Page 258 - Mechanics of Asphalt Microstructure and Micromechanics

250 Ch a p t e r E i gh t

y
k

v i
j

i u i
x i y i

x
FIGURE 8.2 Illustration of the symbol scheme for a triangle element.

Conveniently, one can solve for the coefficients (a 1 , b 1 , c 1 , a 2 , b 2 , c 2 ) and represent the
interpolation function in the format by Zienkiewicz (1977).

1
+
u = [(α + β x + γ y u + (α + β x + γ y u + (α + β x + γ yu)]
)
)
2 A i i i i j j j j k k k k
1
+
v = [(α + β x + γ y v + (α + β x + γ y v + (α + β x + γ yv)] (8-37)
)
)
2 A i i i i j j j j k k k k
α xy − x y
i j k k j
β y − y
i j k
γ x − x
i k j
N = (α + β x + γ y)/2 A
i i i i
N = (α + β x + γ y)/2 A
j j j j
N = (α + β x + γ y)/2 A
k k k k (8-38)
Where A is the area of the triangle.
The matrix format of the interpolation functions can be represented as:
⎧u ⎫
⎪ i ⎪
v ⎪
⎪ i
⎪
⎧ u⎫ ⎡ N ,,0 N ,,0 N ,0 ⎤ u j ⎪ ⎪
⎪
i
k
j
⎨ ⎬ ⎢ ⎥ ⎥ ⎨ ⎬
⎩ v ⎭ ⎢ ,0 N , ,0 N , ,0 N ⎥ ⎪ v j ⎪
⎣
k ⎦
i
j
⎪ ⎪
⎪ u k ⎪
⎪ v ⎪
⎩ k ⎭