We consider a moving boundary problem with kinetic condition that describes the diffusion of solvent into rubber and study semi-discrete finite element approximations of the corresponding weak solutions. We report on both a priori and a posteriori error estimates for the mass concentration of the diffusants, and respectively, for the a priori unknown position of the moving boundary. Our working techniques include integral and energy-based estimates for a nonlinear parabolic problem posed in a transformed fixed domain combined with a suitable use of the interpolation-trace inequality to handle the interface terms. Numerical illustrations of our FEM approximations are within the experimental range and show good agreement with our theoretical investigation. This work is a preliminary investigation necessary before extending the current moving boundary modeling to account explicitly for the mechanics of hyperelastic rods to capture a directional swelling of the underlying elastomer.
翻译:我们考虑的是具有动能条件的移动边界问题,它描述溶剂在橡胶中的扩散,并研究相应的微弱解决办法的半分解有限元素近似值。我们报告的是,关于悬浮剂大规模集中的先验和事后误差估计,分别报告移动边界的先验和先验未知位置。我们的工作技巧包括:对在已改变的固定域内形成的非线性抛物线问题进行综合和基于能源的估计,同时适当使用内推-递不平等处理界面术语。我们FEM近似值的数值插图在实验范围之内,并表明我们同意我们的理论调查。这项工作是在扩大目前的移动边界模型以明确说明超弹性杆的机械以捕捉基本弹性体的脉冲膨胀之前必须进行的初步调查。