A boundary face method for potential problems in three dimensions.

*(English)*Zbl 1176.74212Summary: This work presents a new implementation of the boundary node method (BNM) for numerical solution of Laplace’s equation. By coupling the boundary integral equations and the moving least-squares (MLS) approximation, the BNM is a boundary-type meshless method. However, it still uses the standard elements for boundary integration and approximation of the geometry, thus loses the advantages of the meshless methods. In our implementation, here called the boundary face method, the boundary integration is performed on boundary faces, which are represented in parametric form exactly as the boundary representation data structure in solid modeling. The integrand quantities, such as the coordinates of Gauss integration points, Jacobian and out normal are calculated directly from the faces rather than from elements. In order to deal with thin structures, a mixed variable interpolation scheme of 1-D MLS and Lagrange Polynomial for long and narrow faces. An adaptive integration scheme for nearly singular integrals has been developed. Numerical examples show that our implementation can provide much more accurate results than the BNM, and keep reasonable accuracy in some extreme cases, such as very irregular distribution of nodes and thin shells.

##### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

##### Keywords:

meshless methods; boundary node method; moving least-squares approximation; boundary face method
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\textit{J. Zhang} et al., Int. J. Numer. Methods Eng. 80, No. 3, 320--337 (2009; Zbl 1176.74212)

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