Numerical approximation of a 3D mechanochemical interface model for skin patterning

• Published in: Journal of computational physics Vol. 384; pp. 383 - 404
• Main Authors: De Oliveira Vilaca, Luis Miguel, Milinkovitch, Michel C., Ruiz-Baier, Ricardo
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.05.2019; Elsevier Science Ltd
• Subjects: Adaptive algorithms; Adaptive time stepping; Advection; Computational physics; Coupling (molecular); Deformation; Diffusion layers; Elasticity; Elasticity-diffusion problem; Finite element methods; Galerkin method; Interface coupling; Mass transfer; Mathematical models; Mechanical properties; Nonlinear equations; Numerical methods; Pattern formation; Reaction-diffusion equations; Robustness (mathematics); Skin; Skin appendage modelling; Three dimensional models; Pattern formation; Interface coupling; Adaptive time stepping; Finite element methods; Elasticity-diffusion problem; Skin appendage modelling
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2019.01.023

We introduce a model for the mass transfer of molecular activators and inhibitors in two media separated by an interface, and study its interaction with the deformations exhibited by the two-layer skin tissue where they occur. The mathematical model results in a system of nonlinear advection-diffusion–reaction equations including cross-diffusion, and coupled with an interface elasticity problem. We propose a Galerkin method for the discretisation of the set of governing equations, involving also a suitable Newton linearisation, partitioned techniques, non-overlapping Schwarz alternating schemes, and high-order adaptive time stepping algorithms. The experimental accuracy and robustness of the proposed partitioned numerical methods is assessed, and some illustrating tests in 2D and 3D are provided to exemplify the coupling effects between the mechanical properties and the advection-diffusion–reaction interactions involving the two separate layers. •A new model for the interaction of morphogens and deformation of two-layered tissue.•Coupling of the elasticity and reaction–diffusion blocks through non-overlapping Schwarz domain decomposition.•High-order adaptive time stepping based on local indicators.•Computational tests evaluating the spatio-temporal accuracy and efficiency of the finite element schemes.•Numerical tests illustrating the properties of the model.