Domain Decomposition Analogies and Extensions

Abstract

One can consider the multiple sample testing introduced in the last chapter as at ypeo fd omaind ecomposition. In order to see this, we consider a " par-titioning " approachw hereby ag lobal domain, under arbitrary loading, is di-vided into nonoverlapping subdomains.O nt he interiors ubdomain partitions an approximate globallyk inematically admissible solution is projected. This allows the subdomains to be mutually decoupled, and therefore separately solvable. The subdomain boundary value problems are solved with the exact microstructural representationc ontained withint heirr espectiveb oundaries, but with approximate displacement boundary data.The resulting microstruc-tural solution is the assembly of the subdomain solutions, eachrestricted to its correspondings ubdomain. The approximate solutioni sf ar morei nexpensive to compute than the direct problem. The work followsf romr esults foundi n Zohdi et al. [234]. 8.1 Boundary valuep roblem formulations The globally exact solution, u ,i sc haracterizedb yt he following virtual work formulation: Find u ∈ H 1 (Ω) , u | Γ u = d , sucht hat Ω ∇ v : σ d Ω = Ω f · v d Ω + Γ t t · v d A ∀ v ∈ H 1 (Ω) , v | Γ u = 0 , (8.1) where σ is the Cauchy stress. In the infinitesimal strain, linearly elastic, case σ = IE : ∇ u .I no rder to construct approximate solutions, nextw e consider the subdomain boundary value problems, whichh avet he exact mi-crostructural representationc ontained withint he domain, but approximate displacement boundary data on thei nterior subdomain boundaries.