To construct ${\mathcal{C}}^1$-continuous basis functions for the numerical solutions of two dimensional biharmonic equations on non-convex domains with clamped and/or simply supported boundary conditions, we use B-spline basis functions instead of conventional Hermite finite element basis functions. The ${\mathcal{C}}^1$-continuous B-spline approximation functions constructed on the master patch are moved onto a physical domain by a geometric patch mapping. However it is difficult to cover a non-convex polygonal domain by one smooth patch mapping. Hence we decompose a non-convex domain into several overlapping rectangular subdomains and use the Schwartz alternating method to assemble local solutions for the global solution. Furthermore, in order to handle the corner singularities arising in non-convex domains, we introduce the implicitly enriched Galerkin method, which is similar to the explicit enrichment techniques used in XFEM, G-FEM, PUFEM, that are successful in handling the singularity problems arising in second-order differential problems.

构建 ${\mathcal{C}}^{\mathrm{1个}}$-对于具有约束和/或简单支持边界条件的非凸域上的二维双调和方程数值解的连续基函数，我们使用B样条基函数代替常规的Hermite有限元基函数。的${\mathcal{C}}^{\mathrm{1个}}$通过几何补丁映射将构建在主补丁上的连续B样条近似函数移到物理域上。但是，通过一个平滑的贴图映射很难覆盖一个非凸的多边形域。因此，我们将一个非凸域分解为几个重叠的矩形子域，并使用Schwartz交替方法为全局解组装局部解。此外，为了处理非凸域中出现的角奇点，我们引入了隐式富集的Galerkin方法，该方法类似于XFEM，G-FEM，PUFEM中使用的显式富集技术，成功地解决了奇异问题产生于二阶微分问题。