1,042 research outputs found
A numerical methodology for enforcing maximum principles and the non-negative constraint for transient diffusion equations
Transient diffusion equations arise in many branches of engineering and
applied sciences (e.g., heat transfer and mass transfer), and are parabolic
partial differential equations. It is well-known that, under certain
assumptions on the input data, these equations satisfy important mathematical
properties like maximum principles and the non-negative constraint, which have
implications in mathematical modeling. However, existing numerical formulations
for these types of equations do not, in general, satisfy maximum principles and
the non-negative constraint. In this paper, we present a methodology for
enforcing maximum principles and the non-negative constraint for transient
anisotropic diffusion equation. The method of horizontal lines (also known as
the Rothe method) is applied in which the time is discretized first. This
results in solving steady anisotropic diffusion equation with decay equation at
every discrete time level. The proposed methodology for transient anisotropic
diffusion equation will satisfy maximum principles and the non-negative
constraint on general computational grids, and with no additional restrictions
on the time step. We illustrate the performance and accuracy of the proposed
formulation using representative numerical examples. We also perform numerical
convergence of the proposed methodology. For comparison, we also present the
results from the standard single-field semi-discrete formulation and the
results from a popular software package, which all will violate maximum
principles and the non-negative constraint
A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity
Mathematical models for flow through porous media typically enjoy the
so-called maximum principles, which place bounds on the pressure field. It is
highly desirable to preserve these bounds on the pressure field in predictive
numerical simulations, that is, one needs to satisfy discrete maximum
principles (DMP). Unfortunately, many of the existing formulations for flow
through porous media models do not satisfy DMP. This paper presents a robust,
scalable numerical formulation based on variational inequalities (VI), to model
non-linear flows through heterogeneous, anisotropic porous media without
violating DMP. VI is an optimization technique that places bounds on the
numerical solutions of partial differential equations. To crystallize the
ideas, a modification to Darcy equations by taking into account
pressure-dependent viscosity will be discretized using the lowest-order
Raviart-Thomas (RT0) and Variational Multi-scale (VMS) finite element
formulations. It will be shown that these formulations violate DMP, and, in
fact, these violations increase with an increase in anisotropy. It will be
shown that the proposed VI-based formulation provides a viable route to enforce
DMP. Moreover, it will be shown that the proposed formulation is scalable, and
can work with any numerical discretization and weak form. Parallel scalability
on modern computational platforms will be illustrated through strong-scaling
studies, which will prove the efficiency of the proposed formulation in a
parallel setting. Algorithmic scalability as the problem size is scaled up will
be demonstrated through novel static-scaling studies. The performed
static-scaling studies can serve as a guide for users to be able to select an
appropriate discretization for a given problem size
On mesh restrictions to satisfy comparison principles, maximum principles, and the non-negative constraint: Recent developments and new results
This paper concerns with mesh restrictions that are needed to satisfy several
important mathematical properties -- maximum principles, comparison principles,
and the non-negative constraint -- for a general linear second-order elliptic
partial differential equation. We critically review some recent developments in
the field of discrete maximum principles, derive new results, and discuss some
possible future research directions in this area. In particular, we derive
restrictions for a three-node triangular (T3) element and a four-node
quadrilateral (Q4) element to satisfy comparison principles, maximum
principles, and the non-negative constraint under the standard single-field
Galerkin formulation. Analysis is restricted to uniformly elliptic linear
differential operators in divergence form with Dirichlet boundary conditions
specified on the entire boundary of the domain. Various versions of maximum
principles and comparison principles are discussed in both continuous and
discrete settings. In the literature, it is well-known that an acute-angled
triangle is sufficient to satisfy the discrete weak maximum principle for pure
isotropic diffusion. An iterative algorithm is developed to construct
simplicial meshes that preserves discrete maximum principles using existing
open source mesh generators. Various numerical examples based on different
types of triangulations are presented to show the pros and cons of placing
restrictions on a computational mesh. We also quantify local and global mass
conservation errors using representative numerical examples, and illustrate the
performance of metric-based meshes with respect to mass conservation
A monolithic multi-time-step computational framework for first-order transient systems with disparate scales
Developing robust simulation tools for problems involving multiple
mathematical scales has been a subject of great interest in computational
mathematics and engineering. A desirable feature to have in a numerical
formulation for multiscale transient problems is to be able to employ different
time-steps (multi-time-step coupling), and different time integrators and
different numerical formulations (mixed methods) in different regions of the
computational domain. We present two new monolithic multi-time-step mixed
coupling methods for first-order transient systems. We shall employ unsteady
advection-diffusion-reaction equation with linear decay as the model problem,
which offers several unique challenges in terms of non-self-adjoint spatial
operator and rich features in the solutions. We shall employ the dual Schur
domain decomposition technique to handle the decomposition of domain into
subdomains. Two different methods of enforcing compatibility along the
subdomain interface will be used in the time discrete setting. A systematic
theoretical analysis (which includes numerical stability, influence of
perturbations, bounds on drift along the subdomain interface) will be
performed. The first coupling method ensures that there is no drift along the
subdomain interface but does not facilitate explicit/implicit coupling. The
second coupling method allows explicit/implicit coupling with controlled (but
non-zero) drift in the solution along the subdomain interface. Several
canonical problems will be solved to numerically verify the theoretical
predictions, and to illustrate the overall performance of the proposed coupling
methods. Finally, we shall illustrate the robustness of the proposed coupling
methods using a multi-time-step transient simulation of a fast bimolecular
advective-diffusive-reactive system
A deep learning modeling framework to capture mixing patterns in reactive-transport systems
Prediction and control of chemical mixing are vital for many scientific areas
such as subsurface reactive transport, climate modeling, combustion,
epidemiology, and pharmacology. Due to the complex nature of mixing in
heterogeneous and anisotropic media, the mathematical models related to this
phenomenon are not analytically tractable. Numerical simulations often provide
a viable route to predict chemical mixing accurately. However, contemporary
modeling approaches for mixing cannot utilize available spatial-temporal data
to improve the accuracy of the future prediction and can be compute-intensive,
especially when the spatial domain is large and for long-term temporal
predictions. To address this knowledge gap, we will present in this paper a
deep-learning (DL) modeling framework applied to predict the progress of
chemical mixing under fast bimolecular reactions. This framework uses
convolutional neural networks (CNN) for capturing spatial patterns and long
short-term memory (LSTM) networks for forecasting temporal variations in
mixing. By careful design of the framework -- placement of non-negative
constraint on the weights of the CNN and the selection of activation function,
the framework ensures non-negativity of the chemical species at all spatial
points and for all times. Our DL-based framework is fast, accurate, and
requires minimal data for training
Physics-Informed Machine Learning Models for Predicting the Progress of Reactive-Mixing
This paper presents a physics-informed machine learning (ML) framework to
construct reduced-order models (ROMs) for reactive-transport quantities of
interest (QoIs) based on high-fidelity numerical simulations. QoIs include
species decay, product yield, and degree of mixing. The ROMs for QoIs are
applied to quantify and understand how the chemical species evolve over time.
First, high-resolution datasets for constructing ROMs are generated by solving
anisotropic reaction-diffusion equations using a non-negative finite element
formulation for different input parameters. Non-negative finite element
formulation ensures that the species concentration is non-negative (which is
needed for computing QoIs) on coarse computational grids even under high
anisotropy. The reactive-mixing model input parameters are a time-scale
associated with flipping of velocity, a spatial-scale controlling small/large
vortex structures of velocity, a perturbation parameter of the vortex-based
velocity, anisotropic dispersion strength/contrast, and molecular diffusion.
Second, random forests, F-test, and mutual information criterion are used to
evaluate the importance of model inputs/features with respect to QoIs. Third,
Support Vector Machines (SVM) and Support Vector Regression (SVR) are used to
construct ROMs based on the model inputs. Then, SVR-ROMs are used to predict
scaling of QoIs. Qualitatively, SVR-ROMs are able to describe the trends
observed in the scaling law associated with QoIs. Fourth, the scaling law's
exponent dependence on model inputs/features are evaluated using -means
clustering. Finally, in terms of the computational cost, the proposed SVM-ROMs
and SVR-ROMs are times faster than running a high-fidelity
numerical simulation for evaluating QoIs
Modeling flow in porous media with double porosity/permeability: A stabilized mixed formulation, error analysis, and numerical solutions
The flow of incompressible fluids through porous media plays a crucial role
in many technological applications such as enhanced oil recovery and geological
carbon-dioxide sequestration. The flow within numerous natural and synthetic
porous materials that contain multiple scales of pores cannot be adequately
described by the classical Darcy equations. It is for this reason that
mathematical models for fluid flow in media with multiple scales of pores have
been proposed in the literature. However, these models are analytically
intractable for realistic problems. In this paper, a stabilized mixed
four-field finite element formulation is presented to study the flow of an
incompressible fluid in porous media exhibiting double porosity/permeability.
The stabilization terms and the stabilization parameters are derived in a
mathematically and thermodynamically consistent manner, and the computationally
convenient equal-order interpolation of all the field variables is shown to be
stable. A systematic error analysis is performed on the resulting stabilized
weak formulation. Representative problems, patch tests and numerical
convergence analyses are performed to illustrate the performance and
convergence behavior of the proposed mixed formulation in the discrete setting.
The accuracy of numerical solutions is assessed using the mathematical
properties satisfied by the solutions of this double porosity/permeability
model. Moreover, it is shown that the proposed framework can perform well under
transient conditions and that it can capture well-known instabilities such as
viscous fingering
Modelling fluid deformable surfaces with an emphasis on biological interfaces
Fluid deformable surfaces are ubiquitous in cell and tissue biology,
including lipid bilayers, the actomyosin cortex, or epithelial cell sheets.
These interfaces exhibit a complex interplay between elasticity, low Reynolds
number interfacial hydrodynamics, chemistry, and geometry, and govern important
biological processes such as cellular traffic, division, migration, or tissue
morphogenesis. To address the modelling challenges posed by this class of
problems, in which interfacial phenomena tightly interact with the shape and
dynamics of the surface, we develop a general continuum mechanics and
computational framework for fluid deformable surfaces. The dual solid-fluid
nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian
descriptions of deforming bodies. Here, we extend the notion of Arbitrarily
Lagrangian-Eulerian (ALE) formulations, well-established for bulk media, to
deforming surfaces. To systematically develop models for fluid deformable
surfaces, which consistently treat all couplings between fields and geometry,
we follow a nonlinear Onsager formalism according to which the dynamics
minimize a Rayleighian functional where dissipation, power input and energy
release rate compete. Finally, we propose new computational methods, which
build on Onsager's formalism and our ALE formulation, to deal with the
resulting stiff system of higher-order of partial differential equations. We
apply our theoretical and computational methodology to classical models for
lipid bilayers and the cell cortex. The methods developed here allow us to
formulate/simulate these models for the first time in their full
three-dimensional generality, accounting for finite curvatures and finite shape
changes.Comment: 58 pages, 21 figures, submitted for publication to the Journal of
Fluid Mechanic
Inference, prediction and optimization of non-pharmaceutical interventions using compartment models: the PyRoss library
PyRoss is an open-source Python library that offers an integrated platform
for inference, prediction and optimisation of NPIs in age- and
contact-structured epidemiological compartment models. This report outlines the
rationale and functionality of the PyRoss library, with various illustrations
and examples focusing on well-mixed, age-structured populations. The PyRoss
library supports arbitrary structured models formulated stochastically (as
master equations) or deterministically (as ODEs) and allows mid-run
transitioning from one to the other. By supporting additional compartmental
subdivision ad libitum, PyRoss can emulate time-since-infection models and
allows medical stages such as hospitalization or quarantine to be modelled and
forecast. The PyRoss library enables fitting to epidemiological data, as
available, using Bayesian parameter inference, so that competing models can be
weighed by their evidence. PyRoss allows fully Bayesian forecasts of the impact
of idealized NPIs by convolving uncertainties arising from epidemiological
data, model choice, parameters, and intrinsic stochasticity. Algorithms to
optimize time-dependent NPI scenarios against user-defined cost functions are
included. PyRoss's current age-structured compartment framework for well-mixed
populations will in future reports be extended to include compartments
structured by location, occupation, use of travel networks and other attributes
relevant to assessing disease spread and the impact of NPIs. We argue that such
compartment models, by allowing social data of arbitrary granularity to be
combined with Bayesian parameter estimation for poorly-known disease variables,
could enable more powerful and robust prediction than other approaches to
detailed epidemic modelling. We invite others to use the PyRoss library for
research to address today's COVID-19 crisis, and to plan for future pandemics.Comment: Code and updates at https://github.com/rajeshrinet/pyross 75 pages,
14 figures, and 1 tabl
Computational Methods for Multi-Scale Temporal Problems: Algorithms, Analysis, and Numerical Experiments
A major challenge in numerical simulation of most natural phenomena is the presence of disparate temporal and spatial scales. Capturing all the fine features can be computationally prohibitive. Hence, development of efficient and accurate multi-scale numerical algorithms has gained immense attention from engineers and scientists. Typically, a single numerical method cannot efficiently capture all the aforementioned features. Due to the assumptions made in construction of numerical methods and mathematical models, the range of applicability to various length and time-scales is often limited. A direction in resolving this issue is to apply different numerical methods in different regions of the computational domain. This strategy enables computation of necessary details as desired by the user. In this work, we propose numerical methodologies based on domain partitioning techniques that allow different time-steps and time-integrators in different regions of the computational domain. The first problem of interest is elastodynamics, which can pose various temporal scales in impact, contact and wave propagation problems. A monolithic (strong) coupling algorithm based on non-overlapping domain partitioning is proposed. The proposed algorithm is based on the theory of differential/algebraic equations and its numerical stability, energy conservation and accuracy is studied in detail. Following these findings, we extend this algorithm to advection-diffusion-reaction problems. The proposed algorithm proves useful especially in cases where the relative strength of the involved processes changes dramatically with respect to spatial coordinates. Numerical stability and accuracy of this method is studied and its application to fast bimolecular chemical reactions is showcased. Further on, we confine our attention to single and multiple-relaxation-time lattice Boltzmann methods for the advection-diffusion equation and study their performance in preserving the maximum principle and the non-negative constraint. Finally, a computational framework based on overlapping domain decomposition methods is proposed. This framework is designed for advection-diffusion problems and allows coupling of the finite element method and lattice Boltzmann methods with different time-steps and grid sizes. Additionally, a new method for enforcing the Dirichlet and Neumann boundary conditions on the numerical solution from the lattice Boltzmann method is proposed. This method is based on maximization of entropy and ensures non-negativity of the discrete distributions on the boundary of the domain. We study the performance of this framework through numerical experiments and showcase its application to fast and equilibrium chemical reactions.Civil and Environmental Engineering, Department o
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