Assessment of the Robustness of Flexible Antennas to Complex Deformations

Wearable antennas can suffer from a variety of mechanical deformations that are induced by the body dynamic. The article analyses how these complex deformations impact the performance of a flexible antenna operating in the 5–6 GHz band. The Green Coordinates (GC) spatial manipulation technique is used to generate a range of complex 2-D deformations, namely spherical, saddle, and twisting deformation. Generating full geometries is a key enabler in this study. The results offer valuable insight into the stability of antenna performance under in situ deformations.

Abstract-Wearable antennas can suffer from a variety of mechanical deformations that are induced by the body dynamic. The article analyses how these complex deformations impact the performance of a flexible antenna operating in the 5-6 GHz band. The Green Coordinates (GC) spatial manipulation technique is used to generate a range of complex 2-D deformations, namely spherical, saddle, and twisting deformation. Generating full geometries is a key enabler in this study. The results offer valuable insight into the stability of antenna performance under in situ deformations.
Index Terms-Computer graphics, flexible antenna, microstrip antenna, numerical modeling.

I. INTRODUCTION
T HE next-generation wireless communication systems have increased the demand for flexible and wearable antennas for a variety of applications ranging from sports to biomedical. Considerable research has been focused on developing flexible substrates [1], [2], and intrinsically stretchable conductors [3], [4], [5], [6]. It is now well established that the polydimethylsiloxane (PDMS) polymer is one of the most promising substrate materials for wearable electronics due to its high conformability compared to other polymer types, adjustable relative permittivity, acceptable dielectric losses, and low-cost fabrication methods [2], [7], [8]. In addition to PDMS substrates, textile substrates, such as e-textile, woven, and knit textile structures, are favorable alternatives [9], [10], [11], [12] but exhibit lower stability in combination with conductive prints and limited design performance predictability due to complex geometrical details of conductive threads [9], [10], [11], [12], [13], [14].
Integration of antennas into clothing means that the antennas will be affected by the dynamics of the human body, which can cause a variety of deformations [15], [16], [17], [18], [19], which in turn, can degrade the overall system's performance and present a challenging design task for wearable antenna design.
Although there are numerous, unfortunately often conflicting, results on the impact of cylindrical bending on antenna performance, it is now widely accepted that bending antennas along antenna length is more critical for performance as it increases the resonant frequency of antennas [20], [21], [24]. The bending does not significantly affect the bandwidth of the antenna or the far-field radiation pattern [20], [21], [24]. More recently, the impact of twisting on flexible interconnects [27] and wideband dipole antennas [17] was reported as showing a small impact on their performance.
The impact of other types of deformations, in particular more complex 2D deformations, such as spherical, saddle, and twisting deformations, as shown in Fig. 1, have, however, not been considered. The focus on simpler cylindrical deformations is due to the fact that it is relatively straightforward to generate these geometries in standard computational electromagnetic (EM) software, such as Ansys and CST, which are based on constructive solid geometry (CSG) primitives and Boolean geometry. It is commented here that the subsequent choice of modeling method, finite elements, finite integration technique, method of moments, or their implementation within particular software packages is not the issue; rather, it is the geometrical data upon which they operate that has the failing. The main difficulty when generating arbitrary deformations is in obtaining well behaved interfaces between the constitutive parts of the antenna, free of misalignments and microscopic gaps, as discussed in [28], [29], and [30], which can undermine subsequent EM numerical simulations. Our recent work [28] proposed the use of a computer graphics spatial manipulation technique based on Green Coordinates (GC) for generating arbitrary deformations of antennas [31], [32]. The GC technique belongs to a class of cage-based methods that enclose the object of interest by a so-called "bounding cage" that has a similar but less complicated shape than the object of interest. The GC method stands out from other similar spatial manipulation techniques (Mean Value Coordinates [33], [34], and Harmonic Coordinates [33]) as it is the most shape-preserving, i.e., introduces a minimal amount of unphysical distortions and is computationally efficient as is implemented using a closed-form analytical expression [31], [32]. Our recent work demonstrated that the GC method provides a robust approach to generating highly variable geometries of deformed antennas without introducing disruptive CAD artifacts that can either block or seriously undermine EM characterization.
The systematic distortions that are introduced by the GC method can be effectively compensated for by using an iterative prescaling approach, as shown in [28], [29], and [30], that guarantees the physical reality of the final geometries. This approach was calibrated against the simple case of cylindrical bending for which no distortion is expected to occur and proved that the performance of the GC-generated cylindrically bent antenna agrees well with the CSG-generated antennas opening the way for robust generation of more general antenna deformations [28]. A few representative complex, double curved antenna geometries have been demonstrated in [28] but a more general analysis of antenna performance under varying degrees of these complex deformations is needed in order to 1) assess the general sensitivity of an antenna to a variety of deformations, 2) identify which class of deformations have the most disruptive impact on the antenna performance, and 3) investigate whether simpler cylindrical deformations can indeed be used to predict the impact of more complex deformations.
This article extends the approach of [28] to consider a more systematic parameter sweep of the problem space for several cases of doubly curved deformations, namely, spherical and saddle bending and twisting deformation. Although these still do not span all "irregular" deformations, they are, in nature, more complex than the simpler case of cylindrical bending. Specifically, the article explores how these deformations affect the performance of flexible antennae fabricated on PDMS substrates operating in the 5.5-6 GHz wireless band.
The article is structured as follows. Section II briefly overviews the GC method in the context of arbitrary antenna generation and defines the parameters of the warped cages for spherical, saddle, and twisting deformations. Section III assesses the impact of spherical, saddle, and twisting deformation on antenna performance, namely S 11 , bandwidth, and far-field profile. This section further investigates whether the impact of 2D antenna deformations can be predicted by simpler cylindrical deformations. Section IV gives the overall conclusions of the article.

II. COMPLEX ANTENNA DEFORMATIONS
In this section, the main principle of the GC method is overviewed and applied to a variety of complex 2D antenna deformations.
The GC technique belongs to a type of cage-based method that encloses the object of interest by a bounding cage that has a similar but less complicated shape than the object of interest, and as shown in Fig. 2(a), for the case of patch antenna the bounding cage is a simple polygonal tube [28], [29], [30]. All spatial points within the interior of the cage can be expressed in terms of the cage's geometry, e.g., vertices and face normals of the cage's surface triangulations [28], [29], [30]. The vertices of the simple cage are then manipulated to generate a wanted deformation, as shown in Fig. 2(b) for the case of saddle deformation. The deformation of the cage is then mapped onto the geometry of the enclosed object, as shown in Fig. 2(c). Removing the warped cage leaves the deformed object, as shown in Fig. 2(d). It is a highly valuable feature of the GC method that it maps relatively crudely defined deformations of the warped cage onto the smooth deformations of the final geometry beyond the scope of explicit manual intervention.
For the patch antenna with a coaxial feed, two approaches can be adopted when constructing the flat cage, namely, as follows.
1) Construct the cage that follows the shape of the antenna and antenna cross section that includes the extrusion around the coaxial cable. That part of the cage that surrounds the coaxial feed does not experience any deformation, but the rest of the cage is deformed as required. 2) Define the flat cages, as shown in Fig. 2(a), i.e., a simple polyhedron, deform the cage as required, and then add the coaxial cable to the deformed antenna using Boolean "through" operation. In this article, the latter approach is used due to the controlled nature of deformations; however, in case of more general or irregular deformations, the former approach may be better suited. The GC method opens a way of generating more complex antenna deformations, and these will now be investigated on a typical example of a wearable patch antenna fabricated on a flexible PDMS substrate and operating in the wireless 5.5-6 GHz band. The schematic of the flat patch antenna is shown in Fig. 3. The substrate's length L sub , width W sub , and thickness h are 36, 36, and 1.5 mm, respectively. The metallic patch has a width of W = 18 mm, and a length of L = 14.3 mm. The feed position is offset by z o = 2.8 mm in the z-direction from the middle of the patch and is designed to give an optimal S 11 parameter of the antenna and 3.5% of fractional bandwidth. The coaxial cable feed has an inner radius r in , and outer radius r out of 0.625 and 2.15 mm respectively. The cable dielectric constant is 2.2. The dielectric constant of the PDMS substrate is 2.7, and its dielectric losses are neglected. The radiating patch and the ground plane are assumed to be perfect conductors of 35 µm thickness. Fig. 4 illustrates the cages that are considered in this article. In all cases, the flat cage width, length, and height are 56, 56, and 7.5 mm, respectively, and the antenna is positioned in the center of the flat cage. The concave and convex spherical deformations are characterized by a spherical radius, r . To generate these spherical deformations, a cross section of a polygonal tube is defined and then extruded along the width or the length of the cage [see Fig. 4(a)]. By changing the origin of the sphere, i.e., above or below the top antenna surface, concave and convex deformations can be obtained, and the radius of the sphere defines the strength of imposed deformation.
Similarly, to generate saddle deformations, a cross section of a polygonal tube is defined across the width (length) of the cage as a sinusoidal parabola, and the tube is extruded along the length (width) along a defined parabola. Fig. 4(c) shows  an example of a saddle cage where the saddle deformation of the cage is controlled by the half period of the sine function A W sin(x) along the width of the antenna and A L sin(z) along the length of the antenna. The strength of the deformation is controlled by the parameter A W and A L , with the peak value centered at the midpoint along the width and length of the cage, respectively. Saddle deformations are obtained when amplitudes A W and A L have opposite signs. Finally, the twisting deformation is generated by defining a cross section of the cage, which is then extruded and twisted along the length or width of the antenna, Fig. 4(d), with a twisting deformation controlled by a mean twist per meter parameter.
To reassure the reader of the accuracy of the GC method in the context of 2D deformations, Fig. 5 shows the relative distortion in the length and width of the radiating patch. The iterative prescaling approach described in [28], [29], and [30] is applied to each deformation, namely, the spherical convex (r = 30 mm), saddle (A = A W = A L = 20 mm), and twisting (0.0125 twist/m) deformations. It can be seen that relative distortion is initially high but rapidly reduces after only a few iterations. Fig. 5 also shows that distortion errors for the spherical and twisting deformations reduce more rapidly than for the case of saddle deformation.

III. DEFORMED ANTENNA PERFORMANCE
In this section, the impact of complex deformations on antenna performance, namely, on antenna resonant frequency, reflection coefficient, and bandwidth, is investigated. The change in material properties of the antenna substrate and the metallic patch due to deformations is not included in the model.
The generated geometries of deformed antennas are imported into an in-house time-domain EM solver based on the transmission line modeling (TLM) method [37] and tetrahedral Delaunay meshing [38]. The tetrahedral TLM method [38] is a well-established extension of the Cartesian TLM method [37] and has been demonstrated to be second-order accurate with respect to wavelength, provides both smooth boundary and graded mesh capabilities and has been industrially characterized and deployed for a range of applications, including EMC and aerospace [39], [40], [41], [42]. Space precludes a more detailed description of the method, and readers are specifically referred to [38] and [39] for further details.
In all cases, the whole problem is meshed with a hybrid mesh that is a combination of a 2.5 mm cubic mesh and a tetrahedral mesh. The antenna in the near-field region is meshed more finely with 0.625 mm hybrid mesh, as described in [40]. The antenna in the near-field region is captured in a fictitious box of size 0.92ε × 0.92ε × 0.55ε, where ε is the operating wavelength of 5.5 GHz. A free space impedance boundary condition is imposed on the boundaries of the computational box. An example of the meshed computational problem is given in Fig. 6, where the inset in Fig. 6 gives a closer view of the sampled antenna surface within the computational box. In all cases, the antenna is excited with the fundamental TEM mode of the coaxial feed modulated by a time-domain pulse with 3 dB frequencies of 4.6 and 7 GHz. The fundamental TEM mode is obtained as an eigen solution of the discretized 2D cross section of the coaxial cable [43]. All simulations are run on eight processor cores of a commodity cluster for 2 million time steps. The threshold for forming cell clusters is 5 µm, and the timestep is 0.018 ps [39]. The performance of the flat patch antenna is taken as a reference with a resonant

A. Spherical Convex and Concave Deformations
In this section, the impact of spherical convex and concave deformations on antenna performance is considered. The flat antenna is generated using Boolean geometry. To achieve desired spherical deformation, the deformed cage is generated by placing the origin of the sphere below (concave) or above (convex) the antenna ground plane, and the cage cross section is extruded along the length of the antenna. The deformation of the cage is controlled by the radius parameter, as shown in Fig. 4(a) and (b), with the radius varying from 50 to 20 mm for the concave case and from 50 to 30 mm for the convex case. Fig. 7 shows the impact of the spherical concave and convex deformation on the change of the resonant frequency compared to the resonant frequency of the flat antenna, f 0 . The dB values in the figure denote the value of the S 11 parameter at the resonant frequency for the given spherical radius, and the shaded region defines the 3.5% fractional bandwidth of the flat antenna. The inset of the figure shows the deformed antenna geometries for selected spherical radii, namely r = 50, 30, and 20 mm, where the reduction of the radius corresponds to increased deformation. It can be seen that in the case of both convex and concave bending, the resonant frequency increases with increased deformation. The concave deformation can cause up to a 5% shift of the antenna resonant frequency, while the antenna resonant frequency is less sensitive to the convex spherical deformation. In all cases, the S 11 value at the resonant frequency stays below −24 dB, as indicated in Fig. 6. Furthermore, it can be seen that for a deformation radius smaller than 30 mm, the concave antenna performance moves outside the operating bandwidth of the flat antenna, which is a key practical observation. This can be intuitively explained by the fact that the spherical bending acts to reduce the effective length of the antenna. This effect is stronger in the case of concave bending, while in the case of convex  bending, the fringing fields at radiating edges act to counteract this effective length reduction resulting in a smaller change in resonant frequency. Fig. 8 shows the impact of the concave and convex spherical deformations on the fractional bandwidth of the antenna. The blue horizontal line denotes the 3.5% fractional bandwidth of the flat antenna. It can be seen that concave deformation marginally increases the bandwidth of the antenna (up to 0.2%), while convex deformation has the opposite effect on antenna bandwidth and results in bandwidth reduction (up to 0.5%). This may be explained by the fact that the fringing fields in the case of concave bending essentially decrease the effective dielectric constant, which, in turn, increases the bandwidth of the antenna while the opposite is happening in the case of convex bending.
The impact of the spherical deformation on the far-field radiation pattern is shown in Fig. 9. Fig. 9(a) compares the E-plane radiation pattern of the flat patch antenna (solid line) with the convex and concave spherically deformed antenna for the radius r = 30 mm (dotted lines). Fig. 9(b) shows the same information but for the H-plane radiation pattern. It can be seen that convex and concave deformations do not significantly affect the main beam in the E-plane, with the main change being in the increased sidelobes. For the H-plane radiation pattern, however, the concave deformation increases both the main beam and the sidelobes, while the convex spherical deformation slightly reduces the width of the main beam and increases the width and strength of the sidelobe. Again, the concave deformation is shown to affect the radiation pattern more, which can be explained by the fact that fringing fields contribute to the increased sidelobe radiation.

B. Saddle Deformations
In this section, the impact of the 2D saddle deformations on antenna performance is considered. The cage for the saddle deformation is constructed, as in Fig. 4(c), and the strength of the deformation is controlled by the parameter |A W | = |A L | = A that is varied from 0 mm (flat case) to 20 mm. Two cases are considered, namely when convex bending is along the width of the antenna (concave bending is along the length of the antenna) and when convex bending is along the length of the antenna (concave bending is along the width of the antenna).
The impact of saddle deformations on the change of resonant frequency compared to the resonant frequency of the flat case is shown in Fig. 10. As before, the dB values in the figure indicate the valued of the S 11 parameter for given deformation and the shaded region defines the 3.5% fractional bandwidth of the flat antenna. It can be seen that convex bending along the width increases the antenna resonant frequency by up to 2%, while the convex bending along the length of the antenna causes a reduction in the antenna resonant frequency by up to 4% and moves it outside the operating bandwidth of the flat antenna. This can be explained by the fact that convex bending along the antenna width (and concave along the antenna length) acts to reduce the effective length of the antenna resulting in increased resonant frequency. On the other hand, convex bending along the antenna length (and concave along the antenna width) means that fringing fields can act to increase the effective length of the antenna Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. and consequently decrease the antenna's resonant frequency. Moreover, the value of the S 11 parameter at the antenna resonant frequency decreases to −9.9 dB resulting in the antenna not meeting the bandwidth requirements. This implies that the convex nature of the deformation along the length of the antenna is more critical for antenna resonant frequency. Fig. 11 shows the impact of saddle deformations on the fractional bandwidth of the antenna. It can be seen that convex deformation along the width of the antenna generally reduces the operating bandwidth of antenna, while the convex deformation along the length slightly increases the bandwidth. Fig. 10 also, however, shows the increase in the S 11 value at the resonant frequency that can be attributed to the changes in the bandwidth of the antenna. Particularly in the case of convex deformation along the length of the antenna, the value of S 11 at the resonant frequency increases with deformation, as shown in Fig. 10, and results in a complete loss of bandwidth for stronger deformations.
The impact of saddle deformations on the far-field radiation pattern is shown in Fig. 12. Fig. 12(a) compares the E-plane radiation pattern of the flat patch antenna (solid line) with the deformed antenna for A = 20 mm (dotted lines). Fig. 12(b) shows the same information but for the H-plane radiation pattern. It can be seen that saddle deformation with convex deformation along the length of the antenna is more critical as it contributes to an increase of the main beamwidth, as well as the sidelobe level. Saddle deformation with convex deformation along the width of the antenna is seen to reduce the main radiation beam in the E-plane compared to the flat antenna. This can be explained by the complex interplay between fringing fields at antenna radiating edges that act to increase the radiated power in the sidelobes and also contribute to asymmetric radiation profile.

C. Twisting Deformations
In this section, the impact of the twisting deformation on antenna performance is investigated. Twisting deformations are imposed on the warped cage by specifying mean rotating  per meter along the width and length of the antenna ranging from 0.0050 to 0.0125 twist/m. Fig. 13 shows the ratio of the resonant frequency of the deformed antenna to that of the flat antenna for twisting deformation along the length and width of the antenna. It can be seen that increasing the twisting along the width of the antenna increases the resonant frequency by about 1% while twisting along the length of the antenna decreases the resonant frequency by about 2%. This implies that twisting deformation along the length of the antenna tends to increase the effective length of the antenna resulting in a higher resonant frequency while the opposite happens when antenna is twisted along its width. These results agree with our previous investigation of the impact of twisting on a microstrip patch antenna operating at 2.45 GHz and fed by a microstrip line [29]. As the dimensions of the 5.68 GHz antenna are smaller, the impact of the twisting, however, results in a smaller shift of the resonant frequency compared to the 2.45 GHz antenna.
The shaded region depicts the fractional bandwidth of 3.5% of the flat antenna, and it can be seen that the deformed antenna resonant frequency stays within the bandwidth of the flat antenna. In all cases, the S11 values at the resonant frequency tend to increase, but more so for the case of twisting along the length of the antenna.  Twisting deformation along the length of the antenna decreases the overall bandwidth of the antenna and has a larger impact than twisting deformation along the width of the antenna, as shown in Fig. 14. In all cases considered, the deformed antenna predominantly operates within the band defined by the flat antenna, which is depicted by the shaded region in Fig. 13. Twisting deformation of the antenna tends to average the effective dielectric constant and the impact of the fringing fields so that the overall impact of the deformation on the bandwidth and the resonant frequency of the antenna is smaller compared to other deformations investigated in this article, which is also confirmed in [17] and [27].
The impact of the twisting deformation on the far-field radiation pattern is shown in Fig. 15. Fig. 15(a) compares the E-plane radiation pattern of the flat patch antenna (solid line) with the deformed antenna for maximum twisting parameter considered in this article; i.e., 0.0125 twist/m. Fig. 15(b) shows the same information for the H-plane radiation pattern. Comparing Fig. 15(a) and (b), it can be seen that twisting the antenna along its length has a much bigger impact on the antenna radiation pattern, both in terms of the main beam and also in the increased back radiation compared to the case when twisting is induced along the width of the antenna.

D. Comparison With the Cylindrical Deformations
An important question to consider is how the results from Section III-A compare with those of representative 1D cylindrical deformations along the cylinder plane and whether it is enough to just consider simpler 1D deformations in order to assess antenna stability under deformations. To answer this question, this section compares the impact of 2D spherical deformations on antenna performance against the cylindrical deformation along both E-and H-plane. GC cage of the cylindrical deformation is controlled by the cylinder radius that is varied from 20 to 50 mm. Four cases are considered, namely concave and convex cylindrical bending, each along both the length and the width of the antenna. Fig. 16 compares the impact of the spherical concave deformations on antenna resonant frequency and S 11 parameter against the concave cylindrical deformations along the length and width of the antenna. Our results for concave cylindrical bending agree with majority of published literature, which state that bending along the length of antenna results in the increase of resonant frequency due to the reduced effective length of the antenna [20], [21], [24]. Concave bending along the width of the antenna does not have much impact on the resonant frequency and has a stronger impact on the bandwidth of the antenna and is in agreement with [21], [24]. Fig. 16 shows that the impact of concave spherical deformation is more significant than representative cylindrical deformations. While there seems to be a correlation between the results in the frequency shift between the cylindrical deformation along the antenna length and spherical deformation, 1D deformations underestimate the amount of frequency shift. Furthermore, the values of the S 11 parameter at the resonant frequency in the case of cylindrical bending tend to increase for all cases but more significantly in the case of cylindrical bending along the width of the antenna, mostly due to the introduced curvature in the excitation plane. This will have most consequence for the bandwidth of the antenna, as explored in Fig. 17, which compares the impact of concave spherical and cylindrical deformations on the antenna fractional bandwidth. The cylindrical bending along the length of the antenna acts to decrease the antenna fractional bandwidth [21], [24], which can be explained by the fact that the fringing fields at radiating ends of the antenna are in different Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  planes tending to decrease the bandwidth of the antenna. The cylindrical bending along the width of the antenna acts to increase the antenna bandwidth [21], [24] as the fringing fields at radiating ends are in the same plane. Results for cylindrical bending along the width of the antenna are more strongly correlated with the results of spherical bending but again underestimate the change in bandwidth.
Comparisons of the impact of the concave spherical and cylindrical bending on the far-field radiation pattern are shown in Fig. 18 for the deformation radius r = 20 mm. Fig. 18(a) compares the far-field radiation patterns in the E-plane of the flat antenna with a cylindrically deformed antenna along the length and the width axis and a spherically deformed antenna. Fig. 18(b) gives the same information but for the far-field radiation pattern in the H-plane. It can be seen that in both cases, the spherical and cylindrical deformations have a similar impact on the main lobe but the spherical deformation has a much stronger impact on the back lobes. Fig. 19 compares the impact of the convex spherical deformation and the simpler cylindrical deformations on the antenna resonant frequency and the value of S 11 at the resonant frequency. According to Fig. 18, convex cylindrical bending along the antenna length acts to reduce the resonant frequency of the antenna, which is due to the larger effective length of the antenna. Convex cylindrical bending along the width of the antenna again does not have much impact on the resonant frequency. In both cases 1D convex cylindrical bending  underestimate the change in the resonant frequency of the antenna due to convex spherical bending. Furthermore, the S 11 value at the resonant frequency in the case of spherical bending remains largely unaffected, while in the case of cylindrical bending the S 11 values increase and more so in the case of cylindrical bending along the antenna width [22], which we believe is caused by the curvature in the excitation plane. This can have potential consequences on the antenna bandwidth, and this is explored in Fig. 20, where the impact of convex spherical and cylindrical deformations on antenna fractional bandwidth is presented. Fig. 20 demonstrates that both spherical and cylindrical deformation in the H-plane predicts a reduction of the antenna fractional bandwidth, while the cylindrical bending in the Eplane, results in gradual increase in the antenna bandwidth. Overall, cylindrical bending tends to overestimate the impact on the antenna bandwidth compared to the spherical case, and this is more pronounced as the deformations are increased.
The impact of convex spherical and cylindrical bending on the antenna far-field radiation pattern is shown in Fig. 21(a) and (b) for the case of deformation r = 30 mm Fig. 21.
Comparison of the far-field radiation pattern at 5.6 GHz in (a) E-plane and (b) H-plane of a flat patch antenna with antenna deformed using a convex deformation with radius 30 mm, and respective length and width axis deformation of the antenna. and compared against the result of the flat antenna. It can be seen that the major differences are in the back lobes and that all deformations similarly impact the main radiation beam.
Overall, comparing Figs. (16)- (21) it can be concluded that when concave deformations are considered, the spherical deformation is better correlated with the cylindrical bending along antenna length when the impact on the resonant frequency is considered but better correlated with the cylindrical bending along the width of the antenna when fractional bandwidth is considered. For the case of convex deformations, the spherical deformation is better correlated with the cylindrical bending along the width of the antenna when the impact on both the resonant frequency and the bandwidth is considered. In each case, the simpler cylindrical deformations, however, either underestimate or overestimate the changes in antenna performance parameters by up to 4%. This is a significant observation fundamental to the design. And finally, spherical convex and concave antenna deformations predict similar changes to the main beam pattern as simpler cylindrical deformations, with the main changes being in the back lobes.

IV. CONCLUSION
This article investigates the impact of complex deformations imposed on practical, flexible antennas designed on the PDMS substrate and operating in the wireless 5.5-6 GHz region. A wide range of 2D deformations of varying strengths have been considered, namely, spherical deformations, both convex and concave in nature, saddle-type deformations, and twisting deformations. The parameters of deformations are defined in terms of warped cages of the GC method.
Our results show that the spherical bending of the antenna shifts the resonant frequency of the antenna to higher frequencies and that the concave spherical bending has a particularly strong impact on the resonant frequency. Concave spherical deformation can act to move the bandwidth of a deformed antenna outside the operating bandwidth of the flat antenna.
Saddle and twisting deformation tend to reduce the antenna resonant frequency and are more detrimental to antenna operation when deformations are applied along the length of the antenna. In particular, saddle deformations where convex bending is induced along the length of the antenna can reduce the resonant frequency of the antenna and move it outside the operating band designed for the flat antenna.
Our analysis and comparison of the impact of spherical bending with simpler cylindrical bending along the width and length of antenna have found that, although some correlation between results can be made when resonant frequency or bandwidth is considered, it is found that simpler cylindrical bending can underestimate or overestimate the prediction of antenna performance by up to 4%. This is deemed to be sufficiently high, and the antenna can be out of the operating band designed for the flat antenna.
The presented analysis of deformed antennas shows that the GC method coupled with the EM simulator is a promising design toolkit that can be confidently used to assess the impact of deformations on antenna performance and give a more accurate prediction of antenna stability under deformations.