## 1 Introduction
Alleviating these difficulties is possible in specific situations using the methods such as high-dimensional model representation (HDMR) [44], least-angle regression [45], or incorporation of variable-fidelity models (two-stage GPR [46], co-kriging [47], Bayesian model fusion [48]). Model domain confinement has been recently suggested as an alternative approach to computationally efficient surrogate modeling [49; 50; 51]. By focusing the modeling process on the region containing designs that are optimum with respect to the selected performance figures (e.g., operating frequencies of a multi-band antenna) or material parameters (e.g., relative permittivity of the dielectric substrate the antenna is implemented on), reliable surrogate can be rendered over wide ranges of operating conditions using small training data sets. Identification of such regions is realized with the help of pre-existing reference designs [50], which could be available from the previous design work with the same structure or obtained as a part of the modeling procedure. One of the most recent methods adopting the domain confinement approach is the nested kriging framework [51]. The two kriging metamodels are utilized: the first-level one to establish the region of interest (model domain), and the second-level model being the actual surrogate. The important advantage of the technique is a straightforward arrangement of uniform training data sampling as well as model optimization, both enabled by a one-to-one mapping between the unity interval and the surrogate domain being a part of the formulation of the procedure [51]. The critical (scalar) control parameter of nested kriging is the thickness coefficient determining the ratio between the lateral and tangential size of the domain. Its value establishes the trade-off between the model predictive power and the coverage of the regions containing potentially optimum antenna designs for various sets of performance specifications. This paper proposes a variable-thickness domain for nested kriging, where the thickness parameter is a function of the objective space vectors. The keystones of the approach are the reference design triangulation and an appropriate mapping defined over the expansion coefficients of the objective vector with respect to the vertices of the simplexes being a result of the triangulation process. Using this mapping, the domain thickness is maintained small in the vicinity of the reference designs (which are optimal by definition) and increases towards the simplex centers (where the first-level model deviates the most from the optimum design manifold). The variable-thickness domain method permits the construction of reliable surrogates using significantly (by up to seventy percent) smaller data sets than required by the fixed thickness version. The computational benefits are achieved without compromising the model coverage in terms of geometry parameter and operating condition ranges. These properties are validated using two antenna examples as well as benchmarking against conventional modeling methods (kriging, RBF) and the original nested kriging. The major novelties and technical contributions of the paper, beyond what has been presented in the literature so far, include: (i) introduction of the concept of variable-thickness domain including its rigorous analytical formulation, (ii) incorporation of the concept into the nested kriging modeling framework, (iii) demonstration of significant computational benefits that can be obtained (up to seventy percent savings in terms of the training data acquisition cost) as compared to the state-of-the-art approaches, without compromising the model predictive power, (iv) demonstration of the model efficacy as antenna design tool. To the best of authors' knowledge, the presented approach is the first modeling framework that permits construction of fast and reliable surrogates of antenna structures within broad ranges of geometry and operating parameters while using such a small number of data samples.