This paper aims to propose a wavelet boundary element method (WBEM) to study the three-dimensional elasticity problem and potential problem. In contrast with conventional polynomial interpolation in the BEM, the scaling functions of the B-spline wavelet on the interval (BSWI) are applied to derive calculated formats of the WBEM, construct BSWI elements, discretize the geometric shape, and form BSWI BEM algebraic equations. Because BSWI scaling functions have specific expressions, the integration related to scaling functions can be evaluated directly without other complicated calculation steps like applying other wavelet bases. The Gauss quadrature scheme based on the background cell is implemented to evaluate the integration involved in the WBEM. Furthermore, the singular integration problem appearing in the WBEM can be transformed into that appearing in traditional BEM to solve. Unlike the WBEM in the published literature, arbitrary boundary conditions can be solved directly as the wavelet-based elements are employed to discrete the geometry of structures. Some typical examples with several boundary conditions are provided to verify this method. The numerical solutions illustrate the good convergence, reliability, and flexibility of WBEM by comparison with the traditional BEM and exact solutions.

本文旨在提出一种小波边界元方法（WBEM）来研究三维弹性问题和潜在问题。与 BEM 中的常规多项式插值不同，B 样条小波在区间 (BSWI) 上的标度函数用于导出 WBEM 的计算格式、构造 BSWI 元素、离散几何形状并形成 BSWI BEM 代数方程. 由于BSWI标度函数有特定的表达式，与标度函数相关的积分可以直接求值，无需应用其他小波基等复杂的计算步骤。实施基于背景单元的高斯正交方案来评估 WBEM 中涉及的积分。此外，WBEM中出现的奇异积分问题可以转化为传统BEM中出现的问题来解决。与已发表文献中的 WBEM 不同，由于采用基于小波的元素来离散结构的几何形状，因此可以直接求解任意边界条件。提供了一些具有多个边界条件的典型例子来验证该方法。通过与传统边界元法和精确解法的比较，数值解说明了WBEM 具有良好的收敛性、可靠性和灵活性。