We summarise three applications of the obstacle problem to membrane contact, elastoplastic torsion and cavitation modelling, and show how the resulting models can be solved using mixed finite elements. It is challenging to construct fixed computational meshes for any inequality-constrained problem because the coincidence set has an unknown shape. Consequently, we demonstrate how $h$-adaptivity can be used to resolve the unknown coincidence set. We demonstrate some practical challenges that must be overcome in the application of the adaptive method.

翻译：本文总结了障碍问题在薄膜接触、弹塑性扭转及空化建模中的三项应用，并展示了如何通过混合有限元方法求解所得模型。由于重合集具有未知形状，为任何不等式约束问题构建固定计算网格均具挑战性。为此，我们论证了如何利用$h$-自适应方法解析未知重合集，并揭示了应用该自适应方法时必须克服的若干实际挑战。