Finite Element Method Magnetics : ImprovisedABCs

Improvised Asymptotic Boundary Conditions

"Open Boundary" problems are finite element problems that are solved on a finite domain but emulate a solution in an unbounded domain. Recent papers (here and here) describe "Improvised Asymptotic Boundary Conditions" for magnetic and electrostatic problems. Improvised ABCs are a new approach to solving "open boundary" problems.

The paper derives Improvised ABCs and reports the relative permeabilities needed to implement IABCs through third order for both 2D and axisymmetric problems. However, even higher-order IABCs are possible. The purpose of this page is to report the relative permeabilities needed to implement IABCs through seventh order and to provide Mathematica notebooks that document the computation of these results.

2D Improvised Asymptotic Boundary Conditions for Vector Potential Problems

2D Mathematica Notebook

PDF Printout of 2D Mathematica Notebook

δ	μ1/μo	μ2/μo	μ3/μo	μ4/μo	μ5/μo	μ6/μo	μ7/μo	μ8/μo	μ9/μo	μ10/μo
1⁄10	10.5238
1⁄20	0.145314	32.1429
1⁄30	5.1494	0.053269	54.1064
1⁄40	0.241961	13.6813	0.033014	76.075
1⁄50	3.46611	0.0920926	21.7973	0.0240632	98.0545
1⁄60	0.333693	9.02367	0.057897	29.5733	0.0189665	120.04
1⁄70	2.6499	0.129533	14.399	0.0428826	37.1975	0.0156642	142.03
1⁄80	0.419421	6.73965	0.0807328	19.4093	0.0342553	44.7374	0.0133469	164.022
1⁄90	2.17517	0.167434	10.8857	0.0598784	24.2374	0.0286004	52.2252	0.0116295	186.016
1⁄100	0.498229	5.35441	0.102918	14.6965	0.0480031	28.9563	0.024587	59.6785	0.0103052	208.011

Axisymmetric Improvised Asymptotic Boundary Conditions for Vector Potential Problems

Axisymmetric Mathematica Notebook

PDF Printout of Axisymmetric Mathematica Notebook

δ	μ1/μo	μ2/μo	μ3/μo	μ4/μo	μ5/μo	μ6/μo	μ7/μo	μ8/μo	μ9/μo	μ10/μo
1⁄10	10.0634
1⁄20	0.188706	39.9975
1⁄30	4.37479	0.075118	85.8557
1⁄40	0.285096	12.4933	0.051874	147.652
1⁄50	3.06077	0.112378	22.0265	0.0423167	225.43
1⁄60	0.374816	8.17242	0.0740027	33.0192	0.0371234	319.234
1⁄70	2.40892	0.149299	13.8176	0.0582172	45.656	0.0338673	429.046
1⁄80	0.457719	6.15238	0.0951954	19.795	0.0495821	60.002	0.0316392	554.889
1⁄90	2.01871	0.187242	10.323	0.0732287	26.2901	0.0441395	76.0845	0.0300195	696.75
1⁄100	0.533241	4.9312	0.116425	14.5221	0.0612364	33.375	0.0403993	93.9198	0.028790	854.64

2D Improvised Asymptotic Boundary Conditions for Scalar Potential Problems

2D Mathematica Notebook

PDF Printout of 2D Mathematica Notebook

δ	ε1/εo	ε2/εo	ε3/εo	ε4/εo	ε5/εo	ε6/εo	ε7/εo	ε8/εo	ε9/εo	ε10/εo
1⁄10	0.0950226
1⁄20	6.88167	0.0311111
1⁄30	0.194197	18.7726	0.0184821
1⁄40	4.1329	0.0730927	30.2898	0.0131449
1⁄50	0.288508	10.8586	0.0458772	41.5572	0.0101984
1⁄60	2.99677	0.11082	17.2721	0.0338143	52.7245	0.00833053
1⁄70	0.377373	7.72003	0.0694494	23.3195	0.0268835	63.8397	0.00704076
1⁄80	2.38424	0.148376	12.3865	0.0515218	29.1926	0.0223527	74.9238	0.00609673
1⁄90	0.459734	5.97251	0.0918638	16.7005	0.0412586	34.9645	0.0191478	85.988	0.00537587
1⁄100	2.00711	0.186762	9.71645	0.0680432	20.832	0.0345348	40.6719	0.0167565	97.0386	0.00480743

Axisymmetric Improvised Asymptotic Boundary Conditions for Scalar Potential Problems

Axisymmetric Mathematica Notebook

PDF Printout of Axisymmetric Mathematica Notebook

δ	ε1/εo	ε2/εo	ε3/εo	ε4/εo	ε5/εo	ε6/εo	ε7/εo	ε8/εo	ε9/εo	ε10/εo
1⁄10	0.0909091
1⁄20	10.0505	0.0434961
1⁄30	0.16015	31.2123	0.0356179
1⁄40	5.07522	0.0674798	58.8308	0.0324964
1⁄50	0.249837	13.9595	0.0485094	93.6268	0.0308341
1⁄60	3.43867	0.100076	23.6832	0.0405888	135.819	0.0298002
1⁄70	0.338611	9.11831	0.0674884	34.5315	0.0362401	185.492	0.0290973
1⁄80	2.63643	0.134865	15.0974	0.0539561	46.7532	0.0334921	242.665	0.0285874
1⁄90	0.422731	6.77301	0.0874553	21.3014	0.0465142	60.4415	0.0316	307.363	0.0282013
1⁄100	2.16756	0.171316	11.1989	0.0680376	27.9482	0.0418045	75.638	0.0302177	379.587	0.0278984

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