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Abstract We study numerical schemes for incompressible Navier–Stokes equations using IMEX temporal discretizations, finite element spatial discretizations, and equipped with continuous data assimilation (a technique recently developed by Azouani et al. (2014)). We analyze stability and accuracy of the proposed methods, and are able to prove well-posedness, long time stability, and long time accuracy estimates, under restrictions of the time step size and data assimilation parameter. We give results for several numerical tests that illustrate the theory, and show that, for good results, the choice of discretization parameter and element choices can be critical. [ABSTRACT FROM AUTHOR]

We adapt a continuous data assimilation scheme, known as the Azouani–Olson–Titi (AOT) algorithm, to the case of moving observers for the 2D incompressible Navier–Stokes equations. We propose and test computationally several movement patterns (which we refer to as "the bleeps, the sweeps and the creeps"), as well as Lagrangian motion and combinations of these patterns, in comparison with static (i.e. non-moving) observers. In several cases, order-of-magnitude improvements in terms of the time-to-convergence are observed. We end with a discussion of possible applications to real-world data collection strategies that may lead to substantial improvements in predictive capabilities. • 2D Navier–Stokes flow is captured well by data assimilation with moving observers. • Moving observers during data assimilation dramatically improves convergence rates. • Far fewer observers are needed if observers move in certain patterns. • Several movement patterns clearly outperform others; reasons for this are explored. [ABSTRACT FROM AUTHOR]

A general and fast convolution-based method (FCBM) for peridynamics (PD) is introduced. Expressing the PD integrals in terms of convolutions and computing them by Fast Fourier Transform (FFT), the computational complexity of PD models drops from O(N 2) to O(N log 2 N), with N being the number of discretization nodes. Initial neighbor identification and storing neighbor information is not required, and this means memory allocation scales with O(N) instead of O(N 2), common for existing methods. FCBM is applicable to bounded domains with arbitrary shapes and boundary conditions via an "embedded constraint" (EC) approach. The formulation is shown for certain bond-based and state-based, linear and nonlinear, PD models of elasticity and dynamic brittle fracture, as applications. The method is verified on a 3D elastostatic problem and it is shown that the FCBM-PD reduces the computational time from days to hours and from years to days, compared with the original meshfree discretization for PD models. Large-scale computations of PD models are feasible with the new method, and its versatility is demonstrated by simulating, with ease, the difficult problem of cascading crack branching in a brittle plate. [Display omitted] • Fast computations of PD models on arbitrary domains with Fourier transforms. • Convolutional structure of nonlocal operators leads to O(NlogN) computations from FFT. • PD models of linear/nonlinear elasticity and fracture demonstrated. • A new energy-based damage model; dynamic brittle fracture with branching cascades example. • Eliminates crack path grid dependencies: using many nodes inside horizon is now affordable. [ABSTRACT FROM AUTHOR]

We introduce a fast convolution-based method (FCBM) for solving linear and a certain class of nonlinear peridynamic (PD) transient diffusion problems in 1D, 2D, and 3D. The method exploits the convolutional structure of the PD diffusion operator to compute it efficiently by using the fast Fourier transform (FFT). A new "embedded constraint" (EC) strategy allows using the Fourier transform on irregular domains and imposing arbitrary nonlocal boundary conditions. The complexity of the new method is O (N log 2 N) , compared with O (N 2) for the conventional peridynamic meshfree or finite element solvers of the same problem. We find quadratic convergence rates in terms of spatial discretization to the exact nonlocal solutions in 1D and 2D problems. Numerical tests show substantial efficiency gains when using the FCBM-EC method compared to the meshfree discretization method for problems with a larger number of nodes inside the nonlocal region. Further speedup is achieved with Matlab's intrinsic FFT functions for computations on GPUs or multiple CPUs. An example in 3D with over 1 billion degrees of freedom over tens of thousands of time-steps, is solved by the new method on a single CPU in a matter of days. The same problem would have required over a century to complete with the commonly used meshfree discretization. • A fast convolution-based method for transient 3D peridynamic diffusion with arbitrary B.C.'s. • We show that method is O (N log 2 N) and it convergence to exact nonlocal solutions in 1D and 2D. • Diffusion in a 3D sample with an insulated cut-out compared with FEM-based ABAQUS solution. • Linear and certain nonlinear peridynamic diffusion problems solved efficiently. • Simulation of transient heat transfer in 3D with billions of discretization nodes are now possible. [ABSTRACT FROM AUTHOR]