Systematic Generation of Arbitrary Antenna Geometries

—Applications for conformal, wearable antennas are growing for consumer electronics. Hence, it is important to assess to what degree antenna performance can be tolerant to in situ deformations that can take the form of bending, crumpling, and twisting and combinations of these effects. However, generating geometries of arbitrary antenna deformations such as bending, crumpling, and twisting, which can be processed by standard electromagnetic (EM) software, is a major challenge that signiﬁcantly complicates full assessment of in situ antenna performance. Constructive solid geometry methods of generating geometries is difﬁcult to robustly apply to nonconformal antennas and more ﬂexible techniques required to progress the antenna studies further. To address this challenge, this article investigates the utility of the green coordinate (GC) method for spatial manipulation of 3-D objects. First, calibration of a straightforward application of the GC method against a reference case of a patch antenna bent over a cylindrical surface, which can also be generated exactly, is undertaken. This article shows that systemic scaling distortions are introduced by the GC method and introduces a compensation method that can overcome these distortions. Subsequently, the compensated method is used to obtain new predictions of the EM performance of patch antennas with deformations.


I. INTRODUCTION
W EARABLE and foldable electronics can receive, store or transmit digital information, and are the focus of increased attention in many areas, including healthcare, entertainment, and the sports industry. Individual components, such as RFIDs, medical sensors, and antennas, can be integrated into a variety of substrates, including knits, embroidery, or nonwoven materials [1]- [3]. More generally, nonflat antennas are routinely found mounted on curved surfaces, such as airplane wings and fuselages [4] and 5G antenna arrays on cylindrical surfaces [5]. The design and integration of individual antennas is often a crucial aspect in the overall system design. One of the most challenging cases for predicting in situ performance is that of wearable textile antennas fabricated on woven and nonwoven materials that will experience bending, twisting, and crumpling when pulled in different directions by human activity, as shown in Fig. 1. While woven and nonwoven materials are more stable than knits [1], it is still expected that all textile antennas will, to some extent, experience elongation and compression when in use.
To date, there has been an increasing effort to predict performance changes caused by antenna deformations, but these have primarily focused on patch antennas and antenna arrays [6]- [16] bent over cylindrical surfaces. The results reported were mainly obtained from practical measurements [6], [7] and only a few predictive results have been reported using the modeling techniques embodied in CST's and Ansys software [8], [9], [14], [15]. Both the experimental and simulated results demonstrate that bending has a significant influence on antenna performance; specifically, increased curvature decreases the overall gain of antenna and shifts the resonant frequency, although the bandwidth does remain approximately the same across all measured cases [6], [8], [9], [11], [14]- [16].
The virtually exclusive focus of previous work on purely cylindrical deformations can be attributed to the difficulty in robustly generating arbitrary deformed antenna geometries suitable for predictive simulations. Clearly, to advance understanding and develop practical design guidelines, the problems associated with generating arbitrary geometries need 0018-926X © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
to be fully understood and resolved and this motivates the contribution presented in this article. The specific and critical CAD issue for the generation of deformed antenna geometries is to obtain well-behaved and consistent interfaces between the different elements that make up the antenna. This is clearly illustrated by a finite thickness metal patch on a substrate: Either the patch and the substrate are generated separately as deformed parts and stitched together or a flat patch-substrate combination is generated with a well-behaved interface, which is easy to do, and this combination is deformed to the final desired geometry. This article discusses the superiority of the latter approach and then demonstrates its practical implementation in a straightforward manner.
It is important to explain the phrase a well-behaved interface. Stitching together noncoordinate aligned and separately created parts inevitably leads to dirty interfaces that do not perfectly align and typically have microscopic nonphysical gaps and overlaps, which are disastrous for subsequent mesh generation and electromagnetic (EM) simulations. At best, dirty interfaces result in unrealistically small mesh cells that severely compromise the computational efficiency and, at worst, mesh generation proves impossible. It is commented here that the subsequent choice of modeling method, finite elements, finite differences, method of moments, or their implementation within particular software packages is not the issue, and rather, it is the geometrical data upon which they operate that has the failing.
Experience shows that even the CAD models generated using mechanical software need CAD repair techniques to clean these interfaces lacks robustness and require significant manual intervention.
Overall, the number of trial cases a designer can easily consider is severely curtailed, which explains the existing focus on purely cylindrical geometries that are amenable to exact generation [8]- [11], [15], [17]. To avoid the issue of dirty interfaces, the strategy pursued here is to generate flat antenna geometries and deform them holistically using a technique borrowed from computer graphics.
Computer graphics offers a number of techniques for spatial manipulation of 3-D objects, for example, mean value coordinates (MVCs), harmonic coordinates (HCs), and green coordinates (GCs) [18]- [21]. The common feature of these techniques is to enclose the object of interest by a so-called bounding cage that has a similar, but less complicated shape than the object of interest. All spatial points within the interior of the cage can be expressed in terms of the cage's geometry (e.g., vertex locations in the case of MVC) and by subsequently manipulating the simple cage, for example, interactively dragging one of its vertices, smooth deformations are induced into the complex geometry of the enclosed object. The appeal of fast and simple modification of characters for animations is exactly the feature that is also sought here for systemic performance studies on deformed antennas.
Each of the particular methods listed above differs in: 1) the extent to which they preserve the physically realistic nature of practical deformations of the object's shape (shape-preserving property) and 2) the efficiency of their numerical implementa-tion, i.e., either exploiting closed-form formulas or requiring substantive numerical processes [20]. Shape preservation in the context of this work is a measure of whether the object deformation follows the expected behavior of stress-strain physics, which is clearly desirable. Two points to note here are that: 1) it would be difficult to theoretically express the complexity of such deformations in the general case with a view to explicitly code geometry generation, vertex by vertex, as one can obviously do for the purely cylindrical case and 2) the scope of this work does not extend to the further complexities of the materials, for example, the opening and closing of woven fabrics.
The GCs approach, described in more detail next, stands out from the alternatives listed above as that which best preserves the shape of 3-D objects and is implemented using only closed-form analytical formulas [20], [21]. The good shape preservation properties trace to the fact that the 2-D and 3-D GC methods induce purely conformal and quasi-conformal maps, respectively, with minimal anisotropic scaling [20], [21].
Earlier work has demonstrated the use of the GC method to generate arbitrarily deformed antenna geometries [22], [23]. However, while successfully, the approach does introduce systemic global distortion of the resulting antenna geometry and removing this is a first objective here. For clarity, we use the term deformation for the desired change of shape of an antenna from the flat to the in situ case and distortion to denote these undesirable systemic errors.
A suitable benchmark geometry for assessing and then removing the GC distortions is the purely cylindrical case as it is a developable surface that ought to be generatable without any distortion. This work strongly emphasizes the quantification of antenna deformations in the context of the subsequent impact on performance, characterized by reflection coefficient S 11 and radiation pattern, and therefore, the next few paragraphs will describe the meshing and simulation approaches providing this context.
As the antennas under investigation are clearly not coordinate aligned, discretization with an unstructured mesh is advantageous for minimizing discretization errors and managing computational effort. An in-house 3-D Delaunay mesher and an unstructured mesh EM transmission line modeling (TLM) simulator have been used [24]- [26].
The Delaunay mesher [26] is a robust 3-D implementation of the techniques described in [27] augmented by many features that ensure that the meshes produced are sympathetic to the needs of subsequent large-scale EM simulation and has been widely used for the purpose [28]- [34].
The unstructured TLM method is a well-established extension to the Cartesian mesh TLM method, the latter having over 50 years of practical use for EM simulations [35], [36]. The TLM approach has the distinct advantages that all field component samples are colocated in time and space, and crucially, the stability is provable on a cell-by-cell basis without resorting to estimators such as the Courant condition. For large-scale simulations, this is a critical advantage and late time instability has never been observed with TLM. The unstructured mesh variant of the TLM algorithm 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 [29], [31], [37], and readers are specifically referred to [24] and [25] for further details.
The remainder of this article is structured as follows. Section II provides an overview of GCs method. Section III investigates the impact of the GC parameters on antenna distortions and Section IV introduces a simple GC compensation method. The effectiveness of this is assessed by comparing the EM performance of compensated-GC antennas with the special case of cylindrical bending whose geometries are in closed form. Finally, Section VI demonstrates the performance of antennas bent over double-curved surfaces compared to the flat case, which are new results beyond the possibility of using constructive solid geometry (CSG), to straightforwardly generate the geometries. Section VII gives the overall conclusions of this article.

II. GC METHOD FOR GENERATING CYLINDRICALLY BENT ANTENNAS
GCs are a method for spatial manipulation of objects that belongs to the class of methods where a cage of a simple polyhedral shape encloses the object of interest [20], [21].
In the case of a flat microstrip antenna, the undeformed cage is a segmented cuboid and the antenna is positioned in the middle of it surrounded by padding of length , as shown in Fig. 2(a). Here, the same padding is used in all directions. The theoretical basis of the GC method is that the spatial coordinate variable r within the cage satisfies Laplace's equation, and thus, the introduction of Green's function and use of Green's third identity expresses the value of any particular point r p within the cage as an integral of the values of r and its normal derivative over the cage surface. For a triangulated cage surface, these integrals can be evaluated in closed form such that r p is expressed as a discrete and finite extent superposition of the cage's vertices, r v , face normals, n f , total number of vertices, N v , and faces, N f , with suitable coefficients g V v and g F f [20], [21] The key point to note is that the continuous space within the cage (or in practice here, the relatively large number of topologically related vertices defining the object being deformed) is expressed in terms of a relatively few discrete quantities. The GC method deforms the enclosed space by making a one-to-one mapping of the set r v → r v (and consequently n f → n f ) and using the above superposition with the same weight coefficients as before, i.e., By definition, the flat cage is chosen to be a simple shape capturing only the global spatial characteristics of the enclosed object so that it is easy for a user to manipulate its overall shape by relocating (dragging and dropping) a few of its defining vertices to yield a deformed cage [see Fig. 2 . This is to be compared with attempting to manually deform the antenna, vertex by vertex, in a topologically consistent and physically realistic manner, which is clearly untenable in general. The full detail of the GC method is not given here and the reader is referred to the available literature on the method and the particular closed-form expressions used [20], [21]. The GC methodology is implemented in our in-house geometry kernel (GK) [26], which also contains an extension to the published GC approach in order to avoid self-intersection of the deformed object. As the GC method does not embody an affine transformation, deforming long straight edges defined by a pair of end points, for example, the sides of the patch, cannot be achieved by simply deforming the end vertices. An iterative splitting of such edges is required, which terminates when the deformed midpoint of the edge coincides, within a given tolerance, with the linear interpolation of the deformed end points. For reference, we will refer to this tolerance as the edge split tolerance and all the results presented below used a value of 0.01 mm. In this section, the distortions from the target deformation introduced by the GC method are quantified. The degrees of freedom examined are first the complexity of the cage used. This is quantified by the number of piecewise segments used to describe the undeformed cuboidal cage, each of which deforms to one of the piecewise sectors of angle θ seen in the deformed cage of Fig. 2(b) and more dramatically in Fig. 3(a) and (b). The second degree of freedom is the extent of the padding, , as shown in Fig. 2(a).
The figure of merit used to assess the undesirable distortion of the geometry is chosen to be consistent with one of the simple parameters used to design standard patch antennas, the radiating surface area. In fact, both the total (top + bottom + sides) surface area of the radiating patch and the total surface area of the substrate are observed, noting that the intention is to deform a flat antenna of prescribed patch area into curved antenna with the same patch area (which is only exactly possible for developable surfaces). The relative distortion surface error, ρ, is thus defined as ρ = (A GC − A Flat )/A Flat , where A indicates the surface area of either the metallic patch or the substrate.
The microstrip patch antenna used for this study is designed to resonate at 10.4 GHz and has patch width, W, of 10.830 mm and length, L, of 8.177 mm, as shown in Fig. 4(a) and (b). The substrate is Rogers Corporation Duroid 5880 of relative permittivity 2.2, width, W sub , length, L sub , of 20 mm, and thickness, h, 1.575 mm. The antenna is fed from below by a coaxial cable positioned at (W /2, y o ), where y o = 1.75 mm. The coaxial cable has inner radius, r in , of 0.625 mm, outer radius, r out , of 2.16 mm, and the relative permittivity of the sheath is 2.2. The metallic patch and the ground plane of the antenna both have a thickness of 0.035 mm and are treated as perfect conductors. The radius of curvature in the following results is defined as the bottom plane of the patch.
There are two potential causes of undesirable distortion: poor choice of method parameters and any systematic spatial stretching introduced by the GC method. This section seeks to identify the latter by converging to eliminate the former.
The relevant GC method parameters are the padding, , and the number of piecewise segments, N, that are used to describe the cross section of the cage, as shown for N = 2 and N = 10 in Fig. 3. An understanding of the likely impact of these parameters may be gauged from the description of the space within the cage by the GC method in terms of solutions to Laplace's equation and these solutions exhibit the usual smoothing behavior with distance from sharp features such as corners. Therefore, although the cage is actually polygonal, at sufficient distance, from its boundaries, the Laplacian solutions will be more dominated by the large-scale arc geometry that the cage approximates, rather than the exact detail of its piecewise linear nature. This behavior is clear in Fig. 3 for the case of N = 2.
It is also noted that using more segments in the cage, N, to approximate a fixed length antenna of given bending radius or increasing the bending radius for fixed N reduces the angular steps in the cage's boundary, θ in Fig. 3, that further smoothens the deformed space around the antenna.
It is tempting to conclude that one should use a large value of padding as this has no impact on the computational effort of the approach and ought to reduce the error. However, in the more general case, this would also mean loss of control of the detail of the antenna's deformation, so in fact, there is a conflicting driver to minimize the padding. Similarly, although using a larger value for N would appear advisable, this does linearly impact on the computational cost. Fig. 5(a) and (b) shows the impact of the padding, , on the relative surface distortion error, ρ, for both the metal patch and the substrate of the antenna, as the radius of curvature is varied with N = 10 piecewise segments used to describe the cross section of the cage. Similarly, Fig. 6(a) and (b) fixes the padding to = 3 mm and varies N between 2 and 12 segments.
Together, these figures reveal that for bending radii above 50 mm, the choice of and N has negligible impact, but there remains a systematic error introduced by the GC method. For smaller radii, the errors do increase reaching a peak in some cases before rapidly increasing again with the opposite sign, i.e., the areas shrink rather than expand. In this regime, the choice of a larger is appropriate, and however, there remains a remarkable insensitivity to the choice of N.
Overall, it is noted that, given an antenna length of 20 mm, bending radii below 50 mm are certainly severe deformations in shape, and the systematic errors of Figs. 5 and 6 for all radii are deemed to be unacceptably large.
In fact, this is partially a positive result as the intention is to keep the cage as simple as possible. The explanation of the systematic error is actually found in a subtlety of the GC algorithm, which is used to ensure that in the 3-D case, least deforming and quasi-conformal spatial deformations are produced [20], [21] compared to other space manipulation techniques. A local stretch factor appears in the 3-D algorithm, which although generally successful in preserving shape, does not decouple to the known 2-D stretch factor and an axial stretch factor if the geometry happens to be separable as is the case for a cylinder. Therefore, it has been shown that the GC method can simply produce bent antennas, but a spatially varying stretch factor in its algorithm introduces an error that must be managed. In the context of textile antennas, in Section IV describes how this systemic error can be successfully compensated for on average.

IV. METHODS FOR COMPENSATING GC-INDUCED DISTORTIONS
From an engineering viewpoint, the resonant frequency and radiation pattern are key performance metrics of an antenna and these are, for the above case of patch antenna, predominantly determined by the size of the radiating patch. Since the GC method does preserve the overall shape, prescaling of the size of the patch ought to minimize the distortions just observed. This section describes a straightforward prescaling method referred to as the width and length (WL) compensation method.
The compensation method is based on comparing the WLs of the radiating patch before and after deformation. The WL of the deformed patch is extracted by fitting an (x, y) order polynomial function, where x and y indicate the order of the polynomial function used along the WL of the patch. The actual WL of the deformed patch is then analytically evaluated by integrating over the polynomial approximation.
Each of the ratios of the WL of the radiating patch before and after deformation gives a scaling factor, e.g., s W = W /W and s L = L /L, with which the flat antenna can be scaled This process is iterated until a prescribed relative difference in the width and/or length of the patch is satisfied and typically this converges within 3-5 iterations. Fig. 7 compares the relative surface distortion error of the radiating patch with and without the prescaling compensation for a range of bending radii from 4.2 to 60 mm, the padding, = 1 mm and N = 10 [i.e., as in Fig. 2(a)]. It is clear that this relatively simple prescaling approach is highly effective at removing the undesirable patch distortion over a wide range of radii.
It is important to adopt a good surface fitting approach for the GC compensation without incurring a notable computational cost. Fig. 8 confirms that the adequacy of a (3, 3) order polynomial surface approximation. Slight improvement is obtained for higher order models for small radii (noting that symmetry pairs, the curves), with virtually no impact on the approximately 3 min calculation time of the overall deformation calculation. Fig. 9 further explores how the area distortions decompose into WL distortions and confirms that, as the bending is applied along the width of antenna, the largest distortion is occurring along the width of the patch.

V. EM PERFORMANCE OF CYLINDRICALLY BENT ANTENNAS
The previous results demonstrate the success of a simple distortion compensation scheme quantified in terms of a geometrical measure of error. This section now extends this by exploring its effect on the more practically important figures of merit of the antenna's performance. The return loss, S 11 , and the far-field radiation pattern are presented for bending along both the width (H-plane bending) and the length (E-plane bending) of the antenna. Throughout, results are compared to both the flat antenna and an ideal cylindrical version of the antenna. The geometry of the latter is generated directly in cylindrical coordinates and as such may Relative area error of WL approach for different degrees of polynomials. The notation (poly x, y) indicates the order of the polynomial function used along the WL of the patch. be regarded as a perfect model without any CAD artifacts for a given sampling resolution and is hence used as the benchmark when comparing against GC-generated versions. We show both the results for the GC geometry antenna with and without the compensation applied. Generally, the success of the compensation scheme can be gauged by how close the compensated GC results agree with the ideal cylindrical results in Figs. 10-13. The padding value in the GC method is fixed to 6 mm. The WL compensation method uses order (3, 3) polynomial surface fitting and iterates until the agreement between the flat and deformed patch WL agrees to a relative tolerance of 0.03%. Fig. 10 compares the return loss, S 11 , for H-plane bending obtained with the reference antenna geometry and with the uncompensated and compensated GC method along with the flat antenna case for comparison. Two bending radii, R = 6.5 mm, Fig. 10(a), and R = 10 mm, Fig. 10(b), are shown. First, it is seen that the overall impact of H-plane bending on the antenna's resonant frequency is not significant, which agrees with published results [13]- [15]. Second, the use of the WL compensated GC geometry predicts a return loss that is in much closer agreement with the reference result than does use of the uncompensated GC geometry. Fig. 11 repeats the analyses of the case of E-plane bending. Now, the shift of the reference antenna frequency to higher frequencies is more significant compared to the case of H-plane bending as already reported in [13]- [15]. It is seen that the effect of the uncompensated GC geometry actually leads to a reduction in resonant frequency, i.e., the wrong physical behavior, but again, the compensation method has essentially removed this error, yielding excellent agreement with the reference case results. In fact, with the smaller bending radii, the relative error in resonant frequency is 0.552% with no discernable error in bandwidth, and with the larger bending radii, the relative difference in the resonant frequency is 0.0552% and 2.2% in the bandwidth. Finally, it is noted that in both H-and E-plane bending cases, the bandwidth is preserved, which is in agreement with other published results for cylindrically bent antennas [13]- [15].
Figs. 12 and 13 complete this validation assessment by presenting the far-field radiation patterns for all cases considered in Figs. 10 and 11. The main beam characteristics of the antenna are not significantly affected by bending in either the E or H-planes, but there are some discrepancies in the minor sidelobes.

VI. EM PERFORMANCE OF ARBITRARY DEFORMED ANTENNAS
As stated in Section I, the driver for this work is not to reinvestigate the particular case of cylindrical patch antenna deformations, rather to be able to study the impact of more general shape deformations. However, to reiterate, methodically generating robust geometries has been a significant hurdle to achieving this goal and now that the previous section has successfully validated the use of the WL compensated GC method for this purpose, and the remaining space in this article is dedicated to illustrating its use for the greater challenge. Several canonical antenna deformations that cannot be generated using CSG approaches are considered. These cases are shown in Fig. 14 and are labeled as twisted, convex, concave, and saddle deformations.
Results using uncompensated GC geometries will still be shown here to emphasize the importance of compensation of the GC distortion errors when assessing the EM performance of such antennas.
For all examples shown, the flat cage is defined using N = 10 cage segments along both its width and its length  and the padding, , is fixed at 3 mm. The reader is reminded that a key attraction of the GC method is that the desired deformations are then simply produced by relocating a few vertices of the flat cage. Here, the convex, concave, and saddle examples are all produced using different parts of a double-curved polygonal deformed cage that is separable; for all three cases, the part-cylindrical cage with bending radius of R = 20 mm used as in Fig. 2 and is further deformed by a half a period of sinusoidal variation of amplitude 10 mm along its length. The twisted case is obtained by applying a simple twisting function of 0.01 rad/m to the cuboidal undeformed cage along the length of antenna and sinusoidal deformation of amplitude 10 mm along the width of antenna.
The resulting antenna geometries are presented in Fig. 14 and it is noted here that the GC method is not applied to the coaxial feed.
The EM performance, in terms of return loss S 11 and radiation patterns, of the convex, concave, saddle, and twisted antennas generated using the uncompensated and compensated GC method is shown in Figs. 15 and 16. These results are compared with the flat case to show the impact of the deformation.
In all cases, the uncompensated GC method produces antenna geometries that significantly underestimate or overestimate resonant frequency compared to the GC compensation method. We note that all deformations are similar in size but different in their nature. Space precludes an exhaustive discussion of these results here, but it is clear that the size of the resonant frequency shifts is similar in all cases. In fact, comparing convex and concave deformations, we can conclude that convex deformation is more detrimental to antenna performance as it leads to a larger frequency shift.
It is acknowledged here that the compensation scheme developed above is only addressing the average distorting effect of the GC method and naturally more work could  be undertaken to further tune it for the arbitrarily deformed antenna case. Space precludes expanding on this further here.

VII. CONCLUSION
Modern applications in wearable and foldable electronics are creating increased interest in the performance of antennas subject to in situ deformations. Unlike traditional conformal antennas whose geometry remains fixed in service, textile antennas must successfully operate while suffering a wide range of deformations. This poses new challenges for the antenna designer trying to ensure robust performance. Even systematically defining the, in principle, continuous problem space of the deformations is difficult, and from this, identifying a finite set of suitable candidates for a testing program will require a clear understanding of the impact of many possible deformations. It is imperative that complete and systematic studies are undertaken to inform future designs and this article has addressed a notable hurdle preventing this activity, namely, the generation of robust and highly variable antenna geometries from a flat prototype.
The GC method provides an approach to holistically deformed structures such as patch antennas without introducing the disruptive CAD artifacts that can either block or seriously undermine EM characterization. The method is ideally adapted to a simple drag and drop user interface and yields geometries that provide certain guarantees of physical reality in the deformations, albeit not for all the subtleties of woven materials.
This article has shown that naïve use of the GC method introduces a systematic distortion expressed as an undesirable area scaling. However, this has successfully been compensated for using a straightforward prescaling approach, which although not completely general works well for the widely used class of patch antennas considered here. Furthermore, with the compensation in place, a good degree of insensitivity to method parameters, such as the padding and complexity of the cage used, has been shown, which is important for undertaking broad range design studies.
After calibration against a known canonical case whose geometry can be exactly generated, the final GC-generated antenna performance in terms of return loss and radiation patterns has been examined and shown good agreement with published results. Finally, the scope to address the grander challenge of undertaking a systematic design sweep over the problem space of possible doubly curved deformations has been demonstrated and results from this activity will be the subject of a future contribution.