Page 283 - Numerical Analysis and Modelling in Geomechanics

264 THE 2-D CHEBYSHEV SPECTRAL ELEMENT METHOD

(9.18)

with U(0)=U , (0)= as initial conditions, where the unknown vector U contains
0
0
the values of the discrete solution ũ at all Chebyshev points , for i, j=0,…, N and
for all e=0,…, n . A dot above a variable denotes differentiation with respect to
e
time. In equation (9.18), M is the mass matrix, K is the stiffness matrix, and F is
the force vector obtained after a global nodal renumbering and assembly of all
the elemental matrices and force vector contributions. They can be computed by
using the following expressions:

(9.19)

(e)
where Σ≥ denotes the matrix element summation over all the elements, and M ,
K (e) and F (e) are the elemental matrices and force vector, respectively. The
contributions from nodes that are common to an element pair are summed—this
approach is called stiff summation (Hughes, 1987)—to enforce the continuity
requirement of the solution on the element boundaries. The elemental matrices
and force vector are given by

(9.20)

(9.21)

(9.22)

where and are the nodal submatrices and vector respectively.
The global matrices M, K and F are sparse, symmetric and positive-definite.
Equation (9.18) is a linear, second-order ordinary differential equation with
constant coefficients, which must be integrated over the time interval [0, T].
Time integration is performed using the “three-point-recurrence weighted
residual” scheme (Zienkiewicz and Wood, 1987), which is a two-step finite
difference scheme belonging to the Newmark family. This scheme is implicit,
unconditionally stable and accurate to the second order. The solution u is
computed solving a symmetric positive-definite sparse linear system at each time
step.