Relating phase field and sharp interface approaches to structural topology optimization

Abstract

✭✷✳✶✸✮ ✇❤❡r❡ t❤❡ ✐♥❞❡① s❡ts ❛r❡ ❞❡✜♥❡❞ ❛s ▼♦r❡♦✈❡r✱ ✇❡ ♦❜t❛✐♥✱ ✉s✐♥❣ ✭✸✳✶✮✱ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C s✉❝❤ t❤❛t ❚❤❡ ❛❜♦✈❡ ❞✐s❝✉ss✐♦♥ s❤♦✇s ✭✸✳✶✽✮❚❤❡r❡❢♦r❡ (u, ϕ) ∈ H

Molding Direction Constraints in Structural Optimization via a Level-Set Method

“…Among the several methods that appeared in the literature, such as SIMP (Solid Isotropic Material with Penalization) method [14,49,16], the homogenization method [3,13,15], the phase field method [18,41,51,17] or the Soft Kill Option [31,23], the level-set method for shape and topology optimization [10,11,35,38,44] seems to fulfill industrial requirements in a satisfying way. Using a level-set function to describe implicitly the boundary of a shape [36,37] allows topological changes to appear in an easy way, while the geometric nature of the method is a benefit for the study of problems where the position of the interface plays a significant role (stress constraints, thermal problems with flux across the boundary, etc.).…”

A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

“…(1.4e) Similar to [23], we define 5) so that (1.4d) becomes 6) which allows us to decouple the chemotaxis mechanism that was appearing in (1.4c) and (1.4d). We point out that it is possible to neglect the effects of fluid flow by sending K → 0 in the case 1 ⋅ U = 0.…”

“…For example, the choice W = I − 1 ⊗ 1, where I is the identity matrix, is used in [5,26,44]. One can check that ζ ⋅ I − 1 ⊗ 1 ζ = ζ 2 for any ζ ∈ TG.…”

Structural optimization of contact problems using Cahn–Hilliard model

“…Due to material science inspiration topology optimization problems may be also considered as multiphase problems [7][8][9][10] where smooth enough material density function describes the distribution of two phases consisting from solid material and void. The areas occupied by these phases, indicated by suitable thin layers, are separated by sharp interfaces.…”