3D-1D coupling on non conforming meshes via three-field optimization based domain decomposition

Abstract

A new numerical approach is proposed for the simulation of coupled threedimensional and one-dimensional elliptic equations (3D-1D coupling) arising from dimensionality reduction of 3D-3D problems with thin inclusions. The method is based on a well posed mathematical formulation and results in a numerical scheme with high robustness and flexibility in handling geometrical complexities. This is achieved by means of a three-field approach to split the 1D problems from the bulk 3D problem, and then resorting to the minimization of a properly designed functional to impose matching conditions at the interfaces. Thanks to the structure of the functional, the method allows the use of independent meshes for the various subdomains.

A gradient based resolution strategy for a PDE-constrained optimization approach for 3D-1D coupled problems

“…For simplicity, mesh refinement is denoted by means of a unique parameter h, representing the maximum diameter of the tetrahedra for the 3D mesh of Ω. The refinement level of the 1D meshes is related to h as follows: called N i the number of intersection points between the faces of the tetrahedra of the 3D mesh and segment Λ i , we build on Λ i a mesh made of N i equally spaced nodes for variable Û and 1 2 N i equally spaced nodes for variables Ψ and Φ. Clearly different refinement levels could be chosen on each segment and for each 1D unknown.…”

“…Matrix M is symmetric positive definite, as it follows from the structure of functional ( 27) and from the equivalence of this formulation with the well posed problem (28) [1]. The minimum of ( 29) is given by condition…”

“…We briefly recall here the derivation of the reduced 3D-1D coupled problem from the original equi-dimensional formulation, referring to [1] for a more comprehensive discussion.…”

“…This work presents a conjugate gradient based resolution strategy for a recently developed numerical scheme for the coupling of three-dimensional and one-dimensional elliptic equations (3D-1D coupling) [1]. Coupled problems with such dimensionality gap arise, in particular, when small tubular inclusions embedded in a much wider domain are dimensionally reduced to 1D manifolds for computational efficiency.…”

“…The present work is based on a re-formulation of the original 3D-3D problem into properly defined functional spaces, thus paving the way for a well posed formulation of the reduced 3D-1D problem [1]. The numerical resolution is further obtained through a PDE-constrained optimization based approach [17,18,19,20], in which problems in the 3D bulk domain and in the 1D inclusions are decoupled using a three-field based domain decomposition method.…”