Elastic Cosserat Models
Newton
Application
Minimization on Riemannian manifolds
and the application to Cosserat models
Wolfgang Müller and Christian Wieners
Faculty of Mathematics
Universität Karlsruhe (TH)
15th April 2005
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
Infinitesimal Cosserat Model
Compute displacement u and infinitesimal rotations id +A with
T
A = −A, such that
Z
W (u, A) − `(u) = min
where the stored energy is
λ
tr(sym(∇u))2
2
+ µc k skew(∇u) − Ak2 + µL2c k∇Ak2 .
W (u, A) = µ k sym(∇u)k2 +
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
Finite Cosserat Model
T
Compute displacement u and rotations R with R R = id, such
that
Z
W (u, R) − `(u) = min
where the stored energy is
λ
T
tr(sym(R F − id))2
2
q/2
T
T
+ µc k skew(R F )k2 + µ 1 + L2c kR DRk2
T
W (u, R) = µ k sym(R F − id)k2 +
depending on the deformation gradient F = id +∇u.
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
Simplified Cosserat Model
T
For given u : [0, 1] −→ R3 compute rotations R with R R = id,
such that
Z
f (R) := W (u, R) − `(u) = min
where for F = id +∇u
λ
T
tr(sym(R F − id))2
2
q/2
T
T
,
+ µc k skew(R F )k2 + µ 1 + L2c kR DRk2
T
W (u, R) = µ k sym(R F − id)k2 +
T
Goal: Minimize f (R) on the manifold M = {R : R R = id} .
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
The Newton method for minimization on manifolds
Let M be a Riemannian manifold with metric h·, ·i, and let
f : M → R be a smooth functional on M. We consider the
minimization problem
p ∈ M:
f (p) = min!
For the definition of the Newton algorithm, we linearize the
problem in the tangent space Tp M. Therefore, for a vector
v ∈ Tp M, let γv : [0, T ] −→ M be the geodesic curve starting at
p = γv (0) with γ̇v (0) = v . This defines expp (v ) := γv (1) on
Dp = {v ∈ Tp M : γv is defined at least for t ∈ [0, 1]}.
Furthermore, we set Vp = expp (Dp ) ⊂ M.
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
The Newton method for minimization on manifolds
Starting at p0 ∈ M, in every step n = 0, 1, 2, ... we consider the
local problem
v ∈ Dpn :
f (exppn (v )) = min!
For the computation of the Newton increment, we define
fpn : Dpn −→ R by
fpn (v ) := f (exppn (v )),
consider the Taylor expansion
1 (2)
(0)
(1)
fpn (v ) = fpn + fpn [v ] + fpn [v , v ] + · · ·
2
and solve the linear problem
vn ∈ Tpn M :
(2)
(1)
fpn [vn , w ] = −fpn [w ] ,
w ∈ Tpn M .
If vn ∈ Dpn , this gives the new iterate pn+1 = exppn (vn ) .
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
The theorem of Newton-Kantorovich on manifolds
Assume that for p0 ∈ M holds:
a) |hgrad f (p0 ), w ip0 | ≤ η kw kp0 for w ∈ Tp0 M ;
b) hHess f (p0 )w , w ip0 ≥ β kw k2p0 for w ∈ Tp0 M ;
c) K ⊂ Vp for p ∈ K , where K := {p ∈ M : d(p, p0 ) ≤ r } and
r = 2η/β ;
d) Hess f is locally Lipschitz in the following sense:
for any geodesic curve γ : [a, b] −→ K with γ(a) = p and
γ(b) = q we have for all w , w 0 ∈ Tp M
hHess f (q)Pγ,a,b w , Pγ,a,b w 0 iq − hHess f (p)w , w 0 ip ≤ λ kw kp kw 0 kp d(p, q) ;
e) 2ηλ ≤ β 2 .
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
The theorem of Newton-Kantorovich on manifolds
Then, we have: the Newton sequence pn builds a well defined
Cauchy sequence in K with limit p ∗ ∈ K , and we have the estimate
d(p ∗ , p0 ) ≤
2η
β+
p
β 2 − 2ηλ
≤r
and
d(p ∗ , pn ) ≤
β 2λη 2n
.
λ β2
Moreover, we have
hHess f (p ∗ )w , w ip∗ ≥ β − λ d(p ∗ , p0 ) kw k2p∗
for
w ∈ Tp∗ M .
If, in addition, 2ηλ < β 2 , Newton’s method converges
quadratically and p ∗ is a isolated local minimum of f in K .
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
Application to the manifold SO(3)N
We compute R 0 = Ra , R 1 , ..., R N = Rb at prescribed points
ϕ(z0 ), ..., ϕ(zN ) ∈ R3 with a = z0 < z1 < · · · < zN = b. Replacing
∂z R by a difference quotient in the interior respectively a simple
one at the boundary. Using trapezoidal rule, we obtain the
minimization problem
f (R 1 , ..., R N−1 ) = min!
depending on R 0 , R N and F 0 = ∇ϕ(z0 ), ..., F N = ∇ϕ(zN ):
f (R 1 , ..., R N−1 ) =
N−1
2
λX
hm−1/2 tr((R m )T F m ) − 3
2
m=1
+µ
N−1
X
m T
hm−1/2 k sym((R ) F
m
2
− I )k +
µL2c
m=1
N−1
X
m=0
1
kR m+1 − R m k2
hm
(hm = zm+1 − zm , hm−1/2 = (hm−1 + hm )/2).
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
Numerical experiment (Globalized Newton method)
Newton defect
number of steps
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
5e02
0
...
...
1e-01
54
1e-05
55
1e-12
56
Universität Karlsruhe (TH)
Elastic Cosserat Models
Newton
Application
Summary, outlook and future work
Already realized:
I
Finite element implementation for linear infinitesimal Cosserat
models in 2D and 3D
I
Nonlinear Newton analysis for minimization problems on
Riemannian manifolds (e. g., Cosserat rotations)
Next steps:
I
Multigrid methods for Cosserat models
I
Elasto-plastic Cosserat models
I
Full nonlinear Cosserat models
Wolfgang Müller and Christian Wieners
Minimization on Riemannian manifolds
Universität Karlsruhe (TH)