The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems

Main, A.; Scovazzi, G.

doi:10.1016/j.jcp.2017.10.026

Title: The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems

Abstract

We propose a new finite element method for embedded domain computations, which falls in the category of surrogate/approximate boundary algorithms. The key feature of the proposed approach is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary, and to appropriately modify the shifted boundary conditions, enforced weakly, in order to preserve optimal convergence rates of the numerical solution. This process yields a method which, in our view, is simple, efficient, and also robust, since it is not affected by the small-cut-cell problem. Although general in nature, here we apply this new concept to the Poisson and Stokes problems. We present in particular the full analysis of stability and convergence for the case of the Poisson operator, and numerical tests for both the Poisson and Stokes equations, for geometries of progressively higher complexity.

Authors:

Main, A. ^[1];

^[1]

Duke Univ., Durham, NC (United States)

Publication Date:: Mon Oct 23 00:00:00 EDT 2017

Research Org.:: Duke Univ., Durham, NC (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1611740

Alternate Identifier(s):: OSTI ID: 1702367

Grant/Contract Number:: SC0012169

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 372; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Computer Science; Physics; Embedded methods; Immersed boundary method; Small cut-cell problem; Approximate domain boundaries; Weak boundary conditions; Finite element method

Citation Formats


                    Main, A., and Scovazzi, G. The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems.  United States: N. p., 2017. 
Web.  doi:10.1016/j.jcp.2017.10.026.

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                    Main, A., & Scovazzi, G. The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems.  United States.  https://doi.org/10.1016/j.jcp.2017.10.026

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                    Main, A., and Scovazzi, G. Mon .  
"The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems".  United States.  https://doi.org/10.1016/j.jcp.2017.10.026.  https://www.osti.gov/servlets/purl/1611740.

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@article{osti_1611740,

  title        = {The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems},

  author       = {Main, A. and Scovazzi, G.},

  abstractNote = {We propose a new finite element method for embedded domain computations, which falls in the category of surrogate/approximate boundary algorithms. The key feature of the proposed approach is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary, and to appropriately modify the shifted boundary conditions, enforced weakly, in order to preserve optimal convergence rates of the numerical solution. This process yields a method which, in our view, is simple, efficient, and also robust, since it is not affected by the small-cut-cell problem. Although general in nature, here we apply this new concept to the Poisson and Stokes problems. We present in particular the full analysis of stability and convergence for the case of the Poisson operator, and numerical tests for both the Poisson and Stokes equations, for geometries of progressively higher complexity.},

  doi          = {10.1016/j.jcp.2017.10.026},

  journal      = {Journal of Computational Physics},

  number       = C,

  volume       = 372,

  place        = {United States},

  year         = {Mon Oct 23 00:00:00 EDT 2017},

  month        = {Mon Oct 23 00:00:00 EDT 2017}

}

Copy to clipboard

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Works referencing / citing this record:

The Shifted Interface Method: A flexible approach to embedded interface computations
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Li, Kangan; Atallah, Nabil M.; Main, G. Alex
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Equal Higher Order Analysis of an Unfitted Discontinuous Galerkin Method for Stokes Flow Systems
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Aretaki, Aikaterini; Karatzas, Efthymios N.; Katsouleas, Georgios
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$\phi$-FEM: A Finite Element Method on Domains Defined by Level-Sets
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Title: The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems

Abstract

Citation Formats

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A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) journal, July 1999

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An Interior Penalty Finite Element Method with Discontinuous Elements journal, August 1982

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Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem journal, April 2014

A bubble-stabilized finite element method for Dirichlet constraints on embedded interfaces journal, January 2006

A fixed-grid b-spline finite element technique for fluid-structure interaction: FIXED-GRID b-SPLINE FINITE ELEMENT TECHNIQUE journal, December 2013

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On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems journal, September 2005

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Shift boundary material point method: an image-to-simulation workflow for solids of complex geometries undergoing large deformation journal, May 2019

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Regularity estimates for elliptic boundary value problems in Besov spaces
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A new finite element formulation for computational fluid dynamics: VII. The stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces
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A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier–Stokes equations
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Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
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A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains
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An adaptive, formally second order accurate version of the immersed boundary method
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A robust Nitsche’s formulation for interface problems
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Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension
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Flow patterns around heart valves: A numerical method
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A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method)
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journal, August 1982

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Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem
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journal, May 2019

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text, January 2019

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Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated
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Cut finite element error estimates for a class of nonlinear elliptic PDEs
preprint, January 2020

Analysis of the Shifted Boundary Method for the Poisson Problem in General Domains
preprint, January 2020

Comparison of Shape Derivatives using CutFEM for Ill-posed Bernoulli Free Boundary Problem
preprint, January 2020

Convergence Analysis for Computation of Coupled Advection-Diffusion-Reaction Problems
preprint, January 2020

Stability and conditioning of immersed finite element methods: analysis and remedies
preprint, January 2022