smart technologies for safety engineering

292 Smart Technologies for Safety Engineering
Ɣ p , is an assembly of the boundaries of all poroelastic subdamains, Ɣ p(N) (N ∈{1,...,Np}), i.e.
Ɣ p =
Np
N=1
Ɣ p(N) =
Np
N=1
Ɣ u
∪ Ɣp =
p(N) p(N)
Np
N=1
Ɣ u p(N) ∪
Np
N=1
Ɣ p
p(N) = Ɣu p ∪ Ɣp p
and, as shown here, this can be divided into two kinds, namely where the value of displacement
or the value of pressure is prescribed, Ɣ u p
(76)
or Ɣp
p , respectively. In fact, two other types could
have been distinguished but, like previously, they are left out because of their low practical
importance. In other words, only piston displacement or acoustic pressure can be imposed on
to the boundaries of poroelastic media.
Similarly, the boundary of the acoustic domain, Ɣ a , incorporates all boundaries of acoustic
subdomains, Ɣ a(N) (N ∈{1,...,Na}), i.e.
Ɣ a =
Na
N=1
Ɣ a(N) =
Na
N=1
Ɣ p
a(N) ∪ Ɣu
= a(N)
Na
N=1
Ɣ p
a(N) ∪
Na
Ɣ
N=1
u a(N)
= Ɣp
a ∪ Ɣu a (77)
Here, as well, two parts are distinguished where pressure or displacement boundary conditions
are applied, Ɣ p
a or Ɣu a , respectively. This time they are clearly the Dirichlet (essential) and
Neumann (natural) boundary conditions.
The boundaries of elastic subdomains, Ɣ (N ∈{1,...,Na}), make up the total boundary
a(N)
of elastic domain, i.e.
Ɣ e =
Ne
N=1
Ɣ e(N) =
Ne
N=1
Ɣ u e(N) ∪ Ɣt
= e(N)
Ne
N=1
Ɣ u e(N) ∪
Ne
N=1
Ɣ t e(N) = Ɣu e ∪ Ɣt e
where, again, a part referring to the prescribed displacement (the Dirichlet condition), Ɣu e
a part referring to the prescribed load (the Neumann condition), Ɣt e , are discriminated.
(78)
, and
The piezoelectric domain represents a slightly more complicated case. The total boundary
of the piezoelectric domain, Ɣ pz , consists of the boundaries of all piezoelectric subdomains,
Ɣ pz(N) (N ∈{1,...,Npz}); this time, however, there is a need to distinguish the two independent
subdivisions relevant for the mechanical and electrical conditions:
Ɣ pz =
Npz
N=1
⎧
Npz
⎪⎨
Ɣ
N=1
Ɣ = pz(N)
⎪⎩
u pz(N) ∪ Ɣt
= pz(N)
Npz
Ɣ V pz(N) ∪ ƔQ
= pz(N)
N=1
Npz
N=1
Npz
N=1
Ɣ u pz(N) ∪
Ɣ V pz(N) ∪
Npz
N=1
Npz
N=1
Ɣ t pz(N) = Ɣu pz ∪ Ɣt pz
Ɣ Q pz(N) = ƔV pz ∪ ƔQ pz
Eventually, the essential boundary conditions can formally be written for the coupled system
as follows:
ui =
⎧
⎪⎨ ûi on Ɣu p ,
⎪⎩
ûe i on Ɣu e ,
û pz
i on Ɣu pz ,
p =
ˆp on Ɣ p
p ,
ˆp a on Ɣ p
a ,
V = ˆV pz on Ɣ V pz
(79)
(80)