A great reference to the theory that has been implemented in practice through WIPL‑D code can be found in the book “Electromagnetic Modeling of Composite Metallic and Dielectric Structures” written by Professor Branko M. Kolundzija and Professor Antonije R. Djordjevic.


Title: Electromagnetic Modeling of Composite Metallic and Dielectric Structures

Authors: Branko M. Kolundzija and Antonije R. Djordjevic

Publisher: Artech House

Copyright: 2002

Pages: 424

ISBN: 9780890063606

Contents: Introduction. Formulation of the Problem. Method of Moments. Geometric Modeling of Structures. Approximation of Currents. Treatment of Excitation. Solution of Equations for Current Distribution. Post Processing of Data. Numerical Examples.


This resource provides a much wider choice of analytical solutions to the everyday problems encountered in electromagnetic modeling. The book enables usage of cutting‑edge method‑of‑moments procedures, with new theories and techniques that helps to optimize computer performance in numerical analysis of composite metallic and dielectric structures in the complex frequency domain.

For the first time, comparisons and unique combinations of techniques bring the elements of flexibility, ease of implementation, accuracy, and efficiency into clear view. Numerous examples are given – from simple to complex – including scatterers, antennas and microwave circuits. One can get an in-depth presentation of intricate models, including TV UHF panels, horn, parabolic, microstrip patch antennas, and many others. More than 800 equations and 150 illustrations support key topics.

Publications describing theoretical basis, technical aspects of using the code, application papers and finally papers referencing the use of WIPL-D products before 2012 can be downloaded from the links below.

Theoretical Papers                            Technical Papers

Application Papers                            References to WIPL-D


Most relevant IEEE publications that describe numerical methods/algorithms embedded into our software are shown below.

Method of Moment Weighted Domain Decomposition Method for Scattering From Large Platforms


Authors: M. S. Tasic and B. M. Kolundzija


Publisher: IEEE Transactions on Antennas and Propagation (vol. 66, no. 7, pp. 3577-3589, July 2018.)


General theory of the Method of Moments weighted domain decomposition method (MoM weighted DDM) is proposed for time-harmonic scattering from composite objects using surface integral equations (SIEs). The MoM weighted DDM is a triple-step iterative method applied to the SIE problem decomposed into subdomain problems. In the first step of each iteration, the subdomain problems are solved without mutual coupling taken into account. In the second step of each iteration, all subdomain solutions from the current and all previous iterations are linearly combined with weights that are determined by the MoM. In the third step of each iteration, the residual of the original SIE problem is found …

Using Ultra-High Expansion Orders of Max-Ortho Basis Functions for Analysis of Axially Symmetric Metallic Antenna

Authors: A. J. Krneta and B. M. Kolundzija


Publisher: IEEE Transactions on Antennas and Propagation (vol. 66, no. 7, pp. 3696-3699, July 2018.)


Implementation of max-ortho basis functions is proposed in a method for analysis of axially symmetric metallic antennas based on exact kernel of electric field integral equation in combination with Galerkin testing. High-precision evaluation of matrix elements is enabled by: a) representing them as a linear combination of impedance integrals due to the Legendre polynomials and their first derivatives; b) using the singularity cancelation techniques; and c) evaluating the Legendre polynomials and their first derivatives by well-known recurrent formulas. Applicability of max-ortho bases up to expansion order of n =128 is illustrated on a full-wave thick dipole antenna.

Evaluation of Potential and Impedance Integrals in Analysis of Axially Symmetric Metallic Structures to Prescribed Accuracy Up To Machine Precision

Authors: Aleksandra J. Krneta, Branko M. Kolundzija


Publisher: IEEE Transactions on Antennas and Propagation ( Volume: 65Issue: 5, May 2017 )


This paper presents a novel method for evaluating potential and impedance integrals appearing in the method of moment analysis of arbitrary axially symmetric metallic structures based on exact wire kernel and higher order bases. Due to new variable transforms proposed for singularity cancellation and smoothing the integrands, high accuracy up to machine precision is achieved using relatively small number of integration points. Simple formulas are determined for predicting a number of integration points needed for prescribed accuracy. Benefits of high-precision evaluation of impedance integrals are illustrated on a number of numerical examples.

Maximally Orthogonalized Higher Order Bases Over Generalized Wires, Quadrilaterals, and Hexahedra


Authors: Milan M. Kostic, Branko M. Kolundzija


Publisher: IEEE Transactions on Antennas and Propagation (Volume: 61, Issue: 6, June 2013)


This paper presents a general theory of maximally orthogonalized div- and curl-conforming higher order basis functions (HOBFs) over generalized wires, quadrilaterals, and hexahedra. In particular, all elements of such bases, necessary for fast and easy implementation, are listed up to order n=8. Numerical results, given for div-conforming bases applied in an iterative method of moments solution of integral equations, show that the condition number and the number of iterations are a) much lower than in the case of other HOBFs of polynomial type and b) practically not dependent on the applied expansion order.


Efficient Analysis of Large Scatterers by Physical Optics Driven Method of Moments


Authors: Tasic, M.S., Kolundzija, B.M.


Publisher: IEEE Transactions on Antennas and Propagation (Volume: 59, Issue: 8, Aug 2011)


A new iterative procedure is presented that enables method of moment (MoM) solution of scattered field from electrically large and complex perfectly conducting bodies using significantly reduced number of unknown coefficients. In each iteration the body is excited by a plane wave and by the currents, which are obtained as an approximate solution in the previous iteration. The physical optics (PO) and modified PO techniques are used to determine the PO and the correctional PO currents, which are expressed in terms of original MoM basis functions and grouped into macro-basis functions (MBFs). Weighting coefficients of all MBFs are determined from the condition that mean square value of residuum …

Power Conservation in Method of Moments and Finite-Element Method for Radiation Problems


Authors: Kolundzija, B.M., Petrovic, V.V.


Publisher: IEEE Trans. on Antennas and Propagation (vol. 53, no. 8, part 2, pp. 2728 – 2737, Aug. 2005)


Conservation of energy and power can be, under certain conditions, exactly satisfied in an approximate numerical method. In this paper necessary and sufficient conditions for this property are rigorously derived for the finite-element method (FEM) and the method of moments (MoM). Two boundary formulations of FEM (strong and weak) and three formulations of MoM (MoM/VIE, MoM/SIE for metallic and MoM/SIE for dielectric bodies) were considered for radiation problems in the frequency domain. The concept of error generators-fictitious generators that produce the difference between the approximate and the exact solution-was introduced to state the power conservation property …

WIPL-D: Electromagnetic modeling of composite metallic and dielectric structures, software and user manual

Authors: Kolundzija, B. M., Ognjanović, J. S., Sarkar T. K.

Publisher: Artech House 2000



This software seeks to make the job easier, cut design time, and reduce costs for designers developing an antenna embedded in a material body, passive microwave circuit components, or determining electromagnetic scattering from complex, lossy/dielectric structures. Now featuring a Windows-based interface, it delivers a powerful program for analysis of electromagnetic radiation and scattering from composite metallic and/or finite-sized dielectric/magnetic structures.




Electromagnetic modeling of composite metallic and dielectric structures


Author: Kolundzija, B. M.

Publisher: IEEE Transaction on Microwave Theory and Techniques (vol. MTT-47, no. 7, pp. 1021-1032, May 1999)


A new, general, and very efficient method for analysis of arbitrary composite metallic and dielectric structures, based on the PMCHW formulation and Galerkin method, is presented in this paper. Flexible geometrical modeling is performed by isoparametric surfaces (i.e., by bilinear surfaces in the particular case). Efficient approximation of currents is achieved by using polynomial entire-domain expansions (i.e., rooftop subdomain expansions in the particular case) that automatically satisfy the continuity equation, assuming that there are no line charges along surface edges. Special care is devoted to the treatment of arbitrary multiple metallic and/or dielectric junctions. Numerical results for …

Spiral Super-Quadric Generatrix and Bodies of Two generatrices in automated Parameterization of 3-D Geometries

Authors: Kolundzija, B. M., and Djordjević, R. A.


Publisher: IEEE Trans. on Microwave Theory and Techn. (vol. MTT-45, no. 5, pp. 864-866, May 1997)


Most of the methods that solve the surface integral equation (SIE) by the method of moments (MoM) use triangles and flat quadrilaterals for geometrical modeling. Many complex structures can be easily modeled by quadrilaterals combining spiral super-quadric generatrices and the concept of the body of two generatrices (BoTG). A BoTG is any body that can be obtained from two generatrices by applying a certain rule. Four simple rules for obtaining BoTG’s are: (1) generalized rotation; (2) translation; (3) constant cut; and (4) connected generatrices. Spiral super-quadric generatrices enable efficient modeling of circles, arcs, ellipses, squares, rectangles, spirals, etc. Thus, a simple but fairly general algorithm …

WIPL: a program for electromagnetic modeling of composite-wire and plate structures


Authors: B.M. Kolundzija, J.S. Ognjanovic, T.K. Sarkar, R.F. Harrington

Publisher: IEEE Antennas and Propagation Magazine (Volume: 38, Issue: 1, Feb 1996)


WIPL is a program which allows fast and accurate analysis of antennas. The geometry of any metallic structure (even a very large structure) is defined as a combination of wires and plates. WIPL’s analysis features include evaluations of the current distribution, near and far field, and impedance, admittance and s-parameters. The program uses an entire-domain Galerkin method. Efficiency of the program is based on the flexible geometrical model, and sophisticated basis functions. In this paper, the basic theory implemented in the program, and some results concerning TV UHF panel antennas and large horn antennas are given.


Comparison of a class of sub-domain and entire-domain basis functions automatically satisfying KCL


Authors: Kolundzija, B. M.


Publisher: IEEE Trans. on Antennas and Propagat. (vol. AP-44, pp. 1362-1366, 1996)


Accuracy, number of unknowns, and CPU time are compared for piecewise linear subdomain basis functions and polynomial entire domain basis functions. Both types of functions automatically satisfy a continuity equation at wire ends and junctions, according to Kirchoff’s current law (KCL). The relative root mean square (RMS) current deviation is chosen as the error metric. An electrically short scatterer, a crossed wire scatterer and an electrically long scatterer are used for comparison. Currents are obtained by solving the electric field integral equation (EFIE), by means of the Galerkin method. It was shown that in most cases, for the same accuracy required, the entire domain approximation …

Entire-domain Galerkin method for analysis of metallic antennas and scatterers


Authors: B. M. Kolundzija, B. D. Popovic


Publisher: IEE Proceedings H – Microwaves, Antennas and Propagation (Volume: 140, Issue: 1, Feb. 1993)


The antenna and scatterer surfaces are approximated by generalised quadrangles. Surface currents are expanded in such local coordinate systems and the general form of the corresponding electric field integral equation is derived. A procedure is described for obtaining entire-domain basis functions which satisfy automatically the continuity equation along the surface element interconnections and free edges, and the expressions are derived for the impedance matrix elements in this case. Starting from the general theory, two new particular methods are presented. The first is intended for the analysis of general structures, and is based on application of truncated cones and bilinear surfaces …