With the exception of a few simple cases, Equation 16–3 cannot be solved analytically, and so a numerical scheme must be used. As described previously, the problem of linear elasticity is of the elliptic type, that is, the behavior is mainly dictated by boundary values (instead of initial values). Therefore, it is rather natural to use a finite element method for solving the stress and deformations in the solid body. In the finite element context, the structure being analyzed is discretized (approximated) as an assembly of N discrete regions (elements) connected at a finite number (M) of points, or nodes. These elements are triangles or quadrilaterals for 2D cases, while tetrahedra, hexahedra, wedges or pyramids are used for 3D cases.
Displacement unknowns are defined at nodes and connected with appropriate interpolation
functions at the level of each element. These interpolation functions are continuous at the
level of each element, as well as the first derivative; they are selected in such a way that
continuity is also guaranteed at the interface between adjacent elements. In many
applications, continuous piecewise linear interpolation functions are considered. There are
as many such interpolation functions
as unknown nodal displacement vectors
. and in particular each interpolation function is such that
=1 at the node
and =0 at the other nodes.
In the first instance, let us omit the acceleration term in
Equation 16–3
. Let
be the approximation for the displacement field, described in terms of
nodal values
and interpolation functions
. The interpolation functions and the approximation
are used for transforming the partial differential
Equation 16–3
into a linear algebraic system of
M
vector equations with
M
vector unknowns. For
this, the following procedure is applied:
M equations are created by multiplying Equation 16–3 with the M interpolation functions.
The part integration rule and Green's theorem are applied to decrease the level of the derivative by one order and to easily embed force boundary conditions into the system.
Eventually, the selected approximation for the displacement field described in terms of nodal values
and interpolation functions is introduced into the system.
The discretized form of Equation 16–3 now becomes
 | (16–6) |
where
is the vector of external forces applied on the boundary
. For the sake of facility,
Equation 16–6
can be rearranged as
 | (16–7) |
In
Equation 16–7
.
is the total stiffness matrix,
is the vector of nodal displacement unknowns
. while
results from the external (continuous or nodal) forces applied on the
boundary of the solid body. In particular, in an FSI simulation, they incorporate the
forces originating from the fluid motion. The additivity property allows the integration
in
Equation 16–6
to be performed at the level
of each individual element in order to obtain a local stiffness matrix, which is
subsequently assembled into the total stiffness matrix
of the linear system. After the assembly of the total stiffness matrix
is completed, displacement boundary conditions are applied by removing the corresponding
unknowns and equations from the system. These methods are well documented (for example,
see Zienkiewicz
[732]). For solving the system in the present context,
an iterative solver is used.
In linear structural dynamics systems, the internal load is linearly proportional to the nodal displacement, and the structural stiffness matrix remains constant. However, inertia terms should be taken into account into the momentum equation, as well as a possible term for structural damping. The discretized Equation 16–7 now becomes
 | (16–8) |
where
.
. and
are the structural mass matrix, the structural damping matrix, and the
stiffness matrix, respectively, while
.
. and
are the vectors of nodal accelerations, velocities, and displacements,
respectively, and
is the applied load vector. In a linear structural dynamics system, the
matrices are constant, while the vectors are a function of time. Presently, the structural
damping matrix
is not considered, however, for the sake of generality, it is kept in
the equations.
Among direct time integration methods for numerically solving the finite element semi-discrete equation of motion given Equation 16–8 . several methods are available in literature; two such methods are the Newmark method (Newmark [475]) and the backward Euler method, which are available in Fluent and are described in the sections that follow.
The Newmark family of time integration algorithms (Newmark [475]) is one of the most popular time integration methods as a single-step algorithm. As noted by Hughes [261]. the semi-discrete Equation 16–8 can be rewritten as:
 | (16–9) |
where
.
. and
are the vectors of nodal accelerations, velocities, and displacements
at time
. respectively, while
is the load vector applied at time
. In addition to
Equation 16–9
. the Newmark family of time integration algorithms requires the displacement and
velocity to be updated as
 | (16–10) |
 | (16–11) |
where
and
are Newmark's integration parameters and
.
. and
are the vectors of nodal accelerations, velocities, and displacements
at time
. respectively, while
is the time step. By making use of the three algebraic equations
(
Equation 16–9
.
Equation 16–10
. and
Equation 16–11), a single-step time integrator in
terms of the unknowns
and the three known quantities can be written as
 | (16–12) |
where the several coefficients are given by
.
.
.
.
. and
.
The calculation procedure is as follows. First,the vector of nodal displacements
is calculated using
Equation 16–12
. Then, the program computes the
vectors of nodal velocities
and nodal accelerations
with the following relationships:
 | (16–13) |
 | (16–14) |
The most important factors in choosing an appropriate time integration scheme for the finite element semi-discrete equation of motion ( Equation 16–7) are accuracy, stability, and dissipation. In conditionally stable time integration algorithms, stability is affected by the selected time step; whereas in unconditionally stable time integration algorithms, a time step can be selected independently of stability considerations.
In the Newmark method, the amount of numerical algorithm dissipation can be
controlled by one of the Newmark's parameters (
) as follows:
 | (16–15) |
With the Newmark parameters satisfying the above conditions, the Newmark family of
methods is unconditionally stable (Hughes
[261]). By introducing
the amplitude decay factor
. the above conditions can be written as:
 | (16–16) |
Consequently, Fluent provides you with the Newmark integration procedure, which is
unconditionally stable through the input of the amplitude decay factor
.
The backward Euler method requires velocity and acceleration to be updated as follows:
 | (16–17) |
 | (16–18) |
Combining
Equation 16–17
and
Equation 16–18
with the semi-discrete equation of
motion given in
Equation 16–9
results in a
single-step time integrator in terms of the unknown
as:
 | (16–19) |
Rayleigh damping is available in the structural model for transient intrinsic fluid-structure interaction (FSI) cases. As described in The Newmark Method . the final form of the equation of transient motion can be written as
 | (16–20) |
where the several coefficients are given by
.
.
.
.
. and
.
Additionally, the Rayleigh model for structural damping matrix is as follows:
 | (16–21) |
where
is the mass-proportional damping coefficient, and
is the stiffness-proportional damping coefficient.