The document describes a method for coupling isogeometric analysis (IGA) plate models with 3D finite element models using a discontinuous Galerkin method. The method allows for mixed-dimensional analysis, with plate/shell descriptions used in some areas and full 3D models used in "hot-spots" of high stress. The coupling is done efficiently using minimal data transfer from the CAD model for the IGA analysis. Numerical examples are presented to demonstrate the non-conforming coupling of an IGA plate with an embedded 3D solid model.
The predominant technology that is used by CAD (Com- puter Aided Design) to represent complex geometries is non-uniform rational b-splines (NURBS). This allows cer- tain geometries to be represented exactly including conic and circular sections. There is a vast array of literature focused on NURBS and as a result of several decades f research, many efficient computer algorithms exist for their fast evaluation and refinement. The key concept outlined by Hughes et al. was to employ NURBS not only as a geometry discretisation technology, but also as a discretisation tool for analysis, attributing such methods to the field of ‘isogeometric analysis’ (IGA).
This Report presents a review of the definition of the equivalent crack concept by defining the relationship between two correlated theories, which are fracture mechanics and damage mechanics based on Mazars static damage model and derived in the framework of the thermodynamics of irreversible processes, an energetic equivalence between the two descriptions is proposed. This paper also presents an example of combined damage and fracture calculation on a concrete specimen, the energy consumption during crack propagation, modelled with damage mechanics, is computed. Finally, the paper provide a comparison for the fracture energy according to the damage model with experiments and linear elastic fracture mechanics
The predominant technology that is used by CAD (Com- puter Aided Design) to represent complex geometries is non-uniform rational b-splines (NURBS). This allows cer- tain geometries to be represented exactly including conic and circular sections. There is a vast array of literature focused on NURBS and as a result of several decades f research, many efficient computer algorithms exist for their fast evaluation and refinement. The key concept outlined by Hughes et al. was to employ NURBS not only as a geometry discretisation technology, but also as a discretisation tool for analysis, attributing such methods to the field of ‘isogeometric analysis’ (IGA).
This Report presents a review of the definition of the equivalent crack concept by defining the relationship between two correlated theories, which are fracture mechanics and damage mechanics based on Mazars static damage model and derived in the framework of the thermodynamics of irreversible processes, an energetic equivalence between the two descriptions is proposed. This paper also presents an example of combined damage and fracture calculation on a concrete specimen, the energy consumption during crack propagation, modelled with damage mechanics, is computed. Finally, the paper provide a comparison for the fracture energy according to the damage model with experiments and linear elastic fracture mechanics
Transient response of delaminated composite shell subjected to low velocity o...University of Glasgow
Transient dynamic response of delaminated composite shell subjected to low velocity oblique impact - a finite element method is proposed and new results are discussed
First order shear deformation (FSDT) theory for laminated composite beams is used to study free vibration of
laminated composite beams, and finite element method (FEM) is employed to obtain numerical solution of the
governing differential equations. Free vibration analysis of laminated beams with rectangular cross – section for
various combinations of end conditions is studied. To verify the accuracy of the present method, the frequency
parameters are evaluated and compared with previous work available in the literature. The good agreement with
other available data demonstrates the capability and reliability of the finite element method and the adopted beam
model used.
Three dimensional static analysis of two dimensional functionally graded platesrtme
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional
theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The
effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated.
The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it
shows very good agreement
Three dimensional static analysis of two dimensional functionally graded platesijmech
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated. The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it shows very good agreement.
Transient response of delaminated composite shell subjected to low velocity o...University of Glasgow
Transient dynamic response of delaminated composite shell subjected to low velocity oblique impact - a finite element method is proposed and new results are discussed
First order shear deformation (FSDT) theory for laminated composite beams is used to study free vibration of
laminated composite beams, and finite element method (FEM) is employed to obtain numerical solution of the
governing differential equations. Free vibration analysis of laminated beams with rectangular cross – section for
various combinations of end conditions is studied. To verify the accuracy of the present method, the frequency
parameters are evaluated and compared with previous work available in the literature. The good agreement with
other available data demonstrates the capability and reliability of the finite element method and the adopted beam
model used.
Three dimensional static analysis of two dimensional functionally graded platesrtme
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional
theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The
effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated.
The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it
shows very good agreement
Three dimensional static analysis of two dimensional functionally graded platesijmech
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated. The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it shows very good agreement.
Tree roots are anchors that prevent blowdown and mechanically reinforce soils. Roots on the windward side of trees have different strengths if compared with those on the leeward side; the difference in strength is attributable to the shape, the diameter or the cross sectional area and the number of roots.
This project characterises root behaviour using fracture mechanics and histological approaches from wood science to examine root tissue structure in relation to mechanical behaviour. On the windward side of trees, roots were hypothesised to be tougher, stronger and have greater Young’s modulus than roots on the leeward side due to thigmomorphogenesis. This means that erratic changes in wind direction would be expected to exacerbate windfall damage.
Root samples from windblown trees with different direction of fall were mechanically loaded in either tensile, compression or bending to assess different failure mechanisms that affect root systems during windthrow.
Advances in fatigue and fracture mechanics by grzegorz (greg) glinkaJulio Banks
Professor Grzegorz (Greg) Glinka has made substantial contributions to the field of stress concentration evaluation using linear FEA results using the ESED (Equivalent Striain Energy Density). ESED aka Glinka methods allows the determination of strain-stress state at a point of local concentration by equating the strain energy from the linear FEA area in the material strain-stress curve to that of the actual strain-stress of the material using a models such as Ramberg-Osgood. The ESED method is more accurate than the Neuber requiring the equating of SED (Strain Energy Densities) of linear FEA results that Stress is proportional to strain even when the FEA predicts a stress greater than the ultimate strength of the material. One easy method of remember when to use ESED versus Neuber is that ESED, more accurate, should be use on the stress analysis of rocket structures and Neuber delegated to aerospace engines and components.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
My name is Spenser K. I am associated with mechanicalengineeringassignmenthelp.com for the past 12 years and have been helping the mechanical engineering students with their Microelectromechanical Assignment. I have a Ph.D. in Mechatronics Engineering from RMIT University Australia.
Effect of lamination angle on maximum deflection of simply supported composit...RAVI KUMAR
In this project a composite laminated beam is studied with glass-epoxy and graphite-epoxy combination. The beam is composed of four layers of different combination of composite material (glass epoxy and graphite epoxy composite). The beam is simply supported at both the ends and is subjected to uniformly distributed load along the length. Transverse deflection is computed for different lamination angle (0^0-〖90〗^0) by using Euler- Bernoulli’s theory (or CLPT). Maximum transverse deflection analysis is carried out using derived analytical expressions. The research carried out in this project will enable to determine the beam strength due to bending loads. The importance of fibre reinforcement in the manufacturing of the beam is studied in terms of bending strength of the beam. MATLAB codes are generated to implement analytical expiations of the composite beam.
The main objective of the paper is to find out the lamination angle at which minimum deflection is obtained & to find out the effect of lamination angle on maximum transverse deflection of the beam.
Comparative study of results obtained by analysis of structures using ANSYS, ...IOSR Journals
The analysis of complex structures like frames, trusses and beams is carried out using the Finite
Element Method (FEM) in software products like ANSYS and STAAD. The aim of this paper is to compare the
deformation results of simple and complex structures obtained using these products. The same structures are
also analyzed by a MATLAB program to provide a common reference for comparison. STAAD is used by civil
engineers to analyze structures like beams and columns while ANSYS is generally used by mechanical engineers
for structural analysis of machines, automobile roll cage, etc. Since both products employ the same fundamental
principle of FEM, there should be no difference in their results. Results however, prove contradictory to this for
complex structures. Since FEM is an approximate method, accuracy of the solutions cannot be a basis for their
comparison and hence, none of the varying results can be termed as better or worse. Their comparison may,
however, point to conservative results, significant digits and magnitude of difference so as to enable the analyst
to select the software best suited for the particular application of his or her structure.
Transient three dimensional cfd modelling of ceilng fanLahiru Dilshan
Ceiling fans are used to get thermal comfort, especially in tropical countries. With the increment of the usage of air conditioners, the emission of CO2 is increased. But ceiling fans are a limited solution, that saves much energy compared to air conditioners. Ceiling fans generate a non-uniform velocity profile, so that, there is a non-uniform thermal environment. That non-uniform environment does not imply lower thermal comfort, that will give enough thermal comfort with low energy cost by air velocity. Hence, there will be difficulties of analysing with simple modelling techniques in that environment. So, to predict the performance of the ceiling fan required more accurate models.
The accurate model of a ceiling fan will generate complex geometry that makes difficulties for the simulation process and requires higher computational power. Because of that, there are several methods used to predict the performance of the ceiling fan using mathematical techniques but that will give an estimated value of properties in the surrounding.
Similar to Coupling_of_IGA_plates_and_3D_FEM_domain (20)
Transient three dimensional cfd modelling of ceilng fan
Coupling_of_IGA_plates_and_3D_FEM_domain
1. Ins$tute
of
Mechanics
&
Advanced
Materials
Coupling of IGA plates and 3D FEM domains by
a Discontinuous Galerkin Method
V.P.
Nguyen1,
P.
Kerfriden1,
S.
Claus2,
S.P.-‐A.
Bordas1
1School
of
Engineering,
Cardiff
University,
UK
2Department
of
Mathema$cs,
University
College
London,
UK
2. Ins$tute
of
Mechanics
&
Advanced
Materials
Aim:
local/global
analysis
of
thin
panels
http://www.supergen-wind.org.uk
Stress
analysis
with
minimium
data
transfer
from
CAD
model
Hot-‐spot
(stress
concentra$ons,
damage)
3. Ins$tute
of
Mechanics
&
Advanced
Materials
• Mixed-‐dimensional
analysis:
§ use
shell/beam
descrip$ons,
homogenisa$on
§ Full
microscale
3D
in
“hot-‐spots”
• Isogeometric
analysis:
minimum
CAD
to
analysis
data
processing
• Efficient
coupling
of
heterogeneous
models
/discre$sa$on
• (efficient
local/global
solver)
• (find
“hot-‐spots”
with
goal-‐oriented
model
adap$vity)
Building
blocks
Figure 30: Cantilever beam subjects to an end shear force: discretisation of the solid
Figure 31: Cantilever beam subjects to an end shear force: von Mises stress di
6.2.3. Non-conforming coupling of a square plate
We consider a square plate of dimension L ⇥ L ⇥ t (t denotes the thickness) in which there
dimension Ls ⇥ Ls ⇥ t as shown in Fig. 32. In the computations, material properties are taken
the geometry data are L = 400, t = 20 and Ls = 100. The loading is a gravity force p = 10
is fully clamped. The stabilisation parameter was chosen empirically to be 1 ⇥ 106
. We us
NURBS plate elements for the plate and NURBS solid elements for the solid. In order to m
[Nguyen
et
al.
2013]
6.2.2. Cantilever plate: non-conforming coupling
A mesh of 32 ⇥ 4 ⇥ 5/ 32 ⇥ 2 cubic elements is utilized for the mixed dimensional model, cf. F
the continuum part in the continuum-plate model is 175 mm. The contour plot of the von Mises s
where void plate elements were removed in the visualisation.
Figure 30: Cantilever beam subjects to an end shear force: discretisation of the solid an
Figure 31: Cantilever beam subjects to an end shear force: von Mises stress distrib
6.2.3. Non-conforming coupling of a square plate
We consider a square plate of dimension L ⇥ L ⇥ t (t denotes the thickness) in which there is a
CAD
model
Analysis
mesh
IGA
/
plate
Solid
FE
???
4. Ins$tute
of
Mechanics
&
Advanced
Materials
Figure 8. Stress contours in 3D–2D mixed-dimensional cantilever model loaded by a terminal
shear force Fz. (2D contours illustrated relate to top surface of model). Abaqus C3D20R brick
elements and S8R shell elements.
Figure 9. Transverse shear stresses 13 ( xz) obtained by method of Reference [5].
Copyright ? 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 49:725–750
• Reference
coupling
§ displacement-‐recovery,
Stress-‐recovery
§ Equality
of
work
provides
coupling
on
dual
quan$ty
• Discrete
treatment
§ Mul$-‐point
constraints
[Monaghan et al 1998,
McCune et al. 2000, Shim et al. 2002, Song et al. 2012]
§ Transi$on
elements
[Surana 1979, Cofer 1991,
Gmur et al, 1993, Dohrmann et al. 1999, Wagner et al. 2000,
Garusi et al. 2002, Chavan et al. 2004]
§ Mortar
methods
- Penalty
formula$ons
[Blanco et al. 2007]
- Lagrange
mul$plier-‐based
mortar
[Rateau et al. 2003, Combescure et al. 2005]
- Hybrid
itera$ve
method
[Guguin et al. 2013]
Some
coupling
methods
in some situations.
3 Finite element formulation
Based on the above described kinematical assumptions the element is d
node in the transition cross–section is called ’reference node’. Furtherm
A2 and A3 define the orientation of the cross section. It is assumed th
couple (’coupling nodes’) lie in this plane. The vectors A2 and A3 are
section coordinates, see eq. (3). In the current configuration the base
the beam element together with the convective coordinates (0, ξ2, ξ3) a
define the coupling nodes.
The mechanical model of the cross section can be considered as a sum
allow only for axial deflections. The boundary conditions are clamped a
and jointed at the coupling node, see Fig. 3.
clamped bounded
rigid beam, axial free
hinged bounded
Transition elements
Fig. 3: Transition elements in a beam cross–section
The implementation of the constraint equation (7) in a transition elem
Penalty and the Augmented Lagrange Method. Furthermore a consi
derived for the element with respect to finite rotations. The transition is
Adapted
from
[Wagner
et
al.
2000]
lumique, qui occupent respectivement l’adh´erence des ouverts conn
commodit´e, nous d´esignons par !coq la surface moyenne du premier
sous-domaine correspondant de !0
. En outre, comme au §5.1.2.1, n
voisinage de la condition d’encastrement est repr´esent´e par le mod`ele
ωcoq
ω3d
sc
Fig. 5.8 – Mod´elisation Arlequin
Les relations de comportement sont celles des paragraphes 5.1.1.3 et 5.
champs de d´eplacement cin´ematiquement admissibles du mod`ele trid
par (5.17), tandis que celui du mod`ele coque est donn´e par l’expressio
W coq =
n
vcoq = v0
+ ⇠3(v1
⌧1 + v2
⌧2) ; v0
2 H1
(!0
coq), v1
, v2
2 H1
(!
106
[Rateau
et
al.
2003]
[McCune
et
al.
2000]
5. Ins$tute
of
Mechanics
&
Advanced
Materials
• Introduc$on
• Automa$c
coupling
§ Problem
statement
§ IGA
§ Discrete
coupling
strategy
• Numerical
examples
• Conclusion
6. Ins$tute
of
Mechanics
&
Advanced
Materials
§ Kinema$cs:
§ Equilibrium:
§ Cons$tu$ve
rela$on:
§ Primal
vibra$onal
formula$on:
Problem
statement:
uncoupled
solid
Figure 2: Coupling of a two dimensional solid and a beam.
as
(us
, us?
) :=
Z
⌦s
✏(u) : Cs
: ✏(us?
) d⌦ = ls
(us?
)
KA0
Z
⌦s
s
: ✏s
(us?
) d⌦ =
Z
⌦s
b · us?
d⌦ +
Z
t
¯t · us?
d
s
= Cs
: ✏s
in ⌦s
✏s
=
1
2
(rus
+ rT
us
)
us
= ¯u on s
u
7. Ins$tute
of
Mechanics
&
Advanced
Materials
§ Kinema$cs:
§ Equilibrium:
§ Cons$tu$ve
rela$on:
§ Primal
VF:
Problem
statement:
uncoupled
beam
Figure 2: Coupling of a two dimensional solid and a beam.
Z
⌦b
✓
N
M
◆
·
✓
v?
,¯x
w?
,¯x¯x
◆
dl =
Z
⌦b
0
@
¯p,¯x
¯p,¯y
¯m
1
A ·
0
@
v?
w?
w?
,¯x
1
A dl +
X
P 2Pntm
0
@
¯N
¯T
¯M
1
A
|P
·
0
@
v?
w?
w?
,¯x
1
A
|P
✓
N
M
◆
=
✓
ES 0
0 EI
◆ ✓
v,¯x
w,¯x¯x
◆
in ⌦b
v = ¯v on b
v
w = ¯w on b
w
w,¯x = ¯✓ on b
✓
ab
(⇥, ⇥b?
) :=
Z
⌦b
✓
v?
,¯x
w?
,¯x¯x
◆T ✓
ES 0
0 EI
◆
·
✓
v,¯x
w,¯x¯x
◆
dl = lb
(⇥b?
)
KA0
8. Ins$tute
of
Mechanics
&
Advanced
Materials
• Solid:
• EB-‐Beam,
use
as
test
and
trial
in
2D
VF
Primal
coupling
(strong/weak)
as
(us
, us?
) = ls
(us?
) +
Z
?
us?
· ( s
(us
) · ns
) d
ab
(⇥b
, ⇥b?
) = lb
(⇥b?
) +
Z
?
ub
(⇥b?
) · ( b
(⇥b
) · nb
) d
ub
(⇥b
) =
✓
v w,¯x ¯y
w
◆
¯R
9. Ins$tute
of
Mechanics
&
Advanced
Materials
• Solid:
• EB-‐Beam,
use
as
test
and
trial
in
2D
VF
• Primal
coupling
§ Kinema$cs:
- For
us:
§ V.
Work
equality
for
any
KA
field:
Primal
coupling
(strong/weak)
as
(us
, us?
) = ls
(us?
) +
Z
?
us?
· ( s
(us
) · ns
) d
ab
(⇥b
, ⇥b?
) = lb
(⇥b?
) +
Z
?
ub
(⇥b?
) · ( b
(⇥b
) · nb
) d
Choice
for
space?
For
us
Z
?
ub?
· ( s
(us
) · ns b
(⇥b
) · ns
) d = 0
ub
(⇥b
) =
✓
v w,¯x ¯y
w
◆
¯R
Figure 2: Coupling of a two dimensional solid and a beam.
Z
?
us
ub
(⇥) · ?
d
us
= ub
(⇥)
10. Ins$tute
of
Mechanics
&
Advanced
Materials
• Introduc$on
• Automa$c
coupling
§ Problem
statement
§ IGA
§ Discrete
coupling
strategy
• Numerical
examples
• Conclusion
12. Ins$tute
of
Mechanics
&
Advanced
Materials
Descrip$on
of
geometry
by
B-‐splines
00.10.20.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.05
N3,3(ξ)
M3,3(η)
Figure 4: A bivariate cubic B-spline basis function with knots vectors Ξ = H = {0, 0, 0, 0, 0.25, 0.5
ξ
η
x
y
z
(ξ, η)
0, 0, 0
0,0,0
1, 1, 1
1,1,1
0.5
0.5
Figure 5: A bi-quadratic B-spline surface (left) and the corresponding parameter space (right).
Ξ = H = {0, 0, 0, 0.5, 1, 1, 1}. The 4 × 4 control points are denoted by red filled circles.
12
⌅ = {⇠1, ⇠2, . . . , ⇠n+p+1}
x(⇠) =
nX
i
Ni,p(⇠) Bi
x(⇠, ⌘) =
nX
i
mX
j
Ni,p(⇠)Mj,p(⌘) Bij
13. Ins$tute
of
Mechanics
&
Advanced
Materials
IsoGeometric
Analysis
(IGA)
−0.5 0 0.5 1
−0.5
0
0.5
1
1.5
00.511.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
ariate cubic B-spline basis function with knots vectors Ξ = H = {0, 0, 0, 0, 0.25, 0.5, 0.75, 1, 1, 1, 1}.
ξ
η
x
y
z
(ξ, η)
0, 0, 0
0,0,0
1, 1, 1
1,1,1
0.5
0.5
uadratic B-spline surface (left) and the corresponding parameter space (right). Knot vectors are
0.5, 1, 1, 1}. The 4 × 4 control points are denoted by red filled circles.
12
⌅1
= {0, 0, 0, 0.5, 1, 1, 1}
⌅2
={0,0,0,0.5,1,1,1}
N2,3(⇠)
u(x(⇠, ⌘)) =
X
i
X
j
Ni,p(⇠)Mj,p(⌫)Uij
1
1
1
1 ¯⇠
ˆ⌦1 ˆ⌦2
ˆ⌦3 ˆ⌦4
¯⌘
⌘
⇠
0 0.5 1
1
0.5
Parametric
domain
Physical
domain
Parent
domain
(integra$on)
x(⇠, ⌘) =
nX
i
mX
j
Ni,p(⇠)Mj,p(⌘) Bij M2,2(⌘)
(⇠, ⌘)|ˆ⌦i = ˜((¯⇠, ¯⌘))
References:
[Kagan
et
al.
1998,
Cirak
et
al.
2000,
Hughes
et
al.
2005,
Cofrell
et
al.
2009]
14. Ins$tute
of
Mechanics
&
Advanced
Materials
• Introduc$on
• Automa$c
coupling
§ Problem
statement
and
reference
§ IGA
§ Discrete
coupling
strategy
• Numerical
examples
• Conclusion
15. Ins$tute
of
Mechanics
&
Advanced
Materials
• Penalty
formula$on
§
Mortaring
non-‐conforming
discrete
spaces
JXK = Xs
Xb
us
= ub
(⇥)
In
discrete
space,
poten$ally
discon$nuous
a ⌘ as
+ ab
u ⌘ (us
, ⇥b
)ap
(uh
, u?
) =: a(uh
, u?
) + ↵
Z
?
Juh
K · Ju?
K d = l(u?
)
16. Ins$tute
of
Mechanics
&
Advanced
Materials
• Penalty
formula$on
§
§ Lack
of
consistency:
Mortaring
non-‐conforming
discrete
spaces
JXK = Xs
Xb
hXi = Xs
+ (1 )Xb
us
= ub
(⇥)
In
discrete
space,
poten$ally
discon$nuous
J( · ns
) · u?
K = J · ns
K · hu?
i + Ju?
K h · ns
i
a ⌘ as
+ ab
u ⌘ (us
, ⇥b
)
=
1
2
ap
(uex
, u?
) l(u?
) =
Z
?
J (uex
) · ns
· u?
K 6= 0
ap
(uh
, u?
) =: a(uh
, u?
) + ↵
Z
?
Juh
K · Ju?
K d = l(u?
)
17. Ins$tute
of
Mechanics
&
Advanced
Materials
• Penalty
formula$on
§
§ Lack
of
consistency:
§ Nitsche
method:
add
to
penalty
formula$on
and
symmetrise
Mortaring
non-‐conforming
discrete
spaces
JXK = Xs
Xb
hXi = Xs
+ (1 )Xb
us
= ub
(⇥)
In
discrete
space,
poten$ally
discon$nuous
an
(uh
, u?
) = a(uh
, u?
)
Z
?
Ju?
K
⌦
(uh
) · ns
↵
d
Z
?
Juh
K h (u?
) · ns
i d + ↵
Z
?
Juh
K · Ju?
K d = l(u?
)
J( · ns
) · u?
K = J · ns
K · hu?
i + Ju?
K h · ns
i```
a ⌘ as
+ ab
u ⌘ (us
, ⇥b
)
=
1
2
ap
(uex
, u?
) l(u?
) =
Z
?
J (uex
) · ns
· u?
K 6= 0
ap
(uh
, u?
) =: a(uh
, u?
) + ↵
Z
?
Juh
K · Ju?
K d = l(u?
)
18. Ins$tute
of
Mechanics
&
Advanced
Materials
• Penalty
formula$on
§
§ Lack
of
consistency:
§ Nitsche
method:
add
to
penalty
formula$on
and
symmetrise
Mortaring
non-‐conforming
discrete
spaces
JXK = Xs
Xb
hXi = Xs
+ (1 )Xb
us
= ub
(⇥)
In
discrete
space,
poten$ally
discon$nuous
an
(uh
, u?
) = a(uh
, u?
)
Z
?
Ju?
K
⌦
(uh
) · ns
↵
d
Z
?
Juh
K h (u?
) · ns
i d + ↵
Z
?
Juh
K · Ju?
K d = l(u?
)
J( · ns
) · u?
K = J · ns
K · hu?
i + Ju?
K h · ns
i
J( · ns
) · u?
K = J · ns
K · (I ⇧b
) hu?
i + Ju?
K h · ns
i
```
a ⌘ as
+ ab
u ⌘ (us
, ⇥b
)
=
1
2
ap
(uex
, u?
) l(u?
) =
Z
?
J (uex
) · ns
· u?
K 6= 0
ap
(uh
, u?
) =: a(uh
, u?
) + ↵
Z
?
Juh
K · Ju?
K d = l(u?
)
19. Ins$tute
of
Mechanics
&
Advanced
Materials
• Coercivity:
Stability
an
(uh
, u?
) = a(uh
, u?
)
Z
?
Ju?
K
⌦
(uh
) · ns
↵
d
Z
?
Juh
K h (u?
) · ns
i d + ↵
Z
?
Juh
K · Ju?
K d = l(u?
)
˜t(u) := (us
) + (ub
) · ns
in
discrete
space,
poten$ally
discon$nuous
Related work:
[Griebel et al. 2002,
Dolbow et al. 2009]
an
(u, u) = a(u, u) + ↵
Z
?
JuK · JuK d
Z
?
JuK · ˜t(u) d
an
(u, u) Cc
kuk2
X
20. Ins$tute
of
Mechanics
&
Advanced
Materials
• Coercivity:
§ Parallelogram
ineq.
:
§ ``Trace
inequality”
(assump$on)
→
Stability
an
(uh
, u?
) = a(uh
, u?
)
Z
?
Ju?
K
⌦
(uh
) · ns
↵
d
Z
?
Juh
K h (u?
) · ns
i d + ↵
Z
?
Juh
K · Ju?
K d = l(u?
)
˜t(u) := (us
) + (ub
) · ns
k˜t(u)k2
? C2
a(u, u)
an
(u, u)
✓
1
C2
✏
2
◆
a(u, u) +
✓
↵
1
2✏
◆
kJuKk2
?
in
discrete
space,
poten$ally
discon$nuous
✏ =
1
C2
) ↵ >
C2
2
Related work:
[Griebel et al. 2002,
Dolbow et al. 2009]
an
(u, u) a(u, u) + ↵kJuKk2
?
✓
1
2✏
kJuKk2
? +
✏
2
k˜t(u)k2
?
◆
an
(u, u) = a(u, u) + ↵
Z
?
JuK · JuK d
Z
?
JuK · ˜t(u) d
an
(u, u) Cc
kuk2
X
21. Ins$tute
of
Mechanics
&
Advanced
Materials
• Solve
numerically
for
regularisa$on
parameter
s.
t.
→
→
Eigenvalue
problem
for
regularisa$on
parameter
↵ >
1
2
Kuncoupled 1
H
a(u, u) = [u]T
Kuncoupled
[u]
k˜t(u)k2
? =
Z
?
( (us
) + (ub
)) · ns
· ( (us
) + (ub
)) · ns
d = [u]T
H [u]
1
largest
eigenvalue
of
Related work:
[Griebel et al. 2002,
Dolbow et al. 2009]
k˜t(u)k2
? < 2↵ a(u, u)
References on embedded interfaces and implicit boundaries using Nitsche [Hansbo et al. 2002,
Dolbow et al. 2009, Sanders et al. 2011, Burman et al. 2012, Chouly et al. 2013]
1
2
[u]T
H [u]
[u]T Kuncoupled [u]
< ↵
22. Ins$tute
of
Mechanics
&
Advanced
Materials
• Introduc$on
• Automa$c
coupling
§ Problem
statement
and
reference
§ IGA
§ Discrete
coupling
strategy
• Numerical
examples
• Conclusion
23. Ins$tute
of
Mechanics
&
Advanced
Materials
Examples
ux(x, y) =
Py
6EI
(6L 3x)x + (2 + ⌫)
✓
y2 D2
4
◆
uy(x, y) =
P
6EI
3⌫y2
(L x) + (4 + 5⌫)
D2
x
4
+ (3L x)x2
(88)
tresses are
xx(x, y) =
P(L x)y
I
; yy(x, y) = 0, xy(x, y) =
P
2I
✓
D2
4
y2
◆
(89)
tions, material properties are taken as E = 3.0 ⇥ 107
, ⌫ = 0.3 and the beam dimensions are D = 6 and
hear force is P = 1000. Units are deliberately left out here, given that they can be consistently chosen
In order to model the clamping condition, the displacement defined by Equation (88) is prescribed as
ary conditions at x = 0, D/2 y D/2. This problem is solved with bilinear Lagrange elements (Q4
high order B-splines elements. The former helps to verify the implementation in addition to the ease of
Dirichlet boundary conditions (BCs). For the latter, care must be taken in enforcing the Dirichlet BCs
on (88) since the B-spline basis functions are not interpolatory.
Figure 13: Timoshenko beam: problem description.
continuum-beam model is given in Fig. 14. The end shear force applied to the right end point is F = P.
Figure 14: Timoshenko beam: mixed continuum-beam model.
ments In the first calculation we take lc = L/2 and a mesh of 40 ⇥ 10 Q4 elements (40 elements in the
n) was used for the continuum part and 29 two-noded elements for the beam part. The stabilisation
enforcement of Dirichlet boundary conditions (BCs). For the latter, care must be taken in enforcing the Dirich
given in Equation (88) since the B-spline basis functions are not interpolatory.
Figure 13: Timoshenko beam: problem description.
The mixed continuum-beam model is given in Fig. 14. The end shear force applied to the right end point is
Figure 14: Timoshenko beam: mixed continuum-beam model.
Lagrange elements In the first calculation we take lc = L/2 and a mesh of 40 ⇥ 10 Q4 elements (40 element
length direction) was used for the continuum part and 29 two-noded elements for the beam part. The stab
26
Analy$cal
solu$on
available
?
24. Ins$tute
of
Mechanics
&
Advanced
Materials
parameter ↵ according to Equation (55) was 4.7128 ⇥ 107
. Fig. 15a plots the transverse displacement (taken as nodal
values) along the beam length at y = 0 together with the exact solution given in Equation (88). An excellent agreement
with the exact solution can be observed and this verified the implementation. The comparison of the numerical stress
field and the exact stress field is given in Fig. 15b with less satisfaction. While the bending stress xx is well estimated,
the shear stress xy is not well predicted in proximity to the coupling interface. This phenomenon was also observed
in the framework of Arlequin method [64] and in the context of MPC method [38]. Explanation of this phenomenon
will be given subsequently.
0 10 20 30 40 50
−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
x
w
exact
coupling
(a) transverse displacement
0 5 10 15 20 25
−400
−200
0
200
400
600
800
x
stressesalongy=0.3
sigmaxx−exact
sigmaxx−coupling
sigmaxy−exact
sigmaxy−coupling
(b) stresses
Figure 15: Mixed dimensional analysis of the Timoshenko beam: comparison of numerical solution and exact solution.
Q4
elements
2-‐noded
cubic
elements
Deflec$on
of
neutral
axis
Stress
profile
25. Ins$tute
of
Mechanics
&
Advanced
Materials
32x4
bi-‐cubic
B-‐spline
elements
8
cubic
B-‐spline
elements
(patch
extends
throughout
the
2D
domain)
shenko beam: non-conforming coupling
ction, a non-conforming coupling is considered. The B-spline mesh is given in Fig. 21. Refined meshes are
m this one via the knot span subdivision technique. We use the mesh consisting of 32 ⇥ 4 cubic continuum
d 8 cubic beam elements. Fig. 22 gives the mesh and the displacement field in which lc = 29.97 so that
interface is very close to the beam element boundary. A good solution was obtained using the simple
scribed in Section 5.
Mixed dimensional analysis of the Timoshenko beam with non-conforming coupling. The continuum part
y 8 ⇥ 2 bi-cubic B-splines and the beam part is with 8 cubic elements.
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
w
exact
continuum
beam
orming coupling
g coupling is considered. The B-spline mesh is given in Fig. 21. Refined meshes are
span subdivision technique. We use the mesh consisting of 32 ⇥ 4 cubic continuum
ts. Fig. 22 gives the mesh and the displacement field in which lc = 29.97 so that
to the beam element boundary. A good solution was obtained using the simple
ysis of the Timoshenko beam with non-conforming coupling. The continuum part
es and the beam part is with 8 cubic elements.
0 10 20 30 40 50
−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
x
w
exact
continuum
beam
(b) displacement field
ysis of the Timoshenko beam with non-conforming coupling: (a) 32 ⇥ 4 Q4 elements
ts and (b) displacement field.
26. Ins$tute
of
Mechanics
&
Advanced
Materials
dofs. The total number of dofs of the continuum-beam model is only 5400. The stabilisation parameter is taken to be
↵ = 107
and used for both coupling interfaces. A comparison of xy contour plot obtained with (1) and (2) is given in
Fig. 25. A good agreement was obtained.
Figure 23: A plane frame analysis: problem description.
Remark 6.1. Although the processing time of the solid-beam model is much less than the one of the solid model, one
cannot simply conclude that the solid-beam model is more e cient. The pre-processing of the solid-beam model, if not
automatic, can be time consuming such that the gain in the processing step is lost. For non-linear analyses, where the
processing time is dominant, we believe that mixed dimensional analysis is very economics.
6.2. Continuum-plate coupling
6.2.1. Cantilever plate: conforming coupling
For verification of the continuum-plate coupling, we consider the 3D cantilever beam given in Fig. 26. The material
properties are E = 1000 N/mm2
, ⌫ = 0.3. The end shear traction is ¯t = 10 N/mm in case of continuum-plate model
and is ¯t = 10/20 N/mm2
in case of continuum model which is referred to as the reference model. We use B-splines
elements to solve both the MDA and the reference model. The length of the continuum part in the continuum-plate
model is L/2 = 160 mm. A mesh of 64 ⇥ 4 ⇥ 5 tri-cubic elements is utilized for the reference model and a mesh of
32 ⇥ 4 ⇥ 5/ 16 ⇥ 2 cubic elements is utilized for the mixed dimensional model, cf. Fig. 27. The plate part of the
mixed dimensional model is discretised using the Reissner-Mindlin plate theory with three unknowns per node and the
Kirchho↵ plate theory with only one unknown per node. The stabilisation parameter was chosen empirically to be
5⇥103
. Note that the eigenvalue method described in Section 4.3 can be used to rigorously determine ↵. However since
it would be expensive for large problems, we are in favor of simpler but less rigorous rules to compute this parameter.
Fig. 28 shows a comparison of deformed shapes of the continuum model and the continuum-plate model and in Fig. 29,
the contour plot of the von Mises stress corresponding to various models is given.
31
Figure 24: A plane frame analysis: solid-beam model.
Figure 25: A plane frame analysis: comparison of xy contour plot obtained with solid model (left) a
model (right).
Figure 24: A plane frame analysis: solid-beam model.
xy(normalised)
xy(normalised)
Q4
elements
Q4
elements
Cubic
B-‐spline
elements
27. Ins$tute
of
Mechanics
&
Advanced
Materials
27: Cantilever beam subjects to an end shear force: typical B-spline discretisation.
r beam subjects to an end shear force: comparison of deformed shapes of the continuum model
• 3D/plate
coupling
(Kirchhoff)
§
Figure 25: A plane frame analysis: comparison of xy contour plot obtained with solid mode
model (right).
Figure 26: Cantilever beam subjects to an end shear force: problem setup
32
27: Cantilever beam subjects to an end shear force: typical B-spline discretisation.
10.0 20.0 30.0 40.0 50.0
von Mises stress
0.429 51
(a) reference model
10.0 20.0 30.0 40.0
von Mises stress
5.17 48.6
(b) mixed dimensional model, Mindlin plate
10.0 20.0
von Mis
5.12
(c) mixed dimensional
10.0 20.0 30.0 40.0 50.0
von Mises stress
0.429 51
(a) reference model
10.0 20.0 30.0 40.0
von Mises stress
5.17 48.6
(b) mixed dimensional model, Mindlin plate
10.0 20.0 30.0 40.0
von Mises stress
5.12 48.6
(c) mixed dimensional model, Kirchho↵ plate
32x4x5
tri-‐cubic
B-‐spline
elements
16x2
bi-‐cubic
B-‐spline
elements
Full
3D
MDA
28. Ins$tute
of
Mechanics
&
Advanced
Materials
ure 32: Square plate enriched by a solid. The highlighted elements are those plate elem
undaries. The plate is fully clamped ans subjected to a gravity force.
ments with some geometry entities is popular in XFEM, see e.g., [58]. Fig. 33 plots the de
the solid-plate model and the one obtained with a plate model. A good agreement can be o
w the flexibility of the non-conforming coupling, the solid part was moved slightly to the rig
figuration is given in Fig. 34. The same discretisation for the plate is used. This should ser
del adaptivity analyses to be presented in a forthcoming contribution.
ure 33: Square plate enriched by a solid: transverse displacement plot on deformed configur
ate enriched by a solid. The highlighted elements are those plate elements cut by the s
is fully clamped ans subjected to a gravity force.
eometry entities is popular in XFEM, see e.g., [58]. Fig. 33 plots the deformed configura
el and the one obtained with a plate model. A good agreement can be observed. In orde
the non-conforming coupling, the solid part was moved slightly to the right and the defor
in Fig. 34. The same discretisation for the plate is used. This should serve as a prototype
yses to be presented in a forthcoming contribution.
te enriched by a solid: transverse displacement plot on deformed configurations of plate m
Figure 32: Square plate enriched by a solid. The highlighted elements are those plate elements cut by the solid
boundaries. The plate is fully clamped ans subjected to a gravity force.
elements with some geometry entities is popular in XFEM, see e.g., [58]. Fig. 33 plots the deformed configuration
of the solid-plate model and the one obtained with a plate model. A good agreement can be observed. In order to
show the flexibility of the non-conforming coupling, the solid part was moved slightly to the right and the deformed
configuration is given in Fig. 34. The same discretisation for the plate is used. This should serve as a prototype for
model adaptivity analyses to be presented in a forthcoming contribution.
Load:
weight
Fully
clamped
29. Ins$tute
of
Mechanics
&
Advanced
Materials
• Versa$le
coupling
for
mixed-‐dimensional
analysis
with
non-‐conforming
discre$sa$ons
(IGA/FEM)
• Future
work
§ Weighted
averages
in
the
Nitsche
Plate/3D
coupling
§ Cheap
way
to
evaluate
the
lower
bound
on
the
regularisa$on
parameter
§ Efficient
and
weakly
intrusive
local/global
solver
§ Damage
in
solid
region
Conclusion