A meshfree method for the solution of 2D and 3D second order elliptic boundary value problems in heterogeneous media

Nima Noormohammadi, Danial Afifi, Bijan Boroomand and Omid Bateniparvar

Mathematics and Computers in Simulation (MATCOM), 2023, vol. 213, issue C, 274-301

Abstract: We propose a simple meshfree method for two-dimensional and three-dimensional second order elliptic boundary value problems in heterogeneous media based on equilibrated basis functions. The domain is discretized by a regular nodal grid, over which the degrees of freedom are defined. The boundary is also discretized by simply introducing some boundary points over it, independent from the domain nodes, granting the method the ability of application for arbitrarily shaped domains without the drawback of irregularity in the nodal grid. In heterogeneous media, the governing Partial Differential Equation (PDE) has non-constant coefficients for the partial derivatives, preventing Trefftz-based techniques to be applicable. The proposed method satisfies the PDE independent from the boundary conditions, as in Trefftz approaches. Meanwhile, by applying the weak form of the PDE instead of the strong form through weighted residual integration, the inconvenience of variable coefficients shall be resolved, since the bases will not need to analytically satisfy the PDE. The weighting in the weak form integrals is such that the boundary integrals vanish. The boundary conditions are thus collocated over the defined boundary points. Domain integrals break into algebraic combination of one-dimensional pre-evaluated integrals, thus omitting the numerical integration from the solution process. Each node corresponds to a local sub-domain called cloud. The overlap between adjacent clouds ensures integrity of the solution and its derivatives throughout the domain, an advantage with respect to C0 formulations.

Keywords: Boundary value problem; 2D; 3D; Meshfree; Equilibrated basis functions; Heterogeneous (search for similar items in EconPapers)
Date: 2023
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0378475423002471
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:213:y:2023:i:c:p:274-301

DOI: 10.1016/j.matcom.2023.06.004

Access Statistics for this article

Mathematics and Computers in Simulation (MATCOM) is currently edited by Robert Beauwens

More articles in Mathematics and Computers in Simulation (MATCOM) from Elsevier
Bibliographic data for series maintained by Catherine Liu ().