IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1

Transcription

1 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1 GPU-Based Calculation of Lightning-Generated Electromagnetic Fields in 3-D Problems With Statistically Defined Uncertainties Georgios G. Pyrialakos, Theodoros T. Zygiridis, Member, IEEE, Nikolaos V. Kantartzis, Senior Member, IEEE, and Theodoros D. Tsiboukis, Senior Member, IEEE Abstract A complete computational framework for the efficient study of lightning-induced electromagnetic fields and solution of pertinent problems with uncertainties in realistic environments is presented in this paper. The latter often involve various factors, such as material inhomogeneities, rough terrain surfaces, and irregular lightning channels that may inhibit the utilization of simplified approaches. To deal with these situations of augmented complexity, the finite-difference time-domain method is applied in 3-D curvilinear formulation, ensuring that all the important details are taken into account. As the study of real-life lightning problems involves intense computations, the algorithm is accelerated by exploiting the computing capabilities of contemporary graphics processing units. Our implementation relies on a massive parallelization approach, introduces several new optimized practices, and ensures significant shortening of the simulations duration. Hence, the investigation of configurations with uncertainties and the extraction of statistical features are greatly facilitated. In other words, the proposed approach comprises an instructive contribution toward the foundation of a useful tool for the in-depth investigation of lightning-related phenomena. Index Terms Finite-difference time-domain method (FDTD), graphics processing unit (GPU) computing, lightning, nonorthogonal grids, stochastic properties. I. INTRODUCTION THE reliable calculation of lightning-generated electromagnetic pulses, the assessment of their consequences, and the impact of statistically-varying factors are subjects of considerable importance for the engineering community. In essence, the consistent investigation of pertinent phenomena constitutes a key element of studies in several problems, including overhead transmission lines [1], [2], underground cables [3] [5], humansafety issues [6], electrical wiring [7] and circuitry protection, Manuscript received March 17, 2015; revised May 13, 2015; accepted June 19, This work was supported by the European Union (European Social Fund ESF) and Greek National Funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF) Research Funding Program: Aristeia. Investing in knowledge society through the European Social Fund. G. Pyrialakos, N. Kantartzis, and T. Tsiboukis are with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece ( pyrialak@auth.gr; kant@auth.gr; T. Zygiridis is with the Department of Informatics and Telecommunications Engineering, University of Western Macedonia, Kozani 50100, Greece ( tzygiridis@uowm.gr). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TEMC quality issues in distribution networks [8], grounding systems [9], as well as other significant configurations [10] [13]. Over the past years, various analytical [14] [16] and numerical methods for predicting electromagnetic fields due to lightning strikes have been presented and investigated. The popular Cooray Rubinstein approximate formula [17] can be used for calculating the horizontal electric component over lossy grounds. Results from the formula s time-domain counterpart can be also obtained [18] through a procedure that necessitates the calculation of several integrals. Other methodologies [19], [20] aim directly at the computation of Sommerfeld integrals, which describe the behavior of electric dipoles (treated as building blocks of the lightning channel). Although fast, the aforementioned approaches cannot easily take into account a number of important factors, which are present in real-world problems. Hence, computer simulations are deemed more suitable for modeling lightning in complex environments. Regarding these solutions, the finite-difference timedomain (FDTD) method [21] is the most widespread choice. In the ideal case of geometries with rotational symmetry, a two-dimensional (2-D) implementation suffices [22], even if mixed propagation paths are present [23]. In more advanced contexts [3], the generated fields are computed with a 2-D approach, and then embedded into a three-dimensional (3- D) simulation, to predict induced voltages. FDTD studies in 3-D are also performed in [24] and [25], for calculating the horizontal electric field and assessing the reliability of the Cooray Rubinstein formula over rough grounds. In [26], a treatment of large-scale problems is presented, which incorporates moving windows in the computational domain, and performs parallel computations based on a message-passing interface. Apart from the FDTD technique, other alternatives based on the multiresolution time domain scheme [27], the finite-element technique [28], the transmission-line-matrix method [29], and the constrained-interpolation-profile approach [30] have been applied for problems of similar complexity. Nevertheless, none of these methodologies investigates the statistical properties of the produced electromagnetic fields. Evidently, the development of a computational methodology that delivers all the necessary calculations, both fast and reliably under realistic constraints, remains an open problem. Moreover, available approaches do not tackle the fact that some aspects of lightning problems exhibit a certain level of randomness, which induces uncertainty to the outputs. In this paper, we present a solution that, to the best of our knowledge, for the first time IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See standards/publications/rights/index.html for more information.

2 2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Fig. 1. Typical computational domain of the lightning problem, for setups with inhomogeneous grounds, nonflat terrains, and irregular lightning channels. fulfills these two objectives. Its basic element is the full-wave FDTD simulations, conducted within a 3-D curvilinear framework. This choice is highly flexible: it can model nonsymmetric configurations, rough terrains, and inhomogeneous soils with Cartesian meshes, while the stricter case of tilted or irregular lightning patterns is dealt with nonorthogonal grids. Furthermore, computations are parallelized by implementing and optimizing the algorithm on graphics processing units (GPUs) [31] [33]. Their multitude of cores allows us to exploit the parallelization potential of the FDTD method, while advanced programming practices contribute toward the most efficient source utilization. In this way, marked reduction of computational times is accomplished, which is crucial for the extraction of statistical information. Numerical tests verify the accuracy and speed of the GPU implementations. Also, the impact of one or more factors on the induced fields in several cases is assessed. Overall, we show that the proposed approach can solve both reliably and efficiently real-life applications that incorporate lightninginduced phenomena, even when uncertainties are involved. II. PROBLEM MODELING The typical layout of the lightning problems is sketched in Fig. 1. The 3-D computational domain is occupied by either air or ground material. The sources of the electromagnetic fields appear on the lightning channel only, which either is assumed straight, or may have a more complex pattern (most studies till today consider only the straight-channel case). The current distribution is determined according to the modified transmission line model with exponential decay [34], I(z,t) =I(0,t z/v)e αz u (t z/v) (1) where u(t) is the unit-step function and I(0,t) is the time signature of the channel base current (z =0indicates the earth s surface). The speed of the return stroke is v = m/s and the constant α of the exponential decay is 1/2000 m 1. The above formula can be also used for tilted straight channels, after replacing z with z, which describes the position along the new axis of the skewed mesh. Irregular channels are treated similarly, after defining a length-measuring variable along the channel curve. Complying with a common practice, the base current is represented by two Heidler functions: I(0,t)= 2 l=1 I 0l η l ( t τ l1 ) 2 e t/τ l 2 1+(t/τ l1 ) 2 (2) where η l =exp[ (τ l1 /τ l2 )(2τ l2 /τ l1 ) 1/2 ], l =1, 2. The values of the parameters in (2) for first and subsequent strokes can be found in [20]. As mentioned, realistic factors such as medium inhomogeneities, rough soil surface, random channel geometries, etc., practically necessitate the implementation of 3-D simulations. Herein, spatial steps of 1-m length are sufficient, and the computational domains are surrounded by a 16-cell convolution perfectly matched layer (CPML) [35], capable of matching lossless and lossy media. When a problem s geometry permits so, a perfect magnetic conductor is inserted at the xz plane of the channel, accompanied by image values for certain field components. In this way, redundant calculations due to symmetry are avoided and memory requirements are minimized. Parallel computations are performed with single accuracy, exploiting the full potential of the GPUs. Tests with double-accuracy variables do not produce any noteworthy changes, while some performance loss is observed. Before proceeding, we would like to elaborate further on some aspects that will be explored. The first one refers to the geometry of the lightning channel. As our purpose is to provide a thorough investigation, the cases where: 1) the channel is completely straight, but forms an oblique angle with the horizontal plane, and 2) the channel is not straight, but exhibits a more irregular pattern are considered in the simulations, apart from the completely straight perpendicular channels. Given that the standard FDTD meshes lack the necessary level of flexibility, the algorithm is modified, so that it allows grid deformation according to the channel s geometry. In this way, we avoid staircase approximations in the vicinity of the problem s source that are likely to produce additional errors. At the same time, the implementation remains relatively simple, because the structured nature of the mesh is preserved and only modifications along the z-axis are called for. Fig. 2 depicts the local grid distortion in the general case, so that cell edges change in accordance to the channel pattern. Following [21], the update equations can be derived from the integral form of Maxwell s equations and involve the necessary metric coefficients. For instance, the E x update is E x n+1 i+ 1 2,j,k = C a i+ 1 2,j,k E x n i+ 1 2,j,k + C b i+ 1 ( ) 2,j,k Δy H z n+ 1 2 g 1 i+ 1 2,j,k i+ 1 2,j+ 1 2,k H z n+ 2 i+ 1 2,j 1 2,k C b i+ 1 ( 2,j,k Δz H y n+ 1 2 g i+ 1 2,j,k i+ 1 2,j,k 1 2 ) H y n+ 1 2 i+ 1 2,j,k+ 1 2 where C a and C b are standard spatially varying constants, (3) C a i+ 1 2,j,k = 2ɛ 1 i+ 2,j,k σ i+ 1,j,kΔt 2 2ɛ i+ 1 2,j,k + σ i+ 1,j,kΔt (4) 2 C b i+ 1 2,j,k = 2Δt 2ɛ i+ 1 2,j,k + σ (5) i+ 1 2,j,kΔt. g i+ 1 2,j,k corresponds to the determinant of the metric tensor, Δz =Δz/cos θ k, and θ k is the tilt angle of the individual cell

3 PYRIALAKOS et al.: GPU-BASED CALCULATION OF LIGHTNING-GENERATED ELECTROMAGNETIC FIELDS IN 3-D PROBLEMS 3 the involved field elements, which facilitates the execution of many simultaneous updates. Several studies have pointed out that GPU programming can be very useful in reducing FDTD computing times; hence, its application in repeated 3-D simulations appears to be a consistent choice. The following literature review describes how FDTD implementations have benefited from the development of GPU computing. Fig. 2. Nonorthogonal FDTD grid that conforms to the irregular shape of a lightning channel. with respect to the original z-axis. The necessary projections used in (3) are obtained according to H z n+ 1 2 i+ 1 2,j+ 1 2,k = ( tan 2 θ k +1 ) H z n+ 1 2 i+ 1 2,j+ 1 2,k tan θ k + H x n+ 1 2 i,j+ 1 2,k+ 1 2 ( H x n+ 1 2 i,j+ 1 2,k H x n+ 1 2 i+1,j+ 1 2,k H x n+ 1 2 i+1,j+ 1 2,k 1 2 ). (6) Similar equations are introduced for the CPML updates as well. The modified stability criterion is now described by Δt c max g l,m. (7) l=1 m =1 i,j,k Since the cell distortion in this case is only pertinent to the z- axis, we have g 13 = tan θ k, g 33 = tan 2 θ k +1, g 11 = g 22 =1, while the rest of the metric coefficients are equal to zero. The second factor that needs to be addressed is the modeling of distorted nonflat terrains. In this case, we use simple averaging of material parameters, at positions on the air ground interface. Specifically, if s air and s ground denote either the dielectric constant or the conductivity of the air and the ground, respectively, and v is the percentage of a cell volume occupied by ground material, then the effective value of the parameter assigned to this specific cell is set according to s eff = vs ground +(1 v)s air. (8) III. PARALLEL GPU IMPLEMENTATION GPU computing is nowadays deemed a highly effective solution for the acceleration of scientific applications. This should be considered rather expected, if one bears in mind the constantly growing capabilities of GPUs, their relatively low cost, and expanding utilization in general purpose computing. Due to continuous research and development, modern GPU hardware introduces a multitude of processing cores and various levels in the memory hierarchy. Such features render GPUs most suitable for the parallelization of the FDTD method [36], [37], as its equations exhibit a nontrivial degree of independence of A. Available Parallelization Strategies on GPUs Early implementations of the FDTD method on GPUs were highly complicated, due to the lack of properly structured programming languages. In one of the first attempts [38], texture memory was used for array storage, and a moderate acceleration factor of 7 was accomplished for a 2-D problem. With the availability of a more easy-to-use programming tool ( Brook ) that allowed the incorporation of kernels, GPU exploitation became simpler [39]. In the latter publication, tiling was applied to create 2-D arrays from 3-D ones, and speedups up to 25 times were reported. Later, the advent of today s programming platforms [40], combined with the upgrade of GPU architectures, made room for more efficient implementations. For example, the issue of shared-memory utilization is addressed in [41], along with appropriate schemes for data transfers. The work presented in [42] stresses the significance of coalesced memory accesses, and investigates different organizations of a block s threads in 1-D arrays. Another implementation of the 2-D algorithm is presented in [43], where texture as well as shared memories are utilized, and acceleration up to 100 times is accomplished. The application of the 3-D method to MRI problems is presented in [44], where a 45 acceleration is noted. Three different approaches regarding the mapping of Yee cells to threads are examined in [45] without using shared memory, while the integration of streams is mentioned in [46], to enable concurrency and hide memory latencies during upload and calculation of data. Consequently, it is certain that the parallelization of the standard FDTD scheme on GPUs has reached an adequate level of maturity today. Nonetheless, developing a parallel numerical code should not be considered a trivial task, especially if full optimization and utilization of available resources are sought. In this paper, we have performed a critical compilation of some of the developed best practices, and combined them for Monte Carlo simulations of lightning problems, using curvilinear grids and CPML boundaries. In addition, we describe a number of different programming strategies, concerning specific aspects of the algorithm s parallelization, and provide evidence that verify the optimality of the proposed actions. B. Proposed GPU Implementation In our case, the developed 3-D code is based on the CUDA 5.5 programming platform [40]. A flowchart that describes the implemented algorithm is depicted in Fig. 3. The computational approach consists of a loop in time, during which the main update of the electric- and the magnetic field components is completed. In the case of fully orthogonal grids, the use of one kernel for each update suffices; however, we have found that

4 4 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Fig. 5. Performance differences between single- and multiple-kernel implementations of the CPML updates. Fig. 3. Flowchart describing the GPU implementation of the algorithm that enables the calculation of statistical properties. Fig. 4. Improvement due to the use of different streams for magnetic field update, compared to the single-kernel implementation. a multikernel implementation (exploiting streams, as discussed later) is more beneficial in the case of curvilinear FDTD algorithm. To calculate the field in the surrounding absorbing layers, two more groups of kernels are introduced. Each one comprises 12 CPML kernels and is initiated after the execution of the corresponding main one. This structure of the CPML update is the result of our search for optimum performance, according to which it was determined that each kernel should be responsible for one boundary layer, and one specific component of the electromagnetic field. In particular, the update procedure within each one of the domain s six sides requires four kernels (two for electric and two for magnetic components), in order for the additional calculations (due to the extra CPML terms) to be completed. In the following paragraphs, the selection of the above, as well as other programming strategies is justified and tested. 1) Use of Streams: Aiming at the highest level of parallelization, we have utilized streams at various points. Streams refer to different and independent flow sequences, programmed to be executed at the same time, avoiding serialization. Thanks to them, we have partially achieved the concurrent execution of the main, as well as the CPML kernels (note that such a solution is very rarely analyzed in pertinent publications). To assess their performance, the improvement in the case of the magnetic field update of the curvilinear algorithm is evident in Fig. 4, as the required calculations are completed in 38% less time, when streams are used (the timings refer to a single timestep update). Practically, performance is improved by hiding the kernel and variable reinitialization overhead. As far as the CPML updates are concerned, the implementation of the aforementioned kernel groups with the aid of parallel streams, also allows to a certain degree their concurrent execution, and facilitates memory coalescing, as explained later. Fig. 5 verifies the gain due to this multiple-kernel implementation, since the required time is reduced by a factor of ) Memory Considerations: A successful GPU implementation depends on several factors, and the incorporation of streams may not be sufficient, if other aspects of parallelization are neglected. In essence, one has to take into consideration the type and size of the available hardware memories, in order to exploit their strengths and, at the same time, avoid the corresponding shortcomings. In the proposed implementation, the global memory is used for storing the main field components, as well as the CPML variables. We have paid attention to the proper matrix alignment in memory, which ensures that adjacent threads access similarly placed elements from memory. Only when this action takes place, then transfers of 32 elements from global memory (in the case of floats) are performed in a single memory access cycle (coalesced access). Especially for the CPML updates, specific CUDA grid alignments have to be selected; otherwise, a naive implementation of a unique kernel for all CPML areas would render coalesced access almost impossible. This is due to the utilization of a small grid size along the perpendicular (with respect to the CPML interfaces) directions, which in certain cases does not allow assigning the first dimension of the grid to the first dimension of the CPML auxiliary field matrices. However, the latter action is required for accomplishing the coalescing mechanism. A simple, yet inefficient solution would make use of a full-size grid, which would nevertheless introduce a large number of idle threads. Hence, dividing the algorithm into individual parts with different grid requirements that exploit streams appears as the most appropriate choice. Furthermore, the constant memory is employed for storing the constants of each simulation, while parameter matrices are loaded to global memory and mapped to a surface reference, exploiting texture memory. The latter constitutes a buffer optimized for read-only memory access of this type. In order to identify the parts of the algorithm that can actually benefit from texture utilization, several tryouts have been conducted for the constant matrices. The latter corresponds to material, as well as geometrical parameters. We have found that 10% speedups can be accomplished, in the case of the curvilinear FDTD algorithm. Tests have also pointed out that the use of texture memory can be beneficial, in cases where field matrices are treated as read only. Evidently, this is a kernel-dependent case, i.e.,

5 PYRIALAKOS et al.: GPU-BASED CALCULATION OF LIGHTNING-GENERATED ELECTROMAGNETIC FIELDS IN 3-D PROBLEMS 5 TABLE I COMPUTATIONAL TIMES OF A SIMPLE TEST SIMULATION FOR VARIOUS CHOICES OF THE KERNEL SIZE Size Time (ms) Size Time (ms) Size Time (ms) magnetic field matrices can be treated as read only within any electric-update kernel. 3) Kernel Size and Register Usage: To maintain a high degree of parallelization, the kernels have been optimized in terms of the block size. In essence, a grouping of threadper-block has been found to guarantee very good performance, in the case of Cartesian grids. Numerical evidence is presented in Table I, which displays the computational times in the case of a grid and 250 time-steps, for different kernel dimensions. Note that the GPUs used in this paper can handle a maximum of 1024 threads per block. After performing a similar study, the most suitable kernel size for the curvilinear algorithm was found to be Moreover, it has been ensured that register usage remains below a certain threshold for the entire implementation. This effectively allows more threads to be active at a given time for each streaming processor; thus, increasing the pool of available ready to be executed commands. The complexity of the curvilinear algorithm did, however, require more extensive manipulation of the kernel variables. To achieve operation just below a certain register limit, the GPU implementation has been designed to minimize performance leeching from unnecessary global memory accesses or extra arithmetic commands. 4) Shared-Memory Exploitation: Possible benefits from the on-chip shared memory have been investigated, since such a practice is quite common in relevant studies. Being available only to the threads of the same block, the main principle regarding shared-memory utilization is to either minimize redundant global memory accesses, or to replace those unable to be optimized (via coalesced access). However, for the simple FDTD algorithm (orthogonal grids), shared memory has been proven to provide virtually no benefit. The reason behind this is that our implementation practically accomplishes the maximum number of instructions per cycle (IPC); hence, no room for improvement is left. If shared memory is used, the additional overhead (e.g., loading of halo elements) unavoidably produces performance degradation. On the other hand, this is not the case when implementing the curvilinear FDTD version, as a small but nonnegligible speedup is then achieved. This can be attributed to the fact that elements of the field matrices are requested more than once (in average), due to the algorithm s spatial field averaging requirements. In fact, the maximum number of IPC cannot be reached in this case, and incorporating shared memory contributes positively. 5) Use of Atomics: Due to the selected kernel organization, atomic operators have been applied at some points to assist with Fig. 6. Example of error that may emerge in the case of parallel CPML updates, and solution provided by the use of atomic operations. Ty and Tz denote terms appearing in the update of E x in the CPML corner regions. Fig. 7. Errors due to reflections from the CPML, where implementations with and without atomics are compared. the proper CPML kernel execution. Specifically, atomics are necessary in rare conditions occurring during the simultaneous update of elements at the mesh corners. In general, the incorporation of atomics prevents the access of an element residing on the global memory, until all other operations involving this element have been completed. In this way, certain parallelization errors are prevented. Fig. 6 clarifies the correction provided in our algorithm, when the race between concurrent threads is likely to produce undesirable miscalculations (top and bottom PML layers are considered in this figure). With the proposed solution, correct updating is guaranteed, without any performance degradation. The necessity of atomics incorporation is also verified via another test simulation. The specific computational setup consists of a space with a hard source located at the center. Fig. 7 displays the absolute error after 250 timesteps regarding the H z component, along a line close to the CPML corners. As seen, incorrect handling of the parallelization strategy may result in additional flaws, which in this case are interpreted as artificial reflections from the corners of the surrounding medium. Note that the levels of these errors are not trivial; hence, the application of atomics is not just complementary here, but rather necessary. IV. CODE VALIDATION AND MEMORY UTILIZATION Before proceeding to the investigation of specific lightning problems with uncertainties, we test the reliability of the parallel 3-D FDTD code, by performing comparisons with other numerical as well as analytical solutions in simple configurations. In

6 6 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Fig. 8. Waveforms from serial 2-D and parallel 3-D simulations, for the radial component of the electric field intensity at (r, z) = (150, 12 m) (first stroke). Fig. 9. Waveforms from analytical computations and parallel 3-D simulations, for the radial component of the electric field intensity (subsequent stroke). addition, a convergence study determines the smallest size of the lightning channel that ensures sufficient accuracy. First, parallel 3-D computations are validated by comparing the waveform at a specific point with the one calculated with a serial 2-D CPU code. The latter applies the FDTD discretization in cylindrical coordinates, taking into account the problem s symmetry. For this test, we consider a configuration where the ground parameters are invariant, equal to σ =0.001 S/m, ɛ r =5. Furthermore, the lightning channel is straight and vertical, and the terrain is flat (hence, rotational symmetry is guaranteed). The 3-D computational space comprises cells. Such a large number of cells along the z-axis is not actually needed; however, we also wish to verify the potential of modeling large-scale problems. Fig. 8 compares the results regarding the radial component of the electric field intensity at (r, z) = (150, 12 m), when the first-stroke current is considered. It is observed that 3-D and 2-D computations coincide, thus, verifying the reliability of the GPU implementation. Another validation test is next performed, where the GPU computations are compared to analytical calculations. Specifically, reference solutions are obtained using the technique described in [20], which transforms Sommerfeld integrals by expressing them in terms of Hankel functions, and then follows a modified integration path on the complex plane. Two examination points are considered ((r, z) = (100, 10 m) and (200, 15 m)), for the case of the subsequent lightning stroke. The comparison with the GPU-obtained values is depicted in Fig. 9, where, as previously, agreement is verified. Given that the technique in [20] is an analytical one, it should produce results very close to the exact ones; hence, the accuracy of the proposed numerical model can be considered very promising. As already mentioned, a critical issue in parallel 3-D simulations is the available memory of a single GPU unit. Given that a realistic estimate of a channel s size is of the order of kilometers, domains with a large number of cells in the z-direction cannot be avoided. Hence, it is crucial to determine the minimum channel length that actually needs to be modeled, to ensure both reliable computations and a balanced treatment of memory resources. Fig. 10. Effect on the lightning-channel length on the radial component of the electric field intensity at (r, z) = (150, 12 m) (first stroke). Bearing this in mind, the predicted radial field of the previous examples (first-stroke case) is examined for various lightning channels. According to Fig. 10, the results converge after a certain source size, at least within the time period examined here. Therefore, we safely conclude that the incorporation of at least 1500 m of the channel is necessary, so that the validity of the simulation is not compromised. Complementary to the previous test, the following one reveals the effect of the channel truncation at different observation heights. Specifically, the relative (%) error at positions z =50, 100, and 150 m is depicted in Fig. 11 (r = 100 m, the subsequent stroke is considered, and ground parameters are the same as previously). Reference waveforms are obtained with a much longer channel. It becomes apparent that the effect of reducing the actual channel size appears more pronounced at higher positions. Nevertheless, the corresponding values are practically insignificant, as they remain lower than 0.2%, proving that our assumption for the necessary channel length remains valid not only for positions close to the ground. Additional tests revealed that the error remains lower than 1% even with 10 more time steps.

7 PYRIALAKOS et al.: GPU-BASED CALCULATION OF LIGHTNING-GENERATED ELECTROMAGNETIC FIELDS IN 3-D PROBLEMS 7 Fig. 11. Relative (%) error at different observation heights, due to the truncation of the lightning channel (r = 100 m, ɛ r =5, σ =0.001 S/m). The case of subsequent stroke is considered. TABLE II SIMULATION TIMES (IN SEC) OF SERIALIZED AND PARALLEL FDTD IMPLEMENTATIONS FOR DIFFERENT GRIDS Grid Size CPU time GPU1 time GPU2 time ( speedup) ( speedup) cells 1795 s 51 s ( 35) 20 s ( 90) cells 3651 s 96 s ( 38) 36 s ( 101) 10 7 cells 7722 s 158 s ( 49) 61 s ( 127) cells s 309 s ( 49) 120 s ( 126) cells s 243 s ( 128) cells s 528 s ( 120) V. ASSESSMENT OF GPU ACCELERATION In problems with uncertainties, the extraction of statistical properties requires performing several simulations, which can be extremely time consuming in the case of 3-D problems. This drawback has been our main motivation for the GPU parallelization, as our intention is to perform Monte Carlo analyses for lightning-induced electromagnetic fields. In this section, the potential of GPU-FDTD implementations to solve the aforementioned problems rapidly is investigated. We consider Cartesian meshes with different sizes and perform simulations with iterations. The CPU code is executed on an Intel Core i processor at 3.6 GHz, while parallel computations are carried out on two different GPUs. The first one (GPU1) is a NVIDIA Tesla C2050 unit, with 448 cores and 3 GB of global memory. The second GPU (GPU2) is an NVIDIA GTX Titan, and offers 2688 cores and 6 GB of memory capacity. The timings are compared in Table II. Although the larger problems could not be handled by the older GPU, the obtained results fully justify our decision regarding the parallelization of the FDTD approach. Compared to the serial code, GPU1 is able to attain speedups up to 49. Even greater reduction of the computational times is accomplished with the newer GPU, as we find acceleration factors as high as 128. Consequently, the significant time reduction enables the execution of repeated simulations within reasonable time limits, facilitating the investigation of statistical properties. VI. NUMERICAL RESULTS The rest of the paper is devoted to numerical results, where the stochastic nature of various aspects of the lightning problem Fig. 12. Mean value (black lines) and standard deviation (blue bars) of the horizontal components of the electric- and magnetic field intensities at point (r, z) = (100, 2 m), due to subsequent stroke current. is investigated. Our main goal is twofold. First, to demonstrate that the simplified models used in many studies till today may not correspond reliably to real-life situations. Second, to provide an assessment of the impact that the fluctuations of modeling variables have on the final results. Pertinent statistics regarding the electric field intensity are extracted after multiple simulations, in the context of a Monte Carlo procedure. The quantity of interest is the horizontal (radial) electric field, as this is more likely to cause detrimental effects on specific structures, e.g., power lines [47]. A. Effect of Ground Inhomogeneities Let us, initially, examine the case where the ground has a completely flat surface, but is made up of inhomogeneous lossy material. This is in contrast to many other approaches, where the soil is considered completely homogeneous. Specifically, the mean values for the ground parameters are selected ɛ r =5 and σ = S/m, with their standard deviations being 1 and S/m, respectively. Moreover, the necessary random values have normal distributions and are obtained with the Box Muller technique [48]. Normally, the extraction of reliable statistics calls for a high number of individual tests. Yet, the considered level of randomness in this problem allows us to draw consistent conclusions with a relatively low number of simulations (60 100). The latter have been carried out for 6000 steps, in a computational domain comprising cells. The average values of the horizontal electric and magnetic field at point (r, z) = (100, 2 m), together with error bars denoting the corresponding standard deviations, are depicted in Fig. 12. Evidently, even small spatial fluctuations of the material parameters trigger nontrivial changes of the field magnitude. In this case, we find a maximum absolute value of V/m for the electric field, and a maximum value of V/m for its standard deviation. Similar properties are exhibited by the magnetic field, whose standard deviation becomes as large as 15.5% of the mean value at early times, and reduces to < 3% when t>2 μs.

8 8 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Fig. 13. Representative geometry of a rough terrain and snapshot of the electric field distortion at z =10m. Fig. 15. Radial component of the electric field intensity, for different angular positions of a straight lightning channel (first-stroke case). Fig. 14. Statistical parameters of the radial electric field due to the subsequent stroke at (r, z) = (100, 10 m), when the terrain is not flat. Cases where the correlation length l is set to different values are displayed. B. Effect of Terrain Roughness The next aspect of our problem is the roughness of the ground, which has been taken into consideration in very few instances till today [49], [50]. A technique similar to the one used for generating randomly inhomogeneous material is also applied herein, in order to introduce varying fluctuations on the ground surface [51]. Essentially, the height of the terrain is selected to be locally modified by only ±1 m. A representative portion of the anomalous terrain is given in Fig. 13, together with a snapshot of the electric field distortion on the z =10mplane. The latter is obtained from the calculated field, after subtracting the values that correspond to a completely flat ground surface. Although only one of the various cases examined is depicted, it is evident that the ground irregularity induces local fluctuations to the field distribution that should not be overlooked. The mean value and standard deviation of the radial electric field at a specific observation point are given in Fig. 14, for three different values of the correlation length l [51] and a vertical straight channel configuration. It is noticed that the standard deviation of the field values is quite significant, and may reach levels up to 10% of the mean value at the corresponding time instants. In any case, it becomes evident that even small deviations from a completely flat earth surface can induce nontrivial changes to the magnitude of the lightning-produced pulses. Additionally, we Fig. 16. Radial component of the electric field intensity, for different angular positions of a straight lightning channel (subsequent-stroke case). may conclude that larger correlation lengths trigger higher field levels. This seems expected, since shorter correlation lengths imply rougher surfaces; hence, stronger scattering at random directions occurs in these cases. C. Effect of Straight Channel s Inclination It is a common approach, especially but not only in calculations via integral expressions, to consider the lightning source in a completely vertical position. This practice facilitates analytical computations, but also introduces an unnecessary simplification. To assess the effect of the channel s angular position on the outcome, we present various results with straight lightning channels at tilted placements. We avoid staircase approximations to the channel s geometry by using skewed meshes, where the cells tilt angle is selected according to the direction of the channel. Representative results of the horizontal electric field component are depicted in Figs. 15 (first stroke) and 16 (subsequent stroke). The examined angles, which are positive when the channel is tilted toward the observation point, fall within the

9 PYRIALAKOS et al.: GPU-BASED CALCULATION OF LIGHTNING-GENERATED ELECTROMAGNETIC FIELDS IN 3-D PROBLEMS 9 Fig. 17. Examples of lightning channels with irregular geometries. range from π/15 to π/15 rad. As observed, the magnitude of the induced field waveforms are very sensitive to the channel s slope. For example, a π/15 angular modification may cause up to four times more intense horizontal fields. Compared to the case of a completely vertical channel, the radial electric field now is a combination of two components, one parallel and one perpendicular to the channel, resulting in considerable changes that cannot be neglected. D. Effect of Channel Irregularity Apart from inclined straight channels, a more realistic approach should include lightning geometries with more irregular shapes. Our framework can incorporate nonorthogonal meshes, where cells may be continuously and randomly tilted along the z-axis. This permits us to study several different models of the lightning channel, and evaluate their effect on the excited fields. Some examples of nonstraight channels are depicted in Fig. 17. Such geometries are produced via a randomized algorithm, which initially divides the channel into smaller linear parts (with lengths from 1 to 75 m in the displayed cases). Then, different skew angles are assigned to individual pieces, resulting in more realistic patterns. For the examples in Fig. 17, the average value of the tilt angles (generally forced to be π/10) is selected zero. The impact of the channel geometry on the electric field is shown in Figs. 18 and 19. Specifically, Fig. 18 displays the predicted curves when the first stroke is considered with irregular channels similar to those in Fig. 17. The observation point is (r, z) = (100, 10 m). Two different cases are tested: Case I refers to a zero average segment inclination with its standard deviation set to π/10, while Case II assumes an average of the angular positions equal to π/15 and values within a range of 2π/7.5 rad. Statistics are obtained after 80 simulations. As seen, standard deviations exhibit (now shown as a percentage of the corresponding mean values) trends close to those of the meanvalue curves. Specifically, for the considered period, the field produced by the first-stroke current continues to grow in time, as the mean value has not started reducing to zero yet. Similar observations can be made for Fig. 19, where the subsequent stroke is considered. Now, Case I refers to the same parameter selection as Fig. 18, while Case II assumes an average of the angular positions equal to π/15, and values within a range of 2π/7.5 rad. As mentioned earlier, an overall positive tilt angle re- Fig. 18. Statistical parameters of the radial electric field due to the first stroke at (r, z) = (100, 10 m). Tilt angles of channel segments have values between ±π/10 rad in Case I, and between π/15 ± π/10 rad in Case II. Fig. 19. Statistical parameters of the radial electric field due to the subsequent stroke at (r, z) = (100, 10 m). Tilt angles of channel segments have values between ±π/10 rad in Case I, and between π/15 ± π/7.5 rad in Case II. sults in stronger fields at the point of interest. Regarding the computational cost, distorted meshes require more time-steps due to stability restrictions; thus, rendering GPU parallelization even more valuable. E. Impact of Combined Factors The last numerical result is concerned with the overall impact on the field s statistical properties when the uncertainty of various factors is taken into account within a single configuration. In essence, we perform simulations where the stochastic nature of the ground s electric parameters, the terrain s roughness, and the channel s irregularity are considered to be present at the same time. The standard deviation of the ground s dielectric permittivity is set to 10% of its mean value (equal to 5), its conductivity is selected 0.01 S/m, and the terrain s surface is distorted as described in Section VI-B. In addition, the tilt angles of the channel s individual elements have a zero mean value and a standard deviation equal to π/10. The statistical parameters of both the horizontal electric- and magnetic field intensities due to the

10 10 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY thank Dr. K. Rallis for providing the programming code for analytically calculating field values, according to [20]. REFERENCES Fig. 20. Statistical parameters of the radial electric and horizontal magnetic field due to the subsequent stroke at (r, z) = (100, 10 m), when inhomogeneous ground, rough terrain and irregular channel are combined (left axis: electric field, right axis: magnetic field). subsequent stroke are illustrated in Fig. 20 for the observation point of previous tests ((r, z) = (100, 10 m)). Comparing with Case I of Fig. 19, where only the channel irregularity is considered, the mean value of the electric field displays a similar pattern, but with lower levels (around 25%). This observation implies that among the three individual factors, the produced field is influenced primarily by the channel s geometry and secondarily by the ground s inhomogeneity and roughness. Interestingly, if the obtained standard deviation is expressed as a percentage of the corresponding mean value, it appears higher than all previous cases, which is a conclusion consistent with the presence of uncertainty in various aspects of the lightning problem. VII. CONCLUSION We have presented a parallel computational approach for the statistical study of lightning-generated electromagnetic fields in complex 3-D environments, exploiting the potential of modern GPUs. Compared to standard serialized solutions, the proposed implementation can accelerate computations by factors higher than 100 in cases of large grids. We have also demonstrated the utility of the rapid simulations in problems where modeling uncertainties need to be considered, and statistical parameters can be extracted after performing a multitude of trials in reasonable computing times. Results have verified that the stochastic character of electric and geometric features of lightning problems has a significant impact on the produced electromagnetic fields, and failure to take these into account may degrade the reliability of results. The proposed computational framework has a very broad range and is suitable for the efficient investigation of diverse engineering problems with random characteristics in other EMC applications, as well. ACKNOWLEDGMENT One of the GTX Titan GPUs used for this research was donated by the NVIDIA Corporation. The authors would like to [1] T. Thang, Y. Baba, N. Nagaoka, A. Ametani, J. Takami, S. Okabe, and V. A. Rakov, FDTD simulation of lightning surges on overhead wires in the presence of corona discharge, IEEE Trans. Electromagn. Compat., vol. 54, no. 6, pp , Dec [2] H. Sumitani, T. Takeshima, Y. Baba, N. Nagaoka, A. Ametani, J. Takami, S. Okabe, and V. A. 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Cooray, Horizontal electric field above- and underground produced by lightning flashes, IEEE Trans. Electromagn. Compat., vol. 52, no. 4, pp , Nov [48] G. E. P. Box and M. E. Muller, A note on the generation of random normal deviates, Annu. Math. Stat., vol. 29, no. 2, pp , [49] Q. Zhang, J. Yang, X. Jing, D. Li, and Z. Wang, Propagation effect of a fractal rough ground boundary on the lightning-radiated vertical electric field, Atmos. Res., vol , pp , Feb [50] Q. Zhang, X. Jing, J. Yang, D. Li, and X. Tang, Numerical simulation of the lightning electromagnetic fields along a rough and ocean-land mixed propagation path, J. Geophys. Res., Atmos., vol. 117, no. D20, pp. 1 7, Oct [51] K. Uchida, M. Takematsu, J.-H. Lee, K. Shigetomi, and J. Honda, An analytic procedure to generate inhomogeneous random rough surface, in Proc. 16th Int. Conf. Netw.-Based Inf. Syst., Gwangju, Korea, 4 6 Sep. 2013, pp Georgios G. Pyrialakos was born in Thessaloniki, Greece, in He received the Diploma degree in electrical and computer engineering from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2013, where he is currently working toward the Ph.D. degree at the Applied Computational Electromagnetics Laboratory. His research interests include the advancement of computational electromagnetics, focusing mainly on the FDTD method and its application to realistic EMC problems, and the hardware acceleration of the related algorithms. His other fields of research include the numerical evaluation of graphene and other advanced materials. Theodoros T. Zygiridis (M 13) received the Diploma and Ph.D. degrees in electrical and computer engineering, both from Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2000 and 2006, respectively. He is currently an Assistant Professor at the Department of Informatics and Telecommunications Engineering, University of Western Macedonia, Kozani, Greece. His research interests include the area of computational electromagnetics, and mainly focus on FDTD methods, error-optimized techniques, unconditionally stable schemes, parallelization on GPUs, lightning problems, simulations with uncertainties, etc.

12 12 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Nikolaos V. Kantartzis (SM 12) received the Diploma and Ph.D. degrees in electrical and computer engineering from the Aristotle University of Thessaloniki (AUTH), Thessaloniki, Greece, in 1994 and 1999, respectively. In 1999, he joined the Applied and Computational Electromagnetics Laboratory, Department of Electrical and Computer Engineering, AUTH, as a Postdoctoral Research Fellow, where he is currently an Associate Professor. He has authored or coauthored three books and several refereed journal papers in the area of computational electromagnetics, EMC analysis, metamaterials, and absorbing boundary conditions. His main research interests include EMC modeling, time- and frequency-domain algorithms, metamaterials, advanced microwave components, and antenna applications. Theodoros D. Tsiboukis (SM 99) received the Diploma degree in electrical and mechanical engineering from the National Technical University of Athens, Athens, Greece, in 1971, and the Doctor Eng. degree from the Aristotle University of Thessaloniki (AUTH), Thessaloniki, Greece, in From 1981 to 1982, he was with the Electrical Engineering Department, University of Southampton, Southampton, U.K., as a Senior Research Fellow. Since 1982, he has been with the Department of Electrical and Computer Engineering (DECE), AUTH, where he is currently a Professor. He has served in numerous administrative positions, including Director of the Division of Telecommunications, DECE ( ) and Chairman, DECE ( ). He was the Chairman of the local organizing committee of the 8th International Symposium on Theoretical Electrical Engineering (1995). He has authored or coauthored eight books and over 350 refereed journal and conference papers. He was the Guest Editor of a special issue of the International Journal of Theoretical Electrotechnics (1996). He is currently the Head of the Advanced and Computational Electromagnetics Laboratory, DECE. His main research interests include electromagnetic-field analysis by energy methods, computational electromagnetics (FEM, vector finite elements, MoM, FDTD method, integral equations, absorbing boundary conditions), metamaterials, photonic crystals, inverse and EMC problems. Prof. Tsiboukis received several awards and distinctions. He is a Member of various societies, associations, chambers, and institutions.