动脉血管粘弹性发展流动问题高效数值方法研究

Numerical simulation of arterial vascular viscoelasticity flow is an important issue on biological fluid dynamics, which can compensate for its deficiency in vitro experiments. Here, we pay more attention to the efficient numerical methods on arterial vascular viscoelasticity flow. Variational inequality is studied and developed for the arterial vascular viscoelasticity flow without regard to elastic deformation of the coupling. Under the assumption of elastic deformation, the finite element method is applied in simulating deformation of the tissue and arterial wall and the finite volume method is designed on moving mesh to compute blood development flow. For time and spatial discretization, the adaptive stabilized finite element/volume method and large time step method are used respectively. In order to solve the nonlinear and large scale computing, stabilized methods on the lowest order finite elements, multi-scale method, preconditioning post-processing approach are combined, moreover, spliting and decoupled technique, and domain decomposition method are used to develop some high-precision calculation scheme with non-increasing energy, mass conservation, good stability, and strong flexibility properties. We study on the efficient numerical methods for the arterial vascular viscoelasticity flow in order to unify mathematical problem, physical background, and numerical simulation so as to improve the structure of the presented methods to make thre numerical solution not only maintain its physical properties, explain from theoretical point of numerical methods, but also solve the problems simply and efficiently. Furthermore, we provide the new research approaches to simulate arterial vascular viscoelasticity flow, and to develop the nonlinear scientific research and apply to the fluid dynamics in the relevant industries.

动脉血管粘弹性发展流动问题数值模拟是生物流体力学研究的重要课题，可以弥补局限离体实验的不足。本课题主要研究动脉血管粘弹性发展流动问题高效数值方法：对不考虑血管弹性形变耦合的模型中，分析与之等价的变分不等式问题高效数值方法；对考虑血管弹性形变情况，流固耦合在移动网格分别对不可压缩流模型和弹性力学模型使用有限体积和有限元方法，时间空间离散分别利用大时间步长方法和自适应稳定化方法，结合低次元稳定化方法、多尺度方法、预条件处理设计高效求解非线性大规模问题的算法，运用分裂解耦方法或区域分解算法设计高效稳定、保耗散结构、适应性强的高精度计算格式。完善构造动脉血管粘弹性发展流动问题高效数值方法，使其数值求解既能保持物理性质，又能从数值方法角度解释，并简单高效地求解问题。以此为数值模拟心血管疾病提供理论依据和算法工具，进而为非线性科学的理论探索和流体力学在与之相关行业的应用提供新的研究途径。

本项目主要研究了动脉血管粘弹性发展流动问题高效数值方法，针对相关模型构造稳定高效的算法，完善其有限元/有限体积方法的优化阶理论，主要研究了变分不等式的高效稳定化数值分析，耦合问题大时间步长方法、分裂解耦方法、多物理区域分解方法等高效数值方法。证明了相关有限元/有限体积方法稳定性和收敛性，并积累了相关的程序包。在本项目支持下，在SIAM J. Sci. Comp, Computer Methods in Applied Mechanics and Engineering，《中国科学》等共计发表论文24篇，SCI论文21篇，1篇SCI论文入选ESI；国家发明专利1项；获批国家自然科学基金面上项目1项，获得陕西省科技奖一等奖1项，并提名陕西省2018年度国家自然科学奖。获得全国优秀教师，宝鸡市突出贡献专家等荣誉。