Hydrodynamics of Suspensions of Passive and Active Rigid Particles: A Rigid Multiblob Approach

We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally-resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a standard Krylov solver converges in a modest number of iterations that is essentially independent of the number of particles. For unbounded suspensions and suspensions sedimented against a single no-slip boundary, we rely on existing analytical expressions for the Rotne-Prager tensor combined with a fast multipole method or a direct summation on a Graphical Processing Unit to obtain an simple yet efficient and scalable implementation. For fully confined domains, such as periodic suspensions or suspensions confined in slit and square channels, we extend a recently-developed rigid-body immersed boundary method to suspensions of freely-moving passive or active rigid particles at zero Reynolds number. We demonstrate that the iterative solver for the coupled fluid and rigid body equations converges in a bounded number of iterations regardless of the system size. We optimize a number of parameters in the iterative solvers and apply our method to a variety of benchmark problems to carefully assess the accuracy of the rigid multiblob approach as a function of the resolution. We also model the dynamics of colloidal particles studied in recent experiments, such as passive boomerangs in a slit channel, as well as a pair of non-Brownian active nanorods sedimented against a wall.Comment: Under revision in CAMCOS, Nov 201

Performance Portable Solid Mechanics via Matrix-Free pp-Multigrid

Finite element analysis of solid mechanics is a foundational tool of modern engineering, with low-order finite element methods and assembled sparse matrices representing the industry standard for implicit analysis. We use performance models and numerical experiments to demonstrate that high-order methods greatly reduce the costs to reach engineering tolerances while enabling effective use of GPUs. We demonstrate the reliability, efficiency, and scalability of matrix-free $p$-multigrid methods with algebraic multigrid coarse solvers through large deformation hyperelastic simulations of multiscale structures. We investigate accuracy, cost, and execution time on multi-node CPU and GPU systems for moderate to large models using AMD MI250X (OLCF Crusher), NVIDIA A100 (NERSC Perlmutter), and V100 (LLNL Lassen and OLCF Summit), resulting in order of magnitude efficiency improvements over a broad range of model properties and scales. We discuss efficient matrix-free representation of Jacobians and demonstrate how automatic differentiation enables rapid development of nonlinear material models without impacting debuggability and workflows targeting GPUs

Efficient implicit FEM simulation of sheet metal forming

For the simulation of industrial sheet forming processes, the time discretisation is\ud one of the important factors that determine the accuracy and efficiency of the algorithm. For\ud relatively small models, the implicit time integration method is preferred, because of its inherent\ud equilibrium check. For large models, the computation time becomes prohibitively large and, in\ud practice, often explicit methods are used. In this contribution a strategy is presented that enables\ud the application of implicit finite element simulations for large scale sheet forming analysis.\ud Iterative linear equation solvers are commonly considered unsuitable for shell element models.\ud The condition number of the stiffness matrix is usually very poor and the extreme reduction\ud of CPU time that is obtained in 3D bulk simulations is not reached in sheet forming simulations.\ud Adding mass in an implicit time integration method has a beneficial effect on the condition number.\ud If mass scaling is used—like in explicit methods—iterative linear equation solvers can lead\ud to very efficient implicit time integration methods, without restriction to a critical time step and\ud with control of the equilibrium error in every increment. Time savings of a factor of 10 and more\ud can easily be reached, compared to the use of conventional direct solvers.\ud

Mathematical models and numerical simulation of mechanochemical pattern formation in biological tissues

Mechanical and chemical pattern formation in the development of biological tissue is a fundamental and fascinating process of self-complexation and self-organization. Yet, the understanding of the underlying mechanisms and their mathematical description still lacks in many interesting cases such as embryogenesis. In this thesis, we combine recent experimental and theoretical insights and numerically investigate the capacity of mechano-chemical processes to spontaneously generate patterns in biological tissue. Firstly, we develop and numerically analyze a prototypical system of partial differential equations (PDEs) leading to mechanochemical pattern formation in evolving tissues. Based on recent experimental data, we propose a novel coupling by tensor invariants describing stretch, stress or strain of tissue mechanics on the production of signaling molecules (morphogens). In turn, morphogen leads to piecewise-defined active deformations of individual biological cells. The presented approach is flexible and applied to two prominent examples of evolving tissue: We show how these simple interaction rules (“feedback loops”) lead to spontaneous, robust mechanochemical patterns in the applications to embryogenesis and to symmetry breaking in the sweet water polyp Hydra. Our results reveal that the full 3D model geometry is essential to obtain realistic results such as gastrulation events. Also, we highlight predictive numerical experiments that assess the sensitivity of biological tissue with regard to mechanical stimuli, namely to micropipette aspiration. These numerical experiments allow for a cross-validation with experimental observations. Besides, we apply our modeling approach to growing tips in colonial hydroids and investigate the role of rotational and shearing active deformations by comparison to experimental data. Secondly, we develop an efficient, numerical method to reliably solve these strongly coupled, prototypical systems of PDEs that model mechanochemical long-term problems. We employ state-of-the-art finite element methods, parallel geometric multigrid solvers and present a simple, local mesh refinement strategy to obtain an efficient solution approach. Parallel solvers are essential to deal with the huge problem size in 3D and were modified to keep track of biological cells. Further, we propose a stabilization of the structural equation to deal with the strongly coupled system of equations and the challenges of the different timescales of growth (days) and nonlinear elasticity (seconds). Also, this addresses the instabilities which result form the description of homogeneous Neumann values on the entire boundary that is necessary since the locations of patterns is a priori unknown

Mini-Workshop: Interface Problems in Computational Fluid Dynamics

Multiple difficulties are encountered when designing algorithms to simulate flows having free surfaces, embedded particles, or elastic containers. One difficulty common to all of these problems is that the associated interfaces are Lagrangian in character, while the fluid equations are naturally posed in the Eulerian frame. This workshop explores different approaches and algorithms developed to resolve these issues