Access the full text.
Sign up today, get DeepDyve free for 14 days.
A. Idesman, B. Dey (2020)
The treatment of the Neumann boundary conditions for a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshesComputer Methods in Applied Mechanics and Engineering, 365
A. Main, G. Scovazzi (2017)
The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problemsJ. Comput. Phys., 372
A. Coco, G. Russo (2018)
Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interfaceJ. Comput. Phys., 361
James Cheung, M. Gunzburger, P. Bochev, M. Perego (2020)
An optimally convergent higher-order finite element coupling method for interface and domain decomposition problems, 6
A. Idesman, B. Dey (2019)
A new 3-D numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshesComputer Methods in Applied Mechanics and Engineering
S. May, M. Berger (2015)
An Explicit Implicit Scheme for Cut Cells in Embedded Boundary MeshesJournal of Scientific Computing, 71
E. Burman, P. Hansbo (2010)
Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier methodComputer Methods in Applied Mechanics and Engineering, 199
A. Idesman, B. Dey (2020)
Accurate numerical solutions of 2-D elastodynamics problems using compact high-order stencilsComputers & Structures, 229
A. Idesman, B. Dey (2021)
Optimal local truncation error method for solution of wave and heat equations for heterogeneous materials with irregular interfaces and unfitted Cartesian meshesComputer Methods in Applied Mechanics and Engineering, 384
Kangan Li, N. Atallah, G. Main, G. Scovazzi (2019)
The Shifted Interface Method: A flexible approach to embedded interface computationsInternational Journal for Numerical Methods in Engineering, 121
A. Guittet, M. Lepilliez, S. Tanguy, F. Gibou (2015)
Solving elliptic problems with discontinuities on irregular domains - the Voronoi Interface MethodJ. Comput. Phys., 298
A. Idesman, B. Dey (2020)
A new numerical approach to the solution of the 2-D Helmholtz equation with optimal accuracy on irregular domains and Cartesian meshesComputational Mechanics, 65
Fei Wang, Yuanming Xiao, Jinchao Xu (2016)
High-order extended finite element methods for solving interface problemsComputer Methods in Applied Mechanics and Engineering
Abtin Rahimian, Shravan Veerapaneni, G. Biros, S. Veerapaneni, D. Gue-Yffier, D. Zorin, G. Biros
Author's Personal Copy Journal of Computational Physics Author's Personal Copy
Ruchi Guo, Tao Lin (2019)
A Higher Degree Immersed Finite Element Method Based on a Cauchy Extension for Elliptic Interface ProblemsSIAM J. Numer. Anal., 57
Qinghui Zhang, I. Babuska (2020)
A stable generalized finite element method (SGFEM) of degree two for interface problemsComputer Methods in Applied Mechanics and Engineering, 363
H. Kreiss, N. Petersson (2005)
A Second Order Accurate Embedded Boundary Method for the Wave Equation with Dirichlet DataSIAM J. Sci. Comput., 27
A. Idesman, B. Dey (2020)
New 25-point stencils with optimal accuracy for 2-D heat transfer problems. Comparison with the quadratic isogeometric elementsJ. Comput. Phys., 418
E. Rank, M. Ruess, S. Kollmannsberger, D. Schillinger, A. Düster (2012)
Geometric modeling, isogeometric analysis and the finite cell methodComputer Methods in Applied Mechanics and Engineering, 249
Peter Vos, R. Loon, S. Sherwin (2008)
A comparison of fictitious domain methods appropriate for spectral/hp element discretisationsComputer Methods in Applied Mechanics and Engineering, 197
H. Kreiss, N. Petersson, J. Yström (2004)
Difference Approximations of the Neumann Problem for the Second Order Wave EquationSIAM J. Numer. Anal., 42
S. Vallaghé, T. Papadopoulo (2010)
A Trilinear Immersed Finite Element Method for Solving the Electroencephalography Forward ProblemSIAM J. Sci. Comput., 32
A. Idesman (2020)
A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D casesArchive of Applied Mechanics, 90
H. Johansen, P. Colella (1998)
A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular DomainsJournal of Computational Physics, 147
B. Dey, A. Idesman (2020)
A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—part 2: numerical simulations and comparison with FEMArchive of Applied Mechanics, 90
A. Idesman, B. Dey (2020)
Compact high-order stencils with optimal accuracy for numerical solutions of 2-D time-independent elasticity equationsComputer Methods in Applied Mechanics and Engineering, 360
Ting Song, A. Main, G. Scovazzi, M. Ricchiuto (2018)
The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flowsJ. Comput. Phys., 369
Ruchi Guo, Tao Lin (2019)
An immersed finite element method for elliptic interface problems in three dimensionsJ. Comput. Phys., 414
Ceren Gürkan, A. Massing (2018)
A stabilized cut discontinuous Galerkin framework: I. Elliptic boundary value and interface problemsArXiv, abs/1803.06635
H. Kreiss, N. Petersson (2005)
An Embedded Boundary Method for the Wave Equation with Discontinuous CoefficientsSIAM J. Sci. Comput., 28
Journal of Computational Physics, 230
A. Idesman (2018)
The use of the local truncation error to improve arbitrary-order finite elements for the linear wave and heat equationsComputer Methods in Applied Mechanics and Engineering, 334
E. Rank, S. Kollmannsberger, C. Sorger, A. Düster (2011)
Shell Finite Cell Method: A high order fictitious domain approach for thin-walled structuresComputer Methods in Applied Mechanics and Engineering, 200
P. McCorquodale, P. Colella, H. Johansen (2001)
A Cartesian grid embedded boundary method for the heat equation on irregular domainsJournal of Computational Physics, 173
Qian Zhang, Kazufumi Ito, Zhilin Li, Zhiyue Zhang (2015)
Immersed finite elements for optimal control problems of elliptic PDEs with interfacesJ. Comput. Phys., 298
The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry).Design/methodology/approachThis study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to domains with different material properties (e.g. different inclusions, multi-material structural components, etc.). This study develops OLTEM using compact 9-point and 25-point stencils that are similar to those for linear and quadratic finite elements. In contrast to finite elements and other known numerical techniques for interface problems with conformed and unfitted meshes, OLTEM with 9-point and 25-point stencils and unfitted Cartesian meshes provides the 3-rd and 11-th order of accuracy for irregular interfaces, respectively; i.e. a huge increase in accuracy by eight orders for the new 'quadratic' elements compared to known techniques at similar computational costs. There are no unknowns on interfaces between different materials; the structure of the global discrete system is the same for homogeneous and heterogeneous materials (the difference in the values of the stencil coefficients). The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of OLTEM at a given stencil width. The numerical results with irregular interfaces show that at the same number of degrees of freedom, OLTEM with the 9-points stencils is even more accurate than the 4-th order finite elements; OLTEM with the 25-points stencils is much more accurate than the 7-th order finite elements with much wider stencils and conformed meshes.FindingsThe significant increase in accuracy for OLTEM by one order for 'linear' elements and by 8 orders for 'quadratic' elements compared to that for known techniques. This will lead to a huge reduction in the computation time for the problems with complex irregular interfaces. The use of trivial unfitted Cartesian meshes significantly simplifies the solution and reduces the time for the data preparation (no need in complicated mesh generators for complex geometry).Originality/valueIt has been never seen in the literature such a huge increase in accuracy for the proposed technique compared to existing methods. Due to a high accuracy, the proposed technique will allow the direct solution of multiscale problems without the scale separation.
International Journal of Numerical Methods for Heat and Fluid Flow – Emerald Publishing
Published: Jun 17, 2022
Keywords: Cartesian meshes; Poisson equation with discontinuous coefficients; Irregular interfaces; Local truncation error; Optimal accuracy
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.