Multimaterial Topology Optimization Method of Magnet Structure in NMR Logging Sensor

Nuclear magnetic resonance (NMR) logging is an essential technique in petroleum exploration. The magnet structure, a critical component of NMR sensors, generates gradient static magnetic fields in remote regions of interest (ROI). Current designs, however, rely on sequential optimization of permanent magnet and magnetic material structures through size optimization (SO) and single-material topology optimization (SMTO), which restricts design flexibility and compromises signal strength and signal-to-noise ratio (SNR). To address these limitations, this study proposes a novel multimaterial topology optimization (MMTO) framework based on the variable density method. This approach simultaneously optimizes the spatial distribution of both permanent magnets and magnetic materials, significantly enhancing design freedom while minimizing geometric constraints. The sensor’s 2D cross-sectional design domain is meshed into N elements, each controlled by two density parameters (ρ₁, ρ₂) ranging from 0 to 1. The Method of Moving Asymptotes (MMA) with gradient descent is employed to iteratively update element densities based on the objective function’s gradient information, progressively converging to an optimal solution. A prototype sensor with a 2.5 cm outer radius was fabricated, producing a static magnetic field of 832.9 Gauss - 1,513.5 Gauss and a gradient of 399.8 Gauss/cm - 982.1Gauss/cm in the ROI. Experimental measurements show close agreement with simulation results. Water phantom tests demonstrate an SNR of 62.4, representing a 28.2% improvement over conventional SO and SMTO designs. These results validate the proposed method’s effectiveness in enhancing sensor performance for NMR logging applications. Additionally, the sensor’s capability to detect water mobility in samples was confirmed through transverse relaxation time (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i>₂) measurements of both pure water and quartz sand-water mixtures.