Hector Gomez (University of A Coruna, Spain)
Abstract:
Phase-field modeling refers to a particular mathematical description of a system with evolving interfaces. The key idea is that interfaces are described by a smoothly-changing phase field, defined on a fixed domain. The phase field is governed by a partial differential equation, which tracks the so-called diffuse interfaces and encodes the interfacial physics at once. The phase-field methodology is having a significant impact in the field of complex fluids [1] and holds promise to impact other areas of fluid mechanics and also solid mechanics. Typically, the phase-field equations are strongly nonlinear with higher-order spatial derivatives that account for the interfacial forces. From a numerical perspective, the higher-order partial- differential operators usually present in phase-field equations are difficult to deal with by standard finite element approaches that utilize C0 trial and weighting functions. In the first part of this presentation, I will show how our computational approach, based on Isogeometric Analysis [2], permits simple and efficient discretizations through the use of continuously differentiable splines [3]. I will also introduce our new second-order accurate and nonlinearly stable algorithms for phase field models. I will illustrate the effectiveness of our algorithms, by showing an application example in which we model tumor angiogenesis, that is, the growth of new capillaries from pre-existing ones in the microenvironment of a tumor [4].
References:
[1] R.I. Saye, J.A. Sethian, Multiscale modeling of membrane rearrangement, drainage, and rupture in evolving foams, Science, Vol. 340, pp. 720-724, 2013.
[2] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., Vol. 194, pp. 4135-4195, 2005.
[3] H. Gomez, V.M. Calo, Y. Bazilevs, T.J.R. Hughes, Isogeometric analysis of the Cahn- Hilliard phase-field model, Comput. Methods Appl. Mech. Engrg., Vol. 197, pp. 4333-4352, 2008.
[4] G. Vilanova, I. Colominas, H. Gomez, Capillary networks in tumor angiogenesis: From discrete endothelial cells to averaged phase-field descriptions via isogeometric analysis, Int. J. Numer. Meth. Biom. Eng., Vol 29, pp. 1015-1037, 2013.