The Solution of a FEM Equation in Frequency Domain Using a Parallel Computing with CUBLAS

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1 The Solution of a FEM Equation in Frequency Domain Using a Parallel Computing with CUBLAS R. Dominguez 1, A. Medina 1, and A. Ramos-Paz 1 1 Facultad de Ingeniería Eléctrica, División de Estudios de Posgrado, U.M.S.N.H., Ciudad Universitaria, C.P , Morelia, Michoacán, MEXICO. Abstract - The recent technological computer advances have allowed the use of the Finite Element Method (FEM), to calculate the solution of the Maxwell field equations of electrical machines or devices. In some cases, an axisymmetric or a plane symmetry can be assumed to reduce the complexity of the finite element analysis to be performed. Nevertheless, the large size of the matrix equations derived, could imply a significant computing effort. In this paper, a parallel method of solution in frequency domain of a FEM equation with currents known is proposed. It consists on implementing the LU method using a parallel computing with CUBLAS. A normal and a reduced type of FEM equation proposed by the authors have been solved in the frequency domain using this parallel computing platform. It is shown that a significant reduction in the computing time to solve these FEM equations in the frequency domain is achieved. Keywords: Finite element method, frequency domain analysis, parallel processing 1 Introduction The Finite Element Method (FEM) is a very powerful tool to solve the electric and magnetic equations of electrical machines or devices. The method has been widely used, since the computational technological advances have allowed the application of the method on the modeling and simulation of electrical machines or devices with complex geometries of configurations [1]-[3]. Nevertheless, the method can be difficult to use in devices with 3D geometries or in those which need a detailed geometry model; the reason is the large matrix equations derived by the finite element analysis, which in turn can be difficult to solve in the frequency domain or in the time domain. However, the finite element analysis can be simplified if a planar or axisymmetric assumption is taking into account [2], [3]. In an earlier paper, the authors proposed a new form to solve a FEM equation with currents or voltages known [4]. The method consists on deriving a lesser order equation from a normal FEM equation. The reduced equivalent equation obtained is expressed in terms of the time varying variables, and it can be easily solved in time domain or in the frequency domain [3]. The can be calculated from a normal FEM equation derived from of a finite element analysis performed on a device with a planar or an axisymmetric symmetry [4]. The is easy to derive and solve, since it implies the use of simple matrix operations [4]. These matrix operations can be derived by a parallel computing. Moreover, the normal and the reduced FEM equations can be solved in the frequency domain by a parallel solution. Thus, it is possible to obtain a significant computation time reduction. The FEM equations to be solved correspond to equations that model a device with a planar or axisymmetric symmetry, and whose conductor currents are known. In this paper, the LU method has been implemented in the CUBLAS parallel platform, in order to solve normal and reduced FEM equations in the frequency domain. Specifically, an LU decomposition process was implemented using parallel processing using routines of the CUBLAS library. The proposed parallel solution has been tested in two devices: a planar conductor and a series reactor with an axisymmetric symmetry assumption. The rest of the paper is organized as follows: Section 2 explains the features of the partial differential equations of devices modelled by planar or the axisymmetric symmetries. Section 3 explains the features of the FEM matrix equations, derived from a finite element analysis performed with the partial differential equations shown in Section 2. Section 4 explains how the normal and the reduced FEM matrix are solved in the frequency domain; Section 5 describes how these equations are solved using the CUBLAS computing platform; Section 6 describes a case study which consists of two devices in which the parallel solution has been tested: the first device is a T conductor modelled by a planar symmetry and the second device is an air series reactor modelled by an axisymmetric symmetry. Finally, Section 7 contains the main conclusion drawn from this investigation. 2 Partial Differential Equations of a Device with Planar or Axisymmetric Symmetries This investigation is based on the following assumptions: the frequency of the voltage source of the device to be modelled is low enough to neglect the

2 displacement current in the Maxwell field equations [2], [3], [5]. The permeability and the conductivity of the device are assumed to be constant. Finally, there are no voltage difference at different conductor points [5]. In some cases, the modelling of a device can be simplified by a planar or axisymmetric symmetries [2], [3]. If it is considered that a skin effect exists on the conductors of the devices, and that these conductors are excited by voltage sources, then the partial differential equations for a device with a planar or an axisymmetric symmetries are given by [5]. [6], + = { } + + (1) = { } (2) Where A z and A ϕ are the magnetic vector potential of a device with a planar or an axisymmetric symmetry assumption, respectively; σ and v are the conductivity and reluctivity of the materials, respectively. {U c} is a vector which contains the voltages applied at the conductors of the device. If it is considered that the voltages along the z-axis are constant for a planar symmetry; and that the voltages along the ϕ-axis are constant for an axisymmetric symmetry; then it is possible to derive an equation to relate the voltage, current and the magnetic vector potentials at the conductors of the device [5], [6]. The equation is given by [5], [6], [! ] #$ {% & } () * = {+} (3) Where {I} is a vector that contains the conductors current. The matrix [Δ x] for the planar and the axisymmetric symmetry is defined by the equations (4) and (5), respectively. [! ] = -.* #$ / *.012 s [! ] = [3 & ] = - () #$ / *.012 Where [R c] is the conductor matrix resistance if the device has a planar symmetry assumption. The surface area S c of the equations (4) and (5) varies if the device is modeled by a planar or an axisymmetric symmetry. For the case of the planar symmetry, the surface area S c involves the plane x-y [5]. For an axisymmetric symmetry, the surface area involves the plane r-z [6]. 3 Finite Element Analysis of the Device It is possible to perform a finite element analysis on the partial differential equations defined on (1) and (2). At the (4) (5) same time, a Newton Cotes analysis can be performed on the expression defined in (3). It yields [5], [6], [)]{ } + [4].{ 5 }. [! ] #$ {% & } [7 & ].{ 5 }. = {6}{% & } (6) = {+} (7) Where the matrices [S], [T], [M c] and the vector {f} are obtained from the finite element analysis performed for a planar or axisymmetric symmetry [5], [6]. The vector {I} contains the currents in the conductors of the device, A x is defined in the z-axis and the ϕ-axis for the planar and the axisymmetric symmetry, respectively. If the conductor currents in {I} are known, it is possible to calculate the magnetic vector potentials {A x} and the conductor voltages {U c}. This can be achieved by coupling the equations (6) and (7) in a unique equation that can be easily solved in the frequency domain [3], [7]. It gives, [)] {6} [4] [! ]#$/ + :(2=6) - [7 & ] 0 /?@A B C A%D & C E = F 0 {+B} G (8) Where the vector of magnetic potentials {B }, the conductor voltages {%D & } and the conductor currents {+B} are all harmonic variables defined for frequency f. The equation (8) can be represented as, ([H] + :(2=6)[I]){JK} = A6BC (9) Moreover, (9) can be represented in a simpler way, i.e. []{JK} = {LK} (10) The equation (10) is a normal FEM matrix expression. It is possible to derive a simpler equation from (10) [4]. This allows to express (10) in terms of its time varying variables, e.g. the vector of magnetic potentials of the conductors [4]. The equation is of lesser order than (9) and can be also solved in the frequency domain. The can be represented by, [ M ]{JK M } = {LK M } (11) The equations shown in (10) and (11) have a preprocessing step, where their matrices are formed by a finite element analysis, and by a calculating step in which their solution in the frequency domain is derived. These stages will be discussed next. 4 Solution of the Normal and the Reduced FEM Equations in the Frequency Domain The FEM matrix equation to be solved are the normal (10) and the reduced types (11). For both equations can be recognize two specific steps in the process of calculating

3 their solution in the frequency domain, i.e. a preprocessing and a calculating steps, respectively. These stages will be explained next. 4.1 Preprocessing Step of the FEM Equations The preprocessing step of the normal FEM method consists on deriving the final matrices [K] and [G] and the vector {f} of the (10). The process consists on first calculating the FEM matrices and vectors of one finite element, integrate them into the global matrices and vectors that model the device [2], [3] and apply the required boundary conditions. The preprocessing step of the reduced FEM method consists on deriving sub-matrices and sub-vectors from the final matrices and vectors obtained from the preprocessing step of the normal FEM equation, in order to calculate matrices of lesser order [4]. These FEM matrices permit to formulate a FEM equation of lesser order, which allows to directly solve the time varying variables of the device. The preprocessing step of a normal and a reduced FEM equations can be seen in Fig. 1. Fig. 1 Preprocessing steps of the FEM equations 4.2 Calculating Step of the FEM Equations Once the matrices and vector of the normal and the reduced FEM equations are calculated, it is possible to derive their solution in the frequency domain. The normal and the s have the form of the expressions previously defined in (10) and (11), respectively. It can be seen that these FEM equations have the form of the expression N 2 OAPQ 2 C = ALK 2 C. This matrix equation can be solved by using the LU method. The calculating process of the normal and the reduced FEM equations is performed using the LU method. Thus, the first step consists on performing a decomposition of the matrix [A g] into two matrices [L g] and [U g], respectively. It yields, N 2 O = NR 2 ON% 2 O (12) After having the matrices [L g] and [U g], the solution of [A g]{x g}={b g} can be achieved by triangular decomposition LU; and the normal and reduced FEM equations can be solved. The difference between these equations is the preprocessing step and the order of the FEM matrix equation to be solved by the calculating step. 5 Calculating Process implemented by a Parallel Computing in CUBLAS The calculating process for the normal and the reduced FEM matrix equations are implemented in the CUBLAS computing platform. Some steps of the preprocessing process of the reduced FEM equation can also be implemented by parallel computing. This will be explained next. a) Preprocessing step of the normal FEM equation 5.1 Decomposition LU implemented in CUBLAS Once the complex matrix equation, that corresponds to the normal or the reduced FEM equation, has been formulated, the matrix [A] will be decomposed into the product of matrices [L] and [U]. This can be achieved by using the standard LU decomposition process. This process implies to calculate a pivot located in the main diagonal of [A], performing a modification of the next rows and, finally, eliminating the rows using the Gauss eliminating process. The decomposition process was implemented by a parallel computing in CUBLAS. This process is shown in Fig. 2. b) Preprocessing step of the reduced FEM equation

4 Fig. 2 Decomposition process implemented in CUBLAS equations will be solved in a sequential and a parallel computing platform. 6.1 Device modelled by a Planar Symmetry Assumption It consists on analyzing a T slot-embedded conductor with a copper conductor and an air region in a frequency range of 5Hz to 60Hz with a frequency step of 5Hz. The objective of the example is to analyze how the total source current density J ct of the conductor varies in this frequency range [5]. The source density J ct will be obtained via the calculating process shown in Fig. 3 (b). The FEM model and the geometry of the T conductor is shown in Fig. 3(a). The CUBLAS routines used for the parallel computation ot the LU decomposition, correspond to matrices and vectors composed of single precission complex numbers [8]. Once the matrix [A g] is decomposed int the product of [L g] and [U g], the equation N 2 OAPQ 2 C = ALK 2 C can be easily solved. This will be explained next. Fig. 3 Device with a planar symmetry assumption 5.2 Final Solution achieved by CUBLAS After having the matrixes [L g] and [U g], the solution APQ 2 C can be calculated by solving the next equations in the CUBLAS computing platform, NR 2 OASQ 2 C = ALK 2 C (13) N% 2 OAPQ 2 C = ASQ 2 C (14) a) Geometry and FEM model Equation (13) is solved by using the routine cublascstrv, and specifying that the equation to be solved corresponds to a triangular matrix stored in lower mode [8]; while (14) is also solved using the routine, but specifying that the equation to be solved corresponds to a triangular matrix stored in upper mode [8]. It can be seen that the solution of the complex equation N 2 OAPQ 2 C = ALK 2 C can be easily derived by implementing the LU method by a parallel computing in CUBLAS. The results and the performance of this method of solution were tested for the case study described next. 6 Case Study It consists on analyzing in the frequency domain two devices modelled by a planar and the axisymmetric symmetry assumption. The first device to be analyzed is a T planar conductor. The second device is an air series reactor that can be modelled by an axisymmetric symmetry assumption. The finite element analysis to be performed on these devices involves the solution of the normal and the reduced FEM equations, which have the form of the expressions shown in (10) and (11), respectively. These b) Calculating process of the device 6.2 Device modelled by an Axisymmetric Symmetry Assumption It consists on analyzing in the frequency domain, a small air-cored reactor [6]. The example consists on finding how the reactor inductance ratio (R L=L ca/l cd) varies within a

5 frequency range [6], defined from 20Hz to 1000Hz with a frequency step of 20Hz. L ca is defined as the inductance obtained at a specific frequency; and L cd is the inductance in a near to zero frequency. Here, the inductance ratio will be obtained via the calculating process shown in Fig. 4(b). The FEM model and the geometry of the series reactor is shown in Fig. 4(a). Table I. FEM equations to be solved in a frequency range Device analyzed Planar symmetry No. FEM Eqs 14 Normal FEM equation [ TT ]{PQ TT } = ALK TT C Reduced FEM equation N M,VW OAPQ M,VW C = ALK M,VW C Fig. 4 Device with an axisymmetric symmetry assumption Axisym. symmetry 51 [ XWV ]{PQ XWV } = ALK XWV C N M,$YV OAPQ M,$YV C = ALK M,$YV C a) Geometry and FEM model In order to measure the performance of the method implemented in CUBLAS, the normal and the reduced FEM equations were also solved in a sequential computing platform. Specifically, the LU routines included in the GSL computing platform [9]. In the sequential form of the solution, the preprocessing and the calculating steps were entirely implemented in the GSL platform [9]. For the parallel solution, some stages of the preprocessing step were calculated by a sequential computing in GSL [9], while the calculating steps were completely implemented in the CUBLAS computing platform [8]. Thus, the calculating step of the normal and the reduced FEM equation will be solved for each frequency by the LU method implemented in the CUBLAS. Table II and III describe the specific routines that are used for the sequential and the parallel solutions of the normal and the reduced FEM equations, respectively. Table II. Routines used in the sequential form of solution of the FEM Equations Normal FEM Reduced FEM Stage Equation Equation Preprocessing Step Normal preprocessing step C routines, GSL matrix routines b) Calculating process of the device 6.3 Methods of Solution of the FEM equations The two devices will be solved by the normal and the reduced FEM equations, which have the form of the expressions defined in (10) and (11), respectively. The dimensions and features of both, normal and reduced FEM equations, are listed in Table I. Please notice that the FEM equations of each device are required to be solved several times for the respective frequency range. Deriving Submatrixes for the Not applied Calculating final matrixes for the Not applied Calculating Step Forming equation LU Decomposition [] = [R][%] Solving equation GSL matrix routines gsl_blas_dgemm gsl_blas_dgmev GSL matrix routines gsl_linalg_complex_lu_decomp gsl_linalg_complex_lu_solve

6 Table III. Routines used in the parallel form of solution of the FEM equations Normal FEM Reduced FEM Stage Equation Equation Normal preprocessing step Deriving Submatrixes for the Calculating final matrixes for the Forming equation LU Decomposition [] = [R][%] Solving equation Preprocessing Step C routines, GSL matrix routines Not applied Not applied Calculating Step GSL matrix routines (Matrix inverse calculated using routine defined in [10]) cublassgemm cublassgemv CUBLAS matrix routines See Fig. 2 cublasctsv: ([R]{SQ} = {LK}) ([%]{PQ} = {SQ}) The computing times obtained from solving the normal and the reduced FEM equations in the sequential and the parallel form of solution, it will be shown in the next section. 6.4 Results and Performance Comparison It is important to mention that the results obtained from the solution of the planar and axisymmetric problems, were validated and compared against simulations performed with ANSYS in the frequency domain. The results derived by the normal and the reduced FEM equations are accurate and validate the proposed parallel form of solution of both equations. The normal and the reduced FEM equations were solved in the computing platforms GSL and CUBLAS. The programs were implemented in the same computer and operative system. A Dell Precision R5500 Rack Workstation, GPU NVIDIA Quadro 600, 1 GB RAM and an Ubuntu Operative System were used. The total computation time (CPU time) required to solve the devices with planar and axisymmetric symmetries in the correspondent frequency range was measured. Fig. 5 illustrates the CPU times needed to solve these equations using the sequential and the parallel computing platforms. Fig. 5. CPU times derived for the FEM equations solutions a) CPU time derived for the planar device b) CPU time derived for the axysimmetric device For the case of the device with a planar symmetry, it can be observed that the reduced FEM equation allows to derive a faster solution compared to the normal FEM equation solution. Specifically, when the sequential computing was used, the CPU time of the normal and the are 1.92sec and 0.89sec, respectively. Moreover, when the parallel computing was used, the CPU time of the normal and the are 6.36sec and 0.90sec, respectively. Although the reduced FEM equation allows a faster solution with both computing platforms to be achieved, a reduction of CPU time was not obtained when parallel computing with CUBLAS was used. The reason being is that the reduced and the normal equations of the planar device are of low order, i.e. 205 and 266, respectively. A CPU time reduction cannot be achieved, since the advantage of using the parallel platform is only evident when the size of the equations to be solved is really huge. For the specific case of the device with an axisymmetric symmetry, it can be observed that the reduced FEM equation also permits to derive a faster solution compared to the normal FEM equation solution. For example, for sequential processing, the CPU time of the normal and the are sec and

7 763.89sec, respectively. Moreover, when parallel computation was used, the CPU time of the normal and the were sec and sec, respectively. It can be seen that the parallel processing of the reduced FEM equation requires of only sec. The sequential computation of a normal FEM equation requires a CPU time of sec. The difference between these CPU times is really significant, nearly 6760%. The reason is that the reduced and the normal equations of the planar device are of higher order, i.e and 1270, respectively. 7 Conclusions A method of solution of a FEM equation, using the LU method implemented in the CUBLAS computing platform has been proposed. It has the following advantages: 1) It can be used to solve a normal and a reduced FEM equation that models devices that can be simplified by a planar or an axisymmetric symmetry assumption. 2) Its solution has been compared against a sequential computing platform. It has allowed a significant reduction of computer effort, as compared to the sequential solution, which was implemented by using the LU routines included in the GSL platform. 3) It allows a significant time reduction when the reduced FEM equation is solved. A significant reduction of CPU time to solve larger order FEM equations sets in the frequency domain has been obtained. The CPU time for solving this equation using CUBLAS is times lesser, than the time required for solving the normal FEM equation with GSL. The parallel solutions of the normal and the reduced FEM equations have been successfully tested for a case study where a finite element analysis has been used to analyze planar and axisymmetric devices. The results derived by the parallel and the sequential solutions of these FEM equations have been against those obtained by finite element simulations performed in ANSYS in the frequency domain. An excellent agreement between the results obtained with both approaches has been achieved. A significant time reduction has been achieved with the application of CUBLAS platform for solving the FEM equations in the frequency domain. For the specific case of the device modelled by an axisymmetric symmetry assumption, it has been obtained a CPU time of s which is a significant small time, compared with the CPU time of s, which was derived by the sequential solution with GSL. [2] Bianchi N., Electrical Machine Analysis Using Finite Element Method, 1st ed. Taylor & Francis Group, Boca Raton, [3] A. Arkkio, Analysis of Induction motors based on the numerical solution of the magnetic field and circuit equations, Doctoral thesis, Acta Polytechnica Scandinavica, Electrical Engineering Series, no. 59, September [4] R. Dominguez, A. Medina, A Novel Method for the Solution of a Finite Element-Circuit Coupled Equation using a Reduced Equivalent Equation, Proceedings of the 2013 IEEE International Electric Machines and Drives Conference, pp , May [5] A. Konrad, Integrodifferential Finite Element Formulation of Two-Dimensional Steady-State Skin Effect Problems, IEEE Transactions On Magnetics, vol. mag-18, No.1, pp , January [6] K. Preis, A Contribution to Eddy Current Calculations in Plane and Axisymmetric Multiconductor Systems, IEEE Transactions On Magnetics, Vol. 19, No.6, pp , November [7] S.L. Ho, H.L. Li, W.N. Fu, Inclusion of Interbar Currents in a Network-Field Coupled Time-Stepping Finite- Element Model of Skewed-Rotor Induction Motors, IEEE Transactions on Magnetics, vol. 35, No.5, pp , September [8] CUDA Toolkit 5.0 CUBLAS Library, Nvidia Corporation, 2701 San Tomas Express Way Santa Clara CA, pp , [9] GNU Scientific Library, GNU Project, [10] R. Cisneros-Magaña, A. Medina, V. Dinavahi, Parallel Kalman Filter Based Time-Domain Harmonic State Estimation, Proceedings of the North American Power Symposium (NAPS), pp. 1-6, September References [1] X. Wang, D. Xie, Analysis of Induction Motor Using Field-Circuit Coupled Time-Periodic Finite Element Method Taking Account of Hysteresis, IEEE Transactions on Magnetics, Vol. 45, No.3, pp , March 2009.