## 1 Introduction  
With rapid progress in developing organic and inorganic materials, it becomes more important to determine the quality and performance of the manufactured products alongside with their reproducibility and repeatability in terms of characteristic parameters. These strongly depend on materials' chemical composition, which often reveals itself directly in electromagnetic properties. The electromagnetic waves have been used for material characterization under various settings. Whereas the scanning microwave microscopy (SMM) techniques make it possible to quantitatively characterize, analyze, and categorize materials at microwave frequencies [1; 2; 3; 4; 5; 6], in parallel, microwave resonator techniques are developed, which also focus on pure electromagnetic characterization at bulk level. The resonator based techniques also come with a much better ease-of-use and applicability in different industrial sectors. In particular, the split-post dielectric resonator (SPDR) has been accepted as a standard for characterizing low loss laminar dielectric materials accurately, conveniently, and in a non-destructive way [7; 8; 9; 10]. This work constitutes a part of our research activities performed within the framework of the European Horizon 2020 MMAA project [11], which focuses on the development of various electromagnetic materials characterization techniques including SMM and dielectric resonator setups and their modelling. The modelling of SMM tips and generic dielectric resonators with low loss dielectric samples has already been reported [5; 6], and an extension of dielectric resonator modelling to scanning setups has been presented in [12]. In this paper, we focus on modelling and analyzing a real-life, industrially popular, and commercially available [13] split-post dielectric resonator (SPDR). Its outstanding accuracy for microwave material measurements is acknowledged by the IEC norm [14] and typically attributed to its rigorous design based on electromagnetic field expansion into Bessel functions [7; 10], with the expansion coefficients obtained by dedicated solvers, assuming planar boundaries between the different materials. Our approach is to utilize the general-purpose finite-element (FEM) and finite-difference time-domain (FDTD) methods in cylindrical coordinates. We enforce zero angular dependence of the fields for the considered modes and conduct a numerical solution in the 2D long-section of the resonator. Hence, while the axial symmetry as in [7; 10] is assumed, the boundaries between the materials do not need to be planar, allowing analysis of deformed samples. For the pure electromagnetic modelling of SPDR loaded with a dielectric sample, both FEM and FDTD are applied and quantitatively compared to the measurements. The measurement method and material parameters extraction are based on measuring changes in SPDR's resonant frequency in case of an empty resonator and in the presence of sample under test (SUT). Then, theoretical analysis is conducted to investigate and understand behavior of semiconductor samples under different modes of the SPDR. A time-domain finite element method (TD-FEM) coupled multiphysics solver is adopted from [15] and applied in order to model the nonlinear charge transport in semiconductor samples and to observe their effects in the resonance behavior. The focus has been on combining the two fields, that is, on reformulating the numerical algorithms for a computationally efficient solution of an industrially relevant problem of semiconductor measurements in popular SPDRs. The paper is organized as follows. Section 2 presents the geometry and materials of the SPDR and defines the modelling techniques used to perform eigenvalue analysis for dielectric samples. In Section 3, simulation results have been given and verified with measurements of test samples in two SPDR units. Section 4 describes the multiphysics analysis in order to account for measurement scenarios with homogeneously doped semiconductor samples.