BEM research
and new developments in Brazil
Ney Augusto Dumont
Civil Engineering Department – PUC-Rio
NSF_BEM_workshop: BEM research and new developments in Brazil
a tiny part of
BEM research
and new developments in Brazil
Ney Augusto Dumont
Civil Engineering Department – PUC-Rio
BEM26 – CONVENTIONAL,
AND
BOUNDARY
ELEMENT METHODS
NSF_BEM_workshop:
BEMHYBRID
research
andSIMPLIFIED
new developments
in Brazil
(A view from this tiny part of
Brazil: my office room
room’s
s window)
NSF_BEM_workshop: BEM research and new developments in Brazil
Prominent BEM-researchers in Brazil
• José Cláudio Faria Telles
And large team at COPPE/UFRJ and
other universities
• Webe João Mansur
• Luiz Wrobel (at Brunel University)
• Euclides de Mesquita Neto
• Paulo Sollero
A UNICAMP
At
• Wilson Sérgio Venturini (passed away last July)
• João Batista de Paiva
• Humberto Breves Coda
• Delfim Soares Júnior – at UFJF
• Francisco Célio de Araújo – at UFOP
NSF_BEM_workshop: BEM research and new developments in Brazil
New ggeneration
At São Carlos
(UESC)
ANNOUNCEMENT
• BETEQ 2011 - XII International Conference on
Boundary Element and Meshless Techniques
12 15 July
12-15
J l 2011,
2011 Brasilia,
B
ili B
Brazil
il
A few days in Rio de Janeiro is mandatory!
NSF_BEM_workshop: BEM research and new developments in Brazil
ANNOUNCEMENT
A few days in Rio de Janeiro is mandatory!
NSF_BEM_workshop: BEM research and new developments in Brazil
• Recently graduated student and works in progress at PUC-Rio
Recently
R
tl concluded
l d d Ph.D
Ph D Thesis:
Th i
• M. F. F. Oliveira, Conventional and simplied-hybrid boundary element
methods applied to axisymmetric elasticity problems in fullspace and
halfspace, PUC
PUC-Rio
Rio (2009). Co-advisor:
Co advisor: Patrick Selvadurai (McGill
University)
p g
Ph.D Theses in progress
• C. A. Aguilar M., Comparison of the computational performance of the
advanced mode superposition technique with techniques that use
numerical Laplace transforms (since September 2008).
• D. Huamán M., Gradient elasticity formulations with the hybrid
boundary element method (since September 2008)
M.Sc.
M
S Th
Thesis
i iin progress
• E. Y. Mamani V., Application of a generalized Westergaard stress
function for the analysis of fracture mechanics problems (since January
2010) Co-advisor: A.
2010).
A A.
A O.
O Lopes
NSF_BEM_workshop: BEM research and new developments in Brazil
(Intended) Contents of this presentation
• Variationally-based, hybrid boundary and finite elements
• Developments for time-dependent problems
• Developments in gradient elasticity
• Dislocation
Dislocation-based
based formulations (for fracture mechanics)
• From the collocation ((conventional of hybrid)
y
) boundary
y
element method to a meshless formulation
Or: The expedite boundary element method
NSF_BEM_workshop: BEM research and new developments in Brazil
Approximations on the boundary
σ ji , j + bi = 0
σ p + bi = 0
ji , j
ui = uim d m
u ip = uim d mp
ti = ti t
t ip = ti t p
d m , d mp : nodal atributes
t , t p : surface atributes
•
NSF_BEM_workshop: BEM research and new developments in Brazil
Fundamental solution
σ *ji , j + Δim pm* = 0
*
*
ui* = uim
pm* + cir ≡ uim
pm* + uisr Csm pm* in Ω
*
*
*
σ ij* = σ ijm
pm* = Dijkp ukm
p
,p
m in Ω
*
ti* = tim
pm* = σ *jim η j pm* on Γ
•
Properties
∫
Ω
•
σ *jim , j dΩ = −δ im
∫
Γ
*
tim
dΓ = −δim
NSF_BEM_workshop: BEM research and new developments in Brazil
The conventional, collocation BEM
H (d − d ) ≅ G (t − t )
p
p
Derived from the Somigliana’s identity
≅
means congruence in terms of weighted residuals (collocation method)
(inherent approximation error due to rigid body displacements!)
dp,t p
or
are already an expedite means of taking a particular solution into account
⎛
⎞
⎛
⎞
∗
∗
⎜ ∫Γ σ jimη j uin dΓ ⎟ d n ≡ H mn d n ≅ Gm t ≡ ⎜ ∫Γ uimti dΓ ⎟ t
⎝
⎠
⎝
⎠
d n = nodal displacement attributes
Where:
uin = displacement interpolation functions
ti = boundary traction force attributes
ti = traction force interpolation functions
∗
σ ∗jim , uim
Stress and displacement expressions of the problem’s fundamental solution
NSF_BEM_workshop: BEM research and new developments in Brazil
Linear algebra properties of H
H+H =I
Ω
p =1
∗
p
p
Ω
η
Γ
Γ
η
Spectral properties:
PACAM 2010 – Foz do Iguaçu, Brazil
Spectral properties of the double layer potential matrix H
Consistent formulation of the BEM
H (d − d
p
) = GP ( t − t )
⊥
R
p
Where the orthogonal projector
P = I − PR = I − R ( R R ) R T
⊥
R
−1
T
R s = ∫ ti u dΓ
with
r
is
Γ
comes from
f
the
h ffact that
h there
h
iis an arbitrary
bi
amount off rigid
i id b
body
d di
displacements
l
iin the
h
fundamental solution
u = u p + c ≡ u p + u Csm p
*
i
*
im
*
m
r
i
*
im
*
m
r
is
NSF_BEM_workshop: BEM research and new developments in Brazil
*
m
in Ω
Literature Review – part I
Hybrid formulations:
Coined by Pian in 1967 “... to signify elements which maintain either equilibrium or compatibility in the
element and then to satisfy compatibility or equilibrium respectively along the interelement boundary”
Parallel developments
Hellinger, E., 1914. Die allgemeinen Ansätze der Mechanik
der Kontinua, Enz. math. Wis, 4, 602-694.
Trefftz, E
Trefftz
E., 1926.
1926 Ein Gegenstück zum
Ritzschen Verfahren. Proc. 2nd International
Congress of Applied Mechanics, Zurich, Switzerland.
Reissner, E. 1950. On a variational theorem in elasticity,
J. Math. Phys., 29, 90-95 .
Hu, H.-C., 1955. On some variational principles in the theory
of elasticity and the theory of plasticity, Scientia Sinica, 4, 33-54
Pian, T. H. H., 1964. Derivation of element stiffness matrices
by assumed stress distribution. AIAA J., 2, 1333
1333-1336
1336
Jirousek, J. & Leon, N., 1977. A powerful finite element for
plate bending, Com. Meths. in Appl. Mech. Engng., 12, 77-96.
Dumont, N. A., 1989. The hybrid boundary element method:
p y
an alliance between mechanical consistencyy and simplicity.
Applied Mechanics Reviews, 42, no. 11, Part 2, S54-S63
Qin, Q
Qin
Q. H.,
H 2003.
2003 The Trefftz Finite and Boundary Element
Method, WITPress.
(and variations)
(Many variations)
Present theoretical investigation
Motivation: we are here
Double layer potential matrix
Mechanical assumptions,
p
,
linear algebra consequences
Theoretical tool:
Displacement virtual work
principle (elastostatics):
PACAM 2010 – Foz do Iguaçu, Brazil
The Ouroboros (from Wikipedia)
Spectral properties of the double layer potential matrix H
The Ouroboros!
Collocation
BEM
((Tail?))
Hybrid
BEM
Hybrid
Displacement
BEM
Expedite
(Head?)
Variational issues
Integration issues
Fundamental solutions
Matrix spectral properties
(Generalized inverses)
Consistency issues
Applications
Simplified
H b id
Hybrid
Galerkin
BEM
Symmetric
Galerkin
BEM
Consistent
Collocation
BEM
(Here and
now)
BEM
(from Wikipedia)
BEM
Dumont, N. A.: "The hybrid boundary element method – fundamentals", in preparation to
be submitted to Engineering Analysis with Boundary Elements
BEM/MRM 32 – The boundary element method revisited
The conventional, collocation BEM
NSF_BEM_workshop: BEM research and new developments in Brazil
The hybrid BEM
Dumont, N.
Dumont
N A.:
A : “Variationally-Based
Variationally-Based, Hybrid Boundary Element Methods
Methods”,
Computer Assisted Mechanics and Engineering Sciences (CAMES) Vol 10 pp 407-430, 2003
NSF_BEM_workshop: BEM research and new developments in Brazil
The hybrid displacement BEM
NSF_BEM_workshop: BEM research and new developments in Brazil
Definition of the matrices involved
H mn := ∫ σ ∗jimη j uin dΓ
Γ
p
Gm := ∫ u t dΓ
Γ
Ω
p∗ = 1
∗
im i
p
Ω
η
F := ∫ σ η u dΓ
∗
mn
Γ
∗
jim
∗
j in
Γ
L m := ∫ uimti dΓ
Γ
Simplified
p
HBEM:
F∗ ← HU∗
NSF_BEM_workshop: BEM research and new developments in Brazil
Γ
η
The Ouroboros!
( d, p )
Some isolated virtual work statements
(p , d
H T p∗ = p ⇔ Hd = d∗
Gt = d
∗
∗
⇔ G p =d
T
L t = p ⇔ Ld = d
T
There are more virtual work statements;
Several restrictions apply!
L
∗
L
( t, d
L
External reference
system
)
)
∗
Internal reference
system
System of Lagrange
multipliers
G and L should be rectangular, in
general;
G should be consistent
Dumont, N.A., An assessment of the spectral properties of the
matrix G used in the boundary element methods,
methods Computational
Mechanics, 22(1), pp. 32-41, 1998
NSF_BEM_workshop: BEM research and new developments in Brazil
The Ouroboros!
( d, p )
Some isolated virtual work statements
(p , d
H T p∗ = p ⇔ Hd = d∗
Gt = d
∗
∗
⇔ G p =d
T
L t = p ⇔ Ld = d
T
There are more virtual work statements;
Several restrictions apply!
∗ ∗
F p =d
L
∗
L
( t, d
L
External reference
system
)
)
∗
Internal reference
system
System of Lagrange
multipliers
G and L should be rectangular, in
general;
G should be consistent
∗
Kd = p
Results at internal points: no integral statement actually required
NSF_BEM_workshop: BEM research and new developments in Brazil
Spectral properties for the hybrid BEM
For a finite domain:
PW
Then,
P = I − PW
HPW = 0 ⇒ H PV = 0
T
Definition of
PV
∗
F PV = 0
−1
T
∗
K = H ( F + PV ) H
For consistency,
and
⊥
W
is the orthogonal projector onto the
space of rigid body displacements and
in
Kd = p
Results at internal points: no integral statement actually required!
p
∗
and
solved from either
∗ ∗
F p = Hd
*
ui* = uim
pm* + cir
or
∗
H p =p
T
*
σ ij* = σ ijm
pm*
NSF_BEM_workshop: BEM research and new developments in Brazil
with
∗
PV p = 0
Some applications already implemented
• 2D and 3D potential and elasticity problems
• 2D and 3D acoustics
• 2D general time-dependent problems (in the
frequency domain):
• Advanced mode superposition analysis
• Use of numerical inverse transforms
• 2D sensitivity analysis
• 2D FGM-analysis for potential problems
• 2D fracture-mechanics analysis – evaluation of stress
intensity factors
• Axysimmetric fullspace and halfspace elastostatics
gradient elasticityy analysis
y
• 2D ((and 3D)) g
NSF_BEM_workshop: BEM research and new developments in Brazil
Example of application of non-singular fundamental solutions: two dimensional
transient heat conduction in a homogeneous square plate
y
u (t ) = 1
(0,1)
u (t ) = 0
(0,0)
u ( x, y,0) = 0
u (t ) = 0
k = 1 Conductivity
c = 1 Specific heat
(1,1)
u (t ) = 1
(1,0)
x
Distorced 4x4 mesh
Temperature results along the edge x = 0 for several time instants
Analytic solution
Classical modal analysis.
analysis
Serendipity elements.
Quadratic elements.
Hybrid formulation
3 mass matrices.
Westergaard’s Stress Function as a Fundamental Solution
- Example 8 – inclined edge crack in a rectangular plate
Stress Intensity Factor
S
4.50
3.50
15
3
π
8
2.50
σ=1
10
3
π
8
KI - Numeric
KII - Numeric
KI - Analytic
1.50
KII - Analytic
25
0.50
3
4
5
6
7
8
9
10
11
12
-0.50
-1.50
Number of Elements Along the Crack
13
14
15
Westergaard’s Stress Function as a Fundamental Solution
Example 3 – semicircular crack in an infinite continuum
1,5
1,3
1,1
Stress intens ity factors
S
0,9
KI - Ratio 1.0
0,7
KII - Ratio 1.0
σ=1
0,5
KI - Analytical
2
KII - Analytical
0,3
KI - Ratio 1.06
KII - Ratio 1.06
0,1
-0,1 0
03
-0,3
-0,5
10
20
30
40
50
60
Num ber of elem ents
70
80
90
100
Westergaard’s Stress Function as a Fundamental Solution
Modeling a curved crack
4
3
2
3
5
4
5
2
y
1
6
1
4
x
3
a
0
2
5
Previous development:
superposition of elipsis
1
Current development:
Superposition of
halfcracks
Inspired
p
by:
y
Tada, H., Ernst, H. A., Paris, P. C. (1993), Westergaard stress functions for displacementprescribed crack problems – I, Int. Journal of Fracture, Vol. 61, pp 39-53
Westergaard’s Stress Function as a Fundamental Solution
26
F
47
36
49
45
A
B
9
14
1
Westergaard’s Stress Function as a Fundamental Solution
U potential along the line segment A‐B due to an external source
‐0.43
26
‐0.44
F ‐0.45
36
‐0.46
47
A
45
49
U potential
B
‐0.47
9
‐0.48
14
‐0.49
1
‐0.5
Analityc
‐0.51
numeric
‐0.52
0
5
10
15
20
25
30
35
Number of point along the line dashed A‐B
40
45
50
Westergaard’s Stress Function as a Fundamental Solution
26
qx gradient along the line segment A‐B due to an external source
0.010
F
0.009
36
47
A
45
49
B
qx gradient
9
0.008
14
0 007
0.007
1
0.006
Analityc
numeric
0.005
0
5
10
15
20
25
30
35
Number of point along the line dashed A‐B
40
45
50
Westergaard’s Stress Function as a Fundamental Solution
Stress intensity factors in terms of p* and J integral - node A
KI numeric / KI ana
alytic
1.10
1.05
1 00
1.00
q=1 0.95
p*2
p*1
0.90
0.2 Analytic
b A
0.85
a1 a2 a3 from p
p*1
1
a
1.0
A
0.5
B
1.0
from J
0.2
0.80
0
20
40
60
Number of elements along
g the crack
80
100
Westergaard’s Stress Function as a Fundamental Solution
Stress intensity factors in terms of p* and J integral - node B
1.10
KI nu
umeric / KI anallytic
1.05
1 00
1.00
q=1 0.95
p*n
p*n‐1
0.90
an‐2
0.85
an‐1
an b Analytic
0.2
B
from p
p*n
n
a
1.0
A
0.5
B
1.0
from J
0.2 0.80
0
20
40
60
Number of elements along
g the crack
80
100
Westergaard’s Stress Function as a Fundamental Solution
Gradient qy along the line segment y=0.20
1.5
1.3
Gradient qyy
1.1
09
0.9
Analytic
q=1
q
1 3 Elements
7 Elements
0.7
4 19 Elements
0.20
0.5
2
2 49 Elements
49 Elements
0.3
99 Elements
0.1
‐0.1
‐2
‐1.5
‐1
‐0.5
0
Points along the dashed line
0.5
1
1.5
2
EXAMPLE: gradient qz along a line segment
0.25
Analytic
0.20
48 elements
0.15
qz
76 elements
0.10
0.05
0.00
-0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
R. A. P. Chaves, “The Hybrid Simplified Boundary Element Method Applied to Time-Dependent Problems”, Ph.D. Thesis, PUC-Rio,
Brazil, 2003
Numerical example
Non-convex axisymmetric volume with multiply connected surfaces
Unitary ring sources (10, -5)
BETEQ 2009 - International Conference on Boundary Element Techniques
Athens, July 2009
An expedite formulation of the BEM
Some numerical assessments for 2D potential problems
sol_9 : u ( x, y ) =
(
ln ( x − x1 ) 2 + ( y − y1 ) 2
4π
)
Numerical and analytical values
of potentials at internal points
H T p∗ = p
V T p∗ = 0
*
ui* = uim
pm* + cir
ICCES 2010 – AN EXPEDITE FORMULATION OF THE BOUNDARY ELEMENT METHOD
An expedite formulation of the BEM
Some numerical assessments for 2D potential problems
sol_9 : u ( x, y ) =
(
ln ( x − x1 ) 2 + ( y − y1 ) 2
4π
*
σ ij* = σ ijm
pm*
Numerical and analytical values
of gradients at internal points
ICCES 2010 – AN EXPEDITE FORMULATION OF THE BOUNDARY ELEMENT METHOD
)
Contents of this presentation
• Variationally-based, hybrid boundary and finite elements
• Developments for time-dependent problems
• Developments in gradient elasticity
• Dislocation
Dislocation-based
based formulations (for fracture mechanics)
• From the collocation ((conventional of hybrid)
y
) boundary
y
element method to a meshless formulation
Or: The expedite boundary element method
NSF_BEM_workshop: BEM research and new developments in Brazil
GENERALIZED
HELLINGER-REISSNER POTENTIAL
∫ (− ∫ (δσ
t1
t0
Ω
(
)(
)
(
)
~ dΩ + δσ η u − u~ dΓ +
−
ρδ
u
u
−
u
,
∫Γ ij j i i
ij j
i
i
i
)
)
+ ∫Ω δu~i σ ijj , j + f i − ρui dΩ − ∫Γ δu~i (σ ijjη j − ti )dΓ dt = 0
~
δδuu i = 0
along
δui = 0
at both time interval extremities t0 and t1
Γu
Literature Review – part I (continued)
Przemieniecki, J.S. (1968). Theory of Matrix Structural Analysis,
Dover Publications, New York
(displacement-based free vibration analysis – truss and beam):
K = K 0 − ω 2M 0 − ω 4 (M 2 − K 4 ) − ω 6 (M 4 − K 6 ) + O (ω 8 )
Voss, H., 1987, A New Justification of Finite Dynamic Element
Methods, Numerical Treatment of Eigenvalue Problems, Vol. 4,
232-242, eds. J. Albrecht, L. Collatz, W. Velte, W. Wunderlich,
Int. Series on Num. Maths. 83, Birkhäuser Verlag, Stuttgart
Gupta, Paz: several other displacement-based
displacement based elements
for plane-state problems. Coined finite dynamic elements,
although they only have been applied to free vibration
analysis (better, then: finite harmonic elements!)
?
K = K 0 − ω 2 (M 0 − K 2 ) − ω 4 (M 2 − K 4 ) − ω 6 (M 4 − K 6 ) + O (ω 8 )
Directly based on Pian and Przemieniecki: General , consistent finite and boundary dynamic
element families: acoustics, free vibration, transient problems via an advanced modal analysis.
Dumont, N
Dumont
N. A
A. & Oliveira
Oliveira, R
R., (1993) 1997
1997. The
exact dynamic formulation of the hybrid
boundary element method, Procs. XVIII
CILAMCE, Brasília, 29 - 31 October, 357-364
Dumont, N.
Dumont
N A.
A & Oliveira,
Oliveira R.,
R 2001.
2001 From frequencydependent mass and stiffness matrices to the
dynamic response of elastic systems. Int. J. Sol.
Struct., 38, 10-13, 1813-1830
Literature Review – part II
Lancaster, P., 2002. Lambda-Matrices & Vibrating Systems, Dover Publications
(from 1966)
(F0 + ω F1 + ω F2 + ...)p = ( H 0 + ω H 1 + ω H 2 + ...)d
2
4
*
2
4
( H 0T + ω 2 H 1T + ω 4 H T2 + ...))p* = p
Dumont, N. A.: “On the Inverse of Generalized Lambda Matrices with Singular Leading Term”, IJNME, Vol 66(4),
571-603, 2006
(K 0 − ω M1 − ω M 2 + ...))d = p(ω)
2
(Przemieniecki)
H
T
0
V = 0
4
F0V = 0
H
T
F
−1
H = K
Nonlinear eigenvalue problem:
Dumont, N. A.: “On the solution of generalized non-linear complex-symmetric eigenvalue problems”, IJNME, Vol 71, 1534-1568,
2007
Sleipjen, G. L. G., Van der Horst, H. A., 1996, A Jacobi-Davidson iteration method for linear eigenvalue problems,
SIAM J. Matrix Anal. Appl., 17, 401-425
Arbenz, P, Hochstenbach, M. E., 2004. A Jacobi-Davidson method for solving complex-symmetric eigenvalue
problems, SIAM J. Sci. Comput. 25, 5, 1655-1673
Problems with viscous damping
n
(
)
K 0Φ − ∑ iC j ΦΩ 2 j −1 + M j ΦΩ 2 j = 0
Nonlinear eigenproblem:
j =1
Augmented
formulation:
⎡ iC1
⎢M
⎢ 1
⎢iC
−⎢ 2
⎢M 2
⎢
⎢
⎢⎣M n
M1
iC 2
M2
iC 3
0
⎡K 0
⎢0
⎢
⎢0
⎢
⎢0
⎢
⎢
⎣⎢ 0
iC 2
0
M1
iC 2
M2
M2
iC 3
iC 3
Mn
M2
0
0
0
0
M2
iC 3
0
Mn ⎤
0 ⎥⎥ ⎡ Φ 00
⎢
0 ⎥ ⎢ Φ10
⎥
0 ⎥⎢
⎥ ⎢⎣Φ n −1, 0
⎥
0 ⎥⎦
0 ⎤
M n ⎥⎥ ⎡ Φ 00
⎢
0 ⎥ ⎢ Φ10
⎥
0 ⎥⎢
⎥ ⎢⎣Φ n −1, 0
⎥
0 ⎥⎦
Φ 01
Φ11
Φ n −1,1
…
Φ 01
Φ11
Φ n −1,1
…
Φ 0,n −1 ⎤
Φ1,n −1 ⎥⎥
⎥
⎥
Φ n −1,n −1 ⎦
Φ 0,n −1 ⎤ ⎡Ω 0
Φ1,n −1 ⎥⎥ ⎢ 0
⎢
⎥⎢
⎥⎢
Φ n −1,n −1 ⎦ ⎣ 0
0 …
Ω1 …
0
0 ⎤
0 ⎥⎥
=0
⎥
⎥
Ω n −1 ⎦
2j
⎛ 2 j k −2 T
2 j −k
2 j −k ⎞
k −1 T
⎜
⎟⎟ = I
Ω
Φ
i
C
ΦΩ
+
Ω
Φ
M
ΦΩ
∑
∑
j
j
⎜∑
j =1 ⎝ k = 2
k =1
⎠
2 j −1
n
⎛ 2 j −2 k T
⎞
T
2 j − k −1
Φ K 0Φ + ∑ ⎜⎜ ∑ Ω Φ iC j ΦΩ
+ ∑ Ω k Φ TM j ΦΩ 2 j − k ⎟⎟ = Ω
k =1
j =1 ⎝ k =1
⎠
n
Orthogonality
properties:
0
iC 2
M2
iC 3
Summary
2i
∂
d
i
K 0d − ∑ ( −1) M i 2i = p(t )
∂t
i =1
n
Equation in the time domain
e. g.
(4)
∂2d
∂4d
∂ 6d
K 0d + M 1 2 − M 2 4 + M 3 6 = p(t )
∂t
∂t
∂t
(Set of coupled, higher-order differential equations)
from
d = Φη
Φ
Mode superposition :
Ω η +η = Φ p
2
T
((Set of uncoupled,
p , second-order differential equations)
q
)
(K (ω ) − ω M(ω ))ϕ = 0
2
(15)
(17)
Frequency-domain formulation
σ ji , j +bi + ρk 2ui = 0
k 2 = ω 2 + 2iζω
where
Solution:
(Laplace/Fourier transforms or)
Advanced mode-superposition technique
St t l dynamics
Structural
d
i
(
)
(
Ω 2 η − ηb + η − ηb = Φ T p − p b
Diffusion-type problems
(
)
(
Ω η − ηb + η − ηb = Φ T p − p b
Structural dynamics with viscous damping
(
) (
)
(
)
)
Ω η−η −i η− η = Φ p −p
b
b
T
b
)
Initial displacements and velocities
ηel
= [Φ
T
el
K 0 Φ el
]
−1
Φ K 0d
T
el
ηrig = Φ M1d
T
rig
In case of viscous damping:
Φ 2 ⎤ ⎧ η1 ⎫
⎧ d ⎫ ⎡ Φ1
⎨ ⎬=⎢
⎨ ⎬
⎥
⎩id ⎭ ⎣Φ1Ω1 Φ 2Ω 2 ⎦ ⎩η2 ⎭
⎡ Φ1T K 0 Φ1
Φ1T K 0 Φ 2 ⎤ ⎧ η1 ⎫ ⎧ Φ1T K 0 d ⎫
⎬
⎢ T
⎥⎨ ⎬ = ⎨ T
T
⎣Φ1 K 0 Φ1Ω1 Φ1 K 0 Φ 2 Ω 2 ⎦ ⎩η2 ⎭ ⎩iΦ1 K 0 d ⎭
EVALUATION OF RESULTS AT INTERNAL POINTS
[
∗
p (ω) = S(ω) d (ω) − d (ω)
b
]
S(ω ) = F −1 (ω ) H (ω )
in which
replaced
l d by
b
n
[
p (ω) ≈ ∑ ω S i d (ω) − d (ω)
∗
2i
b
i =0
n
(
p (t ) = ∑ S i ΦΩ 2i η − ηb
∗
i =0
)
]
⎛
⎞
2i
ω S i = ⎜ ∑ ω Fi ⎟
∑
i =0
⎝ i =0
⎠
n
with
2i
n
−1 n
2i
ω
∑ Hi
i =0
EVALUATION OF RESULTS AT INTERNAL POINTS
(
n
p ∗ (t ) = ∑ S i ΦΩ 2i η − ηb
)
m
n
j =0 i =0
i =0
n
∗
i
∗
i =0
(
i
u(t ) = ∑∑ u ∗j S i - j ΦΩ 2i η − ηb
for n = 3
[
n
u(t ) = ∑∑ ω u p ≡ ∑ u ∗i Ω 2i p ∗
2i
j
i =0 j =0
(
)
(
) ΦΩ ] η
)
u(t ) = u ∗0S 0Φ + u ∗0S1 + u1∗S 0 ΦΩ 2 + u ∗0S 2 + u1∗S1 + u ∗2S 0 ΦΩ 4 +
(
+ u ∗0S 3 + u1∗S 2 + u ∗2S1 + u ∗3S 0
S is required for
6
∗
−1
p = F Hd ≡ Sd
)
Example 1: Initial velocity – Displacement at point A
0.0045
ρ = 7,850 kg/m3
E = 210,000 Mpa
(10, 3)
(10
A
0.0005
v0 = -2.5
2 5 m/s
/
10 m
0.0025
Analytic
30 m
34 elements
-0.0015
-0.0035
-0.0055
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
59 equally
ll spaced,
d
linear boundary elements
35 m
g = -9.81
9.81 m/s2
4g sin (k n x ) [1 − cos(ωn t )]
2
(
)
2
n
−
1
π
ω
n =1
n
∞
u=∑
(20, 30)
50 m
ρ = 7,850 kg/m3
E = 210,0000 Mpa
Transient Analysis – Example 2: Gravitational force
ωn = k n E
(0 15)
(0,
kn
(25, 10)
(15, 0)
(35, 0)
υ= 0 (can be solved in the frame of the theory of potential)
ρ
(
2n − 1) π
=
2L
Example 2: Gravitational force – Displacement at point A
0.00002
(0, 15)
-0.00028
0 00028
g = -9.81 m/s2
(20, 30)
50 m
-0.00018
35 m
ρ = 7,8500 kg/m3
E = 210,000 Mpaa
-0.00008
(10, 15)
A
Analytic
((25,, 10))
59 elements
(15, 0)
(35, 0)
-0.00038
-0.00048
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Example 2: Gravitational force – Displacements along a line segment
-0.00001
t 0 004
t=0.004
-0.00011
t=0.008
-0.00021
-0.00051
t=0.012
g = -9.81
9 81 m/s
/2
(20, 30)
(20, 26.2)
50 m
-0 00041
-0.00041
35 m
ρ = 7,850 kg/m3
E = 210,0000 Mpa
-0.00031
t=0.016
(0, 15)
Analytic
(25, 10)
-0.00061
(15, 0)
(20, 1)
59 elements
(35, 0)
t=0.02
-0.00071
0
5
10
15
20
25
Plane frame with 6 members and 12 d.o.f.
submitted
b itt d tto a ttriangular
i
l pulse
l
5
2
2
L
P(t)
4
1
6
1
3
P(t)
1
P1
5
3
3L
0
8
3
7
t1
t2
t
11
2
2 P(t)
4
10
9
12
4
6
3L
y
6
5
(Weaver, W., Jr., and Johnston, P. R.,
1987. Structural Dynamics by Finite
Elements, Prentice-Hall Inc., Englewood
Cliffs, New Jersey )
x
L
L
3L
L
2L
z
•
NSF_BEM_workshop: BEM research and new developments in Brazil
Plane frame with 6 members and 12 d.o.f.
submitted
b itt d tto a ttriangular
i
l pulse
l
Displacement results with 1 and 4 mass matrices, as
compared with values given by Weaver and Johnston.
Eigenvalues obtained for the problem with 1, 2, 3 and 4
mass matrices.
NSF_BEM_workshop: BEM research and new developments in Brazil
Contents of this presentation
• Variationally-based, hybrid boundary and finite elements
• Developments for time-dependent problems
• Developments in gradient elasticity
• Dislocation
Dislocation-based
based formulations (for fracture mechanics)
• From the collocation ((conventional or hybrid)
y
) boundaryy
element method to a meshless formulation
Or: The expedite boundary element method
NSF_BEM_workshop: BEM research and new developments in Brazil
A Family of 2D and 3D Hybrid Finite Elements for
Strain Gradient Elasticity
Ney Augusto Dumont
Daniel Huamán Mosqueira
e-mail: dumont@puc-rio.br
BETeq – International Conference on Boundary Element & Meshless Techniques XI
12-14 July 2010, Berlin
Conclusions
(1)
σ ji = τ ji − g τ ji ,kk
2
Total stress
•
The hybrid BEM / FEM (a two-field formulation) is a natural variational tool to deal with
gradient elasticity
•
Singular and nonsingular fundamental solutions were either redeveloped for BEM or originally
developed for FEM gradient elasticity implementations
•
General families of finite elements were obtained
•
Extension to time
time-dependent
dependent problems in the frequency domain is straightforward
(in progress – already done for truss and beam elements)
•
Simple implementations for truss and beam elements: was a fruitful apprenticeship
((disagreement
g
with some results in the literature - symmetry
y
y and representation
p
of constant
strain state)
•
Treatment of the normal displacement gradient on Г is different from Mindlin’s
proposition (our results still remain to be validated)
•
Numerical examples for 2D problems are being implemented
BETeq – International Conference on Boundary Element & Meshless Techniques XI
12-14 July 2010, Berlin
Conclusions
(2)
σ jji = τ jji − g τ jji ,kk
2
Total stress
•
•
•
.
.
.
•
Treatment of the normal displacement gradient on Г is different from Mindlin’s
proposition (our results still remain to be validated)
•
Numerical examples for 2D problems are being implemented:
•
Orthogonality to rigid body displacements: OK!
•
Symmetry of the flexibility matrix F
F*: OK!
•
Patch tests for linear displacements fields: OK!
•
Patch tests for non-linear displacement fields: convergence still questionable.
BETeq – International Conference on Boundary Element & Meshless Techniques XI
12-14 July 2010, Berlin
Contents of this presentation
• Variationally-based, hybrid boundary and finite elements
• Developments for time-dependent problems
• Developments in gradient elasticity
• Dislocation
Dislocation-based
based formulations (for fracture mechanics)
• From the collocation ((conventional or hybrid)
y
) boundaryy
element method to a meshless formulation
Or: The expedite boundary element method
NSF_BEM_workshop: BEM research and new developments in Brazil
Conclusions
Expedite formulation of the boundary element method (in progress)
• Implementation extremely advantageous for:
• large problems
• problems with complicated fundamental solutions (time-dependent,
axisymmetric, for gradient elasticity, etc)
• Numerical accuracy still under investigation, but comparable to the
conventional BEM
•Evaluation of results at internal points requires no further integrations
•However, results close to the boundary require the knowledge of the null
space V ((which
hi h may b
be obtained
b i d via
i G
Gauss-Seidel
S id l iiteration
i in
i principle
i i l
with a simple pre-conditioning of HT)
Work in progress
•Implementation in Fortran for large 2D potential and elasticity
problems in the frequency domain
Implementation for strain gradient elasticity
•Implementation
ICCES 2010 – AN EXPEDITE FORMULATION OF THE BOUNDARY ELEMENT METHOD