非线性弹性材料在一定的应力分布下会生成空穴。对空穴生成、生长和融合的研究是揭示材料机理的一个重要组成部分。本项目将针对高分子聚合物，如橡胶、凝胶等不可压或几乎不可压非线性软物质弹性材料的空穴生成现象开展数学建模、算法设计与分析、数值模拟等方面的研究。我们将应用非线性偏微分方程、非线性变分学、多尺度分析、非线性最优化等非线性分析的理论和方法以及曲边等参和双参元、DG、非协调元等方面的最新研究成果，克服空穴生成问题固有的 Lavrentiev 现象和分叉现象给建模和算法设计带来的困难以及在超大各向异性奇性形变条件下材料的不可压或几乎不可压性给有限元逼近带来的困难，研究发展相应的计算空穴生成、生长和融合的高效的数值算法，并通过大量的数值实验模拟相应的非线性物理现象。本项目还将探讨应用谱方法和拟谱方法、区域分解和多重网格方法等发展针对包含可压和不可压材料在内的非线性弹性材料空穴生成问题的快速算法。

Cavities occur in certain nonlinear elastic materials subject to large tractions. The study of cavity initiation, growth and merge is an important part of the effort to reveal the damage mechanism of the materials. The present project will study the mathematical modeling, design and analysis of numerical methods as well as the numerical simulation on the cavitation phenomenon for incompressible and nearly incompressible nonlinear soft elastic materials, such as rubbers and gels. We will make use of the theories and methods developed in nonlinear PDEs, nonlinear variational methods, multiscale analysis, nonlinear optimizations as well as some sophisticated finite element methods such as the combination of iso-parametric and dual-parametric elements, discontinuous Galerkin methods and non-conforming FEM, overcoming the difficulties of the Lavrentiev phenomenon, which is inherited from the original cavitation problem, and keeping the determinant of the deformation gradient (nearly) invariant, which is required by the incompressibility and nearly incompressibility of the material, to develop highly efficient numerical methods for the computation of cavity initiation, growth and merge. We will also develop fast algorithms based on spectral methods and pseudo-spectral methods for cavitation computations.