Walter D’Ambrogio and Annalisa Fregolent LAB E LB E T gAB gB =− LAB E LB E T ⎡ ⎣ BAB E T BB E T ⎤ ⎦{ λ}={0} (16) Then [BE] T is the nullspace of [LE] T, and viceversa [LE] is the nullspace of [BE]: ⎧⎪ ⎪⎨ ⎪⎪⎩ BAB E BB E LAB E LB E ={0} LAB E LB E T BAB E BB E T ={0} (17) Since Eq. (16) is always satisfied, the 3-field formulation reduces to: ⎧⎪ ⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ ZAB [0] [0] − ZB uAB uB +⎡ ⎣ BAB E T BB E T ⎤ ⎦{ λ}= f AB f B BAB C BB C uAB uB ={0} (4∗∗) (5) To eliminate {λ}, Eq. (4∗∗) can be written: uAB uB =− ZAB [0] [0] − ZB − 1⎡ ⎣ BAB E T BB E T ⎤ ⎦{ λ}+ ZAB [0] [0] − ZB − 1 f AB f B (18) which substituted in Eq. (5) gives: BAB C BB C ZAB [0] [0] − ZB − 1⎡ ⎣ BAB E T BB E T ⎤ ⎦{ λ}= = BAB C BB C ZAB [0] [0] − ZB − 1 f AB f B (19) fromwhich{λ}is obtained: {λ}=⎛ ⎝ BAB C BB C ZAB [0] [0] − ZB − 1⎡ ⎣ BAB E T BB E T ⎤ ⎦ ⎞ ⎠ + × × BAB C BB C ZAB [0] [0] − ZB − 1 f AB f B (20) 54
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