Immersed Boundary Methods: Finite Elements Redesigned
Ernst Rank, Technical University of Munich
The success of the Finite Element Method in solid mechanics is, to a large extent, due to its conceptual simplicity and huge flexibility. A domain of computation is meshed, i.e. sub-divided into ‘elements’, on each of these elements an ‘Ansatz’, most often based on polynomials is made, these local approximations are ‘glued’ together fulfilling suitable continuity conditions, and in the thus defined function space a best approximation to an unknown solution of a partial differential equation is searched, e.g. by weighted residuals. The very first of these listed steps, i.e. the ‘sub-division into elements’, is yet not only source for flexibility, but also reason for many of the well-known practical problems associated with finite elements. In order to appropriately mesh a domain of computation, a geometric description of this domain needs to be available. Such a model is often generated in CAD-systems, using B-Splines or NURBS as surface elements. As these geometric entities are different from the description of classical finite elements, the transfer from one geometric model to the other is error-prone and often very time-consuming. This has large implications on the engineering effort for a finite element analysis and was the starting point for the success of the isogeometric analysis which uses the same spaces for geometry and approximating functions. Yet many more types of geometric models play important roles, which cannot easily be addressed by isogeometric analsis. These can be volumetric models, e.g. from computer tomograms, models based on point clouds derived from photographs of consumer cameras, and also ‘flawed’ models which are in a strict sense mathematically invalid, and need to be ‘healed’ for a numerical analysis. Last but not least structures may change during the physical process, e.g. by crack propagation or in melting/deposition processes.
Immersed boundary methods avoid meshing the domain of computation. Instead, the structure is embedded in a larger, simply shaped domain which is itself divided into a simple grid. The resulting ‘elements’ are non-conforming, i.e. cut by the boundary of the original domain. This seemingly simple modification of the original principle has many consequences. Several of the classical steps of FEM need to be redesigned, like the integration of element matrices, the definition of boundary conditions or the stabilization of equation systems. All these questions have been addressed with success during the last years in several variants of immersed boundary methods, like the Finite Cell Method (FCM, [1]), the CutFEM or the cgFEM. Some advantages over classical FEM are very significant and will be shown in several examples. Problems from biomechanics with very complex geometry are presented, computations on flawed, ‘dirty’ geometric models are discussed, simulations on point clouds obtained during drone flights are demonstrated and finally the validity of the FCM is shown on a 3D example of brittle fracture.
Reference
[1] The p‐Version of the Finite Element and Finite Cell Methods. A Düster, E Rank, B Szabó, Encyclopedia of Computational Mechanics Second Edition, 1-35
Ernst Rank, Technical University of Munich
The success of the Finite Element Method in solid mechanics is, to a large extent, due to its conceptual simplicity and huge flexibility. A domain of computation is meshed, i.e. sub-divided into ‘elements’, on each of these elements an ‘Ansatz’, most often based on polynomials is made, these local approximations are ‘glued’ together fulfilling suitable continuity conditions, and in the thus defined function space a best approximation to an unknown solution of a partial differential equation is searched, e.g. by weighted residuals. The very first of these listed steps, i.e. the ‘sub-division into elements’, is yet not only source for flexibility, but also reason for many of the well-known practical problems associated with finite elements. In order to appropriately mesh a domain of computation, a geometric description of this domain needs to be available. Such a model is often generated in CAD-systems, using B-Splines or NURBS as surface elements. As these geometric entities are different from the description of classical finite elements, the transfer from one geometric model to the other is error-prone and often very time-consuming. This has large implications on the engineering effort for a finite element analysis and was the starting point for the success of the isogeometric analysis which uses the same spaces for geometry and approximating functions. Yet many more types of geometric models play important roles, which cannot easily be addressed by isogeometric analsis. These can be volumetric models, e.g. from computer tomograms, models based on point clouds derived from photographs of consumer cameras, and also ‘flawed’ models which are in a strict sense mathematically invalid, and need to be ‘healed’ for a numerical analysis. Last but not least structures may change during the physical process, e.g. by crack propagation or in melting/deposition processes.
Immersed boundary methods avoid meshing the domain of computation. Instead, the structure is embedded in a larger, simply shaped domain which is itself divided into a simple grid. The resulting ‘elements’ are non-conforming, i.e. cut by the boundary of the original domain. This seemingly simple modification of the original principle has many consequences. Several of the classical steps of FEM need to be redesigned, like the integration of element matrices, the definition of boundary conditions or the stabilization of equation systems. All these questions have been addressed with success during the last years in several variants of immersed boundary methods, like the Finite Cell Method (FCM, [1]), the CutFEM or the cgFEM. Some advantages over classical FEM are very significant and will be shown in several examples. Problems from biomechanics with very complex geometry are presented, computations on flawed, ‘dirty’ geometric models are discussed, simulations on point clouds obtained during drone flights are demonstrated and finally the validity of the FCM is shown on a 3D example of brittle fracture.
Reference
[1] The p‐Version of the Finite Element and Finite Cell Methods. A Düster, E Rank, B Szabó, Encyclopedia of Computational Mechanics Second Edition, 1-35
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