Research Article

An Approach to Conformal Transformation Using Symbolic Language Facilities: Application in Electrical Engineering

Algorithm 5

Program for calculating the fluxes in terms of the geometrical dimensions of the considered domain in the general case.
#   The symbols which are supplementary to those of Algorithm 1 are expressed below and
represent expressions in the calculation and have been put in a form to facilitate their usage.
#   They are 𝑦 1 , 𝑦 2 , 𝑦 π‘Ž , 𝑦 𝐿 .
#   The expression of 𝑦 π‘Ž is not necessary in usual calculations, but can serve to some
verifications.
#   Another symbol to be added is π‘˜ and serves to establish the sign of the flux in terms of
the used abscissa.
𝑏 0 ∢ = 1 . 5 ;
𝑐 ∢ = βˆ’ 0 . 1 ;
π‘₯ 𝑀 ∢ = βˆ’ 1 ;
π‘˜ ∢ = s i g n u m ( c + 1 ) ;
𝑦 1  ∢ = ( 1 / 2 . ) l n ( 1 . / ( 𝑐 2 βˆ’ 𝑐 𝑏 2 0 βˆ’ 2 . 𝑐 + 1 . βˆ’ 0 . 5 ( βˆ’ 2 . + 𝑐 𝑏 2 0 + 2 . 𝑐 ) ) ) ;
𝑦 2 ∢ = βˆ’ ( 1 / 2 . ) l n ( 𝑐 2  / ( 𝑐 2 βˆ’ 𝑐 𝑏 2 0 βˆ’ 2 . 𝑐 + 1 . + 0 . 5 ( βˆ’ 2 . + 𝑐 𝑏 2 0 + 2 . 𝑐 ) ) ) ;
𝑦 π‘Ž ∢ = 𝑦 1 + 𝑦 2 ;
𝑦 π‘Ž ∢ = a r c t a n h ( ( 1 . / 2 ) ( βˆ’ 2 . + 𝑐 𝑏 2 0  + 2 . 𝑐 ) / 𝑐 2 βˆ’ 𝑐 𝑏 2 0 βˆ’ 2 . 𝑐 + 1 . ) ;
𝑦 𝐿 ∢ = βˆ’ ( 1 / 2 . ) l n ( βˆ’ 8 . / ( 𝑏 2 0 + 4 . ) 𝑏 2 0 ) ;
#   For 𝑐 = 0 , in expression below, 𝑦 2 should be replaced by its limit 𝑦 𝐿 ;
𝜎 ∢ = βˆ’ π‘˜ ( 0 . 3 1 8 3 0 9 8 8 6 1 ( ( 𝑏 2 0 /  βˆ’ 𝑏 2 0 ) l n ( ( βˆ’ 𝑏 2 0 βˆ’ 𝑐 𝑏 2 0  + 2 . βˆ’ 𝑏 2 0  𝑐 2 βˆ’ 𝑐 𝑏 2 0 βˆ’ 2 . 𝑐 + 1 . ) / ( 𝑐 βˆ’ 1 . ) )
  + l n ( βˆ’ ( 1 . / 2 ) 𝑏 2 0  + 𝑐 βˆ’ 1 . + 𝑐 2 βˆ’ 𝑐 𝑏 2 0 βˆ’ 2 . 𝑐 + 1 . ) βˆ’ ( 𝑦 1 + 𝑦 2 )
  βˆ’ ( 𝑏 2 0 /  βˆ’ 𝑏 2 0 ) l n ( ( βˆ’ 𝑏 2 0 βˆ’ 1 . π‘₯ 𝑀 𝑏 2 0  + 2 . βˆ’ 𝑏 2 0  π‘₯ 𝑀 2 βˆ’ π‘₯ 𝑀 𝑏 2 0 βˆ’ 2 . π‘₯ 𝑀 + 1 . ) / ( π‘₯ 𝑀 βˆ’ 1 . ) )
  βˆ’ l n ( ( βˆ’ 1 . / 2 ) 𝑏 2 0  + π‘₯ 𝑀 βˆ’ 1 . + π‘₯ 𝑀 2 βˆ’ π‘₯ 𝑀 𝑏 2 0 βˆ’ 2 . π‘₯ 𝑀 + 1 . )
+ a r c t a n h ( 0 . 5 0 0 0 0 0 0 0 0 0 ( βˆ’ 2 . + π‘₯ 𝑀 𝑏 2 0  + 2 . π‘₯ 𝑀 ) / π‘₯ 𝑀 2 βˆ’ π‘₯ 𝑀 𝑏 2 0 βˆ’ 2 . π‘₯ 𝑀 + 1 . ) βˆ’ l n ( π‘₯ 𝑀 ) ) ) ;
𝜎 ∢ = s i m p l i f y ( 𝜎 ) ∢
#   The total value will be 𝜎 𝑇 ∢ = 2 𝜎 .
#   For to keep the correct sign of the flux, even say for 𝑐 = βˆ’ 1 0 0 0 , the signum function has been used.
#   Results after running the program:
                   𝑏 0 ∢ = 1 . 5
                    𝑐 ∢ = βˆ’ 0 .
                   π‘₯ 𝑀 ∢ = βˆ’ 1
                    π‘˜ ∢ = 1
                 𝑦 1 ∢ = βˆ’ 0 . 4 3 9 8 9 9 4 3 5 8
               𝑦 2 ∢ = 0 . 1 8 8 7 0 9 2 2 6 0 βˆ’ 1 . 5 7 0 7 9 6 3 2 7 𝐼
              𝑦 π‘Ž ∢ = βˆ’ 0 . 2 5 1 1 9 0 2 0 9 8 βˆ’ 1 . 5 7 0 7 9 6 3 2 7 𝐼
             𝑦 𝐿 ∢ = 0 . 2 8 2 0 3 5 0 6 9 2 βˆ’ 1 . 5 7 0 7 9 6 3 2 7 𝐼
              𝜎 ∢ = βˆ’ 0 . 1 3 3 3 7 3 6 8 0 6 βˆ’ 1 . 1 0 βˆ’ 1 0 𝐼             ( 1)