Numerical modeling of slender structures with contact and friction - - PowerPoint PPT Presentation

Numerical modeling of slender structures with contact and friction

from dynamic simulation to inverse static design

Florence Bertails-Descoubes

• Laboratoire Jean Kuntzmann (EPI BiPop)

September 23, 2016, Séminaire PIC, Grenoble

Research Area

“Modeling and Simulating complex mechanical objects, and application to Computer Graphics”

Research Area

“Modeling and Simulating complex mechanical objects, and application to Computer Graphics”

• Visually rich phenomena

Complex shape or motion

• Increasing demand for simulators

Movies & games, virtual prototyping (cosmetology, virtual trying, medical area,...)

• Pluridisciplinary field

Mechanics, numerical analysis, algorithmic

Research Area

“Modeling and Simulating complex mechanical objects, and application to Computer Graphics”

• Visually rich phenomena

Complex shape or motion

• Increasing demand for simulators

Movies & games, virtual prototyping (cosmetology, virtual trying, medical area,...)

• Pluridisciplinary field

Mechanics, numerical analysis, algorithmic

Guiding line → Search for compact models

Research Area

“Modeling and Simulating complex mechanical objects, and application to Computer Graphics”

• Visually rich phenomena

Complex shape or motion

• Increasing demand for simulators

Movies & games, virtual prototyping (cosmetology, virtual trying, medical area,...)

• Pluridisciplinary field

Mechanics, numerical analysis, algorithmic

Guiding line → Search for compact models Realism

Research Area

“Modeling and Simulating complex mechanical objects, and application to Computer Graphics”

• Visually rich phenomena

Complex shape or motion

• Increasing demand for simulators

Movies & games, virtual prototyping (cosmetology, virtual trying, medical area,...)

• Pluridisciplinary field

Mechanics, numerical analysis, algorithmic

Guiding line → Search for compact models Realism + robustness

Research Area

“Modeling and Simulating complex mechanical objects, and application to Computer Graphics”

• Visually rich phenomena

Complex shape or motion

• Increasing demand for simulators

Movies & games, virtual prototyping (cosmetology, virtual trying, medical area,...)

• Pluridisciplinary field

Mechanics, numerical analysis, algorithmic

Guiding line → Search for compact models Realism + robustness + efficiency

Research Area

“Modeling and Simulating complex mechanical objects, and application to Computer Graphics”

• Visually rich phenomena

Complex shape or motion

• Increasing demand for simulators

Movies & games, virtual prototyping (cosmetology, virtual trying, medical area,...)

• Pluridisciplinary field

Mechanics, numerical analysis, algorithmic

Guiding line → Search for compact models Realism + robustness + efficiency + user control

Discrete Elements Modeling

Example : hair modeling Physical phenomenon

Discrete Elements Modeling

Example : hair modeling Physical phenomenon Simulator

Numerical modeling

Discrete Elements Modeling

Example : hair modeling Physical phenomenon Simulator

Numerical modeling 1 Model for a single element 2 Model for an assembly

Discrete Elements Modeling

Example : hair modeling Physical phenomenon Simulator

Numerical modeling 1 Model for a single element 2 Model for an assembly

Model for a Dynamic Fiber

Desired properties

• Fiber : long and very thin structure
• One clamped end, the other free

Model for a Dynamic Fiber

Desired properties

• Fiber : long and very thin structure
• One clamped end, the other free
• Can bend and twist

Model for a Dynamic Fiber

Desired properties

• Fiber : long and very thin structure
• One clamped end, the other free
• Can bend and twist
• Remains inextensible

Model for a Dynamic Fiber

Desired properties

• Fiber : long and very thin structure
• One clamped end, the other free
• Can bend and twist
• Remains inextensible
• Large displacements allowed

Model for a Dynamic Fiber

Desired properties

• Fiber : long and very thin structure
• One clamped end, the other free
• Can bend and twist
• Remains inextensible
• Large displacements allowed
• Natural “curliness”

Model for a Dynamic Fiber

Desired properties

• Fiber : long and very thin structure
• One clamped end, the other free
• Can bend and twist
• Remains inextensible
• Large displacements allowed
• Natural “curliness”

→ Kirchhoff model for thin elastic rods

Geometry of a Kirchhoff Rod

Geometry of a Kirchhoff Rod

• Centerline C(s)

Geometry of a Kirchhoff Rod

• Centerline C(s)
• Material frame R(s)

R(s) = {n0(s), n1(s), n2(s)} with n0(s) = C′(s)

Geometry of a Kirchhoff Rod

• Centerline C(s)
• Material frame R(s)

R(s) = {n0(s), n1(s), n2(s)} with n0(s) = C′(s)

• Degrees of freedom :
• twist κ0(s)
• curvatures κ1(s), κ2(s)

Geometry of a Kirchhoff Rod

• Centerline C(s)
• Material frame R(s)

R(s) = {n0(s), n1(s), n2(s)} with n0(s) = C′(s)

• Degrees of freedom :
• twist κ0(s)
• curvatures κ1(s), κ2(s)
• Darboux vector :

Ω Ω Ω(s) = κ0(s) n0(s)+κ1(s) n1(s)+κ2(s) n2(s)

Geometry of a Kirchhoff Rod

• Centerline C(s)
• Material frame R(s)

R(s) = {n0(s), n1(s), n2(s)} with n0(s) = C′(s)

• Degrees of freedom :
• twist κ0(s)
• curvatures κ1(s), κ2(s)
• Darboux vector :

Ω Ω Ω(s) = κ0(s) n0(s)+κ1(s) n1(s)+κ2(s) n2(s)

• Rotation of the material frame

∀i = 0, 1, 2 dni ds (s) = Ω Ω Ω(s) ∧ ni(s)

Geometry of a Kirchhoff Rod

• Centerline C(s)
• Material frame R(s)

R(s) = {n0(s), n1(s), n2(s)} with n0(s) = C′(s)

• Degrees of freedom :
• twist κ0(s)
• curvatures κ1(s), κ2(s)
• Darboux vector :

Ω Ω Ω(s) = κ0(s) n0(s)+κ1(s) n1(s)+κ2(s) n2(s)

• Rotation of the material frame

∀i = 0, 1, 2 dni ds (s) = Ω Ω Ω(s) ∧ ni(s)

Geometry of a Kirchhoff Rod

• Centerline C(s)
• Material frame R(s)

R(s) = {n0(s), n1(s), n2(s)} with n0(s) = C′(s)

• Degrees of freedom :
• twist κ0(s)
• curvatures κ1(s), κ2(s)
• Darboux vector :

Ω Ω Ω(s) = κ0(s) n0(s)+κ1(s) n1(s)+κ2(s) n2(s)

• Rotation of the material frame

∀i = 0, 1, 2 dni ds (s) = Ω Ω Ω(s) ∧ ni(s)

Kirchhoff Dynamic Equations

Conservation of linear and angular momenta ρS ∂2r ∂t2 (s, t) = ∂T ∂s (s, t) + F(s, t) ∂M ∂s (s, t) + n0(s, t) ∧ T(s, t) =

with ρS the lineic mass, T the tension, M the internal torque, and F the lineic density

• f external force.

Kirchhoff Dynamic Equations

Conservation of linear and angular momenta ρS ∂2r ∂t2 (s, t) = ∂T ∂s (s, t) + F(s, t) ∂M ∂s (s, t) + n0(s, t) ∧ T(s, t) =

with ρS the lineic mass, T the tension, M the internal torque, and F the lineic density

• f external force.

Elastic constitutive law M(s) = EI

• κ(s) − κ0(s)
• with K the stiffness and κ0 the natural curvatures/twist.

Kirchhoff Dynamic Equations

Conservation of linear and angular momenta ρS ∂2r ∂t2 (s, t) = ∂T ∂s (s, t) + F(s, t) ∂M ∂s (s, t) + n0(s, t) ∧ T(s, t) =

with ρS the lineic mass, T the tension, M the internal torque, and F the lineic density

• f external force.

Elastic constitutive law M(s) = EI

• κ(s) − κ0(s)
• with K the stiffness and κ0 the natural curvatures/twist.

Boundary conditions C(0) = C0, R(0) = R0 and T(L) = M(L) = 0

Space-time Discretisation

Challenges

• Nonlinear, stiff PDE with boundary conditions
• Stability issues with finite differences

Strong curvatures impose a very fine spatial discretization

Space-time Discretisation

Challenges

• Nonlinear, stiff PDE with boundary conditions
• Stability issues with finite differences

Strong curvatures impose a very fine spatial discretization

Better : spatial discretization beforehand

• Choice of a finite number of spatial coordinates

Example : the finite elements method

• System of ODE in time, solved with finite differences

Space-time Discretisation

Challenges

• Nonlinear, stiff PDE with boundary conditions
• Stability issues with finite differences

Strong curvatures impose a very fine spatial discretization

Better : spatial discretization beforehand

• Choice of a finite number of spatial coordinates

Example : the finite elements method

• System of ODE in time, solved with finite differences

→ Which choice for spatial coordinates ?

Two Families of Choice of Coordinates

ℓ g m

Two Families of Choice of Coordinates

(x, y) Nodal model coordinates X =

• x

y

• m¨

X = m g X = ℓ

Two Families of Choice of Coordinates

(x, y) Nodal model coordinates X =

• x

y

• m¨

X = m g X = ℓ Equations are simple to write down Additional constraints are needed

Two Families of Choice of Coordinates

θ Reduced model coordinate θ mℓ¨ θ = −mg sin θ

Two Families of Choice of Coordinates

θ Reduced model coordinate θ mℓ¨ θ = −mg sin θ Inextensibility is intrinsically preserved

Two Families of Choice of Coordinates

θ Reduced model coordinate θ mℓ¨ θ = −mg sin θ Inextensibility is intrinsically preserved Inversion gets easier

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number M(q) · ¨ q + K(q, q0) + A(q,˙ q) = F(q,˙ q, t) s.t. C(q) = 0

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix

M(q) · ¨ q + K(q, q0) + A(q,˙ q) = F(q,˙ q, t) s.t. C(q) = 0

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix
• Internal elastic forces

M(q) · ¨ q + K(q, q0) + A(q,˙ q) = F(q,˙ q, t) s.t. C(q) = 0

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix
• Internal elastic forces
• Nonlinear inertial terms

M(q) · ¨ q + K(q, q0) + A(q,˙ q) = F(q,˙ q, t) s.t. C(q) = 0

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix
• Internal elastic forces
• Nonlinear inertial terms
• External forces

M(q) · ¨ q + K(q, q0) + A(q,˙ q) = F(q,˙ q, t) s.t. C(q) = 0

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix
• Internal elastic forces
• Nonlinear inertial terms
• External forces

M(q) · ¨ q + K(q, q0) + A(q,˙ q) = F(q,˙ q, t) s.t. C(q) = 0

• Constraints

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix
• Internal elastic forces
• Nonlinear inertial terms
• External forces

M(q) · ¨ q + K(q, q0) + A(q,˙ q) = F(q,˙ q, t) s.t. C(q) = 0

• Constraints

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix
• Internal elastic forces
• Nonlinear inertial terms
• External forces

Nodal model

M · ¨ q + K(q, q0) = F(q,˙ q, t) s.t. C(q) = 0

• Constraints

Nodal model : M is sparse , constraints , K is nonlinear

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix
• Internal elastic forces
• Nonlinear inertial terms
• External forces

Reduced model

M(q) · ¨ q + K · (q−q0) + A(q,˙ q) = F(q,˙ q, t) Reduced model : M is dense , no constraint , K is linear

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix
• Internal elastic forces
• Nonlinear inertial terms
• External forces

M(q) · ¨ q + K · (q−q0) + A(q,˙ q) = F(q,˙ q, t) → We choose reduced and high-order coordinates : curvatures

More Generally : Spatial Discretization

Choice of coordinates q ∈ Rm : generalized coordinates, finite number

• Inertia matrix
• Internal elastic forces
• Nonlinear inertial terms
• External forces

M(q) · ¨ q + K · (q−q0) + A(q,˙ q) = F(q,˙ q, t) → We choose reduced and high-order coordinates : curvatures N.B. : The centerline will not be explicit

Geometry of a Kirchhoff Rod

• Centerline C(s)
• Material frame R(s)

R(s) = {n0(s), n1(s), n2(s)} with n0(s) = C′(s)

• Degrees of freedom :
• twist κ0(s)
• curvatures κ1(s), κ2(s)
• Darboux vector :

Ω Ω Ω(s) = κ0(s) n0(s)+κ1(s) n1(s)+κ2(s) n2(s)

• Rotation of the material frame

∀i = 0, 1, 2 dni ds (s) = Ω Ω Ω(s) ∧ ni(s)

Geometry : Darboux Problem

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

Geometry : Darboux Problem

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

Exact solution

• Existence of a unique solution
• However, no explicit formula in the general case

→ Numerical integration may be computationally expensive

Discrete Geometry : Super-Helix

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

Discrete Geometry : Super-Helix

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise constant

[Bertails et al. 2006]

Discrete Geometry : Super-Helix

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise constant

[Bertails et al. 2006]

Discrete Geometry : Super-Helix

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise constant

[Bertails et al. 2006]

• On each element, closed-form solution for R(s) and C(s)

Discrete Geometry : Super-Helix

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise constant

[Bertails et al. 2006]

• On each element, closed-form solution for R(s) and C(s)

→ Equations for a circular helix

Discrete Geometry : Super-Helix

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise constant

[Bertails et al. 2006]

• On each element, closed-form solution for R(s) and C(s)

→ Equations for a circular helix

• Continuous connection of R(s) between elements

Discrete Geometry : Super-Helix

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise constant

[Bertails et al. 2006]

• On each element, closed-form solution for R(s) and C(s)

→ Equations for a circular helix

• Continuous connection of R(s) between elements

→ All the kinematics is of closed-form → The centerline C(s) is C1-smooth

Discrete Dynamics : Super-Helix

q = [κ1

0, κ1 1, κ2 2, . . . , κN 0 , κN 1 , κN 2 ]T ∈ R3 N

Discrete Dynamics : Super-Helix

q = [κ1

0, κ1 1, κ2 2, . . . , κN 0 , κN 1 , κN 2 ]T ∈ R3 N

Computing the terms of the ODE M(q) · ¨ q + K · (q−q0) + A(q,˙ q) = F(q,˙ q, t)

Discrete Dynamics : Super-Helix

q = [κ1

0, κ1 1, κ2 2, . . . , κN 0 , κN 1 , κN 2 ]T ∈ R3 N

Computing the terms of the ODE M(q) · ¨ q + K · (q−q0) + A(q,˙ q) = F(q,˙ q, t)

• Closed-form expression in q, ˙

q for each term

Discrete Dynamics : Super-Helix

q = [κ1

0, κ1 1, κ2 2, . . . , κN 0 , κN 1 , κN 2 ]T ∈ R3 N

Computing the terms of the ODE M(q) · ¨ q + K · (q−q0) + A(q,˙ q) = F(q,˙ q, t)

• Closed-form expression in q, ˙

q for each term

• Example : Mi,j = ρS

L ∂C

∂qi (s)

T

· ∂C ∂qj (s) ds

Discrete Dynamics : Super-Helix

q = [κ1

0, κ1 1, κ2 2, . . . , κN 0 , κN 1 , κN 2 ]T ∈ R3 N

Computing the terms of the ODE M(q) · ¨ q + K · (q−q0) + A(q,˙ q) = F(q,˙ q, t)

• Closed-form expression in q, ˙

q for each term

• Example : Mi,j = ρS

L ∂C

∂qi (s)

T

· ∂C ∂qj (s) ds Time-solving

• Mixed implicit/explicit Euler scheme

M v + f = 0 avec v = ˙ qt+1

Discrete Dynamics : Super-Helix

q = [κ1

0, κ1 1, κ2 2, . . . , κN 0 , κN 1 , κN 2 ]T ∈ R3 N

Computing the terms of the ODE M(q) · ¨ q + K · (q−q0) + A(q,˙ q) = F(q,˙ q, t)

• Closed-form expression in q, ˙

q for each term

• Example : Mi,j = ρS

L ∂C

∂qi (s)

T

· ∂C ∂qj (s) ds Time-solving

• Mixed implicit/explicit Euler scheme

M v + f = 0 avec v = ˙ qt+1

• Implicit elastic forces

Discrete Dynamics : Super-Helix

q = [κ1

0, κ1 1, κ2 2, . . . , κN 0 , κN 1 , κN 2 ]T ∈ R3 N

Computing the terms of the ODE M(q) · ¨ q + K · (q−q0) + A(q,˙ q) = F(q,˙ q, t)

• Closed-form expression in q, ˙

q for each term

• Example : Mi,j = ρS

L ∂C

∂qi (s)

T

· ∂C ∂qj (s) ds Time-solving

• Mixed implicit/explicit Euler scheme

M v + f = 0 avec v = ˙ qt+1

• Implicit elastic forces

→ Stable simulations

Discrete Geometry : Super-Clothoïd

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

Discrete Geometry : Super-Clothoïd

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise-linear

Discrete Geometry : Super-Clothoïd

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise-linear

Discrete Geometry : Super-Clothoïd

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise-linear

• On each element, the solution is a 3D clothoïd

Discrete Geometry : Super-Clothoïd

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise-linear

• On each element, the solution is a 3D clothoïd
• But no more closed-form solution...

Discrete Geometry : Super-Clothoïd

κ2 κ1 κ0

∀i dni ds (s) = Ω Ω Ω(s) ∧ ni(s) R(0) = R0

If κ0(s), κ1(s), κ2(s) are piecewise-linear

• On each element, the solution is a 3D clothoïd
• But no more closed-form solution...
• How to integrate both precisely and efficiently ?

→ Power-series computation

[Casati and Bertails-Descoubes 2013]

Inverse Statics of a “Super-Model”

Goal Given q, find q0, E I and ρ S such that q is a stable equilibrium

Inverse Statics of a “Super-Model”

Goal Given q, find q0, E I and ρ S such that q is a stable equilibrium Equilibrium condition K(E I) ·

• q − q0

= F(q, ρ S) → Solve a linear problem of size ∼ 3 N

Inverse Statics of a “Super-Model”

Goal Given q, find q0, E I and ρ S such that q is a stable equilibrium Equilibrium condition K(E I) ·

• q − q0

= F(q, ρ S) → Solve a linear problem of size ∼ 3 N Sufficient condition of stability E I ρ S ≥ A(q)

Inverse Statics of a “Super-Model”

Goal Given q, find q0, E I and ρ S such that q is a stable equilibrium Equilibrium condition K(E I) ·

• q − q0

= F(q, ρ S) → Solve a linear problem of size ∼ 3 N Sufficient condition of stability E I ρ S ≥ A(q) → Compute the eigen values

• f a real symmetric matrix

(Details in [Derouet-Jourdan et al. 2010] )

Partial Conclusion

• The static inversion is trivial for an isolated “Super-Model”

Partial Conclusion

• The static inversion is trivial for an isolated “Super-Model”
• The only one difficulty is “purely” geometric :

How to convert a given curve as a piecewise helix/clothoïd ?

Partial Conclusion

• The static inversion is trivial for an isolated “Super-Model”
• The only one difficulty is “purely” geometric :

How to convert a given curve as a piecewise helix/clothoïd ?

• Robust and fast approximation algorithms can be designed

Example : floating tangents algorithm

[Derouet-Jourdan et al. 2013]

Open Problems

• How to extend to contacting fibers (with friction) ?
• How to generalize to elastic surfaces (plates / shells) ?

Open Problems

• How to extend to contacting fibers (with friction) ?
• How to generalize to elastic surfaces (plates / shells) ?

→ Work in progress...

And for contacting fibers ?

Input : set of curves (q) Output : natural curvatures (q0)

• f Super-Helices

→ Interpret the geometry as a set of Super-Helices at equilibrium under gravity and frictional contacts

Inverse Modeling of Super-Helices

Without contact K · (q − q0) = F(q) q0 = q − K−1F(q)

Inverse Modeling of Super-Helices

Without contact K · (q − q0) = F(q) q0 = q − K−1F(q) With frictional contact

• K · (q − q0) = F(q) + H(q)⊤r

r r r r r ∈ int(Kµ) (Coulomb’s cone) Kµ (P) (A) (B) r r r

• q0 = q − K−1 · (F(q) + H(q)⊤r

r r) r r r ∈ int(Kµ) Underdetermined problem

Decoupling gravity and contacts

Our approach

• Estimate q0 : q0

Our approach

• Estimate q0 : q0
• Find the “best” force r

r r, i.e., such that : min

r r r

1 2

q0

• q − K−1(H⊤r

r r + F) −q02 s.t. r r r ∈ int(Kµ)

Our approach

• Estimate q0 : q0
• Find the “best” force r

r r, i.e., such that : min

r r r

1 2

q0

• q − K−1(H⊤r

r r + F) −q02 + γr r r2 s.t. r r r ∈ int(Kµ)

• γ : regularization parameter

Our approach

• Estimate q0 : q0
• Find the “best” force r

r r, i.e., such that : min

r r r

1 2

q0

• q − K−1(H⊤r

r r + F) −q02 + γr r r2 s.t. r r r ∈ int(Kµ)

• γ : regularization parameter

→ Can be solved by reusing our direct solver for the dynamics !

(Details in [Derouet-Jourdan et al. 2013] )

Heuristics for estimating q0

Heuristics for estimating q0

1 q0 = q(L)

Heuristics for estimating q0

1 q0 = q(L) 2 q0 = q

Heuristics for estimating q0

1 q0 = q(L) 2 q0 = q

Remember that : min

r r r

1 2

q0

• q − K−1(H⊤r

r r + F) −q02 + γr r r2 s.t. r r r ∈ int(Kµ) → Find r r r which minimizes the elastic energy of the rods

Results

3 hairstyles

(a) 8,922 contacts, 5s (b) 30,381 contacts, 19s (c) 14,358 contacts, 15s

Discussion

Many limitations...

• Very simple heuristics to estimate q0
• Large dependence upon the quality of input data
• No stability criterion yet (= isolated case)
• Many parameters are assumed to be known

Discussion

Many limitations...

• Very simple heuristics to estimate q0
• Large dependence upon the quality of input data
• No stability criterion yet (= isolated case)
• Many parameters are assumed to be known

... And yet

• Some plausible results
• The proposed solution is an exact equilibrium
• Very fast inversion (a few seconds)

And for Plates / Shells ?

And for Plates / Shells ?

Case of a developable shell (ongoing work with A. Blumentals)

And for Plates / Shells ?

Case of a developable shell (ongoing work with A. Blumentals)

• Inextensibility yields 2 coupled Darboux problems

And for Plates / Shells ?

Case of a developable shell (ongoing work with A. Blumentals)

• Inextensibility yields 2 coupled Darboux problems
• Constant material curvatures yield a closed-form surface

One single element

And for Plates / Shells ?

Case of a developable shell (ongoing work with A. Blumentals)

• Inextensibility yields 2 coupled Darboux problems
• Constant material curvatures yield a closed-form surface

One single element

• Dynamic equations feature a linear elastic term

(+ a constraint)

And for Plates / Shells ?

Case of a developable shell (ongoing work with A. Blumentals)

• Inextensibility yields 2 coupled Darboux problems
• Constant material curvatures yield a closed-form surface

One single element

• Dynamic equations feature a linear elastic term

(+ a constraint)

Challenging questions

• How to connect elements ?
• How to handle non-developability ?

Perspectives

• Refine the identification process when subject to contact
• Leverage multiple static poses
• Investigate the stability of the equilibrium

Perspectives

• Refine the identification process when subject to contact
• Leverage multiple static poses
• Investigate the stability of the equilibrium
• Continue the extension to the inversion of plates and shells

Leverage the linearity of curvature-based models

Perspectives

• Refine the identification process when subject to contact
• Leverage multiple static poses
• Investigate the stability of the equilibrium
• Continue the extension to the inversion of plates and shells

Leverage the linearity of curvature-based models

• Perform some experimental validation

Ongoing collaboration with Laboratoire Jean le Rond D’Alembert (Paris 6) One of our goals : study the sensitivity of our inversion process to real data

Acknowledgments

Work performed in collaboration with

• Alexandre Derouet-Jourdan

(PhD student, defended in 2013)

• Romain Casati

(PhD student, defended in 2015)

• Gilles Daviet

(PhD student, ongoing)

• Alejandro Blumentals

(PhD student, ongoing)

• Victor Romero

(Postdoc, ongoing)

• Arnaud Lazarus

(MdC, UPMC)

Acknowledgments

Work performed in collaboration with

• Alexandre Derouet-Jourdan

(PhD student, defended in 2013)

• Romain Casati

(PhD student, defended in 2015)

• Gilles Daviet

(PhD student, ongoing)

• Alejandro Blumentals

(PhD student, ongoing)

• Victor Romero

(Postdoc, ongoing)

• Arnaud Lazarus

(MdC, UPMC)

Thank you for your attention !