nitsche’s method for patch coupling · the nitsche contributions are made by work–conjugate...
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Motivation Idea Nitsche Numerical illustrations
Nitsche’s method for patch couplingin isogeometric analysis
Qingyuan HU DUT ,UL,Supervisor: Stéphane BORDAS UL
DUT Dalian University of TechnologyP.R. China
ULUniversity of LuxembourgLuxembourg
June 6, 2017
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 1 / 23
Motivation Idea Nitsche Numerical illustrations
Why do we need patch coupling
Model complex structures
Figure: Connecting rod[V .P.Nguyen et.al. 2013]
Figure: Intersecting tubular shell[Y .Guo et.al. 2017]
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 2 / 23
Motivation Idea Nitsche Numerical illustrations
Why do we need patch coupling
Model complex structures
Assign various materials to sub-structures
Figure: iPHONE 6S[www.visualcapitalist.com]
Figure: Seat frames[aeplus.com]
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 3 / 23
Motivation Idea Nitsche Numerical illustrations
Why do we need patch coupling
Model complex structures
Assign various materials to sub-structures
Calculate using suitable elements (dimensions)
Figure: Mixed-dimensional coupling[Y .Guo and M.Ruess 2015]
Figure: Simulation of metal forming[D.J.Benson et.al. 2012]
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 4 / 23
Motivation Idea Nitsche Numerical illustrations
Why do we need patch coupling
Model complex structures
Assign various materials to sub-structures
Choose reasonable element types (dimensions)
Discrete model into elements of sufficient numbers
Figure: A truck door, left: commercial software results[Marco Brino 2015]
Figure: Saint Venant’s Principle[V .P.Nguyen et.al. 2014]
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 5 / 23
Motivation Idea Nitsche Numerical illustrations
Why do we need patch coupling: from FEM to IGA
IGA use NURBS (non-uniform rational B-spline) instead of polynomials
Figure: Lagrange basis functions in FEM Figure: NURBS basis functions in IGA
Non-interpolatory control points
Mesh
1x4
Mesh
1x6
Figure: Meshes and nodes in FEM Figure: Control meshes and controlpoints in IGA Figure: NURBS curve
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 6 / 23
Motivation Idea Nitsche Numerical illustrations
Stitching two fabrics together
Figure: Stitching fabrics [bigbgsd.blogspot.com]
Where there are displacement gaps,there should be some kind of forces to prevent the two fabrics from separation,and additional work to be done to stitching them together.
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 7 / 23
Motivation Idea Nitsche Numerical illustrations
Analogy
Figure: Stitching fabrics
Ω1
Ω2
u1
u2
fix
pullFdisplacement gap = u1 − u2
F2
F1
average force = 0.5(F1 + F2)
ΓC
Figure: Patch coupling
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 8 / 23
Motivation Idea Nitsche Numerical illustrations
Constraints and additional work
Ω1
Ω2
u1
u2
fix
pullFdisplacement gap = u1 − u2
F2
F1
average force = 0.5(F1 + F2)
ΓC
Figure: Patch coupling
Constraints on interface ΓC
u1 = u2 on ΓC (1a)
F1 = F2 on ΓC (1b)
Define jump and average operators
JuK := u1 − u2
〈F 〉 :=12
(F1 + F2)(2)
Additional work to be done
Wadd = 〈F 〉JuK (3)
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 9 / 23
Motivation Idea Nitsche Numerical illustrations
Problem setup
1Ω
2Ω
1n
2n
1t
2u
1
NΓ
2
DΓC
Γ
Figure: Couple two patches
u1 = u2 on ΓC (4a)
σ1 · N1 = −σ2 · N2 on ΓC (4b)
The jump and average operators are defined as
JuK := u1 − u2
〈σN〉 :=12
(σ1N + σ2N)(5)
here N is chosen to be N1.
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 10 / 23
Motivation Idea Nitsche Numerical illustrations
Different from fabrics stretching
Instead of scalar u, use vector u for generalized cases, e.g. u = (u, v)T in 2D
Instead of forces F, use traction σ(u)N ,where the stress comes from displacement field
σ(u) = Dε(u) = D∇u (6)
and N is the transformation matrix to collect area contribution.
x , u
y , v n = (nx , ny )T σx
σy
τxy
τyx
Figure: u, n and σ
N =
[nx 0 ny0 ny nx
]
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 11 / 23
Motivation Idea Nitsche Numerical illustrations
Nitsche formulation
Start from the classical weak form
a(u,w) = L(w) (7)
and introduce Nitsche contribution into the weak form
a(u,w)−∫
ΓC
〈σ(u)N〉JwKdΓ−∫
ΓC
JuK〈σ(w)N〉dΓ + α
∫ΓC
JuKJwKdΓ = L(w) (8)
Note
Two Nitsche terms are introduced to keep the stiffness matrix symmetric
Additional stabilisation parameter α to guarantee coercive (positive definite)
Boundary integrations are performed along slave boundary
The Nitsche contributions are made by work–conjugate pairs:for membrane element they are displacement and traction force,for thin bending plate they are rotation and bending moment
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 12 / 23
Motivation Idea Nitsche Numerical illustrations
Penalty and Lagrange multiplier
Penalty method:
a(u,w) +α
2
∫ΓC
JuKJwKdΓ = L(w) (9)
where α is the penalty parameter.
Lagrange multiplier method:
a(u,w) +
∫ΓC
λJwKdΓ +
∫ΓC
δλJuKdΓ = L(w) (10)
where λ is the vector of Lagrange multiplier.
Methods Pros Cons
Penalty No increased DOFs Depends on penalty parameterEasy and straightforward sometimes ill-conditioned
Lagrange multiplier λ means traction Increase DOFsStable when satisfies LBB Not positive define
Nitsche No increased DOFs Not parameter-freePositive define, robust Involve constitutive equation
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 13 / 23
Motivation Idea Nitsche Numerical illustrations
How do they work
Penalty method:
a(u,w) +α
2
∫ΓC
JuKJwKdΓ = L(w) (11)
Lagrange multiplier method:
a(u,w) +
∫ΓC
λJwKdΓ +
∫ΓC
δλJuKdΓ = L(w) (12)
Nitsche’s method:
a(u,w)−∫
ΓC
〈σ(u)N〉JwKdΓ−∫
ΓC
JuK〈σ(w)N〉dΓ + α
∫ΓC
JuKJwKdΓ = L(w) (13)
α
Figure: α is spring stiffness
λ
Figure: λ is external traction
σ
Figure: σ is from slave boundary
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 14 / 23
Motivation Idea Nitsche Numerical illustrations
Slave boundary to perform boundary integration
Choose slave boundary that has more elements
1Ω2
Ω CΓ
Choose slave boundary that has shorter edge
Figure: 2D[V .P.Nguyen et.al. 2013]
Figure: 3D[V .P.Nguyen et.al. 2013]
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 15 / 23
Motivation Idea Nitsche Numerical illustrations
Stiffness matrix illustration
Nitsche
2
3
4
5
2
3
4
5
2
3
4
5
1 Calculate stiffness matrixfor slave and masterK =
∑K s +
∑K m
2 Calculate Nitsche contributionalong coupled boundaryK + =
∑K N
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 16 / 23
Motivation Idea Nitsche Numerical illustrations
Stabilisation parameter α
Nitsche
2
3
4
5
2
3
4
5
2
3
4
5
Cu
Cuλ
1 Solve generalized eigenvalues λalong coupled boundaryK Nuc = λKuc
2 α = 2max(λ)ref A.Apostolatos et.al. IJNME. 2013
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 17 / 23
Motivation Idea Nitsche Numerical illustrations
Bending plate
x
y
z
simply supported
simply supported
simply supported
simply supported
P
A
L = 1
h = 10-3
P = 10-2
E = 200•109
ν = 0.3
Figure: Bending plate
-3.9588
-3.1671
-2.3753
-1.5835
-0.7918
-6.3341
-5.5423
-4.7506
0.0000
w
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 18 / 23
Motivation Idea Nitsche Numerical illustrations
Vibration square plate
a = 1
b = 1
h = 10-1
E = 200•109
ν = 0.3
ρ = 7850
x
y a
b
simply supported
simply supported
sim
ply
su
pp
ort
ed
sim
ply
su
pp
orte
d
Figure: Square plateFigure: Meshes
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 19 / 23
Motivation Idea Nitsche Numerical illustrations
Vibration square plate
0.5
-1.0
1.0
W
-0.5
0
0.5
-1.0
1.0
W
-0.5
0
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Motivation Idea Nitsche Numerical illustrations
Clamped plate and errors [X.Du et.al. 2015]
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 21 / 23
Motivation Idea Nitsche Numerical illustrations
Connecting rod [V.P.Nguyen et.al. 2014]
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 22 / 23
Motivation Idea Nitsche Numerical illustrations
Intersecting tubular shell [Y.Guo et.al. 2017]
Qingyuan HU (University of Luxembourg) Nitsche IGA June 6, 2017 23 / 23