| Abstract |
Active matter represents a unique class of non-equilibrium systems, including examples ranging from cellular structures to large-scale biological tissues. These systems exhibit intriguing spatiotemporal dynamics, driven by the constituent particles’ continuous energy expenditure. Such active-matter systems, featuring complex hydrodynamics, are described by sophisticated mathematical models, typically using partial differential equations (PDEs). PDEs modeling hydrodynamics, such as the Navier-Stokes equations, are analytically intractable, and notoriously challenging to study computationally. The challenges include the need for consistent numerical methods along with their efficient and scalable high-performance computer implementation to solve the PDEs numerically. However, when considering new theoretical PDE models, such as active hydrodynamics, conventional approaches often fall short due to the specialization made in the numerical methods to study certain specific models. The inherent complexity and nonlinearity of active-matter PDEs add to the challenge. Hence, the computational study of such active-matter PDE models requires rapidly evolving high-performance computer software that can easily implement new numerical methods to solve these equations in biologically realistic three-dimensional domains. This presents a rich, yet underexplored territory demanding scalable computational frameworks that apply to a large class of PDEs. In this thesis, we introduce a computational framework that effectively allows for using multiple numerical methods through a context-aware template expression system akin to an embedded domain-specific language. This framework primarily aims at solving lengthy PDEs associated with active hydrodynamics in complex domains, while experimenting with new numerical methods. Existing PDE-solving codes often lack this flexibility, as they are closely tied to a PDE and domain geometry that rely on a specific numerical method. We overcome these limitations by using an object-oriented implementation design, and show experiments with adaptive and numerically consistent particle-based approach called Discretization-Corrected Particle Strength Exchange (DC-PSE). DC-PSE allows for the higher-order discretization of differential operators on arbitrary particle distributions leading to the possibility of solving active hydrodynamic PDEs in complex domains. However, the curse of dimensionality makes it difficult to numerically solve three-dimensional equations on single-core architectures and warrants the use of parallel and distributed computers. We design a novel template-expression system and implement it in the scalable scientific computing library OpenFPM. Our methodology offers an expression-based embedded language, enabling PDE codes to be written in a form that closely mirrors mathematical notation. Leveraging OpenFPM, this approach also ensures parallel scalability. To further enhance our framework's versatility, we employ a \textit{separation-of-concerns} abstraction, segregating the model equations from numerics, and domain geometry. This allows for the rapid rewriting of codes for agile numerical experiments across different model equations in various geometries. Supplementing this framework, we develop a distributed algebra system compatible with OpenFPM and Boost Odeint. This algebra system opens avenues for a multitude of explicit adaptive time-integration schemes, which can be selected by modifying a single line of code while maintaining parallel scalability. Motivated by symmetry-preserving theories of active hydrodynamics, and as a first benchmark of our template-expression system, we present a high-order numerically convergent scheme to study active polar fluids in arbitrary three-dimensional domains. We derive analytical solutions in simple Cartesian geometries and use them to show the numerical convergence of our algorithm. Further, we showcase the scalability of the computer code written using our expression system on distributed computing systems. To cater to the need for solving PDEs on curved surfaces, we present a novel meshfree numerical scheme, the Surface DC-PSE method. Upon implementation in our scalable framework, we benchmark Surface DC-PSE for both explicit and implicit Laplace-Beltrami operators and show applications to computing mean and Gauss curvature. Finally, we apply our computational framework to exploring the three-dimensional active hydrodynamics of biological flowing matter, a prominent model system to study the active dynamics of cytoskeletal networks, celluar migration, and tissue mechanics. Our software framework effectively tackles the challenges associated to numerically solving such non-equilibrium spatiotemporal PDEs. We perform linear perturbation analysis of the three-dimensional Ericksen-Leslie model and find an analytical expression for the critical active potential or, equivalently, a critical length of the system above which a spontaneous flow transition occurs. This spontaneous flow transition is a first realization of a three-dimensional active Fr\'eedericksz transition. With our expression system, we successfully simulate 3D active fluids, finding phases of spontaneous flow transitions, traveling waves, and spatiotemporal chaos with increasing active stress. We numerically find a topological phase transition similar to the Berezinskii–Kosterlitz–Thouless transition (BKT transition) of the two-dimensional XY model that occurs in active polar fluids after the spontaneous flow transition. We then proceed to non-Cartesian geometries and show the application of our software framework to solve the active polar fluid equations in spherical domains. We find spontaneous flows in agreement with recent experimental observations. We further showcase the framework to solve the equations in 3D annular domains and a `peanut' geometry that resembles a dividing cell. Our simulations further recapitulate the actin flows observed in \textit{Xenopus} egg extracts within spherical shell geometries, showcasing our framework's versatility in handling complex geometrical modifications of model equations. Looking ahead, we hope our framework will serve as a foundation for further advancements in computational morphogenesis, fostering collaboration and using the present techniques in biophysical modeling. |