Strategies for Accelerating Nonlinear Convergence for T

IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010

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Strategies for Accelerating Nonlinear Convergence for T- Formulation Ping Zhou, Dingsheng Lin, Bo He, Sameer S. Kher, and Zoltan J. Cendes, Fellow, IEEE Ansoft, LLC, Pittsburgh, PA 15219 USA This paper details the derivation of the Jacobian matrix and the residual vector associated with the Newton–Raphson iteration sequence in terms of the - formulation. Then, a scheme is proposed to efficiently find the optimum relaxation factor for improving global convergence. Furthermore, to address some local convergence issues, a local damping factor that damps the updating of the nonlinear material property for the evaluation of Jacobian matrix during nonlinear iteration is introduced.

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Index Terms—Jacobian matrix, Newton–Raphson method, nonlinear convergence, transient finite element analysis (FEA), formulation.

I. INTRODUCTION

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HE Newton–Raphson technique for the solution of nonlinear problems in finite element analysis (FEA) has been widely accepted due to its quadratic convergence characteristics [1]. However, for highly nonlinear problems, the Newton sequence with an arbitrary initial guess may converge at a very slow rate, or oscillate, or even diverge. It is also known that convergence is more difficult with the magnetic scalar potential than with magnetic vector potential [2]. Hence, underrelaxation is commonly used to improve convergence. If an optimum relaxation factor, which minimizes the total square of the 2-norm of the residual obtained from the finite element discretization, is introduced at each step of nonlinear iteration, a convergent solution can always be obtained using a linear search algorithm [3], [4]. However, it may take a very long time to find optimum relaxation factor because a large number of repeated evaluations for the residual are required. [5] proposed a method to reduce the computation cost for determining the optimum relaxation factor by using linearization based on two values of the residual computation. However, the assumption that the square of the 2-norm of the residual changes quadratically with the relaxation factor is far from reality for most practical applications. In addition, it is understandable that the use of the optimum relaxation factor does not guarantee fast convergence and good efficiency because the “optimum” is only in the global sense and thus may not be suitable for some of the variables (elements). This may lead to poor local convergence due to large overshoot correction or possible oscillation. It was observed that even with only a few such “bad” elements, there was a significant impact on the convergence rate of the Newton sequence, particularly in the case of coarse mesh and large field gradient in nonlinear materials. This undesirable local convergence behavior may also have a significant impact on solution accuracy, such as in the case of considering induced eddy current in the nonlinear region. In such a case, a poorly converged solution in

Manuscript received December 11, 2009; accepted February 02, 2010. Current version published July 21, 2010. Corresponding author: P. Zhou (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2043508

T-

those “bad” elements can directly produce unphysical eddy current, and during a transient analysis, the numerical errors will propagate and accumulate with time. In this paper, the Jacobian matrix and the residual vector are formulation. Then, a method first derived in terms of the is proposed to efficiently find the optimum relaxation factor for improving global convergence performance. Furthermore, to address the local convergence issue, a local damping scheme is introduced. This scheme damps the updating of the nonlinear material property for a small portion of elements that exhibit the largest changes in the equivalent dynamic permeability. Finally, the effectiveness of the approach is demonstrated on a benchmark example. II. NEWTON NONLINEAR - FORMULATION In transient - formulation, the basic field equations are (1) , and is the source field compowhere nent. For the sake of conciseness, but without loss of generality, the following derivation will not involve voltage excitation, circuit coupling, and rigid motion [6]. Let us define (2) The corresponding Newton iteration forms are (3) Taking the Frechet partial derivative of (2) yields (4.a) (4.b) (4.c) (4.d) where tensor can be considered as an equivalent dynamic permeability and defined by

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(5)

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In the case of isotropic material,

is expressed as [7] (6)

,

where

, and (7)

with (8)

has off-diagonal It can be seen from (6) that the tensor entries even when and only have diagonal entries. It is these off-diagonal elements that model the physics of the cross effects of magnetic saturation. Next, let and be approximated as

Fig. 1. Flowchart of Newton–Raphson iteration sequence with the use of underrelaxation factor.

(9) III. OPTIMUM RELAXATION FACTOR SEARCHING ALGORITHM where and are scalar and vector basis functions, respectively, defined locally over are the values of at each element. are the values of the mesh nodes, and at the mesh edges. Thus, (3) becomes (10.a) (10.b) By applying the Galerkin method and multiplying both sides in (10.a) by and multiplying both sides in (10.b) by , and then integrating over the problem domain , we get (11) where matrix

,

,

, and

are the blocks of the Jacobian

(12)

and

and

are the residual vector (13)

To consider the material property variations within the element, Gauss quadrature numerical integration is adopted to evaluate the integrals in (12) and (13).

The nonlinearity in the obtained system of (11) arises from the fact that the entries of the obtained global Jacobian matrix and the residual vector are functions of the permeability or the equivalent dynamic permeability assigned to each nonlinear element. These elemental permeabilities are functions of the unknowns to be solved. Therefore, an iterative process is necessary. Fig. 1 shows the flowchart of the Newton–Raphson iteration scheme. When the relaxation factor is equal to unity, Fig. 1 represents the ordinary Newton–Raphson method. Underrelaxation is commonly used to achieve convergence or to improve the convergence rate. As the nonlinear iteration converges, the residual for each unknown should approach zero. Therefore, the optimal underrelaxation factor is the one that with minimizes the square of the 2-norm of the residual each iteration. The search for the optimal relaxation factor would normally require significant computation time since a large number of repeated evaluations of the residual function may be required. In order to reduce computation time, the following searching algorithm for the optimal underrelaxation factor is proposed: Let the relaxation factor (between 1 and zero) be equally sampled with step of 0.2 as shown in Fig. 2. Here, corresponds to corresponds the previous Newton iteration solution, and to the ordinary nonlinear Newton iteration scheme. The residuals at these two values of are already available as part of the regular solution process. Thus, the evaluation of the residual . For each sampled value of , functions can start from after the solution candidate is obtained, we compute the field, update the permeability, and compute the residual. If the current residual is smaller than the previous residual, we continue to the next value of . If the current residual is greater, we use the current and the previous two values of to construct a quadratic

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ZHOU et al.: STRATEGIES FOR ACCELERATING NONLINEAR CONVERGENCE FOR

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FORMULATION

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Fig. 3. One-quarter model of steel plates around a coil with measurement positions for flux densities and induced eddy current densities. Fig. 2. Search for the optimal underrelaxation factor.

polynomial and find the optimal underrelaxation factor associated with the minimum residual value. A special case occurs is greater than that at . In when the residual at such a case, can be simply chosen as 1. If for , the corresponding computed residual value still does not increase, the subregion of between 0.2 and 0 is further subdivided with a step of 0.05, and the search is continued. In order to improve efficiency, a large step size of 0.2 is used in the above algorithm. This has the added advantage of making the algorithm less sensitive to local minimum. However, the large step size may lead to a large error in the computed optimal underrelaxation factor. To address this, we can insert two additional points in between the above obtained three points and compute corresponding residuals. This will effectively reduce the step size to 0.1, and the optimal underrelaxation factor will be determined based on the appropriately selected set of three points: the point with the smallest residual and two neighboring points on each side.

when when

(15)

is the total number of nonlinear elements in the solved where domain and is the nonlinear element size threshold. The local damping algorithm is only applied if the element size is greater than this threshold. Our investigation has shown that is a reasonable choice. For the local damping process, the first step is to identify the elements with the highest change rate out of the entire set of nonlinear elements, where is determined by

(16) in the list is considered to be the refNext, the smallest erence damping rate . The ratio of the reference rate to the actual rate for each of the elements is then computed as (17)

IV. LOCAL MATERIAL UPDATE DAMPING ALGORITHM As mentioned above, the use of the optimum relaxation factor does not necessarily guarantee fast convergence and good efficiency because the “optimum” is measured only in the global sense and some elements may still have poor local convergence. To improve convergence for such cases, a local damping factor is introduced to damp the updating of the equivalent dynamic permeability tensor in the process of computing the Jacobian matrix (14) It should be emphasized that this modification should be applied to only a very small percentage of elements (less than 1% of total nonlinear elements) with the highest rate of change in the equivalent dynamic permeability tensor. The small percentage ensures that the convergence rate and the efficiency of Newton–Raphson approach will not be adversely affected. In our investigation with many test cases, we have arrived at the following empirical formula for determining , the number of elements to be damped

Finally, the damping factor for each of the termined by

elements is de-

(18) The value of the local damping factor computed from (18) is approximately between 0.35 and 0.9. The lower bound is used less than 0.35; the upper bound to avoid over damping for is applied to prevent altering of the convergence property of the Newton–Raphson method due to a trivial modification. In fact, , the local damping step can be for any element with simply skipped. V. BENCHMARK EXAMPLE The benchmark problem No. 10 of TEAM Workshop is used here as an example to check the nonlinear convergence and simulation accuracy [8]. The problem consists of an exciting coil that carries time-varying current, placed between two steel channels, and a steel plate inserted between the channels as


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Fig. 4. Comparison of average flux densities over the cross sections S , S , and S with and without using the local damping algorithm.

TABLE I COMPUTATION TIME COMPARISON

shown in Fig. 3. The material of the steel is nonlinear, and the exciting current varies as

.

(19)

The time-varying flux density will induce eddy current in the nonlinear conducting steel. The purpose of this benchmark is to study the time function of the average flux density over the three cross sections , , and of the channels as well as the time functions of the induced eddy current density at the three points , , and in the nonlinear steel. The investigation includes three scenarios: constant underrelaxation factor, the adaptive optimal relaxation factor without local damping, and the adaptive optimal relaxation factor with local damping. In the constant underrelaxation factor case, in order to make the problem converge, the underrelaxation factor must not be greater than 0.1. Table I shows the comparison of the computation time for the three cases using the same mesh of 37 385 tetrahedral elements. Fig. 4 shows the comparison of computed average flux denbetween the results sities over the cross sections , , and with local damping and the without local damping. Fig. 5 compares the induced eddy current densities at the points , ,

Fig. 5. Comparison of the induced eddy current densities at the points P , P , and P with and without using the local damping algorithm.

of the nonlinear conducting steel channels for the cases and with and without local damping. Measurement profiles for both the average flux densities and the induced eddy current densities can be found in [8] and [9]. It can be seen from the two figures that while the local damping algorithm does not show an obvious effect on the computed average flux densities, it does have a significant impact on the much more sensitive induced eddy current densities. Fig. 5 shows that the local damping algorithm has effectively eliminated the unphysical noises that also appeared in other researches’ results [8], [9]. REFERENCES [1] A. Mohammed and N. A. Demerdash, “An extremely fast technique for nonlinear three dimension finite element magnetic field computations,” IEEE Trans. Magn., vol. MAG-23, no. 5, pp. 3575–3577, Sep. 1987. [2] L. Janicke and A. Kost, “Convergence properties of the Newton–Raphson method for nonlinear problems,” IEEE Trans. Magn., vol. 34, no. 5, pp. 2505–2508, Sep. 1998. [3] T. Nakata, N. Takahashi, K. Fujiwara, N. Okamoto, and K. Muramatsu, “Improvements of convergence characteristics of Newton–Raphson method for nonlinear magnetic field analysis,” IEEE Trans. Magn., vol. 28, no. 2, pp. 1048–1051, Mar. 1992. [4] J. O’Dwyer and T. O’Donnell, “Choosing the releaxation parameter for the solution of nonlinear magnetic field problems by Newton–Raphson method,” IEEE Trans. Magn., vol. 31, no. 3, pp. 1484–1487, May 1995. [5] K. Fujiwara, Y. Okamoto, A. Kameari, and A. Ahagon, “The Newton–Raphson method accelerated by using a linear search—Comparison between energy functional and residual minimization,” IEEE Trans. Magn., vol. 41, no. 5, pp. 1724–1727, May 2005. [6] P. Zhou, Z. Badics, D. Lin, and Z. J. Cendes, “Nonlinear T- formulation including motion for multiply connected 3-D problems,” IEEE Trans. Magn., vol. 44, no. 6, pp. 718–721, Jun. 2008. [7] D. Lin, P. Zhou, Z. Badics, W. N. Fu, Q. M. Chen, and Z. J. Cendes, “A new nonlinear anisotropic model for soft magnetic materials,” IEEE Trans. Magn., vol. 42, no. 4, pp. 963–966, Apr. 2006. [8] T. Nakata and K. Fujiwara, “Results for benchmark problem 10 (steel plates around a coil),” COMPEL—Int. J. Comput. Math. Electr. Electron. Eng., vol. 9, no. 3, pp. 181–190, 1990. [9] O. Biro, K. Preis, and K. R. Richter, “Various FEM formulation for the calculation of transient 3D eddy currents in nonlinear media,” IEEE Trans. Magn., vol. 31, no. 3, pp. 1307–1312, May 1995.
