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Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on the second-order backward differentiation formula and the second-order interpolation formula using the information at previous two temporal steps. Unconditional unique solvability of both methods is proved, with their second-order accuracy verified through numerical examples in both 1D and 3D. The efficiency of both methods is compared to that of another two popular methods. In addition, we test the robustness of both methods for the first benchmark problem with a ferromagnetic thin film material from National Institute of Standards and Technology.

A B S T R A C TMicromagnetic simulation is an important tool to study various dynamic behaviors of magnetic order in ferromagnetic materials. The underlying model is the Landau-Lifshitz-Gilbert equation, where the magnetization dynamics is driven by the gyromagnetic torque term and the Gilbert damping term. Numerically, considerable progress has been made in the past decades. One of the most popular methods is the Gauss-Seidel projection method developed by Xiao-Ping Wang, Carlos García-Cervera, and Weinan E in 2001. It first solves a set of heat equations with constant coefficients and updates the gyromagnetic term in the Gauss-Seidel manner, and then solves another set of heat equations with constant coefficients for the damping term. Afterwards, a projection step is applied to preserve the length constraint in the pointwise sense. This method has been verified to be unconditionally stable numerically and successfully applied to study magnetization dynamics under various controls.In this paper, we present two improved Gauss-Seidel projection methods with unconditional stability. The first method updates the gyromagnetic term and the damping term simultaneously and follows by a projection step. The second method introduces two sets of approximate solutions, where we update the gyromagnetic term and the damping term simultaneously for one set of approximate solutions and apply the projection step to the other set of approximate solutions in an alternating manner. Compared to the original Gauss-Seidel projection method which has to solve heat equations 7 times at each time step, the improved methods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time and second-order accuracy in space are verified by examples in both 1D and 3D. In addition, unconditional stability with respect to both the grid size and the damping parameter is confirmed numerically. Application of both methods to a realistic material is also presented with hysteresis loops and magnetization profiles. Compared with the original method, the recorded running times suggest that savings of both methods are about 2/7 and 4/7 for the same accuracy requirement, respectively.

The design complexity and outsourcing trend of modern integrated circuits (ICs) have increased the chance for adversaries to implant hardware Trojans (HTs) in the development process. To effectively defend against this hardware-based security threat, many solutions have been reported in the literature, including dynamic and static techniques. However, there is still a lack of methods that can simultaneously detect and diagnose HT circuits with high accuracy and low time complexity. Therefore, to overcome these limitations, this paper presents an HT detection and diagnosis method for gate-level netlists (GLNs) based on different machine learning (ML) algorithms. Given a GLN, the proposed method first partitions it into several circuit cones and extracts seven HT-related features from each cone. Then, we repeat this process for the sample GLN to construct a dataset for the next step. After that, we use K-Nearest Neighbor (KNN), Decision Tree (DT) and Naive Bayes (NB) to classify all circuit cones of the target GLN. Finally, we determine whether each circuit cone is HT-implanted through the label, completing the HT detection and diagnosis for target GLN. We have applied our method to 11 GLNs from ISCAS’85 and ISCAS’89 benchmark suites. As shown in experimental results of the three ML algorithms used in our method: (1) NB costs shortest time and achieves the highest average true positive rate (ATPR) of 100%; (2) DT costs longest time but achieve the highest average true negative rate (ATNR) of 98.61%; (3) Compared to NB and DT, KNN costs a slightly longer time than NB but the ATPR and ATNR values are approximately close to DT. Moreover, it can also report the possible implantation location of a Trojan instance according to the detecting results.

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