This work introduces a numerical approach for the implementation and direct coupling of arbitrary complex ordinary differential equation- (ODE-)governed boundary conditions to three-dimensional (3D) lattice Boltzmann-based fluid equations for fluid-structure hemodynamics simulations. In particular, a most complex configuration is treated by considering a dynamic left ventricle- (LV-)elastance heart model which is governed by (and applied as) a nonlinear, non-stationary hybrid ODE-Dirichlet system. The complete 0D-3D solver, including its treatment of the fluid and solid equations as well as their interactions, is validated through a variety of benchmark and convergence studies that demonstrate the ability of the coupled 0D-3D methodology in generating physiological pressure and flow waveforms -- ultimately enabling the exploration of various physical and physiological parameters for hemodynamics studies of the coupled LV-arterial system. The methods proposed in this paper can be easily applied to other ODE-based boundary conditions (such as those based on Windkessel lumped parameter models) as well as to other fluid problems that are modeled by 3D lattice Boltzmann equations and that require direct coupling of dynamic 0D conditions.
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Most research on preconditioners for time-dependent PDEs has focused on implicit multi-step or diagonally-implicit multi-stage temporal discretizations. In this paper, we consider monolithic multigrid preconditioners for fully-implicit multi-stage Runge-Kutta (RK) time integration methods. These temporal discretizations have very attractive accuracy and stability properties, but they couple the spatial degrees of freedom across multiple time levels, requiring the solution of very large linear systems. We extend the classical Vanka relaxation scheme to implicit RK discretizations of saddle point problems. We present numerical results for the incompressible Stokes, Navier-Stokes, and resistive magnetohydrodynamics equations, in two and three dimensions, confirming that these relaxation schemes lead to robust and scalable monolithic multigrid methods for a challenging range of incompressible fluid-flow models.
We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to the second-order elliptic PDE in non-divergence form. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the 2D and 3D settings. In addition, we present a comparison with the existing multivariate spline methods in \cite{ALW06} and \cite{LW17} to show that the new method produces a similar and sometimes more accurate approximation in a more efficient fashion.
In this work we aim to develop a unified mathematical framework and a reliable computational approach to model the brittle fracture in heterogeneous materials with variability in material microstructures, and to provide statistic metrics for quantities of interest, such as the fracture toughness. To depict the material responses and naturally describe the nucleation and growth of fractures, we consider the peridynamics model. In particular, a stochastic state-based peridynamic model is developed, where the micromechanical parameters are modeled by a finite-dimensional random vector, or a combination of random variables truncating the Karhunen-Lo\`{e}ve decomposition or the principle component analysis (PCA). To solve this stochastic peridynamic problem, probabilistic collocation method (PCM) is employed to sample the random field representing the micromechanical parameters. For each sample, the deterministic peridynamic problem is discretized with an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and demonstrate its convergence for a number of benchmark problems, showing that it sustains the asymptotic compatibility spatially and achieves an algebraic or sub-exponential convergence rate in the random space as the number of collocation points grows. Finally, to validate the applicability of this approach on real-world fracture problems, we consider the problem of crystallization toughening in glass-ceramic materials, in which the material at the microstructural scale contains both amorphous glass and crystalline phases. The proposed stochastic peridynamic solver is employed to capture the crack initiation and growth for glass-ceramics with different crystal volume fractions, and the averaged fracture toughness are calculated. The numerical estimates of fracture toughness show good consistency with experimental measurements.
When applied to stiff, linear differential equations with time-dependent forcing, Runge-Kutta methods can exhibit convergence rates lower than predicted by the classical order condition theory. Commonly, this order reduction phenomenon is addressed by using an expensive, fully implicit Runge-Kutta method with high stage order or a specialized scheme satisfying additional order conditions. This work develops a flexible approach of augmenting an arbitrary Runge-Kutta method with a fully implicit method used to treat the forcing such as to maintain the classical order of the base scheme. Our methods and analyses are based on the general-structure additive Runge-Kutta framework. Numerical experiments using diagonally implicit, fully implicit, and even explicit Runge-Kutta methods confirm that the new approach eliminates order reduction for the class of problems under consideration, and the base methods achieve their theoretical orders of convergence.
We introduce a geometric stencil selection algorithm for Laplacian in 3D that significantly improves octant-based selection considered earlier. The goal of the algorithm is to choose a small subset from a set of irregular points surrounding a given point that admits an accurate numerical differentiation formula. The subset serves as an influence set for the numerical approximation of the Laplacian in meshless finite difference methods using either polynomial or kernel-based techniques. Numerical experiments demonstrate a competitive performance of this method in comparison to the finite element method and to other selection methods for solving the Dirichlet problems for the Poisson equation on several STL models. Discretization nodes for these domains are obtained either by 3D triangulations or from Cartesian grids or Halton quasi-random sequences.
With the field of rigid-body robotics having matured in the last fifty years, routing, planning, and manipulation of deformable objects have emerged in recent years as a more untouched research area in many fields ranging from surgical robotics to industrial assembly and construction. Routing approaches for deformable objects which rely on learned implicit spatial representations (e.g., Learning-from-Demonstration methods) make them vulnerable to changes in the environment and the specific setup. On the other hand, algorithms that entirely separate the spatial representation of the deformable object from the routing and manipulation, often using a representation approach independent of planning, result in slow planning in high dimensional space. This paper proposes a novel approach to spatial representation combined with route planning that allows efficient routing of deformable one-dimensional objects (e.g., wires, cables, ropes, threads). The spatial representation is based on the geometrical decomposition of the space into convex subspaces, which allows an efficient coding of the configuration. Having such a configuration, the routing problem can be solved using a dynamic programming matching method with a quadratic time and space complexity. The proposed method couples the routing and efficient configuration for improved planning time. Our tests and experiments show the method correctly computing the next manipulation action in sub-millisecond time and accomplishing various routing and manipulation tasks.
A video autoencoder is proposed for learning disentan- gled representations of 3D structure and camera pose from videos in a self-supervised manner. Relying on temporal continuity in videos, our work assumes that the 3D scene structure in nearby video frames remains static. Given a sequence of video frames as input, the video autoencoder extracts a disentangled representation of the scene includ- ing: (i) a temporally-consistent deep voxel feature to represent the 3D structure and (ii) a 3D trajectory of camera pose for each frame. These two representations will then be re-entangled for rendering the input video frames. This video autoencoder can be trained directly using a pixel reconstruction loss, without any ground truth 3D or camera pose annotations. The disentangled representation can be applied to a range of tasks, including novel view synthesis, camera pose estimation, and video generation by motion following. We evaluate our method on several large- scale natural video datasets, and show generalization results on out-of-domain images.
We develop a novel human trajectory prediction system that incorporates the scene information (Scene-LSTM) as well as individual pedestrian movement (Pedestrian-LSTM) trained simultaneously within static crowded scenes. We superimpose a two-level grid structure (grid cells and subgrids) on the scene to encode spatial granularity plus common human movements. The Scene-LSTM captures the commonly traveled paths that can be used to significantly influence the accuracy of human trajectory prediction in local areas (i.e. grid cells). We further design scene data filters, consisting of a hard filter and a soft filter, to select the relevant scene information in a local region when necessary and combine it with Pedestrian-LSTM for forecasting a pedestrian's future locations. The experimental results on several publicly available datasets demonstrate that our method outperforms related works and can produce more accurate predicted trajectories in different scene contexts.
We study the use of the Wave-U-Net architecture for speech enhancement, a model introduced by Stoller et al for the separation of music vocals and accompaniment. This end-to-end learning method for audio source separation operates directly in the time domain, permitting the integrated modelling of phase information and being able to take large temporal contexts into account. Our experiments show that the proposed method improves several metrics, namely PESQ, CSIG, CBAK, COVL and SSNR, over the state-of-the-art with respect to the speech enhancement task on the Voice Bank corpus (VCTK) dataset. We find that a reduced number of hidden layers is sufficient for speech enhancement in comparison to the original system designed for singing voice separation in music. We see this initial result as an encouraging signal to further explore speech enhancement in the time-domain, both as an end in itself and as a pre-processing step to speech recognition systems.
Networks provide a powerful formalism for modeling complex systems, by representing the underlying set of pairwise interactions. But much of the structure within these systems involves interactions that take place among more than two nodes at once; for example, communication within a group rather than person-to-person, collaboration among a team rather than a pair of co-authors, or biological interaction between a set of molecules rather than just two. We refer to these type of simultaneous interactions on sets of more than two nodes as higher-order interactions; they are ubiquitous, but the empirical study of them has lacked a general framework for evaluating higher-order models. Here we introduce such a framework, based on link prediction, a fundamental problem in network analysis. The traditional link prediction problem seeks to predict the appearance of new links in a network, and here we adapt it to predict which (larger) sets of elements will have future interactions. We study the temporal evolution of 19 datasets from a variety of domains, and use our higher-order formulation of link prediction to assess the types of structural features that are most predictive of new multi-way interactions. Among our results, we find that different domains vary considerably in their distribution of higher-order structural parameters, and that the higher-order link prediction problem exhibits some fundamental differences from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.
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