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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Fig. 2. Construction of computational 2-complex. Superscripts i , j , k are indices of cells in M ; numeric superscripts are indices of cells in K .

Fig. 3. Illustration of a cochain complex in K with Hodge-star operators. Physics-modified Hodge-stars, ˆ p , have the same action as p .

are unique in a combinatorial sense. When M is considered as geometric realisation of a material microstructure, K is a subdivision of this polyhedral assembly, where the barycentres of σ p ∈ M are mapped to 0-cells of K . The construction is illustrated in Fig. 2 on a 2-complex for simplicity, where α p denote the cells of K . Notably, all α 2 ∈ K are quadrilaterals, and all α 3 ∈ K are hexahedra. The implications from working with K are two-fold: (1) the operations with forms on M are replaced by algebraically simpler operations with cochains on K ; and (2) cells of M with di ff erent dimensions are allowed to have di ff erent properties. For the latter, note that there are three types of 1-cells in K , corresponding to pairs ( σ 0 ≺ σ 1 ) , ( σ 1 ≺ σ 2 ) , ( σ 2 ≺ σ 3 ), providing the means to have di ff erent variations of physical quantities along 1-cell, 2-cell, and through 3-cells of the physical complex M . Further, there are two types of 2-cells in K , corresponding to pairs ( σ 0 ≺ σ 2 ) , ( σ 1 ≺ σ 3 ), providing the means to have di ff erent fluxes of physical quantities across 2-cell, and inside 3-cells of the physical complex M . Henceforth, K is referred to as the computational complex, N p denotes the number of p -cells in K , C p the space of cochains in K , and δ p = δ p ( K ). Algebraically, a p -cochain c p ∈ C p is a column vector (one-dimensional array) of dimension N p . The coboundary operators in K form a cochain complex shown in Fig. 3 with straight arrows. Algebraically, δ p is a sparse matrix of dimensions N p + 1 × N p , with 0, 1, -1 entries. Important to note is that δ p does not change the physical dimension of the quantity represented by a cochain c p ∈ C p . The coboundary operators can be considered as generalised, metric-independent analogues of the familiar operations from vector calculus: δ 0 is like a gradient of a scalar field, δ 1 is like a curl of a polar vector field, and δ 2 is like a divergence of an axial vector field. Note that δ 1 ◦ δ 0 = 0 corresponding to curl · grad = 0, and δ 2 ◦ δ 1 = 0 corresponding to div · curl = 0. Exterior product of forms on M is represented by a cup product of cochains in K , denoted by ; its action is given by C p C q → C p + q , while its technical implementation can be found in Arnold (2012); Berbatov et al. (2022). Metric in C p is introduced by a bilinear map, g p : C p × C p → C 0 , which gives a 0-cochain for a pair of p -cochains, consistent with the smooth exterior calculus Berbatov et al. (2022). In the simplest case, it can be assumed that the