288
A. Wittek et al.
In both the finite element TLED algorithm described in Chap. 10 and in the
MTLED framework discussed here, lumped mass matrices are used. Physical
interpretation of such matrices is that the system mass is assigned entirely (lumped)
to the nodes. In the MTLED framework, the mass allocated to the integration point I
is distributed equally to all nodes within the support domain of that integration point
[64]. Therefore, Eq. (11.20) can be rewritten as [64]
λ
I
max =
N I
m I sup
u
u T K I u
u T u
=
N I
m I ρ max
K
I
,
(11.21)
where N I is the number of nodes in the support domain of the integration point I,
m I is the mass allocated to the integration point I and ρ max (K I ) is the maximum
eigenvalue of the stiffness matrix K I for the integration point I.
The stiffness matrix is defined in terms of the strain-displacement matrices B I for
a given integration point I, elasticity matrix C that contains the information about
the constitutive properties and volume V I allocated to the integration point I [64,
66]:
K
I
iJ lK = B
I
jJ
x
I
C ij kl B
I
lK
x
I
• V
I ,
(11.22)
where the subscripts indicate the tensor order, i.e. ijkl and iJlk indicate the fourthorder tensor and jJ and lK the second-order tensor. For the homogenous materials
with the constitutive properties defined using Lame constants (λ and μ), the
maximum eigenvalue of the stiffness matrix ρ max (K I ) for the integration point I
can be estimated as [66]
ρ max
K
I
≤ (λ + μ) • V
I
•
B
I
2
F
,
(11.23)
where B I F is Frobenius norm
B I
2
F
= B jI B jI . Substituting Eq. (11.23) into
Eq. (11.21) leads to the following formulae for the upper bounds of the maximum
eigenvalues of the stiffness matrix for the integration point I [64]:
λ
I
max ≤ N
I
c
I
2 • B
I
jI B
I
jI ,
(11.24)
where c is the dilatational (acoustic) wave speed. By substituting Eq. (11.24)
into Eq. (11.18), the critical time step t crit for the MTLED framework can be
conservatively estimated as
t crit ≈ Min I
⎛
⎝
2
N I
c I
2 • B I
jI B I
jI
⎞
⎠ .
(11.25)
A. Wittek et al.
In both the finite element TLED algorithm described in Chap. 10 and in the
MTLED framework discussed here, lumped mass matrices are used. Physical
interpretation of such matrices is that the system mass is assigned entirely (lumped)
to the nodes. In the MTLED framework, the mass allocated to the integration point I
is distributed equally to all nodes within the support domain of that integration point
[64]. Therefore, Eq. (11.20) can be rewritten as [64]
λ
I
max =
N I
m I sup
u
u T K I u
u T u
=
N I
m I ρ max
K
I
,
(11.21)
where N I is the number of nodes in the support domain of the integration point I,
m I is the mass allocated to the integration point I and ρ max (K I ) is the maximum
eigenvalue of the stiffness matrix K I for the integration point I.
The stiffness matrix is defined in terms of the strain-displacement matrices B I for
a given integration point I, elasticity matrix C that contains the information about
the constitutive properties and volume V I allocated to the integration point I [64,
66]:
K
I
iJ lK = B
I
jJ
x
I
C ij kl B
I
lK
x
I
• V
I ,
(11.22)
where the subscripts indicate the tensor order, i.e. ijkl and iJlk indicate the fourthorder tensor and jJ and lK the second-order tensor. For the homogenous materials
with the constitutive properties defined using Lame constants (λ and μ), the
maximum eigenvalue of the stiffness matrix ρ max (K I ) for the integration point I
can be estimated as [66]
ρ max
K
I
≤ (λ + μ) • V
I
•
B
I
2
F
,
(11.23)
where B I F is Frobenius norm
B I
2
F
= B jI B jI . Substituting Eq. (11.23) into
Eq. (11.21) leads to the following formulae for the upper bounds of the maximum
eigenvalues of the stiffness matrix for the integration point I [64]:
λ
I
max ≤ N
I
c
I
2 • B
I
jI B
I
jI ,
(11.24)
where c is the dilatational (acoustic) wave speed. By substituting Eq. (11.24)
into Eq. (11.18), the critical time step t crit for the MTLED framework can be
conservatively estimated as
t crit ≈ Min I
⎛
⎝
2
N I
c I
2 • B I
jI B I
jI
⎞
⎠ .
(11.25)