Features of CutFEM:
Minimizes the complexity of mesh generation
Handles complex and evolving geometries
Handles non-conform surface mesh representation of the computational geometry
Popular science description
Boundary value problems (expressed as partial dffierential equations) in science and engineering are often solved numerically and several methods are used in practical applications. However, for problems where physical phenomena take place both inside the domain and on its surface, on interfaces between domains, or with domains evolving in time, several important questions remain unsolved. To date computational simulations of such problems are severely limited, since we lack reliable and accurate methods to simultaneously approximate the differential equations in the domain and on the surfaces.
This project focuses on the development, analysis, and application of a novel finite element technique called CutFEM, which provides a general high order method to simultaneously approximate both differential equations and geometry. To this end, we will include the recently emerged concept of approximation on (possibly embedded) surfaces in combination with cut volume elements in a rudimentary form of CutFEM that was developed earlier by the applicants. Fully developed, CutFEM will completely avoid using standard meshing technology. It can therefore handle different geometry descriptions, ranging from implicit level sets to CAD, as well as complex and evolving geometries, with high order accuracy in a way no other currently available technology can.
In the project we develop the CutFEM method for fundamental models of physical phenomena. This will include a theoretical basis that can be used to construct adaptive methods with guaranteed control of the error in both solution and geometry representation. We will also use CutFEM to construct novel shape and topology optimization methods. The new methods enable optimization with respect to maximum stresses, which strongly depends on accurate surface descriptions, and seamless treatment of thin and thick structures.