A collocation technique based on the use of Bernstein polynomials to approximate the field variable is assessed in Boundary Value Problems (BVPs) of beams with governing non-linear differential equations. The BVPs are transformed into unconstrained optimization problems by means of an extended cost function which leverages the properties of the Bernstein basis to enforce the boundary conditions. The minimization of the squared error cost function is conducted by means of the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. The method is tested in benchmarks of various types of non-linearities, including materials with Ludwick stress–strain curves, follower loads and beams on Winkler foundation. The approach is compared with Isogeometric collocation (IGA-c) and straightforward (pseudospectral) Bernstein collocation in terms of performance and computational effort. Moreover, the accuracy and convergence of the method is discussed to ease its successful application to other non-linear beam problems.

在控制非线性微分方程的梁的边值问题（BVP）中，评估了基于使用伯恩斯坦多项式来近似场变量的搭配技术。BVP通过扩展的成本函数转化为无约束的优化问题，该函数利用Bernstein基的属性强制执行边界条件。平方误差成本函数的最小化是通过准牛顿布赖登-弗莱彻-戈德法布-香诺算法（BFGS）进行的。该方法已在各种非线性类型的基准中进行了测试，包括具有Ludwick应力-应变曲线，随动载荷和Winkler基础上的梁的材料。在性能和计算工作方面，将该方法与等几何并置（IGA-c）和简单的（伪谱）Bernstein并置进行了比较。此外，讨论了该方法的准确性和收敛性，以简化其成功应用于其他非线性光束问题的过程。