334 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
Adaptive dynamical networks
It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical networks
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
Development of a Moving Front Kinetic Monte Carlo Algorithm to Simulate Moving Interface Systems
Moving interfaces play vital and crucial roles in a wide variety of different natural, technological, and industrial processes, including solids dissolution, capillary action, sessile droplet spreading, and superhydrophobicity. In each of these systems, the fundamental process behaviour is entirely dependent on the interface and on the underlying physics governing its movement. As a result, there is significant interest in studying and developing models to capture the behaviour of these moving interface systems over a wide variety of different applications. However, the simulation techniques used to model moving interfaces are limited in their application, as the molecular-level models are unable to simulate interface behaviour over large spatial and temporal scales, whereas the large-scale modeling techniques cannot account for the nanoscale processes that govern the interface behaviour or the molecular-scale fluctuations and deviations in the interface. Furthermore, methods developed to bridge the gap between the two scales are prone to error-induced force imbalances at the interface that can result in fictitious behaviour.
In order to overcome these challenges, this study developed a novel kinetic Monte Carlo (kMC)-based modelling technique referred to as Moving Front kMC (MFkMC) to adequately and efficiently capture the molecular-scale events and forces governing the moving interface behaviour over large length and timescales. This framework was designed to capture the movement of transiently-varying interfaces in a kinetic-like manner so that its movement can be described using Monte Carlo sampling. The MFkMC algorithm accomplishes this task by evaluating the behaviour of the interfacial molecules and assigning kinetic Monte Carlo-style rate equations that describe the transition probability that a molecule would advance into the neighbouring phase, displacing an interfacial molecule from the opposing phase and thus changing the interface. The proposed algorithm was subsequently used to capture the moving interface behaviour within crystal dissolution, capillary rise, and sessile droplet spreading on both smooth and superhydrophobic surfaces. The individual system models for each application were used to analyze the behaviour within each application and to tackle challenges within each field.
The MFkMC modelling method was initially used to capture crystal dissolution for applications in pharmaceutical drug delivery. The developed model was designed to predict the dissolution of a wide variety of crystalline minerals, regardless of their composition and crystal structure. The MFkMC approach was compared against a standard kMC model of the same system to validate the MFkMC approach and highlight its advantages and limitations. The proposed framework was used to explore ways of enhancing crystal dissolution processes by assessing the variability from environmental uncertainties and by performing robust optimization to improve the dissolution performance. The approach was used to simulate calcium carbonate dissolution within the human gastrointestinal system. Polynomial chaos expansions (PCEs) were used to propagate the parametric uncertainty through the kMC model. Robust optimization was subsequently performed to determine the crystal design parameters that achieve target dissolution specifications using low-order PCE coefficient models (LPCMs). The results showcased the applicability of the kMC crystal dissolution model and the need to account for dissolution uncertainty within key biological applications.
The MFkMC approach was additionally used to capture capillary rise in cavities of different shapes. The proposed model was adapted to capture the movement of a fluid-fluid interface, such as the moving interface present in capillary action studies, using kMC type approaches based on the forces acting locally upon the interface. The proposed force balance-based MFkMC (FB-MFkMC) expressions were subsequently coupled with capillary action force balance equations to capture capillary rise within any axisymmetric cavity. The developed model was validated against known analytical models that capture capillary rise dynamics in perfect cylinders. Furthermore, the resulting multiscale model was used to analyze capillary rise within axisymmetric cavities of irregular shape and in cylinders subject to surface roughness. These studies highlighted that the FB-MFkMC algorithm can capture the macroscale behaviour of a system subject to molecular-level irregularities such as surface roughness. Furthermore, they highlighted that phenomena such as roughness can significantly affect moving interface behaviour and highlight the need to accommodate for these phenomena.
MFkMC was furthermore extended to capture sessile droplet spreading on a smooth surface. The developed approach adapted the capillary action FB-MFkMC model to capture the spreading behaviour of a droplet based on the force balance acting upon the droplet interface, which was developed using analytical inertial and capillary expressions from the literature. This study furthermore derived a new semi-empirical expression to depict the viscous damping force acting on the droplet. The developed viscous force term depends on a fitted parameter c, whose value was observed to vary solely depending on the droplet liquid as captured predominantly by the droplet Ohnesorge number. The proposed approach was subsequently validated using data obtained both from conducted experiments and from the literature to support the robustness of the framework. The predictive capabilities of the developed model were further inspected to provide insights on the sessile droplet system behaviour.
The developed FB-MFkMC model was additionally modified to capture sessile droplet spreading on pillared superhydrophobic surfaces (SHSs). These adjustments included developing the Periodic Unit (PU) method of capturing periodic SHS pillar arrays and accommodating for the changes necessary to capture the droplet spreading behaviour across the gaps between the pillars (i.e., Cassie mode wetting). The proposed SHS-based FB-MFkMC (SHS-MFkMC) model was furthermore adapted to accommodate for spontaneous Cassie-to-Wenzel (C2W) droplet transitions on the solid surface. The capabilities of the full SHS-MFkMC model to capture both radial sessile droplet spread and spontaneous C2W transitions were compared to experimental results from within the literature. Furthermore, a sensitivity analysis was conducted to assess the effects of the various system parameters on the model performance and compare them with the expected system results
Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra
We present an extension of the summation-by-parts (SBP) framework to
tensor-product spectral-element operators in collapsed coordinates. The
proposed approach enables the construction of provably stable discretizations
of arbitrary order which combine the geometric flexibility of unstructured
triangular and tetrahedral meshes with the efficiency of sum-factorization
algorithms. Specifically, a methodology is developed for constructing
triangular and tetrahedral spectral-element operators of any order which
possess the SBP property (i.e. satisfying a discrete analogue of integration by
parts) as well as a tensor-product decomposition. Such operators are then
employed within the context of discontinuous spectral-element methods based on
nodal expansions collocated at the tensor-product quadrature nodes as well as
modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter
approach resolving the time step limitation associated with the singularity of
the collapsed coordinate transformation. Energy-stable formulations for
curvilinear meshes are obtained using a skew-symmetric splitting of the metric
terms, and a weight-adjusted approximation is used to efficiently invert the
curvilinear modal mass matrix. The proposed schemes are compared to those using
non-tensorial multidimensional SBP operators, and are found to offer comparable
accuracy to such schemes in the context of smooth linear advection problems on
curved meshes, but at a reduced computational cost for higher polynomial
degrees.Comment: 26 pages, 5 figure
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