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中占有重要地位，但由于涉及到强烈的非线性和多物理特性，对这类问题的准确研究仍然具有很大的挑战性。目前的流固耦合计算理论和方法缺乏足够的精度和逼真的材料表示，并且局限于相对简单的结构形式。这个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，以准确和高效地对复杂的FSI问题进行数值研究，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，开发高效的数值算法，确保FSI计算的高精度，并将建议的数学公式和计算方法应用于基于单个细胞的肿瘤生长建模和模拟。这个建议是建立在PI的基础上的吗？扎实的计算和应用数学背景以及在肿瘤建模方面的重要工作。拟议的研究将通过数学和计算相结合的框架提高对高度复杂的FSI问题中的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计算机FSI研究的现状提供坚实的知识基础，而且将使早期肿瘤形成和发展的基本机制有重要的发现。该项目的经验和结果将加强老道明大学和威廉与玛丽学院在计算数学和数学生物学方面的研究协作和课程开发。由于两所学校地理位置接近，通过方便师生之间的沟通和两所学校之间的联合计划，将最大限度地提高项目影响。教育和外展活动将使研究生和本科生学习计算数学的理论、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.