ITR: Analysis and Simulation of Interface Dynamics in Multiphase Fluids and Solids

摘要：研究者和他的同事们正在关注新的计算技术的发展，以模拟流体和固体中的界面问题，这些技术是建立在对底层数学模型分析的最新进展之上的。本研究的主要目标是：(1)基于研究者及其同事提出的边界项的平滑化和高阶正则化，开发具有表面张力控制的运动界面流体的新型计算算法，具有严格的误差界限和收敛证明；(2)扩展现有的实体碰撞算法，使其包括非穿透不变量和廉价的能量近似，从而能够有效地模拟薄壳碰撞问题；(3)基于已有的分析证明，提出了具有表面张力效应的多相流体的相场格式，该格式能够处理拓扑变化并可证明收敛；(4)证明了具有复杂非线性表面能（包括表面活性剂应力和弹性膜能）的流体存在唯一解；(5)利用这些新的复杂表面能分析结果，开发出针对此类问题的收敛算法，重点关注弹性结构中的流体流动，并将这些方法与现有的固体碰撞代码相结合；(6)利用新的分析结果和计算工具来研究涉及碰撞的固体系统解的存在性和可能的唯一性问题。研究者和他的同事们建议建模的多相流的物理机制包括液滴的形成和破裂、喷雾、聚并、波浪破碎、固体碰撞、破碎和破裂。这些现象在许多科学领域，如海洋-大气动力学、飞机和船只的设计和分析，以及许多生物应用，如动脉血流，都具有重要意义。有许多工程应用，如制造从纳米技术到聚合物流体的先进材料。从装甲车辆和建筑物的反弹道设计到地壳对构造板块滑动的反应，固体碰撞、损伤和断裂在许多领域都是至关重要的。即将开发的新的分析和计算工具将适用于这些具有挑战性的物理问题。具有稳定和收敛的数值编码将极大地帮助研究这类复杂系统。

ABSTRACTThe investigator and his colleagues are concerned with the development of new computational techniques for simulating interface problems in fluids and solids that are founded on recent advances in the analysis of the underlying mathematical models. The primary objectives of the research are to: (1) develop novel computational algorithms for fluids with moving interfaces governed by surface tension, with rigorous error bounds and convergence proofs, based on mollifications and higher order regularizations of boundary terms proposed by the investigator and his colleagues; (2) extend their existing solid collision algorithms to include non-interpenetration invariants and inexpensive energy approximations, allowing the efficient simulation of thin-shell impact problems; (3) develop new phase-field schemes for multiphase fluids with surface tensions effects based on their recent analytical proofs, which can handle topological changes and provably converge; (4) prove the existence of unique solutions for fluids with complex nonlinear surface energies, including surfactant stress and elastic membrane energies; (5) use these new analytical results for complex surface energies to develop convergent algorithms for such problems, focusing on fluid flow in elastic structures, and combine these methods with the existing solids collision code; (6) use the new analytical results and computational tools to investigate the problem of the existence and possible uniqueness of solutions to solids systems involving collisions.The physical mechanisms of multiphase flows that the investigator and his colleagues propose to model include droplet formation and breakup, spray, coalescence, wave breaking, solid-body collisions, fragmentation, and fracture. These phenomena are of primary importance in many areas of science such as ocean-atmosphere dynamics, aircraft and hydrocraft design and analysis, and many biological applications such as arterial blood flows. There are a host of engineering applications such as the manufacturing of advanced materials from nanotechnology to polymeric fluids. Solid body collision, damage, and fracture are of paramount importance in host of areas ranging from anti-ballistic design of armored vehicles and buildings to the response of the earth's crust to the sliding of tectonic plates. The new analytical and computational tools that are to be developed will be applicable to these challenging physical problems. Having stable and convergent numerical codes will greatly aid the investigation of such complex systems.