A Fully-Implicit Spectral Boundary Element Algorithm for Capsules and Biological Cells

国家科学基金会-化学和药物运输系统司-颗粒和多相过程计划(1415年)建议编号：0730811首席研究员：Dimitrakopoulos，Panagiotis附属公司：马里兰大学建议标题：胶囊和生物细胞的全隐式光谱边界元算法本建议的目标是开发一种新的三维计算算法，将有助于研究斯托克斯流动中膜包裹的流体体积(如人工胶囊、囊泡和生物细胞)的动力学。这些系统在广泛的工业和生理过程中都会遇到，因此对它们的研究是一个具有重大技术意义和根本意义的问题。然而，由于流体力学和界面膜的固体力学性质的耦合，这些体系的动力学行为是一个相当复杂的计算问题。这清楚地反映在涉及胶囊、囊泡和红细胞(RBC)的三维问题的计算结果相当有限的事实中。现有技术的算法的使用受到限制，因为(A)它们使用低阶内插方案，从而由于密集的网格而导致低精度和/或高计算成本；(B)它们还使用显式时间积分方案来确定界面位置，从而由于稳定性考虑导致小的时间步长；和/或(C)在多处理器计算机上可能表现出有限的可伸缩性。为了克服这些障碍，我们建议将我们的高效、无雅可比、三维全隐式谱边界元算法推广到液滴动力学的膜状界面情况。本质上，我们将用更复杂的膜的界面条件来代替液滴的简单界面条件(由于表面张力)。将这些边界条件结合到我们的算法中是一个简单但乏味的过程，因为膜张力涉及到高阶界面几何导数的复杂非线性函数。我们强调，我们的全隐式算法可以处理任何非线性的边界/界面条件和约束；这些方程通过在某个时刻涉及未知形状的边界摄动以及形状的演变来线性化，然后通过迭代求解满足这些条件/约束的界面形状到任何期望的精度。该算法能够准确和高效地确定膜的所有界面属性(包括剪切、面积膨胀和弯曲张力)，利用大的时间步长，从而对涉及三维胶囊、囊泡和生物细胞的最复杂的问题产生高质量的结果。因此，我们的新方法克服了最先进算法施加的限制，同时它促进了对大量更现实和更复杂的问题的研究，这些问题目前被认为是无法实现的。更广泛的影响该算法有可能显著增加我们对胶囊、囊泡和红细胞动力学的物理知识，这将在制药工业、生理学和生物力学中产生广泛的影响。研究生将参与P.I.S的研究，转移已获得的知识。这一提议的结果将被整合到马里兰大学的教育项目中，并将通过科学会议、期刊出版物和网络广为人知。

National Science Foundation - Division of Chemical &Transport Systems - Particulate & Multiphase Processes Program (1415)Proposal Number: 0730811 Principal Investigators: Dimitrakopoulos, Panagiotis Affiliation: University of Maryland Proposal Title: A Fully-Implicit Spectral Boundary Element Algorithm for Capsules and Biological Cells The goal of this proposal is to develop a novel three-dimensional computational algorithm that will facilitate the study of the dynamics of membrane-enclosed fluid volumes (such as artificial capsules, vesicles and biological cells) in Stokes flow. These systems are encountered in a broad range of industrial and physiological processes, and thus their study is a problem of great technological and fundamental interest. However, the dynamic behavior of these systems is a rather complicated computational problem due to the coupling of the fluid mechanics with the solid mechanics properties of the interfacial membrane. This is clearly reflected in the fact that there exist rather limited computational results for three-dimensional problems involving capsules, vesicles and red blood cells (RBCs). The use of the state of the art algorithms is restricted because (a) they employ low-order interpolation schemes resulting in low accuracy and/or high computational cost due to dense grids; (b) they also employ explicit time integration schemes for determining the interfacial position resulting in small time steps due to stability considerations; and/or (c) may show limited scalability on multiprocessor computers. To overcome these obstacles, we propose to extend our efficient, Jacobian-free, three-dimensional fully implicit, spectral boundary element algorithm for drop dynamics to the case of membrane-like interfaces. In essence we will replace the simple interfacial condition for a droplet (due to surface tension) with the more complicated membrane's interfacial conditions. The incorporation of these boundary conditions in our algorithm is a straightforward but tedious process since membrane tensions involve complicated non-linear functions of high-order interfacial geometric derivatives. We emphasize that our fully-implicit algorithm can handle any non-linear boundary/interfacial condition and constraint; these equations are linearized via boundary perturbations involving the unknown shape at some time instance, coupled with the shape evolution, and then via iterations we solve for the interfacial shape which satisfy these conditions/constraints to any desired precision.Intellectual Merit The proposed algorithm has the ability to determine accurately and efficiently all the membrane's interfacial properties (including shearing, area dilation and bending tensions), utilize large time steps, and thus produce high-quality results for the most complicated problems involving three-dimensional capsules, vesicles and biological cells. Therefore, our novel methodology overcomes the limitations imposed by the state of the art algorithms, while it facilitates the investigation of a vast array of more realistic and complicated problems which currently are regarded as unattainable.Broader Impact This proposed algorithm has to potential to significantly increase our physical knowledge on the dynamics of capsules, vesicles and RBCs which will have a broad impact in the pharmaceutical industry, physiology and biomechanics. Graduate students will participate in the P.I.'s research transferring the acquired knowledge. The results of this proposal will be integrated into the education program of the University of Maryland, and they will become generally known via scientific meetings, journal publications and the web.