• CutFEM provides higher order discretization of both geometry and Partial Differential Equations

    Mats G. Larson

    Umeå University

Features of CutFEM:
  • Minimizes the complexity of mesh generation

  • Handles complex and evolving geometries

  • Handles non-conform surface mesh representation of the computational geometry

Project objectives

This project focuses on shape and topology optimisation using a new finite element technique called CutFEM. The research topics are of significant importance to industry and the project will be carried out in close collaboration with industrial partners. The main objectives of the project are:

    1. To develop CutFEM as a general finite element method for simultaneous high order approximation of both geometry and partial differential equations, in the bulk and on surfaces, completely avoiding standard meshing technology and capable of handling both implicit and parametrised geometry descriptions, including CAD.
    2. To establish CutFEM based shape and topology optimisation algorithms that enable efficient solution of important industrial design optimisation problems.

The project consists of 5 different work packages, all using CutFEM:

    1. Surface evolution and form finding.
    2. Optimisation of thin structures for wave propagation.
    3. Stress, fatigue, and failure constrained topology optimisation.
    4. Multiscale methods for solution and optimisation of eigenvalue problems.
    5. Fundamental development and applications of CutFEM.

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.