1,721,016 research outputs found
Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier–Stokes Equations
Full text linkWe study preconditioners for the iterative solution of the linear systems arising in the implicit time integration of the compressible Navier–Stokes equations. The spatial discretization is carried out using a discontinuous Galerkin method with fourth order polynomial interpolations on triangular elements. The time integration is based on backward difference formulas resulting in a nonlinear system of equations which is solved at each timestep. This is accomplished using Newton's method. The resulting linear systems are solved using a preconditioned GMRES iterative algorithm. We consider several existing preconditioners such as block Jacobi and Gauss–Seidel combined with multilevel schemes which have been developed and tested for specific applications. While our results are consistent with the claims reported, we find that these preconditioners lack robustness when used in more challenging situations involving low Mach numbers, stretched grids, or high Reynolds number turbulent flows. We propose a preconditioner based on a coarse scale correction with postsmoothing based on a block incomplete LU factorization with zero fill-in (ILU0) of the Jacobian matrix. The performance of the ILU0 smoother is found to depend critically on the element numbering. We propose a numbering strategy based on minimizing the discarded fill-in in a greedy fashion. The coarse scale correction scheme is found to be important for diffusion dominated problems, whereas the ILU0 preconditioner with the proposed ordering is effective at handling the convection dominated case. While little can be said in the way of theoretical results, the proposed preconditioner is shown to perform remarkably well for a broad range of representative test problems. These include compressible flows ranging from very low Reynolds numbers to fully turbulent flows using the Reynolds averaged Navier–Stokes equations discretized on highly stretched grids. For low Mach number flows, the proposed preconditioner is more than one order of magnitude more efficient than the other preconditioners considered
Recommended from our members
Efficient solvers for the implicit time integration of matrix-free high-order methods
Full text linkIn this thesis, we develop and study efficient solvers for high-order Galerkin methods applied to fluid flow problems.Many flow problems necessitate implicit time-integration schemes in order to be practical.
Implicit-in-time discretizations require the solution of nonlinear algebraic systems each time step, which are often in-turn solved by linear solvers.
Therefore, the performance of implicit-in-time solvers is largely determined by the performance of the underlying linear solvers.One approach to create efficient methods is to work with matrix-free operators.Because assembling the underlying discretization matrix can be prohibitively expensive in terms of computational complexity and memory, matrix-free operators are an attractive alternative.
These operators replace the matrix-vector products with on-the-fly sum-factorization evaluations of the discretized differential operators instead.
Indeed, their high arithmetic intensity makes these operators particularly well suited for modern graphics processing units (GPU) and GPU-accelerated architectures.These matrix-free operators are particularly challenging to precondition, however, because they by design do not allow access to the underlying matrix entries.We create a suite of efficient matrix-free preconditioners for a range of fluid flow problems that are robust with respect to polynomial degree and mesh size.
The main building block solver extends sparse, low-order refined preconditioners with parallel subspace corrections.
This work tackles Poisson problems, saddle-point Stokes systems, and the incompressible Navier-Stokes equations in two and three spatial dimensions.A different set of problems exhibit geometrically localized stiffness, where convergence rates are degraded in a localized subregion of the mesh.Generic preconditioners do not perform well across the entire domain because of mesh size, mesh anisotropy, highly variable coefficients, or more challenging physics in the subregion.
Therefore, we seek to save costs by utilizing cheap preconditioners for most of the mesh and only focus our effort on the less expensive subregion problem.
Our iterative subregion correction preconditioners correct naive preconditioners with an adaptive inner subregion iteration to reduce the number of costly global iterations.
This work demonstrates performance on basic convection-diffusion problems, high Reynolds number compressible flow problems, and a angle of attack problem with massively separated flow
Recommended from our members
High-Order Discontinuous Galerkin Fluid-Structure Interaction Methods
Full text linkWe present a high-order accurate scheme for fully coupledfluid-structure interaction problems. The fluid is discretized using adiscontinuous Galerkin method on unstructured tetrahedral meshes, andthe structure uses a high-order volumetric continuous Galerkin finiteelement method. Standard radial basis functions are used for the meshdeformation. The time integration is performed using a partitionedapproach based on implicit-explicit Runge-Kutta methods. Theresulting scheme fully decouples the implicit solution procedures forthe fluid and the solid parts, which we perform using two separateefficient parallel solvers. We demonstrate up to fifth order accuracyin time on a non-trivial test problem, on which we also show thatadditional subiterations are not required. We solve a benchmarkproblem of a cantilever beam in a shedding flow, and show goodagreement with other results in the literature.In addition, we create several simulations which are motivated byreal-world phenomena. First, we investigate flow around a thinmembrane at high-angle of attack, demonstrating the ability of theleading edge of the membrane to align with the incident flow. Examplesare provided in both two and three dimensions. Next, we considerbiologically inspired flight, by investigating wing-like structuresdriven in a flapping motion in both two and three dimensions.Finally, we demonstrate how the method may be used in acousticsproblems, simulating a tuning fork in three dimensions. Here weaccurately capture decay rates purely from the fluid-structureinteraction and without any damping coefficients built into thestructure model
Recommended from our members
Discontinuous Galerkin Methods on Moving Domains with Large Deformations
Full text linkWe present two different numerical approaches for solving compressible flows on moving domains with high-order accuracy. The approaches are base on discontinuous Galerkin (DG) methods and are particularly designed for addressing the large deformation problems as the domain moves.A moving-mesh technique is first introduced to improve the mesh quality with the domain deforming. The technique moves the mesh nodes by DistMesh algorithm and locally changes the mesh topology by flipping edges or faces, which can be applied in both 2D and 3D. Moreover, some local density control operations are also developed to add or remove the mesh nodes to change the mesh adaptivity.Our first numerical scheme is formulated on a space-time framework using a nodal DG discretization on space-time domains with appropriate numerical fluxes for the first and the second-order terms, respectively. The scheme is implicit, and we solve the resulting non-linear systems using a parallel Newton-Krylov solver. Along with the numerical scheme, two efficient algorithms for constructing globally conforming space-time slab meshes are given, based on our moving-mesh technique. The second approach employs DG discreatization with arbitrary-Eulerian-Lagrangian (ALE) framework by solving equations based on smooth mappings. An efficient local L2 projection is used for transferring solutions when mesh topology change happens. We test our two approaches by a number of numerical cases in both 2D and 3D. The tests involve convergence tests as well as simulations of laminar flows, which shows that the proposed methods achieve high-order accuracy and are able to handle problems with complex geometric motions
Recommended from our members
High-Order Moment Methods for Thermal Radiative Transfer
Full text linkNumerically modeling the high energy density regimes characteristic of astrophysical phenomena and intertial confinement fusion (ICF) requires simultaneously modeling hydrodynamics and thermal radiative transfer (TRT). Recently, high-order finite element discretizations of the hydrodynamics equations using high-order (curved) meshes have been shown to have improved robustness and computational performance over low-order methods. Due to the tightly coupled nature of these radiation-hydrodynamics simulations, high-order methods compatible with curved meshes are also desired for TRT. This dissertation develops high-order, moment-based methods for solving the radiation transport equation, a crucial component of modeling TRT. Moment methods are a class of scale and model-bridging algorithms for solving kinetic equations, such as the radiation transport equation, in the context of multiphysics simulations. An efficient and robust iterative scheme is found by coupling the transport equation to a reduced-dimensional model derived from its statistical moments. The moment equations are closed such that, upon iterative convergence, the reduced-dimensional model is capable of reproducing the physics of the high-dimensional transport equation.
Moment methods are attractive in the context of high energy density physics simulations as they provide significant algorithmic flexibility, efficient and robust iterative convergence, and a means to isolate the expensive, high-dimensional transport equation from the evolution of the stiff hydrodynamic multiphysics. The Variable Eddington Factor (VEF) method is a moment-based transport algorithm where the choice of closure causes the moment system to have an unusual, non-symmetric structure. This makes the development of discretizations for the VEF moment system and their corresponding scalable preconditioned iterative solvers difficult. The flexibility provided by moment methods is leveraged to design discretizations for the VEF moment system that are capable of employing existing linear solver technology. We present Discontinuous Galerkin (DG), continuous finite element, and mixed finite element discretizations that all have high-order accuracy, compatibility with curved meshes, and efficient preconditioned iterative solvers. When paired with a high-order DG discretization of the Discrete Ordinates (SN) transport equations, the resulting methods form efficient and robust algorithms for solving the radiation transport equation. We also investigate the use of the Second Moment Method (SMM), a class of moment methods closely related to the VEF method. SMMs avoid the difficult-to-solve VEF moment system through a clever choice of closure, leading to an iterative scheme where only radiation diffusion must be inverted at each iteration. By leveraging a mathematical connection between SMM and VEF, the VEF methods presented in this dissertation are converted to SMMs to derive novel DG, continuous finite element, and mixed finite element-based algorithms. The resulting methods also form robust and efficient transport algorithms while avoiding the non-symmetric solvers that VEF methods require. This work demonstrates that the algorithmic flexibility allowed by moment methods can be used to design efficient algorithms for radiation transport. In addition, this dissertation serves as the foundation for the design of efficient, high-order, moment-based radiation-hydrodynamics algorithms
Recommended from our members
High-order Solution Transfer between Curved Meshes and Ill-conditioned Bézier Curve Intersection
Full text linkThe problem of solution transfer between meshes arises frequently in computational physics, e.g. in Lagrangian methods where remeshing occurs. The interpolation process must be conservative, i.e. it must conserve physical properties, such as mass. We extend previous works --- which described the solution transfer process for straight sided unstructured meshes --- by considering high-order isoparametric meshes with curved elements. The implementation is highly reliant on accurate computational geometry routines for evaluating points on and intersecting Bézier curves and triangles.Two ill-conditioned problems that occur evaluating points on and intersecting Bézier curves are then explored. This work presents a family of compensated algorithms to accurately evaluate a polynomial in Bernstein form with floating point coefficients. The principle is to apply error-free transformations to improve the traditional de Casteljau algorithm. The resulting output is as accurate as the de Casteljau algorithm performed in K times the working precision. After compensated evaluation is considered, a compensated Newton's method is described, both for root-finding for polynomials in the Bernstein basis and for Bézier curve intersection
Recommended from our members
High Order Partitioned Fully Implicit Runge-Kutta Solvers for Fluid-Structure Interaction
Full text linkIn this work, we develop and analyze a partitioned implicit time integration method designed for fluid-structure interaction (FSI) modelling using discontinuous Galerkin (DG) discretizations and fully implicit Runge-Kutta (IRK) methods. The discontinuous Galerkin method is a high-order finite element method used for fluid simulations on unstructured meshes. We take a partitioned approach, modelling the structures separately and limiting communication between structure and fluid by use of a prediction-correction framework. We verify the high-order accuracy, the stability, and the performance of our method on simple model problems including a one-dimensional sprung piston and a two-dimensional pitching airfoil with prescribed vertical motion and a torsional restoring force, both with structures of two degrees of freedom. Finally we apply our method to the standard problem of a cantilever beam shedding vortexes with two sets of parameters, and a vibrating tuning fork problem. Both these final applications consist of fluids modelled on fully unstructured meshes of high-order triangular elements deformed by radial basis functions according to the structure motion, and structures modeled using a neo-Hookean formulation and discretized using standard continuous finite elements.Our scheme fully decouples the implicit solutions of the structure and the fluid, with communication limited to boundary conditions and mesh deformation. We present our method with two variations, one using explicit methods for predicted quantities and another using implicit. We implement our method up to seventh order, and compare its cost with standard Diagonally Implicit Runge-Kutta methods where possible. Our findings are that this new method can be significantly cheaper, in particular for the more complex cantilever problem, and it also has better stability properties
Recommended from our members
Curved and anisotropic unstructured mesh generation and adaptivity using the Winslow equations
Full text linkHigh-order methods are receiving considerable interest from the computational community because they can achieve higher accuracy with reduced computational cost compared to traditional low-order approaches. These methods generally require unstructured meshes of non-inverted curved elements, and the generation of high-order curved meshes in a robust and automatic way is an important and challenging open problem.We present a method to generate high-order unstructured curved meshes by solving the classical Winslow equations using a new continuous Galerkin finite element discretization. This formulation appears to produce high quality curved elements, which are highly resistant to inversion. In addition, the corresponding nonlinear equations can be solved efficiently using Picard iterations, even for highly stretched boundary layer meshes. Another challenge that mesh-based methods face is that the discretization of the domain is usually generated before the solution is known, which can lead to large numerical errors or non-convergent schemes. A tool that can be used to overcome this problem is mesh adaptivity. We use the Winslow variable diffusion equations -- which are a variation of the classical form -- to perform curved and anisotropic unstructured mesh adaptivity. We use a range of numerical examples to validate our models including complex geometries and stretched boundary layers. We demonstrate the high quality of the generated meshes and the performance ofthe nonlinear solver. Finally, we present an example of mesh adaptivity for shock capturing when solving the Euler equations of gas dynamics for supersonic flow
Recommended from our members
Reliable and Efficient Algorithms for Spectrum-Revealing Low-Rank Data Analysis
Full text linkAs the amount of data collected in our world increases, reliable compression algorithms are needed when datasets become too large for practical analysis, when significant noise is present in the data, or when the strongest signals in the data are needed. In this work, two data compression algorithms are presented. The main result is a low-rank approximation algorithm (a type of compression algorithm) that uses modern techniques in randomization to repurpose a classic algorithm in the field of linear algebra called the LU decomposition to perform data compression. The resulting algorithm is called Spectrum-Revealing LU (SRLU).Both rigorous theory and numeric experiments demonstrate the effectiveness of SRLU. The theoretical work presented also develops a framework with which other low-rank approximation algorithms can be analyzed. As the name implies, Spectrum-Revealing LU seeks to capture the entire spectrum of the data (i.e. to capture all signals present in the data).A second compression algorithm is also introduced, which seeks to compression graphs. Called a sparsification algorithm, this algorithm can accept a weighted or unweighted graph and produce an approximation without changing the weights (or introducing weights in the case of an unweighted graph). Theoretical results provide a bound on the quality of the results, and a numeric example is also explored
Recommended from our members
Machine Learning Methods to Optimize the Geometry and Topology of Meshes
Full text linkMeshes are used ubiquitously in engineering for representing geometries, performing computational simulations, and generating computer graphics renderings. Automatically generating suitable meshes for downstream applications remains a key bottleneck in many workflows and often requires significant manual intervention. It is challenging to optimize mesh data-structures because they can be highly unstructured in their most general form.A mesh has two fundamental attributes — geometry and topology. Geometry deals with the position and shape of objects in space. Topology is concerned with the connectivity of mesh elements. It is essential to optimize both of these attributes to generate a desirable mesh for a target application such as simulation. This dissertation explores machine learning methods to optimize both of these attributes.The first part of this dissertation is concerned with mesh topology. We will describe a deep reinforcement learning framework to optimize the topology of 2D meshes using elementary mesh editing operations. The framework is trained purely in self-play reinforcement learning to optimize a given user defined objective function. We describe a novel neural network architecture that is able to encode the local topology of a mesh around a given mesh neighborhood. Subsequently, the neural network is trained to predict a probability distribution over the local action space in order to maximize the cumulative reward as prescribed by the given objective function. The agent is trained on randomly generated 2D polygonal shapes. We demonstrate generalization to inputs that were never seen during training. The proposed framework is particularly effective at coarse block decomposition of polygonal shapes where the aim is to minimize the number of irregular vertices in the mesh.We will then tackle the problem of geometry. We describe a deep learning method to automatically generate patient-specific, simulation ready 3D surface meshes of the human heart directly from clinical imaging. The proposed method is a two-stage mesh deformation process that transforms a given template mesh to match the underlying target geometry in the image data. The first stage consists of a learned affine transformation conditioned on the input image. This stage is trained to roughly align the template in terms of scale and orientation to the image data. The second stage consists of a learned local diffeomorphic deformation field conditioned on the image and the current location of the template. This stage improves the accuracy of the prediction by capturing finer details of the target geometry. We describe a novel loss function derived from the kinematics of motion of continuous bodies that penalizes undesirable phenomenon such as surface interpenetration resulting in anatomically accurate, physically realistic, simulation ready meshes. The proposed framework is validated against a large held-out test dataset and compared with prior state-of-the-art along a variety of accuracy and quality metrics
- …
