This paper proposes an enhancement of the treatment of non-homogeneous boundary conditions to improve the boundary element method (BEM) formulation. The standard formulation is modified by introducing the boundary conditions in the integral kernels. The boundary conditions are implicitly defined through known parameters depending on the geometry, rather than by prescribing nodal values as is done in the standard formulation. The main advantage of this procedure is that the right-hand side of the system of equations is integrated taking the exact distribution of loads into account. This approach is implemented in the Bézier–Bernstein space to yield a geometry-independent field approximation. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The application of the proposed method covers the resolution of complex boundary value problems as optimization with uncertain data, material modelling with graded impedance, and the definition of general boundary constraints. The performance of the proposed method is shown by solving the Helmholtz equation in two dimensions. The proposed method is numerically compared to the standard BEM formulation in two benchmark problems. Finally, the application of complex impedance boundary conditions is analysed in a numerical example.

本文提出了一种对非均匀边界条件的处理方法，以改进边界元方法（BEM）的公式化。通过在积分核中引入边界条件来修改标准公式。边界条件是通过已知参数（取决于几何形状）隐式定义的，而不是像标准公式那样通过规定节点值来定义的。此过程的主要优点是，方程组的右侧已集成，同时考虑了载荷的精确分布。这种方法在Bézier-Bernstein空间中实现，以产生与几何无关的场近似。我们使用多项式的Bézier-Bernstein形式作为近似基础来表示几何和场变量。该方法的应用涵盖了复杂边界值问题的解决方案，包括不确定数据的优化，具有渐变阻抗的材料建模以及一般边界约束的定义。通过二维求解亥姆霍兹方程，表明了所提方法的性能。在两个基准问题中，将所提出的方法与标准BEM公式进行了数值比较。最后，在一个数值例子中分析了复阻抗边界条件的应用。在两个基准问题中，将所提出的方法与标准BEM公式进行了数值比较。最后，在一个数值例子中分析了复阻抗边界条件的应用。在两个基准问题中，将所提出的方法与标准BEM公式进行了数值比较。最后，在一个数值例子中分析了复阻抗边界条件的应用。