We are concerned with the computation of magnetic fields from known electric currents in the finite element setting. In finite element eddy current simulations it is necessary to prescribe the magnetic field (or potential, depending upon the formulation) on the conductor boundary. In situations where the magnetic field is due to a distributed current density, the Biot-Savart law can be used, eliminating the need to mesh the nonconducting regions. Computation of the Biot-Savart law can be significantly accelerated using a low-rank QR approximation. We review the low-rank QR method and report performance on selected problems

Fasenfest, B. J.

White, D. A.

In this paper we present a low-rank QR method for evaluating the discrete Biot-Savart law. Our goal is to develop an algorithm that is easily implemented on parallel computers. It is assumed that the known current density and the unknown magnetic field are both expressed in a finite element expansion, and we wish to compute the degrees-of-freedom (DOF) in the basis function expansion of the magnetic field. The matrix that maps the current DOF to the field DOF is full, but if the spatial domain is properly partitioned the matrix can be written as a block matrix, with blocks representing distant interactions being low rank and having a compressed QR representation. While an octree partitioning of the matrix may be ideal, for ease of parallel implementation we employ a partitioning based on number of processors. The rank of each block (i.e. the compression) is determined by the specific geometry and is computed dynamically. In this paper we provide the algorithmic details and present computational results for large-scale computations

White, D

Fasenfest, B

UNT Digital Library

UCRL-JRNL-225446Performance of low-rank QRapproximation of the finiteelement Biot-Savart lawD. White, B. FasenfestOctober 20, 2006IEEE Transactions on MagneticsDisclaimer   This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.  1Performance of low-rank QR approximation of thefinite element Biot-Savart lawDaniel A. White and Benjamin J. FasenfestAbstract—In this paper we present a low-rank QR method for evaluatingthe discrete Biot-Savart law. Our goal is to develop an algorithmthat is easily implemented on parallel computers. It is assumedthat the known current density and the unknown magnetic fieldare both expressed in a finite element expansion, and we wish tocompute the degrees-of-freedom (DOF) in the basis function ex-pansion of the magnetic field. The matrix that maps the currentDOF to the field DOF is full, but if the spatial domain is prop-erly partitioned the matrix can be written as a block matrix, withblocks representing distant interactions being low rank and hav-ing a compressed QR representation. While an octree partitioningof the matrix may be ideal, for ease of parallel implementation weemploy a partitioning based on number of processors. The rank ofeach block (i.e. the compression) is determined by the specific ge-ometry and is computed dynamically. In this paper we provide thealgorithmic details and present computational results for large-scale computations.Index Terms—Maxwell’s equations, Biot-Savart law, eddy currents, electro-magnetic diffusion.I. INTRODUCTIONThe computation of magnetic fields from a prescribed elec-tric current is a common problem in magnetic design and anal-ysis. One approach is to formulate the problem as a Partial Dif-ferential Equation (PDE) for the unknown field with the pre-scribed electric current as the source term. Regardless of theparticular PDE formulation, e.g. a magnetic vector potentialformulation or a mixed B-H formulation, a large volumetricmesh must be employed, and some boundary condition mustbe applied on the outer boundary of the mesh. In contrast tothe PDE approach, the Biot-Savart law can be employed to di-rectly compute the magnetic field due to the prescribed current[1]. The advantage of the Biot-Savart law approach is that a fullvolume mesh is not required, and no boundary conditions needbe applied. The disadvantage of the Biot-Savart approach isthe computational cost, if there are O(N) magnetic field obser-vation points and O(M) current samples the cost is O(N ∗M).In this paper we review a fast low-rank QR method for com-pressing the M×N Biot-Savart matrix. The approach is similarto low-rank QR methods developed for boundary element elec-trostatics [2] [3] and for low frequency electric field integralequations [4]. The key difference with our approach is that weare concerned with volumetric current densities and implemen-tation on parallel computers.This work was performed under the auspices of the U.S. Department of En-ergy by the University of California, Lawrence Livermore National Laboratoryunder contract No. W-7405-Eng-48,Defense Sciences Engineering Division, Lawrence Livermore National Lab-oratory, white37@llnl.govII. FORMULATIONThe law of Biot and Savart is given byB(x) = ∇×A= µ4πZΩ′J(x′)× (x− x′)|x− x′|3 d3x′. (1)where J(x′) is the known current density at the source point x ′,and B(x) is the desired magnetic flux density at the observationpoint x. We assume that we have a finite element representationfor J over the volumeΩ ′, and a finite element representation forB over a surface Γ,J =N∑i=1ξiW 2i ,B =M∑j=1β jW 1j , (2)where ξ j and β j are the ith degree-of-freedom (DOF), and W 2iand W 1j are vector basis functions. Inserting the basis functionexpansions (2) into (1) yields the discrete Biot-Savart lawM¯β = Z¯ξ, (3)whereZi j =ZΓZΩ′µ4πW 2i (x′)× (x− x′) · W1j (x)|x− x′|3 dΩ′ dΓ, (4)andMi j =ZΓW 1i (x) · W 1j (x) dΓ (5)and where ¯ξ and ¯β are the arrays of DOF. The matrix M is a“mass matrix” due to the fact that the basis functions are notorthogonal. The mass matrix is extremely sparse and the com-putational cost for forming and solving this matrix is negligible.In many applications the problem of determining the B-field canbe posed in terms of the magnetic vector potential A = ∇×BwithA(x) =µ4πZΩJ(x′)|x− x′|d3x′. (6)Using a finite element representation for A yields another ver-sion of the discrete Biot-Savart lawJ =N∑i=1ξiW 2i ,A =M∑j=1α jW 1j , (7)Mα¯= Y¯ξ, (8)whereYi j =ZΓZΩ′µ4πW 2i (x′) · W1j (x)|x− x′| dΩ′ dΓ. (9)2We will refer to the M×N matrices Z and Y as Biot-Savart ma-trices. The computation of these matrices involves singular andnear-singular integrals. The surface integration is performedusing standard Gaussian quadrature points for each surface ele-ment. The volume integration uses an adaptive integration rule,which varies the order of Gaussian quadrature based on the dis-tance between the source point x ′ and the observation point x.When the surface element containing x is a face of the volumeelement containing x′, a highly accurate height-based singular-ity cancellation quadrature rule is used [5]. The matrices (4)and (9) are constructed using 2-form or “face elements” for thebasis functions W 2 and 1-form or “edge elements” for the basisfunctions W 1, see [6] for details on the construction of the basisfunctions.Our primary application for the discrete Biot-Savart law isproviding boundary conditions for finite element solution ofmulti-conductor eddy current problems. In each conductor wesolve the time-dependent vector diffusion equation using anedge element based A-φ finite element formulation [7]. Clearlythe B-field in the air surrounding the conductors is critical. Thefinite element formulation requires that either nˆ× A or nˆ× Bbe specified on the conductor boundaries, corresponding to in-homogeneous Dirichlet or Neumann boundary conditions, re-spectively. Our approach for dealing with the B-field in the airsurrounding the conductors is to use the discrete Biot-Savartlaw (3) or (8) as the boundary condition on each conductingsurface.III. PARALLEL IMPLEMENTATIONWe assume that the volume Ω has been partitioned into Kpartitions, where K is the number of computational proces-sors, with each partition having an equal number of volumeelements. The volume elements are distributed via the parti-tioning. The surface Γ is also partitioned into K equally sizedsurface partitions. Note however that the surface elementsare not distributed via the surface partitions, each processorcan access the entire surface mesh. The Biot-Savart matrix isthen decomposed into a K ×K block matrix, with every blockZpq, p ∈ {1 : K},q∈ {1 : K} representing the interaction of sur-face partition Γp with volume Ωq. The qth processor computesblocks Z pq, p = 1 : K, i.e. a column of blocks. Note that thematrix is decomposed via a partitioning of elements, hence thematrices Z pq are overlapping in DOF space. The specific par-titioning algorithm used to partition the elements is not criti-cal, in the examples below we employ a graph-based algorithm[8]. The key point is that if the partitions Γ p and Ωq are well-separated then the sub-matrix Z pq will have a low-rank QR de-composition. The procedure for computing the low-rank QRdecomposition is described below. We define “well-separated”as follows: the bounding spheres for the element partitions Γ pand Ωq are computed, if the bounding spheres do not intersectthen the partitions are considered well-separated and a low-rankQR representation of Z pq is computed. We employ a recursiveprocedure for computing Z pq when partitions Γp and Ωq arenot well-separated. This results in a hierarchical representa-tion for Z. If Γp and Ωq are not well separated, Ωq is dividedinto eight equally sized sub-partitions, Γ p is divided into fourequally sized sub-partitions, and the “well-separated test” isapplied to the sub-partitions Γ pi and Ωq j, i = 1 : 4, j = 1 : 8.A space-filling curve algorithm is used for creating the sub-partitions. The process is applied recursively, with a low-rankQR representation computed for well-separated sub-partitions.The recursion is halted when a volume sub-partition containsfewer than some number of elements, for example 64 elements.If at the lowest level of recursion the interaction is not wellseparated, it is simply represented by a dense matrix. This isillustrated in Figure 1.No parallel communication is required in the construction ofthe hierarchical Biot-Savart matrix, each processor has the el-ements that it needs to perform the integrals. Each processorhas the same amount of work hence the computation of is loadbalanced. Note, however, that in the low-rank QR approxima-tion the rank k is computed dynamically, and the rank k de-pends upon the geometry. Hence the application of the hierar-chical Biot-Savart matrix, i.e. the matrix-vector multiplication¯β= Z¯ξ, may not be perfectly load balanced. Also note that theapplication of the hierarchical Biot-Savart matrix does requireparallel communication. This communication is as follows: (1)each processor q does a gather operation to get the values of ¯ξthat it needs, (2) each processor q loops over the sub-matricesZpq, p = 1 : K and computes ¯βq = Zpq ¯ξq, (3) each processorparticipates in a global reduction on ¯βq.Fig. 1. Hierarchical partitioning of the Biot-Savart matrix. The highest level ofpartitioning is based on the number of processors. Some of the the interactionsat any Level l will be full rank (black boxes), and these interactions are sub-partitioned by decomposing the corresponding sub-volume and sub-surface tocreate Level l+1.IV. LOW-RANK QR DECOMPOSITIONWhen Γp and Ωq are well separated the matrix Z pq will havea low-rank representationZpqm×n ≈ Qm×k ×Rk×n, (10)where k is the rank. We do not want to form the entire Z pqand then compress it, rather we sample the matrix by pickings rows and columns of Z pq, where s is some predeterminednumber based on an estimate of the rank. The procedure forpicking the sampled rows and columns is ad-hoc, the procedurethat we employ is described in [4]. The sampling procedureis solely linear algebra, it does not depend upon the particularGreen’s function, finite element basis functions, etc. For the ad-hoc sampling procedure to be effective we must have s greater3than the expected rank. The algorithm for computing Q m×k andRk×n is as follows:1: Form the sampled column matrix Scm×s and the sam-pled row matrix Srs×n.2: Compute the rank-revealing QR decomposition˜Qm×s ˜Rs×s = Scm×s using LAPACK routines DGEQPFand DORGQR. The rank k is determined by the crite-ria ˜Rkk < thresh· ˜R11 where thresh is a threshold value.Keep only k columns of ˜Q, denote this as Qm×k, anddiscard ˜R.3: Form a new matrix ˆQs×k by taking s rows of Qm×k,the exact same rows as used to construct Sr.4: Compute the least-squares solution to ˆQs×kRk×n =Srs×n using LAPACK routine DGELSS.At this point we have the desired matrices Qm×k and Rk×nwhich approximate Z pqm×n. To perform a matrix-vector mul-tiplication with the compressed matrix it is necessary to in-clude the permutations due to the column and row sampling,Zpqm×n ≈ Pc ×Qm×k ×Rk×n ×Pr, where Pr and Pc are permuta-tion matrices. The quality of the approximation, and the amountof compression (the rank k), are determined by the value ofthresh used in Step 2 above. Our approach, being based onhighly tuned LAPACK routines, is efficient both in terms ofFLOPS and memory usage. The complexity of a single QR de-composition is O(m ·s)+O(s ·n), using a fixed value of s yieldsa linear complexity in m and n.The two key parameters in the QR decomposition are thethreshold used to determine the rank k in Step 2, and the num-ber of sampled rows and columns s in step 1. In practice wehave found 0.01 < thresh < 0.001 to yield acceptable results,meaning the the error in the QR compression is less than the in-trinsic error of the finite element computation. Of course this isapplication dependent. For the parameter s we use a table look-up, where the argument is the normalized distance between thevolume and surface regions. This normalized distance is de-fined as the distance between the centroids of the two boundingspheres divided by the sum of the radii of the spheres, a valueof d = 1 means that the two bounding spheres are just touching.A table of computed ranks vs distance, generated by running adozen different problems, is shown in Figure 2.2 4 6 8 10510152025303540Fig. 2. Rank vs distance. The Black curve is the maximum rank, which isused to determine s. The Gray curve is the minimum rank.V. EXAMPLESIn these examples we compute a hierarchical low-rank QRapproximation of the matrix defined by Equation (9). For thefirst example consider the geometry shown in Figure 4. Thisgeometry consists of 19000 volume elements and is partitionedfor 16 processors. Therefore the Biot-Savart matrix will bea 16× 16 block matrix. Each block Z pq has roughly 1200rows and 4000 columns. Using a value of thresh = 0.005gives the parallel rank map shown in Figure 3. The compres-sion is significant, each 1200× 4000 matrix is compressed toQ1200×k+Rk×4000 where k is the value shown in Figure 3. Notethat the black blocks represent near interactions and have fullrank. These near interaction blocks were decomposed furtheras explained in Section III above. The total compression was60× for this specific example.2 4 6 8 10 12 14 16246810121416Fig. 3. Rank map for the highest level 16 x 16 partitioning of the three coilproblem. Black=full rank, Dark Gray = 20 < k < 30, Light Gray = 10 < k < 21,White = k < 11.The second example is shown in Figure 5. The geometryconsists of three conducting coils, the center coil is driven withan independent current source, and we wish to compute theeddy currents in the coils due to the B-field in the surround-ing air. The problem consists of 20736 volume elements andwas partitioned for 24 parallel processors, therefore the Biot-Savart matrix is a 24×24 block matrix. Each processor had 24matrices to compute at the highest level, on average 19 of thesecorresponded to well-separated regions and were compressedwith an average rank of 10. The remaining 5 full-rank matri-ces were further partitioned into 4 · 8 = 32 sub-matrices, andon average 29 of these corresponded to well-separated regionsand were compressed with an average rank of 25. At the lowestlevel of the hierarchy, the near interactions were represented,on average, by dense matrices of dimension 335× 439, therewere a total of 24 ·3 = 72 of these. The total compression was109×. This compression represents both the memory savingsand the reduction in CPU time required to apply the Biot-Savartinteraction.We performed a scalability study by refining the computa-tional mesh. As the mesh is refined, more processors were usedto keep a constant number of mesh elements per processor. Theresults are shown in Figure 6, with a Nlog2(N) performance4for the memory usage. The CPU time for applying the com-pressed Biot-Savart matrix at every time step is proportionalto the memory usage. The CPU time for computing the com-pressed Biot-Savart matrix is amortized over many thousandsof time steps and is not dominant. Note that in Figure 6 the datafor the uncompressed case is theoretical, since the memory us-age was prohibitive.Fig. 4. Computational mesh for a linear induction motor partitioned for16 parallel processors. This partitioning gave a compression o 60×.Fig. 5. Computational mesh for an inductive coupling application parti-tioned for 24 parallel processors. This partitioning gave a compression of109×.VI. CONCLUSIONSHierarchical low-rank QR compression of far interactionssignificantly decreases the cost of applying a finite element103 104 105 10610610710810910101011Size NMemory Usage  UncompressedCompressedFig. 6. Compression of the hierarchical low-rank QR algorithm. The y-axis iswords, the x-axis is number of volume elements, on a log scale. The solid lineis 50 ·Nlog2(N) for comparison.Biot-Savart law. The advantage of the proposed algorithm,which uses the parallel partitioning of the volume elements asthe first level of the hierarchy, is that no parallel communica-tion is require in the QR computation. Ideally the computa-tional cost of computing the far-interactions will be less thencost of computing the near interactions, resulting in an O(N)algorithm since the cost of the near interactions is O(N). But itcan be shown [3] that the cost of the far interactions will growas O(Nlog2(N), and this was confirmed by our experiments.In addition, for a given mesh the performance will degrade asthe number of processors p is increased. Given a volume par-tition of n elements and a surface partition of m elements, theQR work is n · k+m · k. If each of these partitions is cut intwo (by doubling the number of processors p) the QR work is4(n/2 · k+m/2 · k), the work is increased by a factor of 2.VII. ACKNOWLEDGMENTThis work was performed under the auspices of the U.S. De-partment of Energy by the University of California, LawrenceLivermore National Laboratory under contract No. W-7405-Eng-48.REFERENCES[1] J. D. Jackson. Classical Electrodynamics. 1962.[2] S. Kapur and D. Long. IES3: Efficient electrostatic and electromagneticsolution. IEEE Comp.. Sci. Eng., 5(4):60–67, 1998.[3] D. Gope and V. Jandhyala. PILOT: A fast algorithm for enhanced 3d para-sitic capacitance extraction. Micro. Opt, tech. Lett., 41(3):169–173, 2004.[4] D. Gope and V. Jandhyala. Efficient solution of EFIE via low-rank com-pression of multilevel predetermined interactions. IEEE Trans. Ant. Prop.,53(10):3324–3333, 2005.[5] M. A. Khaya and D. R. Wilton. Numerical evaluation of singular and near-signular potential integrals. IEEE Trans. Ant. Prop., 53(10):3180–3190,2005.[6] P. Castillo, J. Koning, R. Rieben, and D. White. A discrete differentialforms framework for computational electromagnetics. Computer Modelingin Engineering & Sciences, 5(4):331–346, 2004.[7] R. Rieben and D. White. Verification of high-order mixed FEM solution oftransient magnetic diffusion problems. IEEE Trans. Mag., October 2005.article in press.[8] G. Karypis and V. Kumar. A parallel algorithm for multilevel graph parti-tioning and sparse matrix ordering. J. Parallel Distr. Comp., 48(1):71–95,1998.

Performance of low-rank QR approximation of the finite element Biot-Savart law

UCRL-CONF-218238Performance of low-rank QRapproximation of the finiteelement Biot-Savart lawD. A. White, B. J. FasenfestJanuary 18, 200612th Biennial IEEE Conference on Electromagnetic FieldComputationMiami, FL, United StatesApril 30, 2006 through May 3, 2006Disclaimer   This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.  5(b) Abstract— We are concerned with the computation of magnetic fields from known electric currents in the finite element setting. In finite element eddy current simulations it is necessary to prescribe the magnetic field (or potential, depending upon the formulation) on the conductor boundary. In situations where the magnetic field is due to a distributed current density, the Biot-Savart law can be used, eliminating the need to mesh the non-conducting regions. Computation of the Biot-Savart law can be significantly accelerated using a low-rank QR approximation. We review the low-rank QR method and report performance on selected problems. I. INTRODUCTION There are many different finite element formulations for transient eddy current simulation. In this paper we employ a magnetic vector potential formulation as described in [1]. In order to completely specify the problem it is necessary to apply boundary conditions, and for this particular formulation the boundary conditions are either the inhomogeneous Dirichlet condition bcn A A× =  or inhomogeneous Neumann condition bcn B B× = . These boundary conditions on the conductor of interest can be computed using the Biot-Savart law, 4 bcJ dV ARµπ Ω=∫  or 34 bcJ R dV BRµπ Ω× =∫ . The QR procedure is independent of the kernel and can be applied to either law, for brevity we will focus on the former equation that gives Abc. Applying the Galerkin procedure and employing the basis function expansions ( ) ( )bc i iA r W rα=∑  and ( ) ( )i iJ r c N r=∑  yields the discrete Biot-Savart law =Mα Kc , where the bulk of the computational effort is in computing and applying the dense matrix K, where K is given by II. LOW-RANK QR APPROXIMATION The matrix K is compressed using a low-rank QR decomposition. Consider the geometry shown in Figure 1, where the mesh has been decomposed into 24 partitions using a graph-based algorithm [2]. The K matrix is then a block 24 ×  24 matrix. Most blocks represent distant interactions and hence have a low-rank QR decomposition. The low-rank QR decomposition is computed by first selecting r rows and columns of the block, where r is the rank, and applying the modified Gram-Schmidt procedure [3] to compute Q and R. The rank r is adaptive, based on an error tolerance. The row and column selection procedure is somewhat ad-hoc and is a modification of that used in  [4] for electrostatic applications. Blocks of K that represent near interactions are not low-rank, however these blocks can be sub-partitioned, and the low-rank QR decomposition can be applied to the low-rank sub-blocks. This process is applied recursively until the size of the partitions decreases below an a-priori threshold, e.g. 50 elements.  The mesh in Figure 1 has 20736 volume elements. Using the recursive low-rank QR compression, with only 2 levels of recursion, resulted in a compression of 109×  for this specific problem. The user-specified error tolerance for this specific example was 10-3 and the rank of the sub-blocks varied from 4 < r < 25. It is important to note that as the computational mesh is refined, that rank r does not increase, hence the method asymptotically approaches O(n)log(n).  III. REFERENCES [1] R. Rieben, D. White, “Verification of high-order mixed finite element solution of transient magnetic diffusion problems,” IEEE Trans. Magnetics, v. 42, n. 1, pp 25-39, 2006. [2] G. Karypis, V. Kumar, “A parallel algorithm for multilevel graph partitioning and sparse matrix ordering,” J. Parallel Distr. Comput., 48(1), pp 71-95, 1998. [3] G. Golub, C. Van Loan, Matrix Computations, Johns Hopkins University Press, 2nd Ed.,1993. [4] D. Gope and V. Jandhyala, “PILOT: A fast algorithm for enhanced 3D electrostatic parasitic capacitance extraction,” Micro. Opt. Tech. Lett., 41(3), pp. 169-173, 2004.  Performance of Low-Rank QR Approximation of the Finite Element Biot-Savart Law Daniel A. White and Benjamin J. Fasenfest Lawrence Livermore National Laboratory L-154, 5000 East Ave, Livermore CA USA dwhite@llnl.gov ( ) ( )4k jN r W rK dr drkj r rµπ′ •′= ∫ ∫ ′−Γ ΩFigure 1. A finite element mesh of 3 coils, partitioned into 24 partitions for QR compression. 

https://digital.library.unt.edu/ark:/67531/metadc882191/m2/1/high_res_d/907870.pdf

Performance of low-rank QR approximation of the finite element Biot-Savart law

Abstract

Similar works

Full text

Available Versions

UNT Digital Library

UNT Digital Library