Page 217 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

206 Fundamentals of Ocean Renewable Energy

Many types of 2D or 3D elements are common in FEM, such as triangular,
quadrilateral, and tetrahedron. The triangular mesh is the most common type.
After discretization, the state/dependent variable, u (e.g. water depth, velocity,
displacement, stress, etc.) is approximated by a function, ˆu. In other words,
u is the exact solution of the PDE of interest, which in most cases does not
exist, whilst ˆu can approximately satisfy the PDE. ˆu is approximated using basis
functions

ˆ u = u i ψ i (8.18)
i
where ψ i denotes basis functions and u i is the approximate value of the state
variable at node i. For instance, if an element has three nodes, u can be
approximated by
u ≈ˆu = u 1 ψ 1 (x, y) + u 2 ψ 2 (x, y) + u 3 ψ 3 (x, y) (8.19)
Note that if (x 1 , y 1 ) is the coordinate of node 1, then basis functions are selected
such that ψ 1 (x 1 , y 1 ) = 1, ψ 2 (x 1 , y 1 ) = 0, and ψ 3 (x 1 , y 1 ) = 0. The simplest
form of a basis function is the Lagrangian linear function over an element as
follows
ψ i (x, y) = a + bx + cy; i = 1, 2, 3 (8.20)
The coefficients of this function can be found by imposing the conditions
ˆ
explained previously (i.e. u(x i , y i = u i )), that is
ψ i (x k , y k ) =δ(i, k) i, k = 1, 2, 3 (8.21)
δ(i, k) =1 i = k (8.22)
δ(i, k) =0 i = k (8.23)
As a next step, after selecting the basis/shape functions, the unknown values
u i should be estimated. FEM forms enough algebraic equations to compute the
unknown u i at all nodes by minimizing the error. Let us consider the exact form
of a differential equation
L(u) = 0 (8.24)
2
∂ 2 ∂ 2 ∂ u
where L is a general differential operator (e.g. L = + ⇒ L(u) = +
∂x 2 ∂y 2 ∂x 2
2
∂ u ). If we replace ˆu into the differential equation, it will not exactly satisfy it
∂y 2
(because u ≈ˆu), and leads to a residual error (R) as follows
L(ˆu) = R (8.25)
In FEM (weighted residual method), u i is determined such that the residual error
is minimized, therefore,

W j (x, y)L(ˆu) = W j (x, y)R = 0 j = 1, 2, 3, ... , N (8.26)
Ω Ω