Page 281 - Numerical Analysis and Modelling in Geomechanics

262 THE 2-D CHEBYSHEV SPECTRAL ELEMENT METHOD
subdomains. Then, an approximating function is defined on each subdomain as a
truncated expansion of Chebyshev polynomials. For the simpler 1-D case, the
procedure is described in Priolo and Seriani (1991). In the case of two-
dimensional problems, the original spatial domain Ω is decomposed into non-
overlapping quadrilateral elements Ω , where e=l,…, n , and n e is the total
e
e
number of elements. As approximating functions on each element Ω , functions
e
belonging to the space are chosen, i.e., polynomials of degree≥ N 1 in x 1 and of
degree≥ N in x . Then a global approximating function is built up as a sum of the
2
2
elemental approximating functions. The resulting function is a continuous
piecewise polynomial defined on the decomposition of the original domain Ω.
In this case, the polynomial space is constructed by using the Chebyshev
orthogonal polynomials and for simplicity it is assumed that N =N =N, i.e., the
1
2
order of the polynomials is the same in both directions x and x .
2
1
It can be shown (Canuto et al., 1988) that a function f(ξ)=f(ξ , ξ ), defined on
2
1
the square interval [−1, 1]×[−1, 1], can be approximated by a truncated
expansion using a tensor product of Chebyshev polynomials as follows:
(9.11)
where are the grid values of the function f, and are Lagrangian interpolants
satisfying the relation within the interval [−1, 1], and identically zero outside. Here,
δ ik denotes the Kronecker-delta symbol, and ζ stands for ξ 1 or ξ 2 . The
Lagrangian interpolants are given by

(9.12)

where T p are the Chebyshev polynomials and ζ i are the Chebyshev Gauss-
Lobatto quadrature points ζi=cos (π /N) for i=0,…, N. The coordinates ξ ={ξ ,
1i
i
ij
ξ } of the internal nodes for the discretisation of the rectangular domain [−1 , 1]
2j
× [−1, 1] are obtained as Cartesian products of the ζ i points. In order to apply
these interpolants and construct the approximating function space, there needs to
2
(e)
(e)
be defined the mapping Λ (x): x≥ Ωe→ξ ≥ [−1, 1] between the points x of
each element Ω =[a , a e+1 ]×[b , b e+1 ] of the decomposition Ω in the physical
e
e
e
space and the local element coordinate system {ξ , ξ } by
1
2
(9.13)
with and dimensions of the element Ω . Then the global approximating function
e
is formed by the sum of the elemental approximating functions (9.11) defined on
each element.