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A linearly conforming point interpolation method (LC-PIM) is developed for 2D solid problems. In this method, shape functions are generated using the polynomial basis functions and a scheme for the selection of local supporting nodes based on background cells is suggested, which can always ensure the moment matrix is invertible as long as there are no coincide nodes. Galerkin weak form is adopted for creating discretized system equations, and a nodal integration scheme with strain smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method (FEM) using linear triangle elements and the radial point interpolation method (RPIM) using Gauss integration, the LC-PIM can achieve higher convergence rate and better efficiency.

Bismuth (Bi)-based electrodes are highly attractive for potassium-ion batteries (PIBs) while suffering from a short cycle life due to the larger diameter of K ion, leading to unstable solid electrolyte interface (SEI) films during continuous potassiation/depotassiation. Herein, we developed novel ultrathin carbon film@carbon nanorods@Bi nanoparticle (UCF@CNs@BiN) materials for the long cycle life anode of PIBs. Bi nanoparticles are uniformly distributed in carbon nanorods, which not only provides a high-speed channel for ion transport but also accommodates the volume change of Bi nanoparticles during continuous potassiation/depotassiation processes. The UCF@CN matrix can direct most SEI film formation on the surface of the carbon film, not on the surface of individual Bi nanoparticles, avoiding the fracture of the matrix. Benefiting from their unique structure, the UCF@CNs@BiN anodes exhibit an outstanding capacity of ∼425 mAh g–1 at 100 mA g–1 and a capacity decay of 0.038% per cycle over 600 cycles. Even at a higher current density of 1000 mA g–1, there is a capacity decay as low as 0.036% per cycle during 700 cycles. Meanwhile, this work provides a new way of utilizing the metal–organic framework structure and reveals a highly promising PIB anode.

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SUMMARYThis 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.

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