DC6 Mathematical tools for immersed IGA:

Development of mathematical tools for immersed IGA, related, in particular, to higher order phase-field modeling.

Hello World!

#include <iostream>

int main() {
    std::cout << "Hello World!";
    return 0;
}

My name is Lucas and here I will write some highlights of my research within the Gecko Doctoral Network!

During the past months, I have been studying higher order phase-field formulations, notably the Cahn-Hilliard equation, used to simulate phase separation phenomena. The smooth basis functions of Isogeometric Analysis (IGA) simplify the resolution of fourth-order spatial derivatives, enabling enhanced accuracy and computational efficiency compared to second-order models, even with coarser meshes. Local adaptive refinement schemes using truncated hierarchical B-splines (THB-splines) further optimize computational costs by increasing the resolution only in regions where it is needed; in phase-field problems, these regions are the interfaces.

Recently, we developed an IGA-based phase-field implementation of the Cahn-Hilliard equation within G+Smo, an open-source C++ library that brings together mathematical tools for geometric design and numerical simulation. This implementation includes an adaptive scheme for solving the Cahn-Hilliard equation using truncated hierarchical B-spline (THB-spline) basis functions, which are particularly well-suited for adaptive refinement in the context of IGA, as they preserve the properties of hierarchical B-splines, such as linear independence and non-negativity, and also form a partition of unity.

This implementation provides the foundation for introducing other higher-order phase-field models in the library and serves as a prototype for solving the Cahn-Hilliard equation with a locally adaptive scheme on any single-patch geometry parametrization that exists therein.

Stay tuned for more news coming up soon!