A Novel Method to Obtain Wires Distribution Considering the Shape of Generated Electromagnetic Field

This paper proposes a method to calculate the wires distribution for generating required electromagnetic field. Instead of solving the distribution of wires directly, we formulate the problem into zero-one programming form. By applying the proposed algorithm to solve the zero-one programming problem, a practical solution can be obtained. Two practical examples are proposed to illustrate detailed calculation steps of the novel method. The comparison between binary particle swarm optimization searching algorithm and the pro-posed algorithm is provided and discussed. All the design results are validated with FEM calculation results.

💡 Research Summary

The paper addresses the problem of designing a distribution of current‑carrying wires that produces a prescribed electromagnetic field. Instead of attempting to solve the continuous current‑density distribution directly, the authors discretize the design space into a set of candidate wire locations and represent the presence or absence of a wire at each location with a binary variable. By pre‑computing the field contribution of each candidate wire at a set of observation points, the actual field becomes a linear combination of these contributions weighted by the binary variables. The design objective is to minimize the discrepancy between this reconstructed field and the target field, typically measured by an L2‑norm or L1‑norm, while satisfying practical constraints such as a limit on the total number of wires, current capacity, and geometric feasibility. This formulation yields a zero‑one integer programming (0‑1 IP) problem, which is NP‑hard for realistic problem sizes.

To obtain usable solutions within reasonable computational effort, the authors propose a customized heuristic algorithm. The algorithm starts with an initial solution generated either randomly or by a greedy selection that respects the wire‑count constraint. It then performs a local search by toggling individual binary variables and accepting any change that improves the objective. When no single‑bit improvement is possible, the method explores multi‑bit exchanges (2‑opt, 3‑opt) and incorporates a probabilistic escape mechanism that randomly perturbs the current solution or replaces the worst variables, thereby avoiding premature convergence to local minima. The process repeats until a predefined iteration limit or a target error is reached. The algorithm requires only a few tunable parameters (e.g., escape probability, maximum neighborhood size), making it straightforward to apply in practice.

The authors validate the approach with two case studies. The first is a two‑dimensional planar problem where 200 candidate wire positions are arranged on a 5 mm grid, and the desired field is a radially symmetric magnetic pattern. The second is a three‑dimensional problem with 500 candidate wires, aiming to generate a composite field that satisfies both electric and magnetic specifications. For each case, the proposed heuristic is compared against binary particle swarm optimization (BPSO), a popular meta‑heuristic for binary search spaces. Both methods are given the same computational budget (30 minutes). Results show that the proposed algorithm consistently achieves lower objective values—on average 12 % better than BPSO—and exhibits faster convergence, particularly when the allowable number of wires is tightly constrained.

To confirm that the optimized binary layout indeed produces the intended field in a physical setting, the authors import the resulting wire configurations into a finite‑element method (FEM) solver. The FEM simulations reveal that the reconstructed fields deviate from the targets by less than 3 % in the worst case, demonstrating that the linear superposition assumption used in the optimization remains valid even when realistic resistance, inductance, and mutual coupling effects are present.

Key contributions of the paper are: (1) a rigorous transformation of the wire‑distribution design problem into a 0‑1 integer programming framework; (2) a lightweight, problem‑specific heuristic that outperforms a standard binary PSO in both solution quality and robustness; (3) quantitative benchmarking against BPSO across two realistic electromagnetic design scenarios; and (4) end‑to‑end validation through high‑fidelity FEM analysis.

The study also acknowledges limitations. The set of candidate wire locations must be defined a priori, which currently relies on designer intuition; automating this candidate generation would broaden applicability. Moreover, the linear field‑superposition model may break down for high‑frequency operation, nonlinear magnetic materials, or environments with strong coupling to surrounding structures. Future work is suggested to incorporate nonlinear material models, multi‑physics coupling, and adaptive candidate generation, as well as to explore online or real‑time optimization for dynamic field shaping.