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In this work, we propose a phase-field-based lattice Boltzmann method to simulate moving contact line (MCL) problems on curved boundaries. The key point of this method is to implement the boundary conditions on curved solid boundaries. Specifically, we use our recently proposed single-node scheme for the no-slip boundary condition and a new scheme is constructed to deal with the wetting boundary conditions (WBCs). In particular, three kinds of WBCs are implemented: two wetting conditions derived from the wall free energy and a characteristic MCL model based on geometry considerations. The method is validated with several MCL problems and numerical results show that the proposed method has utility for all the three WBCs on both straight and curved boundaries.

Linear triangular elements with three nodes (Tr3) were the earliest, simplest and most widely used in finite element (FE) developed for solving mechanics and other physics problems. The most important advantages of the Tr3 elements are the simplicity, ease in generation, and excellent adaptation to any complicated geometry with straight boundaries. However, it cannot model well the geometries with curved boundaries, which is known as one of the major drawbacks. In this paper, a four-noded triangular (Tr4) element with one curved edge is first used to model the curved boundaries. Two types of shape functions of Tr4 elements have been presented, which can be applied to finite element method (FEM) models based on the isoparametric formulation. FE meshes can be created with mixed linear Tr3 and the proposed Tr4 (Tr3-4) elements, with Tr3 elements for interior and Tr4 elements for the curved boundaries. Compared to the pure FEM-Tr3, the FEM-Tr3-4 can significantly improve the accuracy of the solutions on the curved boundaries because of accurate approximation of the curved boundaries. Several solid mechanics problems are conducted, which validate the effectiveness of FEM models using mixed Tr3-4 meshes.

Linear tetrahedral elements with four nodes (Te4) are currently the simplest and most widely used ones in the finite element (FE) developed for solving three-dimensional (3D) mechanics problems. However, the standard Te4 element cannot be used to simulate accurately the 3D problems with curved boundaries because of the flat surfaces. In this paper, we develop a set of new elements having curved surfaces to properly simulate the curved boundaries. At the same time, additional nodes are put on the curved boundaries to improve the accuracy of the approximation. These novel elements are defined as five-noded, six-noded, and seven-noded tetrahedron elements (Te5, Te6, and Te7) according to the number of the nodes in one element. Based on the Te4 FE mesh, a hybrid mesh can be conveniently built for 3D problems with curved boundaries, in which the standard Te4 elements are used for the interior elements, and Te5, Te6, and Te7 elements are used for the curved boundary elements. Compared with the standard FEM using Te4 elements, our hybrid mesh can significantly improve the accuracy of the solutions at the curved boundaries. Several solid mechanics problems are studied using the hybrid meshes to validate the effectiveness of the present new elements.

We revisit classical Virtual Element approximations on polygonal and polyhedral decompositions. We also recall the treatment proposed for dealing with decompositions into polygons with curved edges. In the second part of the paper, we introduce a couple of new ideas for the construction of Virtual Element Method (VEM)-approximations on domains with curved boundary, both in two and three dimensions. The new approach looks promising, although sound numerical tests should be made to validate the efficiency of the method.