635 research outputs found
Advancements in Fluid Simulation Through Enhanced Conservation Schemes
To better understand and solve problems involving the natural phenomenon of fluid and air flows, one must understand the Navier-Stokes equations. Branching several different fields including engineering, chemistry, physics, etc., these are among the most important equations in mathematics. However, these equations do not have analytic solutions save for trivial solutions. Hence researchers have striven to make advancements in varieties of numerical models and simulations. With many variations of numerical models of the Navier-Stokes equations, many lose important physical meaningfulness. In particular, many finite element schemes do not conserve energy, momentum, or angular momentum. In this thesis, we will study new methods in solving the Navier-Stokes equations using models which have enhanced conservation qualities, in particular, the energy, momentum, and angular momentum conserving (EMAC) scheme. The EMAC scheme has gained popularity in the mathematics community over the past few years as a desirable method to model fluid flow. It has been proven to conserve energy, momentum, angular momentum, helicity, and others. EMAC has also been shown to perform better and maintain accuracy over long periods of time compared to other schemes. We investigate a fully discrete error analysis of EMAC and SKEW. We show that a problematic dependency on the Reynolds number is present in the analysis for SKEW, but not in EMAC under certain conditions. To further explore this concept, we include some numerical experiments designed to highlight these differences in the error analysis. Additionally, we include other projection methods to measure performance. Following this, we introduce a new EMAC variant which applies a differential spatial filter to the EMAC scheme, named EMAC-Reg. Standard models, including EMAC, require especially fine meshes with high Reynold\u27s numbers. This is problematic because the linear systems for 3D flows will be far too large and take an extraordinary amount of time to compute. EMAC-Reg not only performs better on a coarser mesh, but maintains conservation properties as well. Another topic in fluid flow computing that has been gaining recognition is reduced order models. This method uses experimental data to create new models of reduced computational complexity. We introduce the concept of consistency between a full order and a reduced order model, i.e., using the same numerical scheme for the full order and reduced order model. For inconsistency, one could use SKEW in the full order model and then EMAC for the reduced order model. We explore the repercussions of having inconsistency between these two models analytically and experimentally. To obtain a proper linear system from the Navier-Stokes equations, we must solve the nonlinear problem first. We will explore a method used to reduce iteration counts of nonlinear problems, known as Anderson acceleration. We will discuss how we implemented this using the finite element library deal.II \cite{dealII94}, measure the iteration counts and time, and compare against Newton and Picard iterations
A distributed-memory parallel algorithm for discretized integral equations using Julia
Boundary value problems involving elliptic PDEs such as the Laplace and the
Helmholtz equations are ubiquitous in physics and engineering. Many such
problems have alternative formulations as integral equations that are
mathematically more tractable than their PDE counterparts. However, the
integral equation formulation poses a challenge in solving the dense linear
systems that arise upon discretization. In cases where iterative methods
converge rapidly, existing methods that draw on fast summation schemes such as
the Fast Multipole Method are highly efficient and well established. More
recently, linear complexity direct solvers that sidestep convergence issues by
directly computing an invertible factorization have been developed. However,
storage and compute costs are high, which limits their ability to solve
large-scale problems in practice. In this work, we introduce a
distributed-memory parallel algorithm based on an existing direct solver named
``strong recursive skeletonization factorization.'' The analysis of its
parallel scalability applies generally to a class of existing methods that
exploit the so-called strong admissibility. Specifically, we apply low-rank
compression to certain off-diagonal matrix blocks in a way that minimizes data
movement. Given a compression tolerance, our method constructs an approximate
factorization of a discretized integral operator (dense matrix), which can be
used to solve linear systems efficiently in parallel. Compared to iterative
algorithms, our method is particularly suitable for problems involving
ill-conditioned matrices or multiple right-hand sides. Large-scale numerical
experiments are presented to demonstrate the performance of our implementation
using the Julia language
FETI-DP algorithms for 2D Biot model with discontinuous Galerkin discretization
Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) algorithms
are developed for a 2D Biot model. The model is formulated with mixed-finite
elements as a saddle-point problem. The displacement and the Darcy
flux flow are represented with piecewise continuous elements
and pore-pressure with piecewise constant elements, {\it i.e.},
overall three fields with a stabilizing term. We have tested the functionality
of FETI-DP with Dirichlet preconditioners. Numerical experiments show a
signature of scalability of the resulting parallel algorithm in the
compressible elasticity with permeable Darcy flow as well as almost
incompressible elasticity.Comment: Accepted to the 27th International Conference on Domain Decomposition
Methods (DD27), 8 pages. arXiv admin note: text overlap with arXiv:2211.1502
A hybrid probabilistic domain decomposition algorithm suited for very large-scale elliptic PDEs
State of the art domain decomposition algorithms for large-scale boundary
value problems (with degrees of freedom) suffer from bounded strong
scalability because they involve the synchronisation and communication of
workers inherent to iterative linear algebra. Here, we introduce PDDSparse, a
different approach to scientific supercomputing which relies on a "Feynman-Kac
formula for domain decomposition". Concretely, the interfacial values (only)
are determined by a stochastic, highly sparse linear system of size , whose coefficients are
constructed with Monte Carlo simulations-hence embarrassingly in parallel. In
addition to a wider scope for strong scalability in the deep supercomputing
regime, PDDSparse has built-in fault tolerance and is ideally suited for GPUs.
A proof of concept example with up to 1536 cores is discussed in detail
Robust Methods for Multiscale Coarse Approximations of Diffusion Models in Perforated Domains
For the Poisson equation posed in a domain containing a large number of
polygonal perforations, we propose a low-dimensional coarse approximation space
based on a coarse polygonal partitioning of the domain. Similarly to other
multiscale numerical methods, this coarse space is spanned by locally discrete
harmonic basis functions. Along the subdomain boundaries, the basis functions
are piecewise polynomial. The main contribution of this article is an error
estimate regarding the H1-projection over the coarse space which depends only
on the regularity of the solution over the edges of the coarse partitioning.
For a specific edge refinement procedure, the error analysis establishes
superconvergence of the method even if the true solution has a low general
regularity. Combined with domain decomposition (DD) methods, the coarse space
leads to an efficient two-level iterative linear solver which reaches the
fine-scale finite element error in few iterations. It also bodes well as a
preconditioner for Krylov methods and provides scalability with respect to the
number of subdomains. Numerical experiments showcase the increased precision of
the coarse approximation as well as the efficiency and scalability of the
coarse space as a component of a DD algorithm.Comment: 32 pages, 14 figures, submitted to Journal of Computational Physic
A scalable domain decomposition method for FEM discretizations of nonlocal equations of integrable and fractional type
Nonlocal models allow for the description of phenomena which cannot be
captured by classical partial differential equations. The availability of
efficient solvers is one of the main concerns for the use of nonlocal models in
real world engineering applications. We present a domain decomposition solver
that is inspired by substructuring methods for classical local equations. In
numerical experiments involving finite element discretizations of scalar and
vectorial nonlocal equations of integrable and fractional type, we observe
improvements in solution time of up to 14.6x compared to commonly used solver
strategies
Computational modelling and optimal control of interacting particle systems: connecting dynamic density functional theory and PDE-constrained optimization
Processes that can be described by systems of interacting particles are ubiquitous in nature, society, and industry, ranging from animal flocking, the spread of diseases, and formation of opinions to nano-filtration, brewing, and printing. In real-world applications it is often relevant to not only model a process of interest, but to also optimize it in order to achieve a desired outcome with minimal resources, such as time, money, or energy.
Mathematically, the dynamics of interacting particle systems can be described using Dynamic Density Functional Theory (DDFT). The resulting models are nonlinear, nonlocal partial differential equations (PDEs) that include convolution integral terms. Such terms also enter the naturally arising no-flux boundary conditions. Due to the nonlocal, nonlinear nature of such problems they are challenging both to analyse and solve numerically.
In order to optimize processes that are modelled by PDEs, one can apply tools from PDE-constrained optimization. The aim here is to drive a quantity of interest towards a target state by varying a control variable. This is constrained by a PDE describing the process of interest, in which the control enters as a model parameter. Such problems can be tackled by deriving and solving the (first-order) optimality system, which couples the PDE model with a second PDE and an algebraic equation. Solving such a system numerically is challenging, since large matrices arise in its discretization, for which efficient solution strategies have to be found. Most work in PDE-constrained optimization addresses problems in which the control is applied linearly, and which are constrained by local, often linear PDEs, since introducing nonlinearity significantly increases the complexity in both the analysis and numerical solution
of the optimization problem.
However, in order to optimize real-world processes described by nonlinear, nonlocal DDFT models, one has to develop an optimal control framework for such models. The aim is to drive the particles to some desired distribution by applying control either linearly, through a particle source, or bilinearly, though an advective field. The optimization process is constrained by the DDFT model that describes how the particles move under the influence of advection, diffusion, external forces, and particle–particle interactions. In order to tackle this, the (first-order) optimality system is derived, which, since it involves nonlinear (integro-)PDEs that are coupled nonlocally in space and time, is significantly harder than in the standard case. Novel numerical methods are developed, effectively combining pseudospectral methods and iterative solvers, to efficiently and accurately solve such a system.
In a next step this framework is extended so that it can capture and optimize industrially relevant processes, such as brewing and nano-filtration. In order to do so, extensions to both the DDFT model and the numerical method are made. Firstly, since industrial processes often involve tubes, funnels, channels, or tanks of various shapes, the PDE model itself, as well as the optimization problem, need to be solved on complicated domains. This is achieved by developing a novel spectral element approach that is compatible with both the PDE solver and the optimal control framework. Secondly, many industrial processes, such as nano-filtration, involve more than one type of particle. Therefore, the DDFT model is extended to describe multiple particle species. Finally, depending on the application of interest, additional physical effects need to be included in the model. In this thesis, to model sedimentation processes in brewing, the model is modified to capture volume exclusion effects
Reduced Order Modeling based Inexact FETI-DP solver for lattice structures
This paper addresses the overwhelming computational resources needed with
standard numerical approaches to simulate architected materials. Those
multiscale heterogeneous lattice structures gain intensive interest in
conjunction with the improvement of additive manufacturing as they offer, among
many others, excellent stiffness-to-weight ratios. We develop here a dedicated
HPC solver that benefits from the specific nature of the underlying problem in
order to drastically reduce the computational costs (memory and time) for the
full fine-scale analysis of lattice structures. Our purpose is to take
advantage of the natural domain decomposition into cells and, even more
importantly, of the geometrical and mechanical similarities among cells. Our
solver consists in a so-called inexact FETI-DP method where the local,
cell-wise operators and solutions are approximated with reduced order modeling
techniques. Instead of considering independently every cell, we end up with
only few principal local problems to solve and make use of the corresponding
principal cell-wise operators to approximate all the others. It results in a
scalable algorithm that saves numerous local factorizations. Our solver is
applied for the isogeometric analysis of lattices built by spline composition,
which offers the opportunity to compute the reduced basis with macro-scale
data, thereby making our method also multiscale and matrix-free. The solver is
tested against various 2D and 3D analyses. It shows major gains with respect to
black-box solvers; in particular, problems of several millions of degrees of
freedom can be solved with a simple computer within few minutes.Comment: 30 pages, 12 figures, 2 table
Parallel Selected Inversion for Space-Time Gaussian Markov Random Fields
Performing a Bayesian inference on large spatio-temporal models requires
extracting inverse elements of large sparse precision matrices for marginal
variances. Although direct matrix factorizations can be used for the inversion,
such methods fail to scale well for distributed problems when run on large
computing clusters. On the contrary, Krylov subspace methods for the selected
inversion have been gaining traction. We propose a parallel hybrid approach
based on domain decomposition, which extends the Rao-Blackwellized Monte Carlo
estimator for distributed precision matrices. Our approach exploits the
strength of Krylov subspace methods as global solvers and efficiency of direct
factorizations as base case solvers to compute the marginal variances using a
divide-and-conquer strategy. By introducing subdomain overlaps, one can achieve
a greater accuracy at an increased computational effort with little to no
additional communication. We demonstrate the speed improvements on both
simulated models and a massive US daily temperature data.Comment: 17 pages, 7 figure
BDDC preconditioners for virtual element approximations of the three-dimensional Stokes equations
The Virtual Element Method (VEM) is a novel family of numerical methods for
approximating partial differential equations on very general polygonal or
polyhedral computational grids. This work aims to propose a Balancing Domain
Decomposition by Constraints (BDDC) preconditioner that allows using the
conjugate gradient method to compute the solution of the saddle-point linear
systems arising from the VEM discretization of the three-dimensional Stokes
equations. We prove the scalability and quasi-optimality of the algorithm and
confirm the theoretical findings with parallel computations. Numerical results
with adaptively generated coarse spaces confirm the method's robustness in the
presence of large jumps in the viscosity and with high-order VEM
discretizations
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