Lie Group and Homogeneous Variational Integrators: Towards a - - PowerPoint PPT Presentation

Lie Group and Homogeneous Variational Integrators: Towards a Geometrically Exact Model of Elasticity

Melvin Leok

Mathematics, Purdue University.

Joint work with Mathieu Desbrun, Anil Hirani, Taeyoung Lee, Jerrold Marsden, Harris McClamroch, and Alan Weinstein. Discrete Differential Geometry, Berlin, July, 2007. mleok@math.purdue.edu http://www.math.purdue.edu/˜mleok/ Mathematics, Purdue University NSF DMS-0504747 and DMS-0726263

2

The Mathematics of Falling Cats

3

Connections, Curvature, and Geometric Phase

Connections

• Connections provide a means of comparing elements of a fiber

based at different points on the manifold. Holonomy and Curvature

• Geometric Phase is an example
• f holonomy.
• Curvature can be thought of as

infinitesimal holonomy.

area = A finish start

4

Example of Holonomy

Vatican Museum double helical staircase designed by Giuseppe Momo in 1932.

5

Geometric Control of Spacecraft

Geometric Phase based controllers

• Shape controlled using internal momentum wheels and gyroscopes.
• Changes in shape result in corresponding changes in orientation.
• More precise than chemical propulsion based orientation control.

rigid carrier spinning rotors

6

Shape Dynamics of Formations of Satellites

NASA Terrestrial Planet Finder (TPF)

• The NASA Terrestrial Planet Finder (TPF) and the ESA Darwin

missions are examples of a potential application of the geometric formation control of satellite clusters.

Small satellites in arranged in a formation to yield a large effective aperture telescope. Courtesy NASA/JPL-Caltech. Artist’s conception of the ESA Darwin

• flotilla. Courtesy ESA.
• It is quite natural to think of controlling the shape of the cluster,

and its orientation (group) separately.

7

Geometry and Numerical Methods

Dynamical equations preserve structure

• Many continuous systems of interest have properties that are con-

served by the flow:

• Energy
• Symmetries, Reversibility, Monotonicity
• Momentum - Angular, Linear, Kelvin Circulation Theorem.
• Symplectic Form
• Integrability
• At other times, the equations themselves are defined on a mani-

fold, such as a Lie group, or more general configuration manifold

• f a mechanical system, and the discrete trajectory we compute

should remain on this manifold, since the equations may not be well-defined off the surface.

8

Motivation: Geometric Integration

Main Goal of Geometric Integration: Structure preservation in order to reproduce long time behavior. Role of Discrete Structure-Preservation: Discrete conservation laws impart long time numerical stability to computations, since the structure-preserving algorithm exactly conserves a discrete quantity that is always close to the continuous quantity we are interested in.

9

Geometric Integration: Energy Stability

Energy stability for symplectic integrators

Continuousenergy Isosurface Discreteenergy Isosurface Controlonglobalerror

10

Geometric Integration: Energy Stability

Energy behavior for conservative and dissipative systems

200 400 600 800 1000 1200 1400 1600 0.05 0.1 0.15 0.2 0.25 0.3 Time Energy

Variational Runge-Kutta Benchmark

100 200 300 400 500 600 700 800 900 1000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Time Energy

Midpoint Newmark Explicit Newmark Variational non-variational Runge-Kutta Benchmark

(a) Conservative mechanics (b) Dissipative mechanics

11

Geometric Integration: Energy Stability

Solar System Simulation

• Forward Euler

qk+1 = qk + h ˙ q(qk, pk), pk+1 = pk + h ˙ p(qk, pk).

• Inverse Euler

qk+1 = qk + h ˙ q(qk+1, pk+1), pk+1 = pk + h ˙ p(qk+1, pk+1).

• Symplectic Euler

qk+1 = qk + h ˙ q(qk, pk+1), pk+1 = pk + h ˙ p(qk, pk+1).

12

Geometric Integration: Energy Stability

Forward Euler

−30 −20 −10 10 20 30 −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 2144 1980 2000 2020 2040 2060 2080 2100 2120 2140 2160 0.5 1 Energy error

13

Geometric Integration: Energy Stability

Inverse Euler

−30 −20 −10 10 20 30 −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 2077 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 −50 50 Energy error

14

Geometric Integration: Energy Stability

Symplectic Euler

−30 −20 −10 10 20 30 −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 2268 1950 2000 2050 2100 2150 2200 2250 2300 −2 2 x 10

−3

Energy error

15

Introduction to Computational Geometric Mechanics

Geometric Mechanics

• Differential geometric and symmetry techniques applied to the

study of Lagrangian and Hamiltonian mechanics. Computational Geometric Mechanics

• Constructing computational algorithms using ideas from geometric

mechanics.

• Variational integrators based on discretizing Hamilton’s principle,

automatically symplectic and momentum preserving.

16

Discrete Mechanics

Discrete Variational Principle

q a () q b () q t ( ) Q q t ( ) variedcurve q0 qN qi Q qi variedpoint

• Discrete Lagrangian

Ld ≈ h L (q(t), ˙ q(t)) dt

• Discrete Euler-Lagrange equation

D2Ld(q0, q1) + D1Ld(q1, q2) = 0

17

Comparing representations of the rotation group

Euler Angles

• Local coordinate chart, exhibits singularities.
• Requires change of charts to simulate large attitude maneuvers.

Unit Quaternions

• Reprojection used to stay on unit 3-sphere.
• The 3-sphere is a double-cover of SO(3) which causes topological

problems for optimization. Rotation Matrices

• 9 dimensional space (3×3 matrices) with a 6 dimensional constraint

(orthogonality), but the exponential map saves the day.

18

Variational Lie Group Techniques

Basic Idea

• To stay on the Lie group, we parametrize the curve by the initial

point g0, and elements of the Lie algebra ξi, such that, gd(t) = exp

• ξs˜

lκ,s(t)

• g0
• This involves standard interpolatory methods on the Lie algebra

that are lifted to the group using the exponential map.

• Automatically stays on SO(n) without the need for reprojection,

constraints, or local coordinates.

• Order of accuracy of method is independent of the retraction, as

the variational principle is at the level of the Lie group, and the retraction is simply used to locally parametrize the group.

19

Model Problem

3D Pendulum

• A rigid body, with a pivot point

and a center of mass that are not collocated, under gravita- tional forces.

• Three degrees of freedom.
• Exhibits surprisingly rich and

complex dynamics.

20

Physical Realization of a 3D Pendulum

Triaxial Attitude Control Testbed

Attitude Dynamics and Control Laboratory, University of Michigan, Ann Arbor.

21

Example of a Lie Group Variational Integrator

3D Pendulum

• Lagrangian

L(R, ω) = 1 2

• Body
• (˜

ρ)ω

2dm − V (R),

where · : R3→R3×3 is a skew mapping such that xy = x × y.

• Equations of motion

J ˙ ω + ω × Jω = M, where M = ∂V

∂R TR − RT ∂V ∂R.

˙ R = R ω.

22

Example of a Lie Group Variational Integrator

3D Pendulum

• Discrete Lagrangian

Ld(Rk, Fk) = 1 h tr [(I3×3 − Fk)Jd] − h 2V (Rk) − h 2V (Rk+1).

• Discrete Equations of Motion

Jωk+1 = F T

k Jωk + hMk+1,

S(Jωk) = 1 h

• FkJd − JdF T

k

• ,

Rk+1 = RkFk.

23

Example of a Lie Group Variational Integrator

Automatically staying on the rotation group

• The magic begins with the ansatz,

Fk = exp( fk), and the Rodrigues’ formula, which converts the equation,

• Jωk = 1

h

• FkJd − JdF T

k

• ,

into hJωk = sin fk fk Jfk + 1 − cos fk fk2 fk × Jfk.

• Since Fk is the exponential of a skew matrix, it is a rotation matrix,

and by matrix multiplication Rk+1 = RkFk is a rotation matrix.

24

Numerical Simulation

Chaotic Motion of a 3D Pendulum

25

Numerical Simulations

Flyby of two dumbbells

26

Numerical Simulations

Effect of representations (Runge-Kutta)

5 10 15 1 2 3 4 T

T ro t T tran

5 10 15 −4 −2 2 4 t E

T U E

Runge-Kutta with quaternions

5 10 15 1 2 3 4 T

T ro t T tran

5 10 15 −4 −2 2 4 t E

T U E

Runge-Kutta with Euler angles

27

Numerical Simulations

Effect of representations (Runge-Kutta)

5 10 15 1 2 3 4 T

T ro t T tran

5 10 15 −4 −2 2 4 t E

T U E

Runge-Kutta with SO(3)

5 10 15 −1 1 2 3 x 10

−3

∆ E 5 10 15 2 4 6 x 10

−3

SO(3) error t

I − RT R I − RT

2 R2

Runge-Kutta with SO(3) (Integration error)

28

Numerical Simulations

Lie group variational integrator on SO(3)

Trajectory in inertial frame

5 10 15 1 2 3 4 T 5 10 15 −4 −2 2 4 t E

T ro t T tran T U E

Transfer of energy

29

Numerical Simulations

Conservation Properties: Lie Group Integrator

20 40 60 80 100 −2 2 4 6 8 10 12 14 x 10

−4

∆¯ E ¯ t

deviation in total energy

20 40 60 80 100 0.05 0.1 0.15 0.2 norm(I−RT R) 20 40 60 80 100 1 2 3 x 10

−5

norm(I−R2

T R2)

¯ t

error in the rotation matrix

30

Numerical Simulations

Conservation Properties: Lie Group Integrator

20 40 60 80 100 −5 5 x 10

−14

20 40 60 80 100 −5 5 x 10

−14

∆¯ γT 20 40 60 80 100 −5 5 x 10

−14

¯ t

deviation in total linear momentum

20 40 60 80 100 −2 −1 1 x 10

−6

20 40 60 80 100 −5 5 x 10

−6

∆¯ πT 20 40 60 80 100 −5 5 x 10

−6

¯ t

deviation in total angular momentum

31

Numerical Simulations

Comparison with other methods

• Our Lie group variational integrator (LGVI) is a Lie St¨
• rmer–

Verlet method, so it is a second-order symplectic Lie group method.

• We compare it to other second-order accurate methods:
• Explicit Midpoint Rule (RK):

Preserves neither symplectic nor Lie group properties.

• Implicit Midpoint Rule (SRK):

Symplectic but does not preserve Lie group properties.

• Crouch-Grossman (LGM):

Lie group method but not symplectic.

32

Numerical Simulations

Comparison with other methods

10 20 30 −0.1593 −0.159 time E

RK SRK LGM LGVI

Computed total energy for 30 seconds

10

−4

10

−3

10

−2

10

−15

10

−10

10

−5

10 Step size mean |I−RTR|

Mean orthogonality error I − RTR vs. step size

10

−4

10

−3

10

−2

10

−8

10

−6

10

−4

10

−2

Step size mean |Δ E|

RK SRK LGM LGVI

Mean total energy error |E − E0| vs. step size

10

−4

10

−3

10

−2

10

2

10

3

10

4

10

5

Step size CPU time (sec)

CPU time vs. step size

33

Applications to Asteroid Simulations

Computational Considerations

• Asteroids are approximated by rubble piles (hard sphere models)
• r simplicial complexes.
• Force evaluations using fast multipole methods or polyhedral po-

tential techniques.

• Computational cost is dominated by cost of force evaluation.
• Lie St¨
• rmer–Verlet variational integrators use only one force eval-

uation per timestep, and are only implicit in the attitude, and converge in 2-3 Newton steps.

• Used by Scheeres, et al. to accurately simulate binary near-Earth

Asteroid 66391 (1999 KW4).

34

Applications to Asteroid Simulations

35

Extensions to Homogeneous Spaces

Variational Integrators for S2

• SO(3) acts transitively on S2. A Lie group variational integrator

can be used to propagate dynamical systems evolving on S2.

• Constructed using constrained variations, as in the case of

discrete reduction theory.

• Constrain the infinitesimal generators to lie in the plane orthogonal

to the position vector of the point on the sphere, so as to eliminate rotations that are in the symmetry direction. This construction is related to the Hopf fibration S1 ֒ → S3 ֒ → S2.

• By using either the Rodrigues formula or the Cayley transforma-

tion, we can explicitly parametrize the constrained space, so the method is extremely efficient.

36

Under-actuated control with symmetry of a 3D pendulum

−2 −1.5 −1 −0.5 −2 −1.5 −1 −0.5

37

Under-actuated control with symmetry of a 3D pendulum

−12 −10 −8 −6 −4 −2 −12 −10 −8 −6 −4 −2

38

Constraint Distributions

Towards A Unified Numerical Treatment

• Geometric Phases: The conservation of the angular momen-

tum in the gravity direction induces a connection whose holonomy yields geometric phases.

• Nonholonomic Systems: Constraint distributions that are

nonintegrable.

• Homogeneous Spaces: The isotropy group associated with el-

ements of a homogeneous space induces a nonholonomic constraint distribution on the Lie group.

39

Variational Integrators on S2

Constrained variations

• The condition that q ∈ S2 = {q ∈ R3|q · q = 1} yields a con-

strained variation of q in the variational principle. δq = ξ × q, where ξ ∈ R3 is constrained to be orthogonal to q, i.e., ξ · q = 0. Lagrangian

• The configuration space is a cartesian product of two-spheres, Q =

(S2)n, and the Lagrangian has the form, L(q1, . . . , qn, ˙ q1, . . . , ˙ qn) = 1 2

n

• i,j=1

˙ qT

j Mij ˙

qi − V (q1, . . . , qn),

40

Variational Integrators on S2

Continuous equations

• The variation of the action integral is given by,

δG =

n

• i,j=1

ξi · ( ˙ qi × Mij ˙ qj + qi × Mij ¨ qj)

• T

−

n

• i=1

T ξi ·  (qi ×

n

• j=1

Mij ¨ qj) + qi × ∂V ∂qi   .

• The continuous equations of motion are given by,

Mii¨ qi = qi × (qi ×

n

• j=1

j=i

Mij ¨ qj) − ( ˙ qi · ˙ qi)Miiqi + qi ×

• qi × ∂V

∂qi

• Or equivalently in matrix form,

     M11I3×3 −M12ˆ q1ˆ q1 · · · −M1nˆ q1ˆ q1 −M21ˆ q2ˆ q2 M22I3×3 · · · −M2nˆ q2ˆ q2 . . . . . . . . . −Mn1ˆ qnˆ qn −Mn2ˆ qnˆ qn · · · MnnI3×3           ¨ q1 ¨ q2 . . . ¨ qn      =      −( ˙ q1 · ˙ q1)M11q1 + ˆ q2

1 ∂V ∂q1

−( ˙ q2 · ˙ q2)M22q2 + ˆ q2

2 ∂V ∂q2

. . . −( ˙ qn · ˙ qn)Mnnqn + ˆ q2

n ∂V ∂qn

    

41

Variational Integrators on S2

Lagrangian

• The discrete Lagrangian has the form,

Ld(q1k, . . . , qnk, q1k+1, . . . , qnk+1) = 1 2h

n

• i,j=1

Mij(qik+1 − qik) · (qjk+1 − qjk) − h 2Vk − h 2Vk+1.

Discrete equations

• The variation of the discrete action sum is given by,

δGd =

N−1

• k=1

n

• i=1

ξik ·  1 h(qik ×

n

• j=1

Mij(−qjk+1 + 2qjk − qjk−1)) − hqik × ∂Vk ∂qik   .

• The discrete equations of motion are,

1 h(qik ×

n

• j=1

Mij(−qjk+1 + 2qjk − qjk−1)) − hqik × ∂Vk ∂qik = 0.

42

Variational Integrators on S2

Discrete equations using Cayley representation

     

2M11I3×3 1+f1·f1

−2M12ˆ

q1(ˆ q2+q2fT

2 )

1+f2·f2

· · · −2M1nˆ

q1(ˆ qn+qnfT

n )

1+fn·fn

−2M21ˆ

q2(ˆ q1+q1fT

1 )

1+f1·f1 2M22I3×3 1+f2·f2

· · · −2M2nˆ

q2(ˆ qn+qnfT

n )

1+fn·fn

. . . . . . . . . −2Mn1ˆ

qn(ˆ q1+q1fT

1 )

1+f1·f1

−2Mn2ˆ

qn(ˆ q2+q2fT

2 )

1+f2·f2

· · ·

2MnnI3×3 1+fn·fn

           f1 f2 . . . fn      =       M11hω1 − (q1 × n

j=1,j=1 M1j(qj × hωj)) − h2 2 qi × ∂V ∂qi

M22hω2 − (q2 × n

j=1,j=2 M2j(qj × hωj)) − h2 2 q2 × ∂V ∂q2

. . . Mnnhωn − (qn × n

j=1,j=n Mnj(qj × hωj)) − h2 2 qn × ∂V ∂qn

      , Fiqi = 1 1 + fi · fi ((1 − fi · fi)qi + 2 ˆ fiqi).

• The implicit system of equations for the fi’s can be solved using

fixed-point iteration

• These equations automatically evolve the qi’s on the two-sphere.

43

Variational Integrators on S2

Spherical Pendulum

rk+1 =

• hωk + h2g

2l rk × e3

• × rk + rk
• 1 −
• hωk + h2g

2l rk × r3

• 2

ωk+1 =ωk + hg 2l rk × e3 + hg 2l rk+1 × e3

Double Spherical Pendulum

• 2

1+f1·f1I3×3

−

2α 1+f2·f2 ˆ

r1ˆ r2 −

2β 1+f1·f1 ˆ

r2ˆ r1

2 1+f2·f2I3×3

f1 f2

• =
• hω1k − hα(r1k × (r2k × ω2k)) + h2g

2l1 (r1k × e3) + 2αf2·f2 1+f2·f2 ˆ

r1r2 hω2k − hβ(r2k × (r1k × ω1k)) + h2g

2l2 (r2k × e3) + 2βf1·f1 1+f1·f1 ˆ

r2r1

• r1k+1 = (I3×3 + ˆ

f1)(I3×3 − ˆ f1)−1r1k r2k+1 = (I3×3 + ˆ f2)(I3×3 − ˆ f2)−1r2k where α =

m2 m1+m2 l2 l1 and β = l1 l2.

• Implicit system of equations can be solved using fixed-point itera-

tion, and requires about 5-6 iterations to reach machine precision.

44

Variational Integrators on S2

Properties of the double spherical pendulum method

20 40 60 80 100 −0.7505 −0.75 −0.7495 −0.749 −0.7485 −0.748 t E

Energy error

20 40 60 80 100 10

−20

10

−15

10

−10

10

−5

10 t |rT r−1|

S2 error

45

Variational Integrators for S2

Double Spherical Pendulum Simulation

46

Variational Integrators for S2

Elastic Rod

47

Variational Integrators for S2

Magnetic Arrays

48

Discrete Differential Geometry: Constructing the Discrete Lagrangian Density

49

Discrete Curvatures from Geometric Identities

Discrete Gauß Curvature

• The discrete expression for the Gauß curvature can be derived from

the Gauß–Bonnet theorem, and is given by

• D

KdA =

• p∈D

Kp , where Kp is the angle defect at a point p, given by Kp = 2π −

• i

θi .

θ θ θ

50

Discrete Connection Form

Definition A Discrete Connection Form is a continuous map, Ad : Q × Q → G, such that,

• Ad is G-equivariant.

Ad ◦ Lg = Ig ◦ Ad.

This is the discrete analogue of the statement, A ◦ Lg = Adg ◦ A.

• Ad induces a splitting of the Discrete Atiyah sequence.

Ad(iq(g)) = g.

This is the discrete analogue of the statement, A(ξQ) = ξ.

51

Discrete Atiyah Sequence

Discrete Atiyah Sequence

˜

G

1 ˜

G

i

• (π1,Ad)
• (Q × Q)/G

(π,π)

• (·,·)h
• αAd
• S × S

1S×S

˜

G

i1

• π1
• ˜

G ⊕ (S × S)

π2

• i2
• S × S

Maps

• i : ˜

G → (Q × Q)/G, where, i([q, g]G) = [q, gq]G.

• (π, π) : (Q × Q)/G → S × S, where,

(π, π)([q0, q1]G) = (πq0, πq1).

52

Examples of Primal Simplices and Dual Cells

Primal Simplex σ0, 0-simplex σ1, 1-simplex Dual Cell ⋆σ0, 3-cell ⋆σ1, 2-cell

53

Examples of Primal Simplices and Dual Cells

Primal Simplex σ2, 2-simplex σ3, 3-simplex Dual Cell ⋆σ2, 1-cell ⋆σ3, 0-cell

54

Differential Forms and Exterior Derivative

Cochains and Differential Forms

• A Discrete Differential Form is a cochain on the simplicial
• complex. That is,

Ωk

d(K) = Ck(K; R) = Hom(Ck(K), R).

Exterior Derivative

• The Exterior Derivative is defined by using the Generalized

Stokes Theorem, dαk, σk+1 = αk, ∂σk+1. where in the boundary operator, orientation must be carefully taken into account. For example, ∂

• =

.

55

Discrete Levi-Civita Connection

Discrete Riemannian manifold

• Cartan’s persective: Bundle of oriented orthonormal frames over a

manifold as a principal SO(n) bundle.

• Semidiscretize by discretizing manifold with a simplicial complex,

but keeping the group SO(n) continuous.

• Associate with each n-simplex a metric tensor g.

Constructing the Levi-Civita connection

• The Levi-Civita connection is a SO(n)-valued dual one-form.
• This element of SO(n) transforms the frame associated with a n-

simplex into the frame associated with an adjacent n-simplex, and is assigned to the codimension-one face common to both simplices.

56

Discrete Levi-Civita Connection

Curvature

• Curvature is a dual two-form, and is associated with the dual of a

codimension-two simplex, given by ⋆σn−2. Simplicial Complex, K n − 2 primal simplex, σn−2 2 dual cell, ⋆σn−2 1 dual chain, ∂ ⋆ σn−2 n − 1 primal chain, ⋆∂ ⋆ σn−2

• The curvature B of the discrete Levi-Civita connection is given by,

B, ⋆σn−2 = dA, ⋆σn−2 = A, ∂ ⋆ σn−2.

57

Hodge Star and Codifferential

Hodge Star

• The discrete Hodge Star is a map ∗ : Ωk

d(K) → Ωn−k d

(⋆K). For a k-simplex σk and a discrete k-form αk, 1 |σk|αk, σk = 1 | ⋆ σk|∗αk, ⋆σk. Codifferential

• The discrete codifferential operator δ : Ωk+1

d

(K) → Ωk

d(K)

is defined by δ(Ω0

d(K)) = 0 and on (k + 1)-discrete forms to be,

δβ = (−1)nk+1 ∗ d ∗ β .

58

Application

Laplace-Beltrami

• The Laplace-Beltrami operator is a special case of the more

general Laplace-deRham operator ∆ = dδ + δd. 1 |σ0|∆f, σ0 = −δdf, σ0 = −∗d ∗ df, σ0 = − 1 | ⋆ σ0|d ∗ df, ⋆σ0 = − 1 | ⋆ σ0|∗df, ∂(⋆σ0) = − 1 | ⋆ σ0|∗df,

• σ1≻σ0

⋆σ1

V

59

Application

Laplace-Beltrami = − 1 | ⋆ σ0|

• σ1≻σ0

∗df, ⋆σ1 = − 1 | ⋆ σ0|

• σ1≻σ0

| ⋆ σ1| |σ1| df, σ1 = − 1 | ⋆ σ0|

• σ1≻σ0

| ⋆ σ1| |σ1| (f(v) − f(σ0))

V

• This recovers a formula involving cotangents found by Meyer et

al.using a variational approach.

60

Multisymplectic Geometry

Geometry and Variational Mechanics

• Base space X. The independent variables, typ-

ically (n+1)-spacetime, denoted by (x0, . . . , xn).

• Configuration bundle. π : Y → X.
• Configuration q : X → Y . Gives the field variables over each

spacetime point.

• First jet extension J1Y . Consists of the first partials of the

field variables with respect to the independent variables.

• Lagrangian density L : J1Y → Ωn+1(X).
• Action integral S(q) =
• X L(j1q).
• Hamilton’s principle δS = 0.

61

Variational Formulation of Harmonic Functions

Inner Product for Differential Forms

• Need an inner product for forms,
• α, β

=

• M

α ∧ ∗β.

• At a discrete level, this involves a primal-dual wedge product, which

we only have for the case of primal k forms and dual (n−k)-forms, αk ∧ ∗βk, Vσk = |Vσk| |σk|| ⋆ σk|αk, σk∗βk, ⋆σk = 1 n | ⋆ σk| |σk| αk, σkβk, σk.

62

Variational Formulation of Harmonic Functions

Discrete Variational Principle

• A discrete Harmonic function is a stationary point of the following

discrete Lagrangian, L =

• σ1∈K

df ∧ ∗df, Vσ1.

• The corresponding Euler-Lagrange equation is,
• σ1=[v1,v0]≻v0

2 n | ⋆ σ1| |σ1| df, σ1 = 0.

• This means that the variational formulation of discrete Harmonic

functions is equivalent to the formulation in terms of the Laplace- Beltrami operator.

63

Discrete Electromagnetism

Discrete Formulation

• Covariant formulation using the 4-vector potential as the funda-

mental variable.

• 3+1 tensor product discretization, K ⊗ N.
• Lorentzian metric structure causes the Laplace-

Beltrami operator to be a hyperbolic operator as

• pposed to an elliptic operator.
• Equivalent expressions when applying discretiza-

tion at the level of the variational principle, and at the level of the equations.

Space Time

• Discretizing either the Euler-Lagrange equations or the Lagrangian

using DEC yields the same Discrete Euler-Lagrange equations.

64

Discrete Electromagnetism

Hodge Star for Prismal Complexes in Lorentzian Space

• The Hodge star ∗ is defined by the expression,

α ∧ ∗β = α, β v , and depends on the metric in particular.

• The Discrete Hodge Star in Lorentzian Space is given by,

1 | ⋆ σk|∗αk, ⋆σk = κ(σk) 1 |σk|αk, σk, where | · | stands for the volume and κ(σk) is the causality sign

• f the simplex σk.

65

Discrete Electromagnetism

Causality Sign

• The causality sign κ(σk) arises from the pseudo-Riemannian

metric structure of Lorentzian space-time. It is defined to be +1 if all the edges of the simplex are spacelike, and −1 otherwise. σ2 κ(σ2) +1 +1 −1 −1 −1

66

Towards Discrete Geometrically Exact Elasticity

Elastic stress as curvature in a frame bundle

• Introduce functionals elastic stress-energy in the frame bundle by

considering discrete curvature measures.

• Use noncommutative harmonic analysis. Complete basis

for L2(G) using irreducible unitary group representations.

• More explicitly, a group representation ϕ : G → GL(Cn) is a

group homomorphism, i.e., ϕ(g · h) = ϕ(g) · ϕ(h).

• The Peter-Weyl theorem states,

L2(G) =

• ϕ∈ ˆ

G Vϕ,

and g → ej, ϕ(g) · ei form a basis for the vector space Vϕ.

• For compact Lie groups and Lie groups with bi-invariant Haar

measures, tools such as FFTs and Plancherel theorem generalizes.

67

Summary

Lie group variational integrators

• A combination of Lie group ideas with variational integrators, with

the properties:

• global, and singularity-free.
• symplectic, momentum preserving.
• automatically stays on the Lie group without the need for con-

straints, reprojection, or local coordinates. Homogeneous variational integrators

• Based on the transitive Lie group action on a homogeneous space.
• Wide range of potential applications in engineering and scientific

computation.

68

Summary

Towards Discrete Geometrically Exact Elasticity

• Use techniques from discrete differential geometry to construct dis-

crete Lagrangian densities.

• Combine with multisymplectic integrators, and Lie group and ho-

mogeneous variational integrators.

• May involve noncommutative harmonic analysis, global interpola-

tion on Lie groups.

69

Overview

Geometric computational methods

Computational Geometric Mechanics

Discrete Geometry Discrete Mechanics Discrete Exterior Calculus Lie Group Variational Integrators Discrete Principal Connections Nonholonomic Integrators

⇓

Computational Geometric Control Theory

Discrete Controlled Lagrangian Systems Discrete Optimal Control on Lie Groups Discrete Estimation Methods

70

Questions? http://www.math.purdue.edu/˜mleok/