A fast and memory efficient three-dimensional full-wave simulator for analyzing electromagnetic (EM) wave propagation in electrically large and realistic mine tunnels/galleries loaded with conductors is proposed. The simulator relies on Muller and combined field surface integral equations (SIEs) to account for scattering from mine walls and conductors, respectively. During the iterative solution of the system of SIEs, the simulator uses a fast multipole method-fast Fourier transform (FMM-FFT) scheme to reduce CPU and memory requirements. The memory requirement is further reduced by compressing large data structures via singular value and Tucker decompositions. The efficiency, accuracy, and real-world applicability of the simulator are demonstrated through characterization of EM wave propagation in electrically large mine tunnels/galleries loaded with conducting cables and mine carts.

Reliable wireless communication, sensing, and tracking systems in underground mine environments are critically important to ensure workers’ safety and productivity during routine operations and catastrophic events. As mandated by the MINER Act of 2006 [

Present simulation techniques for analyzing EM wave propagation in mine environments are either approximate or full-wave in nature. Examples of approximate techniques include, but are not limited to, single/multimode waveguide models [

This paper presents a fast, full-wave, CPU, and memory-efficient three-dimensional (3-D) SIE technique for analyzing EM wave propagation in electrically large and realistically loaded mine environments. The technique leverages Muller and combined field SIEs to model scattering from mine walls and perfect electrically conducting (PEC) objects residing inside mine tunnels and galleries. The naive iterative solution of such SIEs requires ^{2}) CPU and memory resources. Here ^{4/3} log^{2/3}

near-field interaction matrices;

matrices that characterize far-field signatures of basis functions;

tensors that hold FFT’ed translation operators on a structured grid.

The proposed simulator compresses the first two and third data structures via singular value decomposition (SVD) and its higher-dimensional counterpart, Tucker decomposition [

This section details the Muller and combined field SIEs and their numerical solution via the method of moments (MoM). It also elucidates the proposed SVD and Tucker-enhanced FMMFFT acceleration scheme.

Let _{d}_{1} and permeability _{1} (medium 1). The tunnel or gallery is assumed to be filled by air with permittivity _{0} and permeability _{0} (medium 0) [see _{0}.) Let _{p}_{d}_{d}_{p}^{i}^{i}^{i}_{d}^{i}^{i}_{d}_{p}_{d}_{d}_{d}_{p}_{p}

To compute _{d}_{d}_{p}_{d}_{d}_{d}_{d}

_{d}_{d}_{d}_{a}_{a}/ε_{a}^{0.5} with _{a}_{a}

_{a}_{a} |_{a}_{a}ε_{a}^{0.5},

In the interior problem [see _{d}_{d}^{i}_{p}_{d}_{d}_{p}_{d}_{p}

Here _{p}_{p}_{p}_{d}_{1} and _{0}, respectively, as

Similarly, exterior and interior MFIEs for _{d}_{1} and _{0}, respectively, and combined as

Finally, linearly combining the _{p}_{p}_{p}

_{1} = _{1}, _{0} = −_{0}, _{1} = _{1}, _{0} = −_{0}, and 0 ≤ _{p}_{d}_{d}_{p}_{p}_{d}_{d}_{p}_{n}

_{n}_{d}_{p}_{m}_{d}_{p}_{d}_{p}

_{n}

When analyzing electrically large mine tunnels and galleries loaded with conductors that require large

The FMM-FFT scheme introduces a hypothetical box enclosing the mesh of _{d}_{p}_{x}_{y}_{z}_{u}_{x}, u_{y}, u_{z}_{x}_{x}_{y}_{y}_{z}_{z}_{u}_{n}_{m}^{s}_{u}_{u′}_{u′u}_{u′u}_{u′}_{u}_{u′u}^{s}

Interactions between basis functions in the same group and near-field pairs are directly computed via (

_{a}_{a}^{p}_{a}^{th}–order Gauss–Legendre quadrature rule, ^{q}_{a}_{a}_{a}R^{s}_{10}(1_{1}))^{2/3}(2_{a}R^{s}^{1/3} is the number of multipoles for medium _{1} is the number of desired accurate digits in the FMM approximation [_{c}_{d}_{p}_{d}_{d}_{d}_{p}

_{a}_{a}_{p}_{p}

Here Ψ is the FFT operator,

_{u′u}_{u′u}_{u′u}_{l}

In practice, the circular convolution in (_{x}_{x}_{y}_{y}_{z}_{z}_{x}_{y}_{z}

To execute the FMM-FFT algorithm on high-performance parallel computers for characterizing large-scale mine tunnels/galleries loaded with conductors, a hybrid spatial/angular parallelization strategy, which utilizes hybrid message passing interface/open multiprocessing (MPI/OpenMP) standards, is deployed. This parallelization strategy, described in

To reduce the memory requirement of the SIE simulator leveraging FMM-FFT algorithm, large data structures storing the near-field interactions, far-field (and receiving) patterns of basis functions, and FFT’ed translation operator tensors are compressed viaSVD and its higher-dimensional counterpart Tucker decomposition.

Assume that the near-field interactions between _{s}_{u}_{t}_{u′}_{t} × N_{s}_{mn}_{u′}_{u}

_{t} × r_{s} × r_{i}_{2} times the value of first singular value, i.e., _{i}_{2}_{1},

Assume that one component (_{a}_{a}_{3},

This operation is applied to

The tensor storing the FFT’ed translation operator samples for each
_{u′}_{−}_{u}_{1}
_{2}
_{3} = (2_{x}_{y}_{z}

_{1}
_{2}
_{3}, _{i}_{i} × r_{i}_{i}

The unfolding matrices of 𝓣_{u′}_{−}_{u}_{i}

Given the prescribed tolerance
_{i}

The core tensor can be obtained via

The core tensors and factor matrices of FFT’ed translation operator tensors for all
_{u′}_{−}_{u}

This section presents numerical examples that demonstrate the accuracy, efficiency, and applicability of the proposed FMM-FFT- SIE simulator. In all examples below, the FMM box size is half of the wavelength in ore, FMM accuracy is three digits (_{1} = 3), matrix system (^{−6}, which uses a right diagonal preconditioner, and tolerances _{2}, _{3}, and _{4} for compressing matrices and tensors are 10^{−3}, 10^{−4}, and 10^{−6}, respectively. Furthermore, tunnels and galleries are surrounded by ore with permittivity _{1} = _{0} (_{r,}_{1} − _{1}_{0}) and permeability _{1} = _{0} ; _{r,}_{1}and _{1} are the relative permittivity and conductivity of the ore, respectively. All simulations are performed on a cluster of dual hexacore X5650 Intel processors with 64 GB RAM, launching one MPI process on each processor and distributing the computational load assigned to each processor among its 16 cores via OpenMP. The CPU and memory requirements of the proposed solver for all numerical examples are tabulated in

First, the proposed simulator is used to analyze EM wave propagation in an arched tunnel surrounded by ore with _{r,}_{1} = 8.9 and _{1} =0.15 S/m [see

A 600 m-long tunnel is excited by either a

A 200 m-long tunnel loaded with six PEC mine carts is excited by a

A 650 m-long tunnel loaded with two PEC strips is excited by a

Next, the proposed simulator is used to analyze EM wave propagation in a mine gallery formed by eight rectangular tunnels [see _{r,}_{1} = 3 and _{1} = 0.001 S/m. The gallery is excited by an electric dipole with unit moment that is centered at (7.5, 15.91, 1.12)m

In tunnel 2, power densities at receivers with

Small or large spikes appear in the power density plots near receivers with

The electric current densities on gallery walls computed for each frequency and polarization are shown in

Finally, the proposed simulator is used to analyze EM wave propagation in a rectangular tunnel with rough walls, excited by a _{r,}_{1} = 3 and _{1} = 0.001 S/m. Tunnel walls have a random profile with 0.1 m root-mean-square height and 0.25 m correlation length [

An FMM-FFT-SIE full-wave simulator is presented for characterizing EM wave propagation in electrically large and loaded mine environments. The full-wave simulator rapidly solves the Muller-combined field SIE system using a parallel FMM-FFT acceleration scheme. To reduce the memory requirements, the simulator employs SVDs to compress matrices characterizing near-field interactions and far-field (and receiving) patterns, and Tucker decompositions to compress tensors storing FFT’ed translation operator tensors. Numerical results demonstrate the accuracy, efficiency, and applicability of the proposed simulator. The proposed full-wave simulator is currently being used as a research tool for characterizing wireless channels and predicting radio coverage in underground mine tunnels and galleries. Wireless system manufacturers and network designers can potentially use the simulator to identify best node locations when deploying a wireless communication, tracking, or sensing system in underground mines, especially during an emergency when the mine is obstructed by debris from a cave-in. The coupling of the proposed simulator to fast and accurate uncertainty quantification frameworks [

This work was supported in part by the Alpha Foundation under Grant AFC215-54.

The findings and conclusions in this paper are those of authors and do not necessarily represent the views of the National Institute for Occupational Safety and Health.

From August 2006 to April 2013, he was a Graduate Student Research Assistant with the Radiation Laboratory, University of Michigan, where he was a Research Fellow from May 2013 to December 2015. From January 2016 to December 2017, he was a Post-doctoral Researcher with the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA, and the Computational Electromagnetics Group, King Abdullah University of Science and Technology. Since January 2018, he has been an Assistant Professor with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. His research interests include various aspects of computational electromagnetics with emphasis on uncertainty quantification for electromagnetic analysis on complex platforms, electromagnetic compatibility and interference analysis, nature-based design of electromagnetic devices, and integral equation-based frequency and time domain solvers and their accelerators.

Dr. Yucel was the recipient of the Fulbright Fellowship in 2006, the Electrical Engineering and Computer Science Departmental Fellowship of the University of Michigan in 2007, and the IEEE AP-S Student Paper Competition Honorable Mention Award in 2009.

Since 2013, he has been a Research Assistant with the Radiation Laboratory, University of Michigan. His current research interests include fast solvers of integral equations, numerical methods in computational electromagnetics, optimization algorithms, and uncertainty quantification methods.

He is currently a Senior Research Engineer with the National Institute for Occupational Safety and Health (NIOSH), Washington, DC, USA, part of the Centers for Disease Control and Prevention, under the U.S. Department of Health and Human Services. He was the Project Leader and the Principal Investigator for the wireless communications and tracking project at NIOSH. He also was a Government Contract Officer Representative for NIOSH Broad Agency Announcement contracts. Prior to joining NIOSH in August 2012, he was a Research Fellow with the Disney Research Laboratory, Pittsburgh, PA, USA, where he conducted research in RF ranging based on passive RFID tags. Before he started his career in industry for Disney, he was a Project Research Scientist with the Department of Electrical and Computer Engineering, Carnegie Mellon University. He holds 2 U.S. patents in RFID and has authored or co-authored more than 60 papers in peer-reviewed journals or conference proceedings.

Dr. Zhou is currently an Associate Editor for the

From June 2015 to July 2017, he was a Post-doctoral Fellow with the Radiation Laboratory, University of Michigan. He is currently a Postdoctoral Fellow with the Scalable Solvers Group, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. His research interests include computational electromagnetics, numerical linear algebra, and high-performance computing, with focus on fast iterative and direct solvers for highly oscillatory problems.

Dr. Liu authored the Second Place paper of the Student Paper Competition of the 28th International Review of Progress in Applied Computational Electromagnetics in 2012. He also has coauthored the First Place paper of the Student Paper Competition of the 12th International Workshop on Finite Elements for Microwave Engineering in 2014.

From June 1999 to July 2001, he was an Undergraduate Researcher with the Computational Electromagnetics Group, Bilkent University. From August 2001 to December 2006, he was a Research Assistant with the Center for Computational Electromagnetics and Electromagnetics Laboratory, UIUC. From January 2007 to August 2009, he was a Research Fellow with the Radiation Laboratory, University of Michigan. In August 2009, he joined the Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology, as an Assistant Professor of electrical engineering. His research interests include various aspects of computational electromagnetics with emphasis on time-domain integral equations and their fast marching-on-in-time-based solutions, well-conditioned integral-equation formulations, and development of fast hybrid methods for analyzing statistical EMC/EMI phenomena on complex and fully loaded platforms.

Dr. Bađcý was the recipient of the 2008 International Union of Radio Scientists Young Scientist Award and the 2004–2005 Interdisciplinary Graduate Fellowship from the Computational Science and Engineering Department, UIUC. His paper entitled “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex structures loaded with coaxial cables” was one of the three finalists (with honorable mention) for the 2008 Richard B. Schulz Best Transactions Paper Award given by the IEEE Electromagnetic Compatibility Society. He has authored or co-authored nine finalist papers in the Student Paper Competition of the 2005, 2008, 2010, and 2014 IEEE Antennas and Propagation Society International Symposium and the 2013 and 2014 Applied Computational Electromagnetics Society Conference.

He joined the faculty of the Department of Electrical and Computer Engineering, UIUC, in 1993, becoming a Full Professor in 2002. In 2005, he joined the University of Michigan (UM), as a Professor of electrical engineering and computer science. Since 2013, he has been the University of Michigan’s Associate Vice President for Advanced Research Computing, and also directs the Michigan Institute for Computational Discovery and Engineering. He has authored or co-authored more than 180 journal papers and book chapters and more than 350 papers in conference proceedings. His research interests include all aspects of theoretical and applied computational electromagnetics. His research interests include the development of fast frequency and time domain integral-equation-based techniques for analyzing electromagnetic phenomena, and the development of robust optimizers for the synthesis of electromagnetic/optical devices.

Dr. Michielssen was a recipient of the Belgian American Educational Foundation Fellowship in 1988 and a Schlumberger Fellowship in 1990. He was also the recipient of a 1994 International Union of Radio Scientists (URSI) Young Scientist Fellowship, a 1995 National Science Foundation CAREER Award, and the 1998 Applied Computational Electromagnetics Society Valued Service Award. In addition, he was named 1999 URSI United States National Committee Henry G. Booker Fellow and was selected as the recipient of the 1999 URSI Koga Gold Medal. He also was the recipient of the UIUC’s 2001 Xerox Award for Faculty Research, appointed 2002 Beckman Fellow of the UIUC Center for Advanced Studies, named 2003 Scholar of the Tel Aviv University Sackler Center for Advanced Studies, and selected as UIUC 2003 University and Sony Scholar. In 2011, he was the recipient of the UM College of Engineering David E. Liddle Research Excellence Award and, in 2014, he was the recipient of the IEEE APS Chen-To-Tai Distinguished Educator Award. He is a member of URSI Commission B.

The entries of the unknown expansion coefficient vector _{n}_{d}_{p}

_{S}_{S}

The parallelization strategy used in this paper is built on the hybrid spatial/angular partitioning approach that was originally developed for multilevel FMM[_{p}_{g}/N_{p}_{g}

_{p}

As each processor computes and stores near-field interaction matrices of each group that it is in charge of, it also locally computes the near-field contribution to the matrix–vector multiplication pertinent to

Generic tunnel geometry for Muller-combined field SIE formulation. (a) Original problem. (b) Equivalent exterior problem. (c) Equivalent interior problem.

Partitioning a fictitious box enclosing the mesh of an example structure into small boxes and tabulating near/far-field pairs of a selected group _{(2}_{,}_{2}_{,}_{1)} in an FMM-FFT scheme.

(a) Geometry of an empty 600 m-long arched tunnel (the lateral wall is removed for illustration). The power values on receiver points computed by the proposed FMM-FFT-SIE simulator and obtained by measurements at 455 MHz for (b) vertical and (c) horizontal polarizations and at 915 MHz for (d) vertical and (e) horizontal polarizations. Electric current density on tunnel walls computed by the proposed simulator at 455 MHz for (f) vertical and (g) horizontal polarizations and at 915 MHz for (h) vertical and (i) horizontal polarizations (in dB scale).

(a) Geometry of a 200-m-long arched tunnel loaded with six PEC mine carts (the lateral wall is removed for illustration). (b) Power values on receiver points in empty and loaded tunnels computed by the proposed FMM-FFT-SIE simulator. (c) Electric current density on tunnel walls and mine carts computed by the proposed simulator (in dB scale).

(a) Geometry of a 650-m-long arched tunnel loaded with two parallel PEC strips (the lateral wall is removed for illustration). (b) Power values at receiver points on a line in the middle of the strips and on a line in the middle of the tunnel computed by the proposed simulator. (c) Electric current density on tunnel walls and conductor strips computed by the proposed simulator (in dB scale) (the half of the tunnel is removed to clearly show the currents around the strips).

(a) Geometry of a mine gallery formed by eight tunnels. (b) Power values at receiver points on lines inside tunnel 1, 2, 3, and 4 computed by the proposed FMM-FFT-SIE simulator at 455 MHz for (b) vertical and (c) horizontal polarizations and at 915 MHz for (d) vertical and (e) horizontal polarizations. Electric current density on tunnel walls computed by the proposed simulator at 455 MHz for (f) vertical and (g) horizontal polarizations and at 915 MHz for (h) vertical and (i) horizontal polarizations (in dB scale).

(a) Geometry of a 200-m-long rectangular tunnel with rough walls. (b) Power values at receiver points on a line in the middle of the tunnel computed by the proposed simulator for the tunnel with smooth and rough walls and by the multi-modal decomposition for the tunnel with smooth walls. Electric current density on (c) rough walls and (d) smooth walls of the rectangular tunnel computed by the proposed simulator (in dB scale).

Parallelization strategy in the FMM-FFT scheme for matrix-vector multiplication stage: partitioning of groups and plane wave directions among _{p}

Specifications of Simulations Performed by FMM-FFT-SIE Solver for Each Numerical Example

Numerical Example | Frequency (MHz) | Polarization | Number of processors used for simulation | Average memory required for each processor (GB) | Total simulation time (hours (h)/minutes (min)) | |
---|---|---|---|---|---|---|

| ||||||

Without Compression | With Compression | |||||

Empty arched tunnel (Section III-A.1) | 455 | Vertical | 32 | 28.40 | 14.62 | 5 h/17 min |

Horizontal | 5 h/20 min | |||||

915 | Vertical | 45 | 64.91 | 36.45 | 29 h/54 min | |

Horizontal | 28 h/49 min | |||||

Arched tunnel loaded with carts (Section III-A.2) | 455 | Vertical | 16 | 20.09 | 10.58 | 4 h/27 min |

Arched tunnel loaded with strips (Section III-A.3) | 50 | Vertical | 32 | 2.28 | - | 9 h/22 min |

Mine gallery (Section III-B) | 455 | Vertical | 32 | 21.05 | 10.60 | 4 h/52 min |

Horizontal | 4 h/38 min | |||||

915 | Vertical | 45 | 46.27 | 28.89 | 23 h/52 min | |

Horizontal | 22 h/45 min | |||||

Rectangular tunnel with rough walls (Section III-C) | 455 | Vertical | 32 | 30.04 | 12.73 | 6 h/22 min |

Rectangular tunnel with smooth walls (Section III-C) | 455 | Vertical | 32 | 28.45 | 12.28 | 6 h/45 min |