Mechanical modelling with FEniCS - A wish list - FEniCS Project

Mechanical modelling with FEniCS
A wish list
Garth N. Wells
Faculty of Civil Engineering and Geosciences
Delft University of Technology, The Netherlands
FEniCS ’05, Chicago
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Delft University of Technology

Outline
Interests
Example applications of FEniCS components
Wish list
Issues
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Delft University of Technology

Interests and motivation
◮ Rapid development and efficient solution of mechanical
models.
◮ Minimal simplicity versus computational speed compromise.
◮ Significant productivity increase.
◮ Open source.
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Delft University of Technology

Interest
Licensing: GPL or LGPL?
◮ GPL/LGPL protects me from my project sponsors.
◮ GPL might be too strong for some project sponsors.
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Delft University of Technology

Problem hallmarks
◮ Nonlinear.
◮ Multiple fields.
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Delft University of Technology

Porous media
Classical Biot theory + Darcy flow
p 2
p 1
PSfrag replacements
Mechanical swelling due to fluid penetration
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Delft University of Technology

Porous media
PU = FiniteElement("Vector Lagrange", "triangle", 2)
PP = FiniteElement("Lagrange", "triangle", 1)
ME = PU + PP
(v, q) = BasisFunctions(ME) # test functions
(u, p) = BasisFunctions(ME) # trial functions
.
.
# Bilinear and linear form for solid component
aSolid = dt*theta*dot(grad(v), stress(symgrad(u), p, mu, . . .
LSolidInt = dt*theta*dot(grad(v), stress(symgrad(u0), p0, mu, . . .
LSolidExt = dt*theta*v[1]*(rho*g)*dx
# Bilinear and linear form for fluid component
aFluid = q*comp*p*dx + q*alpha*Sw*div(u)*dx + . . .
LFluidInt = dt*theta*q*comp*pr*dx + dt*theta*q*alpha*Sw* . . .
LFluidExt = dt*theta*q*f*dx + dt*theta*perm*rhow*q.dx(1)*g*dx
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Delft University of Technology

Mechanically-driven diffusion in elastic solids
with Luisa Molari, University of Bologna
Mechanical response
∇ · σ + f = 0
σ = C : ∇ s u − βIc
+ B.C.
Steady-state diffusion
∇ · q = 0
q = −κ∇c + M∇ (∇ · u)
+ B.C.
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Delft University of Technology

Mechanically-driven diffusion in elastic solids
Find u h ∈ V h , e h ∈ S h , c h ∈ X h such that:
(∇w h , C : ∇ s u h) (∇w − h , βIc h) ( )
= w h , f
Ω Ω
∀ w h ∈ V h ,
(v h , e h) Ω − (
v h , ∇ · u h) Ω = 0 ∀ v h ∈ S h ,
(∇q h , κ∇c h) Ω − (
∇v h , M∇e h) Ω = 0 ∀ qh ∈ X h .
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Delft University of Technology

Mechanically-driven diffusion in elastic solids
E1 = FiniteElement("Vector Lagrange", "tetrahedron", 2)
E2 = FiniteElement("Lagrange", "tetrahedron", 1)
E3 = FiniteElement("Lagrange", "tetrahedron", 1)
element = E1 + E2 + E3
.
.
# Elasticity
au = . . .
# Divergence of u
ae = . . .
# Diffusion
ac = . . .
.
.
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Delft University of Technology

Mechanically-driven diffusion in elastic solids
dilation
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Delft University of Technology

Mechanically-driven diffusion in elastic solids
concentration
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Delft University of Technology

Cahn-Hilliard equation
Model for phase separation
◮ Important in material science and other fields.
◮ Diffuse interface model.
◮ Fourth-order in space nonlinear parabolic equation.
ċ = ∇ · M (c) ∇ ( µ c (c) − λ∇ 2 c )
in Ω
Mλ∇c · n = 0 on ∂Ω
M∇ ( µ c − λ∇ 2 c ) · n = 0 on ∂Ω
◮ c: concentration 0 < c < 1
◮ M: mobility
◮ µc : chemical potential (= dΨ c /dc)
◮ λ: surface energy term
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Delft University of Technology

Cahn-Hilliard equation
Chemical free-energy Ψ c
0.06
0.04
0.02
c
Ψ
0
Surface free-energy Ψ s
−0.02
−0.04
−0.06
0 0.2 0.4 0.6 0.8 1
c
Ψ s = 1 2 λ∇c : ∇c 16 / 37
Delft University of Technology

Cahn-Hilliard equation
Mixed formulation:
ċ = ∇ · (M ∇ (µ c − κ)) ,
κ = λ∇ 2 c.
Find c h ∈ S h × [0, T ] and κ h ∈ P h such that
(w h , ċ h) ( (
+ ∇w h , M h ∇ µ h c − κh)) = 0 ∀w h ∈ V h ,
Ω Ω
(v h , κ h) Ω + (
∇v h , λ∇c h) Ω = 0 ∀v h ∈ Q h ,
(
( )
w h , c (x, 0)
)Ω = w h , c 0 (x)
Ω
∀w h ∈ V h .
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Delft University of Technology

Cahn-Hilliard equation
Benchmark
1
0.8
c 1 = 0.71
0.2
1
0.6
0.4
0.2
0
acements
c 2 = 0.69
0.2
1
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Delft University of Technology

Cahn-Hilliard equation
Randomly perturbed initial conditions.
concentration
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Initial condition c = 0.63 + small random fluctuation.
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Delft University of Technology

Cahn-Hilliard equation
concentration
0.9882
0.9059
0.8235
0.7412
0.6588
0.5765
0.4941
0.4118
0.3294
0.2471
0.1647
0.0824
0.0000
concentration
0.9882
0.9059
0.8235
0.7412
0.6588
0.5765
0.4941
0.4118
0.3294
0.2471
0.1647
0.0824
0.0000
low surface energy
higher surface energy
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Delft University of Technology

Cahn-Hilliard equation
C
0.9882
0.9059
0.8235
0.7412
0.6588
0.5765
0.4941
0.4118
0.3294
0.2471
0.1647
0.0824
0.0000
. . . even higher surface energy
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Delft University of Technology

Cahn-Hilliard equation
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Delft University of Technology

Cahn-Hilliard equation
3D plane
DOLFIN
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Delft University of Technology

Cahn-Hilliard equation
Problem: cannot find an iterative solver that works.
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Delft University of Technology

Cahn-Hilliard primal formulation
Find c h ∈ V h × [0, T ] such that
(w h , ċ h) (
(
+ ∇w h , M h ∇µ h c
Ω
)Ω + ∇ 2 w h , M h λ∇ 2 c h) ˜Ω
(
+
(∇w h , ∇M h λ∇ 2 c h)˜Ω − ∇w h 〈
, M h λ∇ 2 c h〉)˜Γ
(〈
− M h λ∇ 2 w h〉 (
,
∇c h)˜Γ − ∇w h · n, M h λ∇ 2 c h) Γ
(
(
)
− M h λ∇ 2 w h , ∇c h · n
)Γ + β∇w h · n, ∇c h · n
Γ
(
+ β ∇w h
, ∇c h)˜Γ = 0 ∀ w h ∈ V h 25 / 37
Delft University of Technology

Modelling discontinuities
Approach:
u h = ū h + H s ũ h Γ s
PSfrag replacements
H s = 0
H1
s = 1
n
Discontinuity surface evolves.
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Delft University of Technology

Modelling discontinuities
Exploit partition of unity property (Melenk & Babuska, Duarte &
Oden, Belytschko & Black)
n∑ m∑
u h = φ i a i +
i=1 j=1
H s φ j b i
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Delft University of Technology

Crack propagation
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Delft University of Technology

Delamination
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Delft University of Technology

Delamination
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Delft University of Technology

Crack propagation
Requirements
◮ Abstract evolving surface representation.
◮ Integration on an evolving surface.
Find . . .
(∇ ¯w h , C : ∇ s ( ū h + H s ũ h)) Ω + (
( (
+ ˜w h , t ũ h)) (
= w h , f
Γ s
( ∇ ˜w h , C : ∇ s ū h + H s ũ h)) Ω
)
+
Ω
∀ ¯w h ∈ ¯V h , ˜w h ∈ Ṽ h 31 / 37
Delft University of Technology

General wishes (FFC)
◮ Complex numbers.
◮ Inter-element boundary integrals.
◮ Simple polar coordinates.
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Delft University of Technology

Compact notation for nonlinear problems
◮ Variational problem compact.
◮ Linearised and time-dependent form very large (decrease in
readability).
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Delft University of Technology

Issues
◮ Students
◮ Loss of transparency.
◮ Don’t develop skills to implement models going beyond the
current capabilities of FIAT/FFC/DOLFIN.
◮ Penetration
◮ Level of abstraction too high for old-style FE users.
◮ Quadrilateral and brick elements.
◮ Developers ready for users?
◮ Preparedness/time to help novice users with trivial questions?
◮ 0800-Anders-and-JohanJansson-help-desk (24 hour).
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Delft University of Technology

DOLFIN needs . . .
Most important aspect for dragging people over the threshold:
simple, accessible demos with immediate postprocessing.
Solver demos to key mechanical problems:
◮ Linear-elasticity.
◮ Classical plasticity.
◮ Advection-diffusion.
◮ Stokes flow / incompressible elasticity.
◮ Incompressible Navier-Stokes.
‘Extras’ section outside of DOLFIN src tree.
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Delft University of Technology

Packaging and distribution
◮ Bloat in number of required components.
◮ Version dependencies becoming an issue.
◮ Bundle components?
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Delft University of Technology

Supporting FEniCS
Details
◮ Sufficient financial structure to make contributions possible.
How to spend it?
◮ Meetings
◮ Hardware, network
◮ ???
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Delft University of Technology