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
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
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 |
