Consider the acoustical problem of a cylindrical expansion chamber with a certain excitation at inlet, a lining l
domain bounded by a surface
Chapter 3 –
99
elements. Although the unknown nodal pressures constitute the degrees of freedom of the discrete model, the solution is obtained not only at each node
domain. In fact a polynomial interpolation of the nodal pressures is carried out,
depends on the typology of the utilized elements and on the number of nodes per each side element. The polynomial interpolation is then related to the notion of shape function
a founding concept of the FEM, and which will be shown afterwards.
Figure 3.1 – Meshed domain of an automotive silencer
approximate FE solution depends on the mesh characteristics, including the size element and the node spacing. In particular, for acoustical problems, a key
r the obtainment of a sufficiently refined mesh is represented by the number of in the investigated frequency range.
A definitely positive aspect of the FEM is the possibility to use
that is irregular meshes where the element size can vary almost arbitrarily. This aspect is an advantage point of FE compared to low-dispersion finite difference schemes, in which the grid points must be aligned in rows and plans. As a consequence the pre
ust deal with a less difficult problem, and the meshing procedure itself is
ENERAL FORMULATION FOR INTERIOR PROBLEMS
Consider the acoustical problem of a cylindrical expansion chamber with a certain excitation at inlet, a lining lateral surface and closed outlet (see Figure 3.2). Let
domain bounded by a surface Γ. This surface can be divided in: Γst
– FEM acoustic model
elements. Although the unknown nodal pressures constitute the degrees of freedom of the discrete model, the solution is obtained not only at each node pj, but in the whole domain. In fact a polynomial interpolation of the nodal pressures is carried out, whose nature depends on the typology of the utilized elements and on the number of nodes per each side shape function, which is
Meshed domain of an automotive silencer.
approximate FE solution depends on the mesh characteristics, including the size element and the node spacing. In particular, for acoustical problems, a key r the obtainment of a sufficiently refined mesh is represented by the number of nodes
A definitely positive aspect of the FEM is the possibility to use un-structured meshes, arbitrarily. This aspect is an dispersion finite difference schemes, in which the grid points must be aligned in rows and plans. As a consequence the pre-processing software problem, and the meshing procedure itself is
OR INTERIOR PROBLEMS
Consider the acoustical problem of a cylindrical expansion chamber with a certain ateral surface and closed outlet (see Figure 3.2). Let Ω the acoustic
st , which is the inlet
node j
element pj
Chapter 3 – FEM: General formulation for interior problems
100
(structural) boundary on where a certain displacement, s, is imposed; Γh , composed by the rigid (hard) surfaces, including the outlet; and ΓZ, the locally reacting surfaces (Z impedance).
The domain Ω is properly discretized, and in Figure 3.2 a little part of the entire mesh, in proximity of the outlet, has been depicted.
Figure 3.2 – The FE model: geometry, mesh, and boundary conditions.
The acoustic pressure p(x,t) is approximated by a trial function p′
( )
x,t of the form:( ) ∑ ( ) ( )
=
′ = n
j
j j t N p t
p
1
, x
x (3.18)
which, in frequency domain, becomes:
( ) ∑ ( ) ( )
=
′ = n
j
j
j N
p p
1
, ˆ
ˆ x ω ω x (3.19)
pj and pˆ represent the nodal values of pressure and pressure amplitude at node j , n is the j
total number of nodes. The generic function Nj(x) is called shape function and it is equal to 1 at node j and 0 at all other nodes (see Figure 3.3). These functions interpolate the nodal quantities all over the domain, although they are defined locally within the single element, as polynomial in physical or spatial coordinates. For this reason the trial solution, expressed by the (3.18) and (3.19), is a summation of these functions weighted by the nodal values of pressure (see Figure 3.4). The shape functions depend on the type of elements and number of nodes: for a simple triangular element with nodes on the corners, they are formed from the basis
{
1,x,y}
and in each triangles they have the form a1+a2x+a3y where a1, a2, a3, are proper constant values chosen in a way that the shape functions take the correct values on theΓst
Γh
Γh
Γh
Γh
Γh
ΓZ
ΓZ
y
x
s
Chapter 3 – FEM: General formulation for interior problems
101
nodes. This means that the number of polynomial terms in the basis set must be the same as the number of nodes in the element topology (three in the considered case).
Figure 3.3 – The FE model: global shape function Nj(xj).
Figure 3.4 – The FE model: trial solution.
x y
( )
xN
( )
j =1N xj
y
x
( )
xp
p p=~ pj
p=
j node
Chapter 3 – FEM: General formulation for interior problems
102 3.4.1WEAK VARIATIONAL FORMULATION
Weak formulations are an important tool for the analysis of mathematical equations, which permit the transfer of concepts of linear algebra to solve problems in other fields, such as partial differential equations [6]. In a weak formulation, an equation is no longer required to hold absolutely and it has, instead, has weak solutions only with respect to certain “test vectors” or “test functions”.
Let χ
( )
x a continuous and differentiable test function. By multiplying the (3.6b) by the( )
xχ and integrating over Ω, one can write:
( )
1 ˆ 2ˆ ˆ 00
0 =
+ −
∇
∫
∇Ω
w p k ρ p
ρ
χ x (3.20)
Applying the divergence theorem and imposing that the normal derivatives of pressure on the boundaries Γh, Γst, ΓZ respectively satisfy the conditions (3.11), (3.12), (3.16), it is possible to obtain:
( )
d d 1 ˆd 01 d
0 2
0 0 0
2
0 2 2
= Ω +
Γ +
′ Γ +
Ω
∇ ⋅∇ ′− ′
∫ ∫ ∫ ∫
Ω Γ Γ Ω
w s
c p i A c p
p n χ
χ ρ ω ρω χ
ω ω χ
ρ χ (3.21)
It is also possible to notice that, if the admittance A is equal to zero, the integral over ΓZ
reduces to zero, so that if, on a certain boundary surface, there is no specified condition, it naturally acts as a hard surface, by default.
When the trial function of (3.18) is substituted in the (3.21), a linear equation is obtained, in the n unknowns pj. By using a set of n test functions χj(with j=1,...n), n linear equations in n unknowns are generated. The best choice for the χjis represented by the shape functions Nj
(with j=1,...n) previously defined: by setting χj
( )
x = Nj( )
x , with j=1,...n, one obtains the following symmetric system of linear equations:[
K+iωC−ω2M] { }
pˆ ={ } { }
fˆst + fˆW (3.22)Chapter 3 – FEM: General formulation for interior problems
103
where M, K and C are the acoustic mass, stiffness and damping matrices, fˆst and fˆW are the forcing terms due to the structural excitation (at inlet, in case of figure 3.2) and to the acoustic sources. They can be expressed as:
( )
{ }
d ,{ }
1 d ,, d
, d ,
d
0 2
0 0
0 2
0 0
∫
∫
∫
∫
∫
Ω Γ
Γ
Ω Ω
Γ
= Γ
=
Γ
=
∇ Ω
⋅
= ∇ Ω
=
w N f
s N f
N c N c A
N K N
c N M N
j j W n
j j st
k j jk
k j jk
k j jk
s Z
ω ρ ρω
ρ ρ
(3.23)
The above integrals are evaluated for each element and then assembled, through a proper procedure [7], in to the global matrices M, K , C and the forcing vectors fst and fW. Numerical integration is generally used within each element.