{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T17:14:44Z","timestamp":1773249284811,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,4]],"date-time":"2024-09-04T00:00:00Z","timestamp":1725408000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this paper, we introduce a new algorithm for calculating the radial and axial magnetic forces between two non-coaxial circular loops with parallel axes. These formulas are derived from a modified version of Grover\u2019s formula for mutual inductance between the coils in question. Utilizing these formulas, we compute the radial and axial magnetic forces between two non-coaxial thick coils of rectangular cross-sections with parallel axes. In these calculations, we apply the filament method and conduct investigations to determine the optimal number of subdivisions for the coils in terms of computational time and accuracy. The method presented in this paper is also applicable to all conventional non-coaxial coils, such as disks, solenoids, and non-conventional coils like Bitter coils, all with parallel axes. This paper emphasizes the accuracy and computational efficiency of the calculations. Furthermore, the new method is validated according to several previously established methods.<\/jats:p>","DOI":"10.3390\/computation12090180","type":"journal-article","created":{"date-parts":[[2024,9,4]],"date-time":"2024-09-04T08:46:22Z","timestamp":1725439582000},"page":"180","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Optimized Calculation of Radial and Axial Magnetic Forces between Two Non-Coaxial Coils of Rectangular Cross-Section with Parallel Axes"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9506-9024","authenticated-orcid":false,"given":"Slobodan","family":"Babic","sequence":"first","affiliation":[{"name":"Independent Researcher, 53 Berlioz 101, Montr\u00e9al, QC H3E 1N2, Canada"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-2641-7601","authenticated-orcid":false,"given":"Eray","family":"Guven","sequence":"additional","affiliation":[{"name":"Poly-Grames Research Center, Department of Electrical Engineering, Polytechnique Montr\u00e9al, Montr\u00e9al, QC H3T 1J4, Canada"}]},{"given":"Kai-Hong","family":"Song","sequence":"additional","affiliation":[{"name":"Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education Anhui University, Hefei 230039, China"}]},{"given":"Yao","family":"Luo","sequence":"additional","affiliation":[{"name":"Electrical Engineering and Automation, Wuhan University, Wuhan 430072, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,4]]},"reference":[{"key":"ref_1","unstructured":"Grover, F.W. 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