Collaborative Research: New Formulation and Algorithms for Fluid-Structure Interaction, with Application to Tumor Growth

流固耦合问题在许多科学和工程领域中发挥着重要作用，但由于其强非线性和多物理场的特点，对流固耦合问题的精确研究仍具有很大的挑战性。目前的计算理论和方法的流体-结构相互作用缺乏足够的精度和现实的材料表示，并限于相对简单的结构配置。该项目的目标是通过建立一种新的数学计算框架来克服这些限制，以便对复杂的FSI问题进行准确有效的数值研究，特别是对于肿瘤生长建模和模拟。PI将推导出新的数学公式，允许研究具有复杂结构设置的一般FSI问题，开发有效的数值算法，确保FSI计算的高精度，并将所提出的数学公式和计算方法应用于基于单个细胞的建模和肿瘤生长模拟。这份提案是建立在PI的基础上的？在计算和应用数学的坚实背景和肿瘤建模的重要工作。拟议的研究将提高高度复杂的FSI问题的非线性动力学的理解，通过结合数学和计算框架。实验测量也将用于验证数值结果。该项目的成功不仅将为推进当前的计算FSI研究提供坚实的知识基础，而且还将使早期肿瘤形成和发展的基本机制的重要发现成为可能。该项目的经验和发现将加强旧自治领大学和威廉和玛丽学院在计算数学和数学生物学方面的研究合作和课程开发。由于两所学校地理位置接近，项目的影响将通过教师和学生之间的方便沟通，并通过两所学校之间的联合计划最大化。教育和推广活动将涉及研究生和本科生的理论，方法和计算数学的应用。

Fluid-structure interaction (FSI) problems play prominent roles in many scientific and engineering fields, yet an accurate study of such problems remains highly challenging due to the strong nonlinearity and multi-physics involved. Current computational theory and methods for fluid-structure interactions lack sufficient accuracy and realistic material representations, and are limited to relatively simple structural configurations. The goal of this project is to overcome such limitations by establishing a new computational mathematical framework for accurate and efficient numerical investigation of complex FSI problems, particularly for tumor growth modeling and simulation. The PIs will derive new mathematical formulation that allows the study of general FSI problems with sophisticated structural settings, develop efficient numerical algorithms that ensure high accuracy in FSI computations, and apply the proposed mathematical formulation and computational methods to the individual-cell-based modeling and simulation of tumor growth. This proposal builds on the PIs? solid background in computational and applied mathematics and significant work on tumor modeling.The proposed research will improve the understanding of the nonlinear dynamics in highly complex FSI problems through a combined mathematical and computational framework. Experimental measurements will also be used for validating the numerical results. The success of this project will not only provide a solid knowledge base for advancing the current state of computational FSI study, but also enable important discoveries in the fundamental mechanism of early tumor formation and development. The project experiences and findings will strengthen the research collaboration and curriculum development in computational mathematics and mathematical biology at both Old Dominion University and the College of William and Mary. Due to the close geographical proximity between the two schools, project impacts will be maximized through convenient communication between faculty and students and through joint programs between the two schools. Education and outreach activities will involve graduate and undergraduate students in the theory, methods and application of computational mathematics.