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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basis p -cochains are mutually orthogonal, so that g p ( α p i , α p

j ) = 0 for i j , and g p ( α p i , α p

i ) > 0 depends on a weight

assigned to the cell α i p . Important element of the metric operations is the volume cochain, a 3-cochain formed by the 3-cell volumes and denoted by vol . Inner (scalar) product of forms is defined using the metric and the volume cochain, c p i , c p j = ( g p ( c p i , c p j ) vol )[ K ], consistent with the smooth exterior calculus. Inner products are used for construction of discrete Hodge-star operators, p : C p → C 3 − p , in a canonical manner Berbatov et al. (2022). Their action is shown in Fig. 3 with curved arrows. Unlike δ p , p changes the physical dimension of the quantity represented by a cochain c p ∈ C p by L 3 − 2 p . Algebraically, p is a sparse matrix with dimensions N 3 − p × N p . It provides geometric relations between Hodge-dual cells. Specifically, [ 0 ] i j 0 and [ 3 ] ji 0 only when α j 0 is on the boundary of α i 3 ; [ 1 ] i j 0 and [ 2 ] ji 0 only when α j 1 and α i 2 intersect at exactly one 0-cell and are on the boundary of one 3-cell. In addition to the geometric relations given by p , the Hodge-dual cells are related by physical parameters that encode constitutive relations between quantities represented by cochains. Such parameters can be mass or charge di ff usivity, thermal conductivity, moduli of elasticity, etc. These relations are collected in matrices, π p , that have the same stencils as p , i.e., the same dimensions and non-zero coe ffi cients at the same positions. It is therefore useful to introduce physics-modified Hodge-star operators, ˆ p , for which [ ˆ p ] i j = [ p ] i j · [ π p ] i j . The action of ˆ p is as shown in Fig. 3 for p . Both p and ˆ p play crucial role in the formulation of balance laws, together with δ p . p . The weight is related to the cell volume and has dimension L A scalar extensive quantity, Q , (mass, heat, charge) can be transported by random motion of the carrier of Q through the matter (di ff usion or conduction) and by transfer of matter containing Q (advection or drift). Recall that the continuum formulation of the balance law is point-wise, written in terms of an intensive quantity q : ∂ q ∂ t = −∇ · f + r , (1) where r is the rate of production of q at a point, and f is the flux (area) density of Q from the point to its surroundings. Generally, f = f d + f a , where f d is the di ff usive, and f a is the advective flux. The discrete framework separates clearly extensive and intensive quantities. Q is represented by a 3-cochain in K denoted by ψ . It is conserved in a 3-cell of K when the rate of change of Q in the 3-cell, ∂ψ/∂ t , equals the influx of Q through its boundary 2-cells plus the rate of production of Q in the 3-cell. The rate of production of Q is a 3-cochain, χ with [ χ ] = [ ψ ] T − 1 , which can be used to incorporate production mechanism(s) operating in 3-cells. For example, chemical or electrochemical reactions in 3-cells can produce or destroy substance or charge, and generate (if exothermic) or consume (if endothermic) heat with known rates. Heat can also be generated by plasticity in 3-cells. The flux of Q across a 2-cell is a 2-cochain, φ , and the influx into 3-cells is given by the 3-cochain − δ 2 φ (see Fig. 3). Since [ ∂ψ/∂ t ] = [ ψ ] T − 1 , it follows that [ φ ] = [ ψ ] T − 1 . The conservation of ψ is therefore ∂ψ ∂ t = − δ 2 φ + χ, (2) which is purely topological; compare with Eq. 1, which involves a background-dependent divergence. The flux can be di ff usive and advective: φ = φ d + φ a . φ d arises from the variation of an intensive quantity, represented by a 0-cochain β . This variation is a 1-cochain given by ϕ d = − δ 0 β (see Fig. 3). φ a arises from the velocity of matter containing Q , which is represented by a 1-cochain γ . The advection of β along 1-cells is a 1-cochain ϕ a = β γ . Both ϕ d and ϕ a are obtained from β by topological (metric-independent) operations. The specific geometric realisation of the 3-complex, which introduces the metric, and the specific physical problem, provide only the relations between β and ψ , and ϕ and φ , via physics-modified Hodge-star operators. Two examples with omitted χ are given here. In di ff usion of substance in structured solids, the extensive quantity is the mass of the substance in 3-cells, rep resented by ψ with [ ψ ] = M , and the intensive quantity is the mass density, represented by β with [ β ] = M L − 3 . In such case, there is no physical property relating 0-cells to 3-cells, and the relations between β and ψ are given simply by 0 with [ 0 ] = L 3 or 3 with [ 3 ] = L − 3 (see Fig. 3). The variation of β is ϕ = − δ 0 β with [ ϕ ] = [ β ] = M L − 3 . 3. Balance laws 3.1. Balance of scalar quantities