Tensile Electroforming Model - Electrically Assisted Forming - page 136

[f1,i 1 ( x 1 , x 2 , ... , x n ) = 0
[f1,i 2 ( x 1 , x 2 , ... , x n ) = 0
[f1,i 3 ( x 1 , x 2 , ... , x n ) = 0

(6.66)

For each equation, a multivariable first-order Taylor series expansion is written.
For example, for the k th equation,

∂ [f1,i k , [f1,i
∂ x 1 +

∂ [f1,i k , [f1,i
∂ x 2

[f1,i k , [f1,i + 1 = [f1,i k , [f1,i + x 1, [f1,i + 1 − x 1, [f1,i x 2, [f1,i + 1 − x 2, [f1,i

(6.67)

x n , [f1,i + 1 − x n , [f1,i

∂ [f1,i k , [f1,i
∂ x n

+···+

where the first subscript ( k ) represents the equation of unknown and the second
subscript denotes whether the value or function is at the present value ( [f1,i ) or at the
next value ( [f1,i + 1).
By setting [f1,i k , [f1,i + 1 to zero, this means that we are looking for the roots of the sys-
tem of equations. This then gives

− [f1,i k , [f1,i + x 1, [f1,i ∂ [f1,i k , [f1,i
∂ x 1

+ x 2, [f1,i ∂ [f1,i k , [f1,i
∂ x 2

+···+ x n , [f1,i ∂ [f1,i k , [f1,i
∂ x n

(6.68)

= x 1, [f1,i + 1 ∂ [f1,i k , [f1,i

∂ x 1 + x 2, [f1,i + 1 ∂ [f1,i k , [f1,i ∂ x 2 +···+ x n , [f1,i + 1 ∂ [f1,i k , [f1,i

∂ x n

By examining Eq. ( 6.68 ), the only unknowns are the x k , [f1,i + 1 terms on the right-hand
side of the equation as all other quantities are known at the present value ( [f1,i ). This
now provides a system of linear equations that can be solved. Matrix notation is
used to simplify the expression. The partial derivatives can be expressed by

∂ [f1,i 1, [f1,i
d x 1

∂ [f1,i 1, [f1,i
d x n

···

.

. . . .

[ J ] =

(6.69)

∂ [f1,i n , [f1,i
d x 1

∂ [f1,i n , [f1,i
d x n

···

where J is commonly called the Jacobian matrix.
The initial and final values in vector form are as follows:

{ X [f1,i } T =

x 1, [f1,i ... x n , [f1,i

(6.70)

and

{ X [f1,i + 1 } T =

x 1, [f1,i + 1 ... x n , [f1,i + 1

(6.71)

The function values in vector form are as follows:

{ F [f1,i } T =

[f1,i 1, [f1,i ... [f1,i n , [f1,i

(6.72)