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

流固耦合（FSI）问题在许多科学和工程领域发挥着突出的作用，但由于其涉及的强非线性和多物理场，对此类问题的精确研究仍然具有很高的挑战性。当前流固相互作用的计算理论和方法缺乏足够的精度和真实的材料表示，并且仅限于相对简单的结构构型。该项目的目标是通过建立一个新的计算数学框架来克服这些限制，以便对复杂的FSI问题进行准确和有效的数值研究，特别是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，并将提出的数学公式和计算方法应用于基于个体细胞的肿瘤生长建模和模拟。该提案建立在pi的基础上。扎实的计算和应用数学背景，在肿瘤建模方面有重要工作。本文提出的研究将通过结合数学和计算框架，提高对高度复杂FSI问题中非线性动力学的理解。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计算FSI研究提供坚实的知识基础，而且将在早期肿瘤形成和发展的基本机制方面有重要的发现。该项目的经验和发现将加强老道明大学和威廉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。由于两所学校之间的地理位置接近，通过方便的教师和学生之间的沟通以及两所学校之间的联合项目，项目的影响将最大化。教育和推广活动将涉及研究生和本科生在计算数学的理论，方法和应用。

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.