The hierarchical finite cell method for multi-scale problems in structural mechanics

• Applicant: Professor Dr.-Ing. Alexander Düster
• Subject Area: Mechanics ; Applied Mechanics, Statics and Dynamics
• Term: from 2010 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 183669279

• Final Report Year: 2016

Final Report Abstract

The FCM is an efficient discretization strategy for problems involving complex geometries. The main characteristic of the FCM is that it offers high convergence rates without the burden of mesh generation. During this project, we aimed to improve and further develop the FCM with regard to different aspects, making the method more robust and effective for real applications. Our main findings and achievements during this project can be summarized as follows: 1. We proposed two different quadrature rules to compute integrals on cut cells – namely the adaptive integration based on spacetrees and the moment fitting method. Both of these methods were thoroughly tested, studied, and compared to other existing numerical integration approaches. The proposed methods are proved to be robust and very effective for different 2D and 3D applications, including the FCM. 2. Further, algorithms to impose homogeneous as well as inhomogeneous Neumann and Dirichlet boundary conditions were developed during the project. 3. Different iterative and direct solvers were also considered for the FCM. In that regard, we proposed a direct solver based on the nested dissection, as well as an iterative solver based on the Additive Schwarz method and the preconditioned conjugate gradient method. Our initial studies reveal that the proposed solvers either outperform the existing solvers or they perform equally well. We were not able to investigate this issue in more detail. Consequently, this issue remains as one of the possible directions of research for the future. 4. We developed an enrichment strategy based on the partition of unity method and the hp-d approach. The proposed enrichment strategy modifies the FCM Ansatz only locally, and it allows to achieve a very high convergence rate. Moreover, it opens up the possibility of treating different types of discontinuities and singularities in a very general way. We also introduced a very efficient technique to consider implicitly defined geometries. The proposed method paves the way for the introduction of an accurate, high-order enrichment function. 5. In this project, the application of the FCM was extended to wave propagation problems. To this end, the spectral cell method was introduced – employing the spectral shape functions, the GLL quadrature, and a novel mass lumping method proposed for the cut cells. The method allows for faster simulation (almost one order of magnitude) as compared to the standard p-FEM. In addition, the mesh in the SCM, likewise the FCM, can be obtained at hardly any cost.

Publications

• Local enrichment of the finite cell method for problems with material interfaces, Computational Mechanics, 52:741–762, 2013
  M. Joulaian and A. Duster
  
• Finite and spectral cell method for wave propagation in heterogeneous materials, Computational Mechanics, 54:661–675, 2014
  M. Joulaian , S. Duczek , U. Gabbert and A. Duster
  
• Numerical analysis of Lamb waves using the finite and spectral cell method, International Journal for Numerical Methods in Engineering, 99:26–53, 2014
  S. Duczek , M. Joulaian , A. Duster and U. Gabbert
  
• Numerical homogenization of hybrid metal foams using the finite cell method, Computers & Mathematics with Applications, 70:1501–1517, 2015
  S. Heinze , M. Joulaian and A. Duster
  
• Numerical integration of discontinuities on arbitrary domains based on moment fitting, Computational Mechanics, 57:979–999, 2016
  M. Joulaian , S. Hubrich and A.Duster