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Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Efficient construction of 2-chains with prescribed boundary and some applications in electromagnetism

Ana Alonso Rodr´ ıguez

Department of Mathematics, University of Trento

Analysis and Numerics of Acoustic and Electromagnetic Problems RICAM, Linz, October 17 - 22, 2016

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Joint work with Enrico Bertolazzi, Department of Industrial engineering, University of Trento; Riccardo Ghiloni, Department of Mathematics, University of Trento; Ruben Specogna, Department of Electrical, Management and Mechanical Engineering, University of Udine.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Outline

Introduction The (extended) dual complex Linking number Construction of homological Seifert surfaces The explicit formula The elimination algorithm Numerical examples Some applications Discrete source field Bases of the second relative homology group

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The (extended) dual complex Linking number

Definition

Let us consider a bounded polyhedral domain Ω ⊂ R3 endowed with a tetrahedral mesh T = (V , E, F, T).

◮ A 1-cycle γ of T is a formal linear combination, with integer

coefficients, of oriented edges of T with zero boundary.

◮ A 2-chain of T is a formal linear combination with integer

coefficients of oriented faces of T .

◮ A 1-cycle γ is said to be a 1-boundary of T if it is equal to

the boundary of a 2-chain S of T . We say that S is an homological Seifert surface of γ in T .

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The (extended) dual complex Linking number

Target

Our aim is to devise a fast and robust algorithm to compute an homological Seifert surface (HSS) S of a given 1-boundary γ of T .

◮ Given an orientation of the edges and the faces of T this

problem is a linear system with as many equations as edges and as many unknowns as faces of T .

◮ The matrix M ∈ Zne×nf of this linear system is the incidence

matrix between edges and faces of T :

◮ each row has as many non zero entries as the number of faces

incident on the corresponding edge.

◮ each column has just three non zero entries.

◮ This kind of problem is usually solved using the Smith normal

form, a computationally demanding algorithm.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The (extended) dual complex Linking number

Other related works

There are few works concerning the computation of homological Seifert surfaces.

◮ The first papers on this subject are those of Allili and

Kaczynski (2001) for a rectangular domain with a cubical subdivision, and Kaczynski (2001) for polyhedral domains with trivial homology. There is an extensive literature concerning the construction of minimal surfaces, but this is a more difficult problem and we are not interested in regularity or minimality.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The (extended) dual complex Linking number

Not uniqueness

Clearly the problem has not a unique solution.

◮ If nt is the number of tetrahedra of T and Γ0, Γ1, . . . Γp are

the conected components of ∂Ω then the dimension of the kernel of M is equal to nt + p.

◮ A natural strategie to obtain an unique solution is to add

nt + p equations by setting equal to zero the unknowns corresponding to suitable faces of T .

◮ To choose such faces we will use a suitable spanning tree of

the extended dual complex of T .

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The (extended) dual complex Linking number

The dual complex

Let T=(V , E, F, T) be a tetrahedral mesh of Ω and denote by T ′ = (V ′, E ′, F ′, T ′) the dual complex of T .

◮ The elements of V ′ are the barycenters of the tetrahedra in T. ◮ The elements of E ′ are associated to faces in F.

They are the line joining the barycenter of the face with the barycenters of the adjacent tetrahedra, two tetrahedra for internal faces and one for boundary faces.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The (extended) dual complex Linking number

The boundary dual complex

Let Tb = (Vb, Eb, Fb) the triangulation of ∂Ω induced by T and denote by T ′

b = (V ′ b, E ′ b, F ′ b) the dual complex of Tb. ◮ The elements of V ′ b are the barycenters of the triangles in Fb. ◮ The elements of E ′ b are associated to edges in Eb.

They are the line joining the barycenter of the edge with the barycenters of the two adjacent boundary faces.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The (extended) dual complex Linking number

The extended dual graph A′ = (V ′ ∪ V ′

b, E ′ ∪ E ′ b).

For the sake of simplicity in the following I assume that ∂Ω is connected, (so for uniqueness we need to add just nt equations).

◮ Let B′ = (V ′ ∪ V ′ b, N′) be a spanning tree of the extended

dual graph A′ such that B′

b = (V ′ b, N′ ∩ E ′ b) is a spanning tree

• f the graph (V ′

b, E ′ b). ◮ In order to have a unique solution we set equal to zero the

unknowns corresponding to the faces with the dual edge belonging to N′.

◮ They are nt. (The number of dual edges in B′ is nt + nfb − 1. If

B′

b = (V ′ b, N′ ∩ E ′ b) is a spanning tree of the graph (V ′ b, E ′ b) then

nfb − 1 dual edges in B′ correspond to boundary edges, hence nt correspond to faces.)

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The (extended) dual complex Linking number

Theorem

If γ =

e∈E ae e is a 1-boundary of T , there exists a unique

homological Seifert surface of γ in T , namely, a 2-chain S =

f ∈F cf f such that ∂S = γ, with

cf = 0 for all f ∈ F with e′(f ) ∈ N′ . (E, F are the set of oriented edges and oriented faces, respectively.)

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The (extended) dual complex Linking number

Linking number

Another important tool in the algorithm is the linking number between two closed and disjoint curves in the three-dimensional space

◮ It is an integer that represents the number of times that each

curve winds around the other.

◮ It is a double line integral.

Given γ and γ′, two 1-cycles in R3 with disjoint supports, we define their linking number by lk(γ, γ′) := 1 4π

• γ
• γ′

x − y |x − y|3 · ds(x) × ds(y) .

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

A 1-cycle associated to each dual edge (face)

Let B′ = (V ′ ∪ V ′

b, N′) be a spanning tree of the dual graph A′

such that B′

b = (V ′ b, N′ ∩ E ′ b) is a spanning tree of the graph

(V ′

b, E ′ b). ◮ Fix a′ ∈ V ′ ∪ V ′ b (the root of B′). ◮ For each v′ ∈ V ′ ∪ V ′ b let C ′ v′ be the unique 1-chain in B′

from a′ to v′.

◮ Given the oriented (dual) edge e′ = (v′, w′) ∈ E ′ ∪ E ′ b we

define D′

e′ the 1-cycle of A′

D′

e′ = C ′ v′ + e′ − C ′ w′ .

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

The graph The spanning tree

1 2 3 4 5 6 7 8 9

An edge in the cotree The associated 1-cycle

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

Theorem

If γ is a 1-boundary of T , there exists an unique homological Seifert surface of γ in T , namely, a 2-chain S =

f ∈F cf f such

that ∂S = γ, with cf = 0 for all f ∈ F with e′(f ) ∈ N′ . Moreover for all f ∈ F with e′(f ) ∈ N′ cf = lk(D′

e′(f ), R+γ) .

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

Preliminaries

If |γ| ∩ ∂Ω = ∅ then D′

e′ and γ could be not disjoint.

If e ∈ Eb then R+e. If e′ ∈ E ′

b then R−e′.

If e ∈ E \ Eb then R+e = e. If e′ ∈ E ′ then R−e′ = e′. By linearity we can define R+γ and R−D′

e′.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

Uniqueness

◮ Let S be an homological Seifert surface of γ without faces

f ∈ F such that e′(f ) ∈ N′. S =

• f ∈ F

e′(f ) ∈ N′

cf f

◮ Then for each g ∈ F such that e′(g) ∈ N′

lk(D′

e′(g), R+γ) = lk(R−D′ e′(g), γ) = lk(R−D′ e′(g), ∂S)

= lk(R−D′

e′(g),

• f ∈ F

e′(f ) ∈ N′

cf ∂f ) =

• f ∈ F

e′(f ) ∈ N′

cf lk(R−D′

e′(g), ∂f ) .

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

Uniqueness

lk(D′

e′(g), R+γ) =

• f ∈ F

e′(f ) ∈ N′

cf lk(R−D′

e′(g), ∂f ) . ◮ For f , g ∈ F such that e′(f ) ∈ N′ and e′(g) ∈ N′

lk(R−D′

e′(g), ∂f ) =

1 if f = g

• therwise.

◮ Hence cf = lk(D′ e′(f ), R+γ).

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

Existence

∂2

• f ∈ F

e′(f ) ∈ N′ lk(D′

e′(f ), R+γ) f

• = γ ?

Two steps:

◮ Setting η = γ − ∂2

• f ∈ F

e′(f ) ∈ N′ lk(D′

e′(f ), R+γ) f

• then

lk(D′

e′(g), R+η) = 0 for all g ∈ F. ◮ If lk(D′ e′(g), R+η) = 0 for all g ∈ F then η = 0.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

Elimination algorithm for the construction of HSS

◮ The direct use of this explicit formula can be very expensive.

BUT

◮ We can apply an easy elimination procedure, inspired in the

• ne proposed in Webb and Forghani (1989) to compute a

discrete vector field with assigned curl, to solve

• f ∈F

cf ∂2f =

• e∈E

aee .

◮ The elimination procedure could stop without having

computed all the coefficients of S.

◮ In this case we can use the explicit formula to restart the

elimination procedure.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

Elimination algorithm for the construction of HSS

• f ∈F

cf ∂2f =

• e∈E

aee . For each edge e ∈ E the equation reads

• f ∈F(e)
• e(f )cf = ae .

being F(e) the set of faces incident to e and oe(f ) = ±1 depending on the orientation of e in ∂f .

B(f) B(t) B(e) D(e) t f e ∂Ω B(f) B(t) B(e) D(e) t f e

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications The explicit formula The elimination algorithm

Elimination algorithm for the construction of HSS

◮ Let us denote by R the set of faces whith cf known.

Initially R = {f ∈ F : e′(f ) ∈ N′}.

◮ Let us denote by C the set of edges with just one incident face

f ∗ ∈ F(e) not in R. While R = F If C is not empty

pick e ∈ C and compute cf ∗ from

f ∈F(e) oe(f )cf = ae;

else

pick f ∗ ∈ F − R and compute cf ∗ = lk(D′

e′(f ∗), R+γ);

R = R ∪ {f ∗}. In many examples the explicit formula is not used: C = ∅ ⇒ R = F .

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Numerical examples

We consider three sets of test problems:

◮ the computational domain is a cube and the considered

1-boundary is a trivial polygonal knot;

◮ the computational domain is a cube but the 1-boundary is a

non-trivial knot or a link;

◮ the computational domain is homologically non-trivial:

◮ A torus with a concentric toroidal cavity; the 1-boundary γ has

two connected components that are circumferences.

◮ A cube with a knotted cavity; the 1-boundary is a trivial

polygonal knot embracing two branches of the cavity.

In all these examples but the last one, the elimination procedure provides the homological Seifert surface. In the last example it is necessary to use once the explicit formula.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Numerical examples

(a) (b) (c)

Figure: (a) The 1-boundary γ . (b) The support of the 2-chain for the coarse

• mesh. (c) The support of the 2-chain for the fine mesh.

Name Tetrah. Faces Edges Vertices |γ| |S| Time Example 1 48 120 98 27 12 28 < 1 Example 2 479,435 973,963 583,183 88,656 341 15,023 233

Table: Column |γ| reports the number of edges in the 1-boundary γ and

column |S| the number of faces in the computed homological Seifert surface. The computational time is expressed in milliseconds.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Numerical examples

(c) (b) (a) (f) (e) (d)

Figure: (a) A non-trivial knot in a cube (top), (b) a zoom on γ that is a 821

knot, (c) the support of the HSS. (d) The Hopf link in a cube (bottom), (e) a zoom on γ that is a Hopf link, (f) The support of the HSS.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Numerical examples

Name Tetrah. Faces Edges Vertices card|γ| card|S| Time 821 knot 87,221 175,317 102,212 14,117 170 2663 37 Hopf link 800,020 1,600,537 937,631 137,115 235 4841 407

Table: A non-trivial knot and the Hopf link in a cube. Column |γ| reports the number of edges in the 1-boundary γ and column |S| the number of faces in the computed homological Seifert surface. The computational time is expressed in milliseconds.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Numerical examples

(a) (b)

Figure: Toric shell: (a) The support of the 1-boundary γ is a pair of disjoint circumferences; (b) the support of the computed homological Seifert surface.

Name Tetrah. Faces Edges Vert. |γ| |S| Time Toric shell 1,851,494 3,871,379 2,419,350 399,465 176 1662 961

Table: The number of geometric elements of the triangulation, the edges belonging to the support of the 1-boundary γ, the faces belonging to the support of the 2-chain S and the computational time in milliseconds.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Numerical examples

(a) (b) (c) (d)

Figure: Knotted cavity: (a) The support of the 1-boundary γ is a trivial polygonal knot embracing two branches of the cavity, (b) a zoom on γ, (c) and (d) two different views of the support of the computed homological Seifert surface.

Name Tetrah. Faces Edges Vert. |γ| |S| Time Knotted cavity 8,267 16,913 10,187 1,551 13 225 < 10 Knotted cavity 529,664 1,065,104 626,566 91,127 52 10,407 309

Table: The number of geometric elements of the triangulation and of the edges belonging to the support of the 1-boundary γ.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications

Numerical examples

Figure: Computational complexity: computational time versus the number of

tetrahedra of the mesh and the regression line for the examples where the elimination procedure success.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

Two applications in computational electromagnetism

We are interested in alternative algorirhms for the construction of

◮ discrete vector fields with assigned curl, namely discrete

source fields;

◮ the so-called cutting surfaces used for the construction of

scalar magnetic potentials in non simply connected domains. The presented algorithm allows to compute bases of the second relative homology group H2(Ω, ∂Ω; Z).

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

Discrete source field

Let (V , L) be a spanning tree of the (V , E) graph. curl He,h = Je,h

• e He,h · τ = 0

∀ e ∈ L , If Ω is simply connected then

◮ for all e ∈ L we have

• e He,h · τ =
• De He,h · ds;

◮ moreover De = ∂2Se

• e

He,h · τ =

• ∂2Se

He,h · ds =

• Se

Je,h · ν . We have an explicit formula once we know Se. (The non simply connected case can be solved in a similar way.)

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

Geometric construction of bases of H2(Ω, ∂Ω; Z)

The key point is Poincar´ e-Lefschetz duality theorem.

◮ First we find g 1-boundaries σ′ 1, . . . , σ′ g with supports

contained in ∂Ω, whose homology classes in R3 form a basis

• f H1(R3 \ Ω; Z).

◮ Then for each m = 1, . . . , g we construct an homological

Seifert surface Sm of σ′

m in T .

The Poincar´ e-Lefschetz duality theorem ensures that the relative homology classes of the Sm’s in Ω modulo ∂Ω form a basis of H2(Ω, ∂Ω; Z).

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

Geometric construction of bases of H2(Ω, ∂Ω; Z)

We just need

◮ an efficient algorithm for the computation of the boundaries

{σ′

m}g m=1; (if ∂Ω is connected see Hiptmair and Ostrowsky

(2002), if it is not see Alonso Rodr´ ıguez, Bertolazzi, Ghiloni and Specogna (2016));

◮ an efficient algorithm for the construction of homological

Seifert surfaces.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

Geometric construction of bases of H2(Ω, ∂Ω; Z)

Figure: The first elementary example is a solid torus with a concentric toric

• cavity. The number of connected components of the boundary of the domain is

2 and also the first Betti number of the domain is 2.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

Geometric construction of bases of H2(Ω, ∂Ω; Z)

Mesh 1 Mesh 2 Mesh 3 Mesh 4 Edges 51521 145963 1321902 10238231 Faces 76330 227314 2177158 17210016 Mesh pre-processing [s] 0.607 1.800 17.76 141.2

• Comp. H1(∂Ω; Z) [s]

0.084 0.216 0.863 3.909 Boundary retrieval [s] 0.012 0.034 0.122 0.513

• Const. of the HSSs [s]

0.061 0.193 2.720 24.52 Total Time [s] 0.764 2.243 21.46 170.1 Table: The torus with a toric cavity: the number of geometric elements

• f the triangulation and the computational time.
• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

Geometric construction of bases of H2(Ω, ∂Ω; Z)

Figure: The domain is the complement of a Hopf link with respect to a two

• torous. The number of connected components of the boundary of the domain

is 3 and the first Betti number of the domain is 4.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

Geometric construction of bases of H2(Ω, ∂Ω; Z)

Mesh 1 Mesh 2 Mesh 3 Mesh 4 Edges 39692 263041 2255753 10152372 Faces 64007 434513 3794183 17148224 Mesh pre-processing [s] 0.857 3.183 30.98 153.1

• Comp. H1(∂Ω; Z) [s]

0.029 0.131 0.657 3.031 Boundary retrieval [s] 0.008 0.034 0.134 0.498

• Const. of the HSSs [s]

0.044 0.415 5.118 27.82 Total Time [s] 0.938 3.763 36.89 184.5 Table: Benchmark Hopf link: the number of geometric elements of the triangulation and the computational time.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

Geometric construction of bases of H2(Ω, ∂Ω; Z)

◮ A specific approach for construction of bases of H2(Ω, ∂Ω; Z)

has been proposed by Kotiuga in a series of works.

◮ The aim is to compute the so-called cutting-surfaces used to

construct scalar magnetic potentials in non simply connected domains.

◮ They are regular surfaces and this justify the substantial

complexity of the procedure.

◮ Our algorithm provides homological Seifert surfaces that could

be not regular.

◮ Our target is to propose an alternative algorithm to make cuts

for scalar potentials in tetrahedral meshes.

◮ We are working on the regularization of the representatives of

a basis of H2(Ω, ∂Ω; Z) constructed in this way.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

References

◮ A. Alonso Rodr´

ıguez, E. Bertolazzi, R. Ghiloni and R. Specogna, Efficient construction of homological Seifert surfaces. arXiv:1409.5487.

◮ M. Allili and T. Kaczynski, Geometric construction of a coboundary of a

cycle, Discrete Comput. Geom., 25 (2001), pp. 125–140.

◮ T. Kaczynski, Recursive coboundary formula for cycles in acyclic chain

complexes, Topol. Methods Nonlinear Anal., 18 (2001), pp. 351–371.

◮ J. P. Webb and B. Forghani, A single scalar potential method for 3D

magnetostatic using edge elements, IEEE Trans. Magn., 25 (1989), pp. 4126–4128.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces Numerical examples Some applications Discrete source field Bases of the second relative homology group

References

◮ A. Alonso Rodr´

ıguez, E. Bertolazzi, R. Ghiloni and R. Specogna, Geometric construction of bases of H2(Ω, ∂Ω; Z). arXiv:1607.05099.

◮ R. Hiptmair and J. Ostrowsky, Generators of H1(Γh, Z) for triangulated

surfaces: construction and classification, SIAM J. Comput., 31 (2002),

• pp. 1405–1423.

◮ P. R. Kotiuga, An algorithm to make cuts for scalar potentials in

tetrahedral meshes based on the finite element method, IEEE Trans. Magn., 25 (1989), pp 4129–4131.

◮ P. W. Gross and P. R. Kotiuga Electromagnetic Theory and Computation:

a Topological Approach, Cambridge University Press, New York, 2004.

• A. Alonso

Efficient construction of 2-chains with prescribed boundary and some