We develop a discontinuous Petrov-Galerkin scheme with optimal test functions (DPG method) for the Timoshenko beam bending model with various boundary conditions, combining clamped, supported, and free ends. Our scheme approximates the transverse deflection and bending moment. It converges quasi-optimally in $L_2$ and is locking free. In particular, it behaves well (converges quasi-optimally) in the limit case of the Euler-Bernoulli model. Several numerical results illustrate the performance of our method.

中文翻译：

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Timoshenko 梁的无锁定 DPG 方案

我们为具有各种边界条件的 Timoshenko 梁弯曲模型开发了一个具有最佳测试函数（DPG 方法）的不连续 Petrov-Galerkin 方案，结合了夹紧端、支撑端和自由端。我们的方案近似于横向偏转和弯矩。它在 $L_2$ 中准最优收敛并且是无锁的。特别是，它在 Euler-Bernoulli 模型的极限情况下表现良好（准最优收敛）。几个数值结果说明了我们方法的性能。