20 research outputs found
Highly scalable solution of incompressible Navier-Stokes equations using the spectral element method with overlapping grids
We present a highly-flexible Schwarz overlapping framework for simulating turbulent fluid/thermal transport in complex domains. The approach is based on a variant of the Schwarz alternating method in which the solution is advanced in parallel in separate overlapping subdomains. In each domain, the governing equations are discretized with an efficient high-order spectral element method (SEM). At each step, subdomain boundary data are determined by interpolating from the overlapping region of adjacent subdomains. The data are either lagged in time or extrapolated to higher-order temporal accuracy using a novel stabilized predictor-corrector algorithm. Matrix stability analysis is used to determine the optimal number of corrector iterations. Stability and accuracy are further improved with an optimal mass flux correction to guarantee mass conservation throughout the domain. The method supports an arbitrary number of subdomains. A new multirate time-stepping scheme is developed (a first for incompressible flow simulations) that allows the underlying equations to be advanced with time-step sizes varying as much as an order-of-magnitude between adjacent domains. All the developments maintain the third-order temporal convergence and exponential convergence of the originating SEM framework. This dissertation also presents a mesh optimizer that has been specifically designed for meshes generated for turbulent flow problems. The optimizer supports surface mesh improvement, which minimizes geometrical approximation errors. The smoother is shown to reduce the computational cost of numerical calculations by as much as 40%. Numerous examples illustrate the effectiveness of these new technologies for analyzing challenging turbulence problems that were previously infeasible.Ope
Spectral element mesh generation and improvement methods
Meshing tools for finite element meshes have been studied extensively over the last few decades. However, relatively less attention has been paid to spectral element meshes. This thesis focuses on mesh generation and mesh improvement methods for spectral element meshes. A mesh smoother, based on a combination of Laplacian smoothing and optimization, has been developed and implemented in Nek5000, an open-source spectral element method based incompressible flow solver. The smoother takes a valid mesh as an input and outputs an improved mesh. Comparison of the original and smoothed mesh has shown that mesh smoothing decreases the iteration count of iterative solvers. This reduction is anticipated from an observed decrease in the ratio of the maximum to minimum eigenvalues of the upper Hessenberg matrix generated by the GMRES method. The mesh smoother was tested on various meshes for complicated geometries, and was found to improve the computational efficiency of calculations by up to 20% which is helping save 100,000s of cpu-hours on high-performance computing machines. A mesh skinning tool has also been developed which adds boundary layer resolving elements of user-specified thickness at user-specified surfaces in an existing mesh. This translates into savings in terms of human time and effort since the user can now robustly add boundary layer resolving elements instead of manually meshing the geometry to add these elements. Additionally, tools have been developed that generate meshes for geometries like turbine blades and random-array of cylinders (to simulate flow in vegetated channels), in a matter of seconds. Finally, a tetrahedral (tet) to hexahedron (hex) mesh converter has been implemented, that generates spectral element meshes for any complicated geometry by taking an all-tet mesh and converting it to an all-hex spectral element mesh. This tool has been developed to quickly generate all-hex meshes with minimal user intervention
Multirate Timestepping for the Incompressible Navier-Stokes Equations in Overlapping Grids
We develop a multirate timestepper for semi-implicit solutions of the
unsteady incompressible Navier-Stokes equations (INSE) based on a
recently-developed multidomain spectral element method (SEM). For {\em
incompressible} flows, multirate timestepping (MTS) is particularly challenging
because of the tight coupling implied by the incompressibility constraint,
which manifests as an elliptic subproblem for the pressure at each timestep.
The novelty of our approach stems from the development of a stable overlapping
Schwarz method applied directly to the Navier-Stokes equations, rather than to
the convective, viscous, and pressure substeps that are at the heart of most
INSE solvers. Our MTS approach is based on a predictor-corrector (PC) strategy
that preserves the temporal convergence of the underlying semi-implicit
timestepper. We present numerical results demonstrating that this approach
scales to an arbitrary number of overlapping grids, accurately models complex
turbulent flow phenomenon, and improves computational efficiency in comparison
to singlerate timestepping-based calculations.Comment: 40 pages, 13 figure
High-Order Mesh Morphing for Boundary and Interface Fitting to Implicit Geometries
We propose a method that morphs high-orger meshes such that their boundaries
and interfaces coincide/align with implicitly defined geometries. Our focus is
particularly on the case when the target surface is prescribed as the zero
isocontour of a smooth discrete function. Common examples of this scenario
include using level set functions to represent material interfaces in
multimaterial configurations, and evolving geometries in shape and topology
optimization. The proposed method formulates the mesh optimization problem as a
variational minimization of the sum of a chosen mesh-quality metric using the
Target-Matrix Optimization Paradigm (TMOP) and a penalty term that weakly
forces the selected faces of the mesh to align with the target surface. The
distinct features of the method are use of a source mesh to represent the level
set function with sufficient accuracy, and adaptive strategies for setting the
penalization weight and selecting the faces of the mesh to be fit to the target
isocontour of the level set field. We demonstrate that the proposed method is
robust for generating boundary- and interface-fitted meshes for curvilinear
domains using different element types in 2D and 3D.Comment: 30 pages, 16 figure
hr-adaptivity for nonconforming high-order meshes with the target matrix optimization paradigm
We present an -adaptivity framework for optimization of high-order
meshes. This work extends the -adaptivity method for mesh optimization by
Dobrev et al., where we utilized the Target-Matrix Optimization Paradigm (TMOP)
to minimize a functional that depends on each element's current and target
geometric parameters: element aspect-ratio, size, skew, and orientation. Since
fixed mesh topology limits the ability to achieve the target size and
aspect-ratio at each position, in this paper we augment the -adaptivity
framework with nonconforming adaptive mesh refinement to further reduce the
error with respect to the target geometric parameters. The proposed
formulation, referred to as -adaptivity, introduces TMOP-based quality
estimators to satisfy the aspect-ratio-target via anisotropic refinements and
size-target via isotropic refinements in each element of the mesh. The
methodology presented is purely algebraic, extends to both simplices and
hexahedra/quadrilaterals of any order, and supports nonconforming isotropic and
anisotropic refinements in 2D and 3D. Using a problem with a known exact
solution, we demonstrate the effectiveness of -adaptivity over both
and -adaptivity in obtaining similar accuracy in the solution with
significantly fewer degrees of freedom. We also present several examples that
show that -adaptivity can help satisfy geometric targets even when
-adaptivity fails to do so, due to the topology of the initial mesh.Comment: 29 pages, 15 figure
DynAMO: Multi-agent reinforcement learning for dynamic anticipatory mesh optimization with applications to hyperbolic conservation laws
We introduce DynAMO, a reinforcement learning paradigm for Dynamic
Anticipatory Mesh Optimization. Adaptive mesh refinement is an effective tool
for optimizing computational cost and solution accuracy in numerical methods
for partial differential equations. However, traditional adaptive mesh
refinement approaches for time-dependent problems typically rely only on
instantaneous error indicators to guide adaptivity. As a result, standard
strategies often require frequent remeshing to maintain accuracy. In the DynAMO
approach, multi-agent reinforcement learning is used to discover new local
refinement policies that can anticipate and respond to future solution states
by producing meshes that deliver more accurate solutions for longer time
intervals. By applying DynAMO to discontinuous Galerkin methods for the linear
advection and compressible Euler equations in two dimensions, we demonstrate
that this new mesh refinement paradigm can outperform conventional
threshold-based strategies while also generalizing to different mesh sizes,
remeshing and simulation times, and initial conditions.Comment: 38 pages, 21 figure
Multi-Agent Reinforcement Learning for Adaptive Mesh Refinement
Adaptive mesh refinement (AMR) is necessary for efficient finite element
simulations of complex physical phenomenon, as it allocates limited
computational budget based on the need for higher or lower resolution, which
varies over space and time. We present a novel formulation of AMR as a
fully-cooperative Markov game, in which each element is an independent agent
who makes refinement and de-refinement choices based on local information. We
design a novel deep multi-agent reinforcement learning (MARL) algorithm called
Value Decomposition Graph Network (VDGN), which solves the two core challenges
that AMR poses for MARL: posthumous credit assignment due to agent creation and
deletion, and unstructured observations due to the diversity of mesh
geometries. For the first time, we show that MARL enables anticipatory
refinement of regions that will encounter complex features at future times,
thereby unlocking entirely new regions of the error-cost objective landscape
that are inaccessible by traditional methods based on local error estimators.
Comprehensive experiments show that VDGN policies significantly outperform
error threshold-based policies in global error and cost metrics. We show that
learned policies generalize to test problems with physical features, mesh
geometries, and longer simulation times that were not seen in training. We also
extend VDGN with multi-objective optimization capabilities to find the Pareto
front of the tradeoff between cost and error.Comment: 24 pages, 18 figure