in , the conducting and dielectric bodies are replaced by fictitious mathematical surfaces, and the entire region is filled with homogeneous material of the exterior medium. We have equivalent electric current Jc on surface Sc and equivalent electric and magnetic currents (Jd, Md) on surface Sd. For the interior equivalent problem, as shown in , the entire region is filled with material of the dielectric medium, and equivalent electric and magnetic currents (&Jd, &Md) are introduced on the surface Sd. The electric and magnetic currents appearing on opposite sides of a dielectric interface in different auxiliary problems are taken to be equal in magnitude and opposite in direction in order to ensure the continuity of the tangential components on this boundary, as they are continuous in the original problem. By enforcing the boundary conditions on the electric and magnetic fields across each interface and combining the interior and exterior equivalent problems [], we obtain the following set of coupled integral equations in terms of the unknown equivalent electric and magnetic currents: where superscripts 1 and 2 represent the medium in which the scattered fields are evaluated and subscripts Sc and Sd represent the surfaces on which the equations are enforced.
is the electric field integral equation (EFIE) for conducting objects (either closed or open), and
constitute the so-called PMCHWT formulation []. (PMCHWT refers to Poggio, Miller, Chang, Harrington, Wu, and Tsai, who originally developed the formulation for dielectric objects [; ; ].) The electric and magnetic fields E and H (suppressing the subscripts and superscripts) are given by where A and F are vector potentials, j is the imaginary unit, & is the angle frequency, ɛ and & are the permittivity and permeability of the medium, respectively, and &P and &P are scalar potentials.Figure&1. Arbitrarily shaped conducting/dielectric body illuminated by a plane wave. (a) Original problem. (b) Exterior equivalent problem. (c) Interior equivalent problem.3.&Application of the Method of Moments[7]&To solve , the conducting and dielectric surfaces are discretized into planar triangular patches, and the unknown current densities Jc, Jd, and Md are approximated in terms of linear combinations of the Rao-Wilton-Glisson (RWG) basis functions [] as follows: where Icn, Idn, and Mdn are the unknown expansion coefficients, Nc and Nd denote the number of edges in the triangulated model of the conducting and dielectric surfaces, respectively, r& is the position vector at the source point, and fn is the basis function. Substituting
and testing
with fm and
with &0fm yields an N & N(N = Nc + 2Nd) matrix equation in the following form: The matrices Z, I, and V can be written in the following partitioned form: Here I is the vector consisting of the unknown coefficients. V is the excitation vector with entries given by Z is the square moment matrix. Elements of the submatrices of the matrix Z can be readily obtained from
and the definitions in . For clarity of the description we provide the following expressions for some of the submatrices: [8]&Expressions for other submatrices can be written in a similar form but are omitted here. The various vector and scalar potential integrals take the following form for i = 1, 2: with Gi(r, r&) denoting the scalar Green's function in medium i defined by where ki = & is the wave number in medium i. In , , , , , , , , , , , , and , fm and fn represent the testing and basis functions, respectively, while Tm and Tn denote the triangular patches supporting them. As we know, the potentials A and &P are produced by the electric current, while F and &P are produced by the magnetic current. When the same basis functions are used for the expansion of the electric and magnetic currents, the vector potential integral Fimn and scalar potential integral &Pimn produced by the nth magnetic current expansion function are the dual of Aimn and &Pimn, respectively, produced by the same function as an electric current expansion function. These duality relationships enable us to simplify the projection operation in the following precorrected-FFT approach. Although both electric and magnetic currents/charges exist in the problem, we need only to construct the projection operators for one current/charge, say, electric current/charge, and then the projection of both the electric and magnetic currents/charges can be performed using the same projection operators.[9]&By solving the resultant matrix
we can obtain the unknown coefficients Icn, Idn, and Mdn. Consequently, the equivalent electric and magnetic currents Jc, Jd, and Md can be computed according to , and the field everywhere can then be evaluated. However, the coefficient matrix Z is fully populated, demanding O(N2) storage. A direct solution to
requires O(N3) operations, and an iterative solution requires O(N2) operations per iteration. These requirements will render the method computationally intractable when the composite structure is electrically large. Therefore we are interested in developing a fast algorithm with less complexity and storage requirements. In this paper, we solve the matrix equation by a block-diagonal preconditioned generalized minimum residual method and use the precorrected-FFT method to speed up the matrix-vector multiplications in iterations. The precorrected-FFT (P-FFT) method is a fast algorithm associated with O(N3/2 log N) (for the surface integral equation) or O(N log N) (for the volume integral equation) complexity. Its basic idea is similar to that of the AIM []. The precorrected-FFT method was proposed by ,
to solve the electrostatic integral equations for capacitance extraction problems and was recently extended by the present authors to calculate the electromagnetic scattering from large conducting objects [] and homogeneous dielectric bodies []. In this paper, we further extend it to the analysis of arbitrarily shaped, finite microstrip structures.4.&Solution by the Precorrected-FFT Method[10]&As in other fast algorithms such as the fast multipole method or AIM, the precorrected-FFT algorithm considers the near- and far-zone interactions separately when evaluating a matrix-vector multiplication. The interaction matrices are partitioned into two parts: one representing the strong near-zone interactions within an appropriate separation between the source point and observation point and the other representing the weak far-zone interactions beyond the said separation. The near-zone elements are computed directly and stored. The far-zone parts are not stored, and their vector multiplications will be calculated using an approximate technique. To compute the far-zone interactions, the entire object is enclosed in a uniform three-dimensional (3-D) rectangular grid, as shown in . The triangular patches are sorted into cells formed by the grid, with each cell containing only a few triangular patches. Equivalent point sources are placed at the cell vertices (grid order p = 2) or at half the spacing of the vertices (grid order p = 3), etc., according to the desired accuracy. Next the matrix-vector product can be approximated in the following four-step procedure:Figure&2. (a) Uniform precorrected-FFT method grid surrounding the discretized object (p = 3). (b) Four steps of the precorrected-FFT algorithm (grid order p = 2).[11]&1. Project the sources distributed on the triangular elements onto the regular grid by matching their vector and scalar potentials at some given test points to guarantee the approximate equality of their far fields.[12]&2. Evaluate the potentials at other grid locations produced by these grid-projected sources by a 3-D convolution.[13]&3. Interpolate the grid point potentials onto the triangular elements. The projection and interpolation operators are represented by sparse matrices, and the convolution can be carried out rapidly using discrete FFTs.[14]&4. Compute the near-field interactions directly and remove the errors introduced by the far-field operators.[15]&The difficulty with steps 1&3 is that although the far-field interactions are approximated very well, the near-field interactions are poorly approximated, since the near fields radiated by the grid sources do not match those radiated by the original sources. Therefore to obtain more accurate results, step 4 is undertaken. This process is referred to as &precorrection.& A pictorial representation of this process is given in . Interactions with nearby elements (shaded area) are computed directly, while interactions between distant elements are computed using the grid. Next, we will describe the steps in detail.[16]&First, we need to construct the projection operators that replace the element source distributions in the cell with an equivalent set of point sources on the grid. As can be seen from , there are three kinds of sources to be projected onto the uniform grid, i.e., the basis function fn, which corresponds to patch currents (either electric or magnetic), the divergence operator & & fn, which corresponds to patch charges, and the curl operator & & fn, which corresponds to & & Aimn or & & Fimn. For the nth RWG basis function fn, the function is assumed to be contained in a given cell k, and a p & p & p (p denotes the grid order) array of grid currents in the cell is used to represent the current distributions on the two adjacent RWG patches. Next, Nc test points are selected on the surface of a sphere of radius rc whose center coincides with the center of the cell k. The vector potential Ai (the subscript i represents the medium in which the vector potentials are computed) produced by the currents at the p3 grid points is enforced to match that produced by the current distributions on the triangular patches at the Nc test points, i.e., where Ai,qpt and i,qgt denote the vector potentials at the qth test point produced by the patch currents and the grid currents, respectively. Their expressions are given by where rqt and rs represent the position vectors at the qth test point and the sth grid point, respectively, and x,s, y,s, and z,s denote the three components of the current at the sth grid point. Substituting
and rearranging, we obtain the following projection operator Wi,u(k, n), which represents the contribution of fn (in cell k) to the u(u = x, y, z) component of the grid currents: where [Pi,ugt]+ indicates the generalized Moore-Penrose inverse of Pi,ugt and Pi,upt,n denotes the nth column of Pi,upt, while Pi,ugt & R&p3 denotes the mappings between the grid currents and the test point potentials, Pi,upt & R&N(k) is the mappings between the patch currents and the test point potentials, and N(k) is the number of the basis functions contained in cell k. The expressions of Pi,ugt and Pi,upt,n can be readily obtained from
and . For any n in cell k this projection operator generates a subset of the grid currents i,u. The contribution to i,u from the currents in cell k can be computed by summing over all the currents in this cell, i.e., Patch currents outside cell k may contribute to some of the elements of Ji,u in the case of shared grids. These matrices are small, and therefore the relative computational cost for this operation is insignificant.[17]&For the divergence operator & & fn we obtain the projection operator Wi,c(k, n) by matching the scalar potential due to the p3 grid charges and that due to the actual patch charge distributions at the test points, where By using the operator Wi,c(k, n) we can project the patch charges In& & fn onto the p3 grid points surrounding cell k. The accuracy of the above projection scheme depends on the proper selection of the test points. To achieve higher accuracy, the test points are chosen to be the abscissas of a high-order quadrature rule, which are shown by .[18]&For the curl operator & & fn introduced by & & Aimn or & & Fimn we employed a simplified approach to compute & & Aimn or & & Fimn directly instead of dealing with & & fn. Considering the evaluation of & & Aimn, the partial derivatives of Aimn can be replaced by the corresponding differences. The required differences of each component of Aimn can be computed through the value of Aimn at several points in the vicinity of the observation point. Knowledge of these potentials can be readily obtained through interpolation once the potentials at grid points have been computed by the FFTs in the following step. This approach only requires several extra interpolations and avoids additional efforts to project the & & fn operator and to perform the corresponding FFTs. It also yields sufficient accuracy since the projection operators for the basis function and the divergence operator have very high accuracy.[19]&Once the source distributions on the triangular patch have been projected onto uniform grids, the relationship between the vector/scalar potentials at the grid points and the grid sources is in fact a 3-D convolution. The convolutions can be efficiently computed using the discrete FFT because of the Toeplitz property of the Green's function matrix. Hence the vector and scalar potentials at the grid points can be computed by where DFT and DFT&1 denote the discrete FFT and inverse FFT, respectively. Gi are the Green's functions between grid points in the corresponding medium i. In practice, each convolution requires one forward and one inverse three-dimensional FFT. The FFT of the kernel matrix Gi needs to be computed only once.[20]&After the grid potentials are computed, the potentials on the triangular patches can be obtained through interpolation. Assume that VT denotes the interpolation operator which interpolates potentials at grid points onto patch coordinates and H denotes the overall inverse and direct FFT operations in
and . Thus the projection followed by convolution and interpolation gives the grid approximation to the potentials as Both the projection and interpolation operators are represented by sparse matrices.[21]&The above projection-convolution-interpolation procedure approximates the far-zone interactions accurately, but interactions between nearby patches are poorly approximated. To obtain more accurate results, it is necessary to calculate the near-field interactions directly and remove the inaccurate contribution of the far-field operator. Assume that Pi(k, l) and Pi,c(k, l) denote the close interactions which are computed directly. Subtracting the inaccurate approximation and then adding the direct contribution produces the accurate result of the vector potential Ai(k) and scalar potential &Pi(k) for each cell k where AiG(k) and &PiG(k) are the respective grid approximations to Ai(k) and &Pi(k). M(k) denotes the indices of the set of cells which are &close to& cell k. Since for each k, M(k) is a small set and each matrix Pi(k, l) or Pi,c(k, l) is also small, this step is also a sparse operation.[22]&Upon obtaining the required vector and scalar potentials the submatrix-vector products required in the iterative solution of , , and
can be computed based on , , , and . In practice, care must be taken to compute the projection operators and the potentials for each region separately, as the Green's functions depend upon the constitutive properties for each medium.[23]&From the analysis described above, the memory requirement of the precorrected-FFT method consists of three parts, i.e., the memory required to store the near-interaction elements, for the projection and interpolation processing, and for the FFTs. Because of the sparsity of the projection, interpolation, and precorrection steps the requirements of these operations are proportional to O(N). The requirement of the FFTs is O(Ng), where Ng denotes the number of the uniform grid points. The CPU time per iteration of the precorrected-FFT method is dominated by the FFT computation of the matrix-by-vector product, which is proportional to O(Ng log Ng). For surface integral equations, Ng is estimated to be asymptotically proportional to N1.5. Hence the total memory requirement of the method scales as O(N1.5), and the CPU time per iteration scales is O(N1.5 log N).5.&Numerical Results[24]&In this section some numerical examples are presented to validate the implementation and to demonstrate the capability of the proposed method. The actual memory reduction and speed up attained by the precorrected-FFT algorithm have already been demonstrated by
and are omitted here.[25]&As the first example we consider the scattering from a disk-cylinder structure illuminated by a normally incident field (&i = 180&). The radius of the cylinder is 0.3&0, and the height of the cylinder is 0.6&0. The cylinder is capped by a conducting disk of radius 0.3&0. The bistatic radar cross sections (RCS) obtained by the conventional MOM and the precorrected-FFT method are presented in
and compared with the results given by . Good agreements are observed, validating both our MOM and precorrected-FFT codes.Figure&3. Bistatic radar cross section (RCS) of a disk-cylinder structure.[26]&The second example is a larger disk-cone structure shown in the inset of . The dielectric cone (ɛr = 2) has a radius of 1.2&0 and a height of 0.6&0. The disk also has a radius of 1.2&0. The structure is discretized into 2896 triangular patches and has 7279 unknowns. The calculated bistatic RCS (&i = 180&) is shown in . It can be seen that the precorrected-FFT solution agrees very well with the MOM solution. In the precorrected-FFT method a grid spacing of 0.113&g is used, and the near-field threshold distance is set to be 0.339&g. The grid order is p = 3. For the conventional MOM the memory requirement is about 422 Mb, and the CPU time per iteration is 24.47 s on a Pentium 2.4G PC. However, the P-FFT uses only 41 Mb of memory and takes 12.71 s per iteration. To achieve a normalized residual error of 10&3, both methods need 103 iterations.Figure&4. Bistatic RCS of a disk-cone structure with a = 1.2&0, h = 0.6&0, and ɛr = 2.[27]&The third example is a 3 & 3 patch array fabricated on a finite substrate backed by a finite ground plane. The array consists of a rectangular element with L = 36.6 mm and W = 26 mm. The distance between two adjacent elements in both the x and y directions is 55.517 mm. The dielectric substrate has a thickness of 1.58 mm and a dielectric constant of 2.17. The size of the ground plane and the substrate is 180 & 180 mm. The patches, ground plane, and dielectric substrate are discretized into a total of 2448 triangles, resulting in 5715 unknowns. The bistatic RCS in the ϕ = 0& plane at 2.2 GHz due to a normally incident -polarized plane wave is shown in . It is observed that the results obtained from the conventional MOM and the P-FFT method agree very well with each other. For the P-FFT method a grid spacing of 0.14&g and a threshold distance of 0.56&g are employed, where &g is the wavelength in the dielectric substrate. The overall FFT grid dimensions are 16 & 16 & 4.Figure&5. Bistatic RCS of a 3 & 3 rectangular-shaped array with L = 36.6 mm, W = 26 mm, a = b = 55.517 mm, d = 1.58 mm, ɛr = 2.17, and f = 2.2 GHz.[28]&The above three examples compare the results of the traditional MOM and the P-FFT method. In the fourth example we consider a 3 & 3 cross-shaped patch array and compare the P-FFT result and the measured data. In this example the element of the array is a cross patch with L = 1.5 cm and W = 0.5 cm. The distance between two adjacent elements in both the x and y directions is 2.0 cm. The thickness of the substrate is h = 0.508 mm, and the dielectric constant is ɛr = 2.98. The dimensions of the ground plane and the substrate are dx = dy = 6 cm. The monostatic RCS &&& in the ϕ = 0& plane at 10 GHz is shown in . Good agreement is observed at most scattering angles. It is also observed that the discrepancy between the measured and calculated results increases as the scattering angle approaches 90&. This is reasonable since & = 90& corresponds to grazing incidence, at which neither accurate measurement nor accurate prediction is possible. In the computation the number of unknowns is 17,117.Figure&6. Monostatic RCS (&&&) versus & for a 3 & 3 cross-shaped array with L = 1.5 cm, W = 0.5 cm, a = b = 2.0 cm, ɛr = 2.98, dx = dy = 6.0 cm, h = 0.508 mm, and f = 10 GHz.[29]&In the next example we consider a single microstrip antenna and investigate its reflection coefficient and far-field radiation pattern. As shown in , a patch of 16 mm & 12.448 mm is placed over a grounded dielectric slab. The slab has a thickness of 0.78 mm and a dielectric constant of ɛr = 2.2. The patch antenna is off-center fed by a 50 &O microstrip line with the width of 2.334 mm. We consider the following three cases: (1) The substrate and ground plane are of infinite extent in the transverse direction, (2) the substrate and ground plane are finite and of 40 mm & 23.34 mm in size, and (3) the patch is bent in the xz plane, while the microstrip line remains planar, as shown in . The radius of curvature r of the bend is 20 mm. We first analyzed the frequency-dependent reflection coefficient ∣S11∣ for the patch antennas under these three cases.
compares our calculated results and the experimental data taken from . It is observed that while ∣S11∣ is not sensitive to changes in the size and shape of the substrate/ground, the radiation pattern for the finite case is quite different from that of the infinite case.
shows the calculated far-field radiation pattern of the patch antenna for the three cases at their resonant frequencies. As the patch was curved in the xz plane, the pattern is visibly tilted in the direction vertical to the patch plane. It is also observed that the bent shape and the edge diffraction caused by the finite substrate clearly affect the cross-polarized component of the field more than the copolarized component.Figure&7. Rectangular microstrip antenna on a (a) planar substrate and (b) curved substrate.Figure&8. . Reflection coefficient ∣S11∣ of a rectangular microstrip antenna.Figure&9. Radiation patterns in (a) ϕ = 0& and (b) ϕ = 90& planes of the microstrip antenna under three cases.[30]&After validating the accuracy and performance of the method, we present some examples to show its capability in analyzing conformal microstrip antennas. First, consider the scattering from a spherical-circular microstrip antenna as shown in . A metallic sphere of radius a is coated with a dielectric substrate of thickness h(= b & a). A circular patch of radius rd is mounted on the substrate and subtends a solid angle 2&0. In this case, there is no incident wave on the surface of the metallic sphere. So the following integral equations are satisfied: where Sc1, Sc2, and Sd represent the surface of the patch, the metallic sphere, and the dielectric substrate, respectively.
shows the bistatic RCS of the composite structure at 7 GHz when the incident wave is in the direction (&i, ϕi) = (180&, 0&) and (&i, ϕi) = (117&, 0&), respectively. It is observed that because of the presence of the circular patch on the top the bistatic RCS of this composite structure is no longer symmetric about the direction of the incident wave for all incident angles in contrast to a homogeneous sphere. However, the nonsymmetry is slight, since the contribution of the circular patch is very small compared to that of the grounded substrate. The electric current distributions on the circular patch and the dielectric surface under these two incidences (& polarized wave) are plotted in . It is seen that the direction of the incident wave is obviously reflected by the magnitude distribution of the & component of the induced electric current.Figure&10. Geometry of a spherical-circular microstrip antenna.Figure&11. Bistatic RCS of a spherical-circular microstrip antenna with a = 3 cm, rd = 7.1 mm, h = 0.7874 mm, ɛr = 2.2, and f = 7 GHz.Figure&12. Current distribution on the patch and dielectric sphere when a & polarized plane wave is incident from two different angles. (a) ∣J&∣ under &i = 180&. (b) ∣Jϕ∣ under &i = 180&. (c) ∣J&∣ under &i = 117&. (d) ∣Jϕ∣ under &i = 117&.[31]&The next example is an 8 & 1 patch array mounted on a section-cylindrical substrate and conducting ground, as shown in . The element of the array is a cylindrical-square patch with dimensions Dz = Dϕ = &0/4. The interelement spacing is also &0/4. The outer radius of the cylindrical substrate is b = 2&0/&, and the thickness of the substrate is t = 0.1&0. The cylinder has a height of h = 4.25&0 and an extension angle of ϕ0 = 67.5&. The permittivity of the substrate is 2.0. The structure is modeled by 7920 triangles, and the total number of unknowns is 17,158.
shows the calculated bistatic RCS as a function of ϕ when the incident wave is along the & axis in the xoy plane (&i = 90&, ϕi = 180&).Figure&13. (a) Geometry of an 8 & 1 patch array on a section-cylindrical substrate. (b) Bistatic RCS versus ϕ in the xoy plane with b = 2&0/&, t = 0.1&0, h = 4.25&0, ϕ0 = 67.5&, Dz = Dϕ = &0/4, d = &0/4, and ɛr = 2.0.[32]&Finally, we consider a dielectric cylinder with radius b = 2&0/&, height h = 4.25&0, and permittivity ɛr = 2.0, covered with a doubly periodic 8 & 8 cylindrical-square patch array. The size of the element and the interelement spacing in the z direction are identical to the previously investigated example. The azimuthal interval is also 0.25&0.
shows the triangulated model of the structure, which has 11,744 triangles and yields 31,648 unknowns.
presents the computed monostatic and bistatic RCS versus ϕ in the xoy plane. The bistatic RCS is for incident wave in the direction of &i = 90& and ϕi = 180&. It is seen that the monostatic RCS changes periodically, since the structure is periodic in the xoy plane. For this example the traditional MOM requires about 7.7 Gb memory, while the present method requires only 386 Mb memory when a grid spacing of 0.15&g and a threshold distance of 0.45&g namely, the present method yields a memory savings of about 95%. The CPU time per iteration is 40.28 s on a Pentium 2.4 Ghz PC, and 289 iterations are required to achieve a normalized residual of 10&3 for the bistatic case.Figure&14. (a) Triangulated model of a dielectric cylinder covered by an 8 & 8 patch array. (b) Monostatic and bistatic RCS versus ϕ in the xoy plane.[33]&In the two examples above, the cylindrical supporting structure is of finite length. Hence the layered media Green's function is not available. In this case the approach based on solving the surface integral equation formulated using the layered media Green's function fails. However, the present method has no difficulty, since it is based on the Green's function in homogeneous medium.6.&Conclusions[34]&A precorrected-FFT accelerated SIE approach was presented for the efficient and accurate analysis of patch antenna/arrays on finite-sized, arbitrarily shaped support structures. In the algorithm the problem is formulated by an EFIE-PMCHWT formulation in terms of the Green's function in homogeneous medium. The integral equations are discretized by the MOM, in which the surfaces are modeled as triangular patches, with the unknown equivalent electric and magnetic currents expanded using the RWG basis functions. The resultant matrix equation is solved iteratively, and the precorrected-FFT method is used to speed up the matrix-vector products and reduce the memory requirement. Numerical results have been presented to validate the implementation and to demonstrate the capability of the algorithm. Since the application of the precorrected-FFT method reduces the memory requirement and CPU time greatly, the present method can analyze microstrip structures of size much larger than could be analyzed by the conventional MOM with the same computer resources. Moreover, this approach includes the dielectric substrate and ground plane in the solution domain, hence giving flexibility to model structures of real size and of nonflat, arbitrarily shaped substrate and ground planes.
Advertisement}