The high-order Shifted Boundary Method and its analysis

The Shifted Boundary Method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods, that has proven efficient in handling partial differential equation problems with complex geometries. The key feature of the SBM is a shift in th...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering 2022-05, Vol.394, p.114885, Article 114885
Main Authors: Atallah, Nabil M., Canuto, Claudio, Scovazzi, Guglielmo
Format: Article
Language:English
Subjects:
Approximate domain boundaries
Approximation
Boundary conditions
Boundary representation
Boundary value problems
Data structures
Differential geometry
High-order method
Immersed boundary method
Interpolation
Mathematical analysis
Partial differential equations
Shifted Boundary Method
Small cut-cell problem
Unfitted finite element methods
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Summary:The Shifted Boundary Method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods, that has proven efficient in handling partial differential equation problems with complex geometries. The key feature of the SBM is a shift in the location where boundary conditions are applied – from the true to a surrogate boundary – and an appropriate modification (again, a shift) of the value of the boundary conditions, in order to reduce the consistency error. This paper presents the high-order version of the method, its mathematical analysis, and numerical experiments. The proposed method retains optimal accuracy for any order of the finite element interpolation spaces despite the surrogate boundary is piecewise linear. As such, the proposed approach bypasses the problematic issue of meshing complex geometries with high-order body-fitted boundary representations, without the need of complex data structures for the integration on cut cells. •The high-order version of the Shifted Boundary Method (SBM) is presented.•The SBM is an unfitted FEM that does not require cut cells.•The SBM is defined on a surrogate domain, with modified boundary conditions.•The SBM is optimally accurate even if the surrogate boundary is piecewise affine.•Proofs of stability, convergence, and L2-error estimates are included.
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2022.114885