The investigators develop and apply efficient boundary integral methods for the motion of interfaces in 3D flow. The methods address a significant difficulty in the numerical computation of fluid interfaces with surface tension or elastic forces in 3D flow. Such forces introduce high order (i.e., high derivative) terms into the evolution equations, which lead to severe stability constraints or `stiffness' for explicit time-integration methods. Furthermore, the high order terms appear in nonlinear and nonlocal operators, making the efficient application of stable implicit methods difficult. The investigators' method relies on using the first and second fundamental coefficients of the surface as dynamical variables, and employs a special parameterization of the interface combined with an analysis of the governing equations at small scales. This enables the efficient application of implicit time-integration methods for 3D flow. The investigators implement the method in canonical interface problems for inviscid fluids, including the Kelvin-Helmholtz, Rayleigh-Taylor, and water wave problems, and study the dynamics of inextensible elastic sheets in inviscid flow and vesicles in 3D viscous flow. Most importantly, they develop a version of the numerical method which uses domain decomposition or overlapping coordinate patches to describe the interface. This has the added benefit of providing a framework for the implementation of spectrally accurate and spatially adaptive methods.
Moving boundary problems occur in many diverse areas in, for example, fluid dynamics, materials science, and biology. Specific examples include traveling ocean waves, growing cancer tumors, beating hearts and moving cells and organisms. The investigators develop accurate and efficient `boundary integral' numerical methods for the simulation of moving boundaries in applications. Boundary integral methods are among the most accurate numerical methods for the simulation of moving interfaces, but are often inefficient when the interface is acted on by surface tension or elastic forces. The development of fast and accurate boundary integral methods for 3D interfacial flow with surface tension or elastic forces will be of great benefit in understanding existing applications and developing technology further.
This project developed fast and accurate numerical methods for 3D multi-phase fluid flow with surface tension or elastic membrane stress. The numerical computation of interfacial fluid flow is a fundamental problem that arises in numerous scientific, technological and industrial applications. Examples include the formation and evolution of rogue ocean waves, the motion of cells and vesicles in a fluid surrounding, the dynamics of heart valves, and motion of ice sheets in the ocean. The focus of the project is on boundary integral numerical methods, which have been widely and successfully used to simulate problems of interfacial fluid dynamics, materials science, and electromagnetic scattering. Compared with other popular numerical methods, boundary integral methods have the advantage that they can be made arbitrarily accurate, even near the interface. They have therefore been used to study phenomena that require high accuracy to resolve, such as fine details of the break-up or pinch-off of fluid threads. Such details can be important in industrial applications, e.g., the operation of an ink jet device. Despite their popularity, significant challenges still exist in the boundary integral simulation of 3D interfacial flow with surface tension or elastic membrane stress. These are: (i) the design of efficient methods to overcome a severe time-step constraint or `numerical stiffness’, which refers to the requirement that very small time steps be used as the numerical mesh is refined, causing prohibitively slow computations, (ii) the fast and accurate evaluation of boundary integral terms in the governing equations, and (iii) the analysis of the stability, convergence, and accuracy of the methods. The main achievement of this project is the design of a fast and accurate, nonstiff numerical method that overcomes the constraint (i) of numerical stiffness. The method includes a novel algorithm to compute singular integrals over surfaces, in the case that the surfaces are periodic. Singular integrals are terms in the governing equations that are particularly difficult to compute, and arise in mathematical models for many applications. The algorithm developed by the investigators requires a time proportional to the number of numerical grid points discretizing the interface to compute singular integrals, which is optimally fast. The investigators have implemented their 3D method to study a model of flow in porous media, which has applications in secondary oil recovery. They have also applied their method to the Rayleigh-Taylor problem with surface tension, which is important in inertial confinement fusion, and the hydroelastic problem, which describes the flapping of flags and the motion of heart valves, among other phenomena. They have analyzed the accuracy, stability, and convergence of their method, giving it a solid theoretical foundation. The investigators have also developed a preliminary version of their algorithm which uses overlapping coordinate patches to describe an interface. Such an approach can have a significant advantage over `global’ parameterizations in the simulation of highly deformed or stretched interfaces. For this reason, it has been used for a number of years in (static) electromagnetic scattering problems, but had not been implemented for time-dependent fluid flow problems. A parameterization using overlapping coordinate patches has the added benefit of providing a framework for the efficient implementation of the investigator’s nonstiff numerical algorithm. Broader impacts: The development of fast and accurate boundary integral methods for 3D interfacial flow with surface tension or elastic membrane stress will be of great benefit in scientific and technological applications. The concepts and methods described here go beyond the context of interfacial fluid flow and extend to other problems featuring moving boundaries with interfacial forces, in which surface-based numerical methods apply. This situation arises, for example, in interface problems modeling tumor growth and in microstructure evolution in materials science. The methods developed here have the potential to impact these and other areas. An important component of this proposal is the education and training of postdoctoral researchers and graduate students. The combination of mathematical and numerical analysis, scientific computing and applications in this project provide valuable preparation for a range of careers in mathematics and science.
