Overview
One aspect of a biological systems that is taken for granted is the fluid shapes. This stands in contrast to manufactured objects, where hard angles and flat faces are the norm. The ability to directly link the geometry to the PDE is very beneficial when trying to determine an optimal design. This is especially important when we know biological shapes have been perfected over time through evolution, so understanding and preserving that geometry in simulation is essential for accurate modeling.
Technical Formulation
Unlike standard finite element methods where the mesh is essentially cut into distinct elements, IGA represents functions through a spline basis. The shapes of splines are manipulated through control points which directly push through to the standard FE stiffness construction. A further benefit is the higher order splines increase the width of support and lower the degrees of freedom on the system, making solutions for continuous PDEs easier to obtain.
Medical Translation
In the medical world, there are two major benefits of leveraging IGA. One is the ability to reduce the computational pipeline by combining the segmentation with the computational mesh generation, where these are typically distinct processes in standard FE simulations. The other benefit is the ability to see how shape effects simulations. Custom medical devices might be created, and we could determine the optimal shape of a device to assist the patient rather than selecting one from a set list.
References & Resources
- Isogeometric Analysis: Toward Integration of CAD and FEA — Cottrell, Hughes, Bazilevs. The primary textbook.
- Introduction to Smooth Manifolds - Lee. Rigorous treatment of manifolds, charts, and atlases