HOMOLOGY IN ELECTROMAGNETIC MODELING Saku Suuriniemi Tampere - - PowerPoint PPT Presentation

HOMOLOGY IN ELECTROMAGNETIC MODELING Saku Suuriniemi

Tampere University of Technology, Institute of Electromagnetics, P.O.Box 692 33101 Tampere, Finland. Email: saku.suuriniemi@tut.fi

1 Intro

• Electromagnetic modeling involves solutions of PDE-BVP’s on

Maxwell’s equations.

• Typical quasi-static electromagnetic BVP over domain Ω

(Riemann manifold with boundary): – First order PDE’s (Maxwell) dh = j, db = 0 – Constitutive equation(s) b = ∗µh – Boundary conditions for disjoint Γ1, Γ2, where ∂Ω = Γ1 + Γ2 holds. th = 0 on Γ1, tb = 0 on Γ2

• May not have unique solution:

– dh = 0 applies in Ω (h is closed), but does not determine circulation

• c h over the dashed 1-chain c.

Ω Ω Ω I c c c

• c h = 0
• c h = 0
• c h = I

th = 0 S – However, boundary condition th = 0 on S does.

• In first and last h can be exact, i.e. admit potential h = dψ.

2 Analysis

Did B.C. really zero out circulations outside S?

• Consider circulation known over Γ1.

Γ2 Γ1 S

• Then for any ∂S,
• Γ1−∂S

h =

• Γ1

h −

• ∂S

h =

• Γ1

h −

• S

dh =

• Γ1

h holds by theorem

• ∂S h =
• S dh and dh = 0.

Yes: if B.C. makes

• Γ1 h = 0, then
• Γ1+∂S h = 0 for any S, too
• Identical circulations ⇒ ignore difference: c and c + ∂S belong

to the same equivalence class [c] for any S. – Division of 1-chains c ∈ C1(Ω) into homology cosets H1(Ω) = ker(∂1)/∂2(C2(Ω)) = (1-cycles)/(1-boundaries). – H1(Ω) topological, invariant under homeomorphisms of Ω. Circulation must be known for each generator of H1(Ω)

• Scalar potential spells “speed”

If

• c h = 0

∀[c] ∈ H1(Ω), is closed 1-form h exact?

• de Rham: For H1

dR(Ω) = (closed 1-forms)/(exact 1-forms) and

H1(Ω), isomorphism H1

dR(Ω) ∼

= H1(Ω) holds. Yes: only nonzero circulations prohibit potentials

• Criterion: closed h exact, if
• c h = 0 for all cycles c.

In practice, check only generators of H1(Ω)!

• Procedure:
• 1. Find H1(Ω)
• 2. Find circulations set by B.C.’s and impose the rest
• 3. Scalar potential for fields whose all circulations vanish

3 Example (concerns relative homology)

• In resistor, PDE’s

de = 0, dj = 0 hold with Ohm’s law j = ∗σe and boundary conditions te = 0 at ends and tj = 0 at side.

S c

• No unique solution unless U =
• c e or I =
• S j is known

(elements of H1(Ω, ends), H2(Ω, side)).

• Traditional strategy: Adopt potential e = dϕ and pose U as

potential difference.

S c

4 Application

• Homology groups useful in posing and checking problems.
• Riemannian manifold makes computation possible.

– FEM mesh needed ⇒ finitely generated chain groups. – Cheap enough in practice.

• Options in everyday modeling software:
• 1. Automatic exhaustive checking: Compute automatically all

homology groups, check that boundary conditions and sources are consistent. Complain for missing or conflicting conditions.

• 2. Checklist: Automatically produce homology groups

(relevant ones), user verifies that appropriate conditions set.

• 3. Toolbox: Groups computed upon request.