Efficient construction of 2-chains with prescribed boundary and some - PowerPoint PPT Presentation

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 The (extended) dual complex Numerical examples Linking number Some applications Definition Let us consider a bounded polyhedral domain Ω ⊂ R 3 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 The (extended) dual complex Numerical examples Linking number Some applications 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 ∈ Z n e × n f 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 The (extended) dual complex Numerical examples Linking number Some applications 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 The (extended) dual complex Numerical examples Linking number Some applications Not uniqueness Clearly the problem has not a unique solution. ◮ If n t is the number of tetrahedra of T and Γ 0 , Γ 1 , . . . Γ p are the conected components of ∂ Ω then the dimension of the kernel of M is equal to n t + p . ◮ A natural strategie to obtain an unique solution is to add n t + 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 The (extended) dual complex Numerical examples Linking number Some applications 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 The (extended) dual complex Numerical examples Linking number Some applications The boundary dual complex Let T b = ( V b , E b , F b ) the triangulation of ∂ Ω induced by T and denote by T ′ b = ( V ′ b , E ′ b , F ′ b ) the dual complex of T b . ◮ The elements of V ′ b are the barycenters of the triangles in F b . ◮ The elements of E ′ b are associated to edges in E b . 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 The (extended) dual complex Numerical examples Linking number Some applications 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 n t equations). ◮ Let B ′ = ( V ′ ∪ V ′ b , N ′ ) be a spanning tree of the extended b , N ′ ∩ E ′ dual graph A ′ such that B ′ b = ( V ′ b ) is a spanning tree of 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 n t . ( The number of dual edges in B ′ is n t + n f b − 1. If b , N ′ ∩ E ′ B ′ b = ( V ′ b ) is a spanning tree of the graph ( V ′ b , E ′ b ) then n f b − 1 dual edges in B ′ correspond to boundary edges, hence n t correspond to faces. ) A. Alonso Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces The (extended) dual complex Numerical examples Linking number Some applications Theorem If γ = � e ∈E a e 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 c f f such that ∂ S = γ , with c f = 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 The (extended) dual complex Numerical examples Linking number Some applications 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 R 3 with disjoint supports, we define their linking number by lk ( γ, γ ′ ) := 1 x − y � � | x − y | 3 · d s ( x ) × d s ( y ) . 4 π γ ′ γ A. Alonso Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces The explicit formula Numerical examples The elimination algorithm Some applications A 1-cycle associated to each dual edge (face) Let B ′ = ( V ′ ∪ V ′ b , N ′ ) be a spanning tree of the dual graph A ′ b , N ′ ∩ E ′ such that B ′ b = ( V ′ 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 ′ v ′ + e ′ − C ′ D ′ e ′ = C ′ w ′ . A. Alonso Efficient construction of 2-chains with prescribed boundary and some

Introduction Construction of homological Seifert surfaces The explicit formula Numerical examples The elimination algorithm Some applications 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 The explicit formula Numerical examples The elimination algorithm Some applications 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 c f f such that ∂ S = γ , with c f = 0 for all f ∈ F with e ′ ( f ) ∈ N ′ . Moreover for all f ∈ F with e ′ ( f ) �∈ N ′ c f = lk ( D ′ e ′ ( f ) , R + γ ) . A. Alonso Efficient construction of 2-chains with prescribed boundary and some