聚合物凝胶非线性分析的维数分裂-无网格方法

Meshless method is another important numerical method for science and engineering computing after finite element method. Because the remeshing technique is not needed, when meshless method is used to solve large deformation and dynamic crack problems, the solution has higher precision than the one of finite element method. For the disadvantage of large amount of computation of the element-free Galerkin method, which is one of the most important meshless methods, for 3-D problems, the dimension split – complex variable element-free Galerkin methods of 3-D elasticity, large deformation and elastoplastic large deformation problems will be discussed in this project. The dimension split – complex variable element-free Galerkin method will have the advantages of high precision and efficiency. Then, considering the large deformation of polymer gel under a very small perturbation, the dimension split – complex variable element-free Galerkin methods of 3-D mass transfer, large deformation and the coupling of mass transfer and large deformation of polymer gels will be presented, and the corresponding theories of error estimates will be discussed. At last, the dimension split – complex variable element-free Galerkin method of the coupling of mass transfer and large deformation of 3-D polymer gel will be applied to the artificial prosthesis structures. This project will promote the development of meshless methods for complicated nonlinear problems, and provide a numerical method with high efficiency for the nonlinear analysis of polymer gels and the corresponding applications in engineering.

无网格方法是继有限元法之后科学和工程计算的又一重要的数值方法，在求解大变形和动态断裂等问题时不需进行网格重构，具有更高的精度。本项目将针对目前最为重要的无网格方法之一——无单元Galerkin方法在求解三维问题时计算量大的问题，研究三维弹性、大变形和弹塑性大变形问题的维数分裂-复变量无单元Galerkin方法，该方法将具有计算精度高和计算速度快的优点。然后，针对聚合物凝胶材料在微小激励下即产生大变形的特点，建立三维聚合物凝胶材料质量迁移、大变形以及质量迁移和大变形耦合等复杂非线性问题的维数分裂-复变量无单元Galerkin方法，并研究相应的误差估计理论。最后，将三维聚合物凝胶材料质量迁移和大变形耦合分析的维数分裂-复变量无单元Galerkin方法应用于人体假肢结构分析。本项目将促进复杂非线性问题的无网格方法的研究进展，并为聚合物凝胶等软材料非线性分析及其工程应用提供高效的数值方法。

本项目建立了三维弹性、弹塑性、大变形和弹塑性大变形问题的维数分裂-无单元Galerkin方法和维数分裂-复变量无单元Galerkin方法。相对于无单元Galerkin方法来说，本项目方法具有计算精度高和计算速度快的优点。针对聚合物凝胶材料在微小激励下即产生大变形的特点，建立了三维聚合物凝胶材料质量迁移、大变形以及质量迁移和大变形耦合等复杂非线性问题的维数分裂-复变量无单元Galerkin方法，并研究了相应的误差估计理论。将三维聚合物凝胶材料质量迁移和大变形耦合分析的维数分裂-复变量无单元Galerkin方法应用于人体假肢结构分析。本项目现已发表期刊论文23篇，会议论文4篇。在23篇期刊论文中，SCI收录21篇，EI收录18篇，1篇论文为ESI热点论文。培养博士生3名和硕士生4名(已毕业)。本项目将促进复杂非线性问题的无网格方法的研究进展，并为聚合物凝胶等软材料非线性分析及其工程应用提供高效的数值方法。