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Meshless local B-spline-FD method and its application for 2D heat conduction problems with spatially varying thermal conductivity. (English) Zbl 1334.80012
Summary: In this paper, a new class of meshless methods based on local collocation and B-spline basis functions is presented for solving elliptic problems. The proposed approach is called as meshless local B-spline basis functions based finite difference (local B-FD) method. The method was straightforward to develop and program as it was truly meshless. Only scattered nodal distribution was required hence avoiding at all mesh connectivity for field variable approximation and integration. In the method, any governing equations were discretized by B-spline approximation in the spirit of FD technique using local B-spline collocation i.e. any derivative at a point or node was stated as neighboring nodal values based on the B-spline interpolants. In addition, as B-spline basis functions pose favorable properties such as (i) easy to construct to any arbitrary order/degree, (ii) have partition of unity property, and (iii) can be easily designed to pose the Kronecker delta property, the shape function construction as well as the imposition of boundary conditions can be incorporated efficiently in the present method. The applicability and capability of the present local B-FD method were demonstrated through several heat conduction problems with heat generation and spatially varying conductivity.

80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
80A20 Heat and mass transfer, heat flow (MSC2010)
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