PSI - Issue 43

Andrey P. Jivkov et al. / Procedia Structural Integrity 43 (2023) 15–22 Author name / Structural Integrity Procedia 00 (2023) 000–000

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real numbers, while δ p transforms p -cochains into ( p + 1)-cochains. One approach to analysis on cell complexes, known as Discrete Exterior Calculus (DEC) Hirani (2003), was developed by assuming that cochains were analogues of continuum di ff erential forms and coboundary operators were analogues of exterior derivatives. DEC was used for problems involving conservation of scalar quantities, e.g. Hirani et al. (2015), and recently for elasticity Boom et al. (2022). However, DEC is not background-independent, as it relies on the existence of external fields, and does not allow for formulating balance laws for materials with components of di ff erent dimensions. A background-independent formulation is possible by a proper intrinsic definition of discrete di ff erential forms and operations with them. The combinatorial di ff erential forms introduced by Forman (2002) meet this requirement, as they are defined as maps from cells to linear combinations of lower-dimensional cells on their boundaries, i.e., without reference to any structures external to the complex. Exterior derivatives on such forms are rigorously defined. To date, this purely combinatorial theory has been used for example for topological analysis of structures Harker et al. (2014) and flows Mrozek and Wanner (2021) on cell complexes, as well as for computing combinatorial Ricci curvatures Forman (2003) and anal ysis of Ricci-flows on such complexes Weber et al. (2016). Remarkably, combinatorial forms and exterior derivatives correspond precisely to cochains and coboundary operators of an extended 3-complex Arnold (2012). This corre spondence has two important consequences. First, it allows for easier construction of additional metric-independent operations, such as exterior products of forms, using the cochains of the extended complex. Second, it allows for processes operating simultaneously on cells of the original / physical complex with di ff erent dimensions. Physical and mechanical problems require metric-dependent operations, such as inner products and Hodge-stars, which have been developed only recently and used for representing conservation of scalar quantities Berbatov et al. (2022). The aims of this paper are to introduce the elements of this new mathematical technique and to demonstrate how these are used in the formulation of physical balance laws for collections of discrete entities. The paper does not provide all mathematical details; interested readers are referred to Berbatov et al. (2022) for the elements omitted here. The base SI units of time T , length L , mass M , and thermodynamic temperature θ will be used. Square brackets around a quantity will denote its units. For shortening some expressions, F = M L T − 2 will be used for force, and E = M L 2 T − 2 = F L will be used for energy. Let M be a 3-complex whose geometric realisation is some material microstructure, and σ p denote an arbitrary p -cell. The relation σ p ≺ σ p + q will mean that σ p is incident on (on the boundary of) σ p + q . A function (map) assigning numbers to all p -cells is called a p -cochain and denoted by c p . The p -cochains in M form a vector space, denoted by C p . The basis cochains of C p , denoted by σ p , map individual p -cells to one, i.e., σ p i : σ j p → 0 for i j and σ p i : σ i p → 1. The cells of the complex are assigned orientations. A standard way is to decide on a consistent orientation of all top-dimensional cells, e.g., to select the positive orientation to be from interior to exterior of the 3-cells and assign arbitrary orientations for all lower-dimensional cells Cooke and Finney (1967). There are exactly three options for the relation between σ p and σ p + 1 in an oriented complex: σ p ⊀ σ p + 1 , encoded by 0; σ p ≺ σ p + 1 and they have consistent orientations, encoded by 1; σ p ≺ σ p + 1 and they have opposite orientations, encoded by -1. These relations are collected in a set of coboundary operators, δ p for p = 0 , 1 , 2, which describe simultaneously the topological structure of M and operations on cochains: δ p transforms p -cochains into ( p + 1)-cochains, δ p : C p → C p + 1 , with the property δ p + 1 ◦ δ p = 0, i.e., the coboundary of a coboundary is empty. A combinatorial di ff erential p -form, ω p , is a map from ( p + q )-cells to linear combinations of the q -cells on their boundaries Forman (2002). In other words, a p -form is a function that assigns numbers to pairs σ q ≺ σ p + q . In particular, a 0-form is a map from q -cells to linear combination of themselves, i.e., it is a function on all cells. The p -forms form a vector space denoted by Ω p . The exterior derivative of forms is not given explicitly here, but can be found in Forman (2002); its action is d p : Ω p → Ω p + 1 . Extending this framework with exterior product of forms and metric operations is challenging. However, it has been shown that there exists a 3-complex, K , such that the space of p -cochains in K is isomorphic to the space of p -forms on M , i.e., C p ( K ) ≡ Ω p ( M ), and the coboundary operators in K represent the exterior derivatives of forms on M , i.e., δ p ( K ) ≡ d p ( M ) Arnold (2012); Berbatov et al. (2022). The construction of K is as follows. All cells σ p ∈ M , for p = 0 , 1 , 2 , 3, are mapped to 0-cells of K , all pairs ( σ p ≺ σ p + 1 ) ∈ M , for p = 0 , 1 , 2, are mapped to 1-cells of K , all pairs ( σ p ≺ σ p + 2 ) ∈ M , for p = 0 , 1 are mapped to 2-cells of K , and all pairs all pairs ( σ p ≺ σ p + 3 ) ∈ M , for p = 0 are mapped to 3-cells of K . The mappings 2. Mathematical basis