A novel technique to formulate arbritrary faceted polyhedral elements in three-dimensions is presented. The formulation is applicable for arbitrary faceted polyhedra, provided that a scaling requirement is satisfied and the polyhedron facets are planar. A triangulation process can be applied to non-planar facets to generate an admissible geometry. The formulation adopts two separate scaled boundary coordinate systems with respect to: (i) a scaling centre located within a polyhedron and; (ii) a scaling centre on a polyhedron’s facets. The polyhedron geometry is scaled with respect to both the scaling centres. Polygonal shape functions are derived using the scaled boundary finite element method on the polyhedron facets. The stiffness matrix of a polyhedron is obtained semi-analytically. Numerical integration is required only for the line elements that discretise the polyhedron boundaries. The new formulation passes the patch test. Application of the new formulation in computational solid mechanics is demonstrated using a few numerical benchmarks.

中文翻译：

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任意面星凸多面体上的双尺度边界有限元公式

提出了一种在三维中制定任意多面体元素的新技术。该公式适用于任意面多面体，前提是满足缩放要求并且多面体面是平面的。三角测量过程可以应用于非平面小平面以生成可接受的几何形状。该公式采用两个独立的缩放边界坐标系，关于：(i) 位于多面体内的缩放中心和；(ii) 多面体小面上的缩放中心。多面体几何形状相对于两个缩放中心进行缩放。多边形形状函数是在多面体小面上使用缩放边界有限元方法导出的。多面体的刚度矩阵是半解析获得的。只有离散多面体边界的线元素才需要数值积分。新配方通过了斑贴试验。使用一些数值基准证明了新公式在计算固体力学中的应用。