Maria Bayard Dühring - Solid Mechanics

8 Chapter 2 Time-harmonic propagating waves
different materials. In cases where an unbounded structure is considered, it is necessary
to apply boundary conditions that do not reflect the waves in order to limit
the modeling domain. In simple cases as in wave propagation in fluids, a condition
as the Sommerfeld radiation condition can be applied [1]. In more complicated
cases as for elastic wave propagation in solid structures and for non-perpendicular
waves, it is more accurate to employ perfectly matched layers, which are employed
for time-harmonic elastodynamic problems in [26]. In this case an extra layer is
applied at the boarders where the disturbances are absorbed gradually before they
reach the outer boundaries. In all the wave problems considered in this work the
driving forces are applied as harmonic varying boundary conditions. If a unit cell
is considered where periodic boundary conditions connect opposite boundaries, the
problem can be solved as an infinite periodic structure.
2.3 Discretizing and solving wave problems
Analytical solutions exist for simple problems as spherical acoustic waves propagating
in an infinite domain in 3D or Rayleigh waves propagating in a homogeneous
half space. For more complex geometries and material distributions the problems
must be solved numerically. In this work the finite element method is employed.
The dependent variables uj in a problem are collected in a field vector defined as
u = [u1, u2, ..., uj, ..., uJ] and are discretized using sets of finite element basis functions
{φj,n(r)}
uj(r) =
N
uj,nφj,n(r), (2.4)
n=1
where r is the position vector. The degrees of freedom corresponding to each field
are assembled in the vectors uj = {uj,1, uj,2, ..., uj,n, ...uj,N} T . In the problems considered
here, a triangular element mesh is employed and for the physical fields either
vector elements or second order Lagrange elements are used depending on the problem.
The governing equations are discretized by a standard Galerkin method [27]
J
j=1
Skjuj =
J
j=1
Kkj + iωCkj − ω 2
Mkj uj = fk, (2.5)
where the system matrix Skj has contributions from the stiffness matrix Kkj, the
damping matrix Ckj and the mass matrix Mkj. fk is the load vector. When material
damping or absorbing boundary conditions are employed the damping matrix is
different from zero and the problem gets complex valued. The material damping can
be introduced to make the problems behave more realistically and can in some cases
be necessary in order to avoid ending up in a local optimum during an optimization
process, see chapter 3.