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Dylan Copeland

Texas A&M University, Mathematics, Alumnus

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Papers by Dylan Copeland

A least-squares method for axisymmetric div–curl systems

Numerical Linear Algebra With Applications, 2006

We present a negative-norm least-squares method for axisymmetric divcurl systems arising from Max... more We present a negative-norm least-squares method for axisymmetric divcurl systems arising from Maxwell's equations for electrostatics and magnetostatics in three dimensions. The method approximates the solution in a two-dimensional meridian plane. To achieve this dimension reduction, we must work with weighted spaces in cylindrical coordinates. In this setting, a stable pair of approximation spaces is developed and analyzed. We also report the results of some numerical experiments, which demonstrate a quasi-optimal convergence rate and a robustness with respect to the domain and coefficients.

Negative-norm least-squares methods for axisymmetric Maxwell equations

A Unified Leak-off and Flow-back Model for Fractured Reservoirs

Domain decomposition solvers for nonlinear multiharmonic finite element equations

Journal of Numerical Mathematics, 2010

In many practical applications, for instance, in computational electromagnetics, the excitation i... more In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure.

Boundary-Element-Based Finite Element Methods for Helmholtz and Maxwell Equations on General Polyhedral Meshes

World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, Sep 20, 2009

We present new finite element methods for Helmholtz and Maxwell equations for general three-dimen... more We present new finite element methods for Helmholtz and Maxwell equations for general three-dimensional polyhedral meshes, based on domain decomposition with boundary elements on the surfaces of the polyhedral volume elements. The methods use the lowest-order polynomial spaces and produce sparse, symmetric linear systems despite the use of boundary elements. Moreover, piecewise constant coefficients are admissible. The resulting approximation on the element surfaces can be extended throughout the domain via representation formulas. Numerical experiments confirm that the convergence behavior on tetrahedral meshes is comparable to that of standard finite element methods, and equally good performance is attained on more general meshes.

Numerical Modeling of Proppant Transport in Complex Hydraulic Fracture Propagation

Hydraulic fracturing is a stimulation technique used in unconventional reservoirs to generate suf... more Hydraulic fracturing is a stimulation technique used in unconventional reservoirs to generate sufficient reservoir contact for viable production. During stimulation, proppant is pumped into the system to keep fractures open when, at a later stage, injection ceases and fluid is recovered as fractures close. Thus, controlling proppant distribution in the fracture is important to improve well productivity. Several factors influence proppant placement and settling such as proppant size, proppant density and carrier-fluid rheology. However, accurately predicting, and consequently optimizing, proppant placement in complex fracture systems remains an industry challenge. The objective of this study is to use advanced modeling techniques of proppant transport inside complex fracture systems using a 3D simulator that couples geomechan ics, fracture mechanics, fluid behavior and proppant transport to assess the optimization of stimulation treatments. In this study proppant transport models incorporate the interaction between hydraulic fractures (HF) and natural fractures (NF) of various sizes and orientations. Treatments were simulated in systematic variations of proppant size, proppant density and fluid viscosity. Fluid rheology, proppant size and density are all major parameters affecting the proppant settling rate. In more complex systems, proppant placement is strongly influenced by the location, size, and orientation of natural fractures. Based on the size of proppant and viscosity of the injected fluid, proppant may or may not enter the natural fractures.

Variable Precision Computing

This report summarizes the activities and major accomplishments of the Variable Precision Computi... more This report summarizes the activities and major accomplishments of the Variable Precision Computing Strategic Initiative project. The overarching goal of this project was to initiate and promote a new paradigm in High Performance Computing (HPC) that would fundamentally change how we represent and make use of representations of real numbers in finite precision and that would establish Lawrence Livermore National Laboratory (LLNL) as a leader in precision-related research for HPC. We pursued three integrated and concurrent research thrusts: new, more dynamic data representations to address data motion and capacity limitations; improved mixed precision algorithms to accelerate computation and to leverage new hardware; and new tools to help developers reason about precision and to automate code transformations. Within these thrust areas, we made significant advances in the use of compressed array data types in numerical calculations, in the theoretical justification, in improved performance, and in the prospects of hardware implementation; in the development of metrics to significantly reduce the storage needs of molecular dynamics simulations; in the development and analysis of efficient mixed-precision algorithms for time-dependent partial differential equations and for orthogonal factorization; and in the capability of tools that can analyze sensitivity of computed results to choices in precision and that can make automated code transformations based on such analysis. We established that there are definite opportunities to realize significant improvements in HPC above and beyond the 2x limits of traditional mixed precision using the technologies we have developed. The project produced numerous papers, presentations, and posters in important venues as well as open-source software contributions to the community, and many of the new capabilities have initiated or are being leveraged in ongoing projects.

Domain Decomposition Solvers for Frequency-Domain Finite Element Equations

Springer eBooks, Oct 5, 2010

The paper is devoted to fast iterative solvers for frequency-domain finite element equations appr... more The paper is devoted to fast iterative solvers for frequency-domain finite element equations approximating linear and nonlinear parabolic initial boundary value problems with time-harmonic excitations. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple linear elliptic system for the amplitudes belonging to the sine-and to the cosine-excitation or a large nonlinear elliptic system for the Fourier coefficients in the linear and nonlinear case, respectively. The fast solution of the corresponding linear and nonlinear system of finite element equations is crucial for the competitiveness of this method.

Local Lagrangian reduced-order modeling for Rayleigh-Taylor instability by solution manifold decomposition

arXiv (Cornell University), Jan 18, 2022

Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various ... more Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various disciplines of science and engineering, including astrophyics, atmospheric sciences and climate, geophysics, and fusion energy. Analytical methods cannot be applied to explain the long-time behavior of Rayleigh-Taylor instability, and therefore numerical simulation of the full problem is required. However, in order to capture the growth of amplitude of perturbations accurately, both the spatial and temporal discretization need to be extremely fine for traditional numerical methods, and the long-time simulation may become prohibitively expensive. In this paper, we propose efficient reduced order model techniques to accelerate the simulation of Rayleigh-Taylor instability in compressible gas dynamics. We introduce a general framework for decomposing the solution manifold to construct the temporal domain partition and temporally-local reduced order model construction with varying Atwood number. We propose two practical approaches in this framework, namely decomposition by using physical time and by penetration distance respectively. Numerical results are presented to examine the performance of the proposed approaches.

Reduced order models for Lagransian hydrodynamics

arXiv (Cornell University), Apr 23, 2021

As a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial ... more As a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial setting, Lagrangian hydrodynamics is characterized by moving meshes, advection-dominated solutions, and moving shock fronts with sharp gradients. These challenges hinder the existing projection-based model reduction schemes from being practical. We develop several variations of projection-based reduced order model techniques for Lagrangian hydrodynamics by introducing three different reduced bases for position, velocity, and energy fields. A time-windowing approach is also developed to address the challenge imposed by the advection-dominated solutions. Lagrangian hydrodynamics is formulated as a nonlinear problem, which requires a proper hyper-reduction technique. Therefore, we apply the over-sampling DEIM and SNS approaches to reduce the complexity due to the nonlinear terms. Finally, we also present both a posteriori and a priori error bounds associated with our reduced order model. We compare the performance of the spatial and time-windowing reduced order modeling approaches in terms of accuracy and speed-up with respect to the corresponding full order model for several numerical examples, namely Sedov blast, Gresho vortices, Taylor-Green vortices, and triple-point problems.

Local Lagrangian reduced-order modeling for the Rayleigh-Taylor instability by solution manifold decomposition

Journal of Computational Physics, 2023

Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various ... more Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various disciplines of science and engineering, including astrophyics, atmospheric sciences and climate, geophysics, and fusion energy. Analytical methods cannot be applied to explain the long-time behavior of Rayleigh-Taylor instability, and therefore numerical simulation of the full problem is required. However, in order to capture the growth of amplitude of perturbations accurately, both the spatial and temporal discretization need to be extremely fine for traditional numerical methods, and the long-time simulation may become prohibitively expensive. In this paper, we propose efficient reduced order model techniques to accelerate the simulation of Rayleigh-Taylor instability in compressible gas dynamics. We introduce a general framework for decomposing the solution manifold to construct the temporal domain partition and temporally-local reduced order model construction with varying Atwood number. We propose two practical approaches in this framework, namely decomposition by using physical time and by penetration distance respectively. Numerical results are presented to examine the performance of the proposed approaches.

S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Models

arXiv (Cornell University), Mar 29, 2022

While projection-based reduced order models can reduce the dimension of full order solutions, the... more While projection-based reduced order models can reduce the dimension of full order solutions, the resulting reduced models may still contain terms that scale with the full order dimension. Hyper-reduction techniques are sampling-based methods that further reduce this computational complexity by approximating such terms with a much smaller dimension. The goal of this work is to introduce a points selection algorithm developed by Shin and Xiu [SIAM J. Sci. Comput., 38 (2016), pp. A385-A411], as a hyper-reduction method. The selection algorithm is originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity S that measures both the column orthogonality and the determinant, we refer to the algorithm as S-OPT. Numerical examples are provided to demonstrate the performance of S-OPT and to compare its performance with an over-sampled Discrete Empirical Interpolation (DEIM) algorithm. We found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy for a given number of indices.

OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Oct 17, 2019

libROM is a collection of C++ classes that compute reduced order models and hyperreduced order mo... more libROM is a collection of C++ classes that compute reduced order models and hyperreduced order models for systems of ordinary differential equations. libROM includes parallel, adaptive methods for proper orthogonal decomposition, and parallel, non-adaptive methods for hyperreduction using the discrete empirical interpolation method.

Energy conserving quadrature based dimensionality reduction for nonlinear hydrodynamics problems

Expanded Mixed Multiscale Finite Element Methods and Their Applications for Flows in Porous Media

Multiscale Modeling & Simulation, 2012

We develop a family of expanded mixed Multiscale Finite Element Methods (Ms-FEMs) and their hybri... more We develop a family of expanded mixed Multiscale Finite Element Methods (Ms-FEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed Multiscale Finite Element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity, and Lagrange multipliers. We use multiscale basis functions for both the velocity and the gradient of pressure. In the expanded mixed MsFEM framework, we consider both separable and non-separable spatial scales. Specifically, we analyze the methods in three categories: periodic separable scales, G−convergent separable scales, and a continuum of scales. When there is no scale separation, using some global information can significantly improve the accuracy of the expanded mixed MsFEMs. We present a rigorous convergence analysis of these methods that includes both conforming and nonconforming formulations. Numerical results are presented for various multiscale models of flow in porous media with shale barriers that illustrate the efficacy of the proposed family of expanded mixed MsFEMs.

Linearized Coulomb Collision Operator for Simulation of Interpenetrating Plasma Streams

IEEE Transactions on Plasma Science, 2019

We present an extension of a linearized Coulomb collision operator, previously used in several Eu... more We present an extension of a linearized Coulomb collision operator, previously used in several Eulerian kinetic codes for like-species collisions and unlike-species collisions in the case where the backgrounds about which the linearization is made all are in collisional equilibrium, to the situation of interpenetrating plasma streams. In the latter case, the backgrounds can not be taken to be in equilibrium and a significant generalization is required. Our development is targeted towards the Eulerian kinetic plasma code LOKI, which evolves the Vlasov-Poisson or Vlasov-Maxwell system in a Cartesian "2+2dimensional" phase space. The extended operator has been implemented in a test code, and results of both quantitative verification and qualitative "realizability" tests are presented.

Helmholtz and Maxwell equations on general

polyhedral meshes

S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Models

Cornell University - arXiv, Mar 29, 2022

While projection-based reduced order models can reduce the dimension of full order solutions, the... more While projection-based reduced order models can reduce the dimension of full order solutions, the resulting reduced models may still contain terms that scale with the full order dimension. Hyper-reduction techniques are sampling-based methods that further reduce this computational complexity by approximating such terms with a much smaller dimension. The goal of this work is to introduce a points selection algorithm developed by Shin and Xiu [SIAM J. Sci. Comput., 38 (2016), pp. A385-A411], as a hyper-reduction method. The selection algorithm is originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity S that measures both the column orthogonality and the determinant, we refer to the algorithm as S-OPT. Numerical examples are provided to demonstrate the performance of S-OPT and to compare its performance with an over-sampled Discrete Empirical Interpolation (DEIM) algorithm. We found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy for a given number of indices.

$Research paper thumbnail of Simulation de fuite de fluide et reflux dans une région souterraine fracturée$

Simulation de fuite de fluide et reflux dans une région souterraine fracturée

Selon certains aspects, un ensemble d'equations d'ecoulement de regulation peut etre defi... more Selon certains aspects, un ensemble d'equations d'ecoulement de regulation peut etre defini pour un fluide de systeme de puits representant un modele d'ecoulement unidimensionnel dans une region souterraine fracturee. Un premier sous-ensemble des equations d'ecoulement de regulation peut representer un ecoulement dans une fracture, et un second sous-ensemble des equations d'ecoulement de regulation peut representer un ecoulement dans un milieu de reservoir adjacent a la fracture. Un ensemble reduit d'equations d'ecoulement de regulation peut etre genere par l'elimination du second sous-ensemble de l'ensemble d'equations d'ecoulement de regulation sur la base d'un accouplement fluidique entre la fracture et le milieu de reservoir. L'ecoulement de fluide de systeme de puits dans la region souterraine fracturee peut etre simule sur la base de l'ensemble reduit d'equations d'ecoulement de regulation.

libROM is a collection of C++ classes that compute reduced order models and hyperreduced order mo... more libROM is a collection of C++ classes that compute reduced order models and hyperreduced order models for systems of ordinary differential equations. libROM includes parallel, adaptive methods for proper orthogonal decomposition, and parallel, non-adaptive methods for hyperreduction using the discrete empirical interpolation method.

A least-squares method for axisymmetric div–curl systems

Numerical Linear Algebra With Applications, 2006

We present a negative-norm least-squares method for axisymmetric divcurl systems arising from Max... more We present a negative-norm least-squares method for axisymmetric divcurl systems arising from Maxwell's equations for electrostatics and magnetostatics in three dimensions. The method approximates the solution in a two-dimensional meridian plane. To achieve this dimension reduction, we must work with weighted spaces in cylindrical coordinates. In this setting, a stable pair of approximation spaces is developed and analyzed. We also report the results of some numerical experiments, which demonstrate a quasi-optimal convergence rate and a robustness with respect to the domain and coefficients.

Negative-norm least-squares methods for axisymmetric Maxwell equations

A Unified Leak-off and Flow-back Model for Fractured Reservoirs

Domain decomposition solvers for nonlinear multiharmonic finite element equations

Journal of Numerical Mathematics, 2010

In many practical applications, for instance, in computational electromagnetics, the excitation i... more In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure.

Boundary-Element-Based Finite Element Methods for Helmholtz and Maxwell Equations on General Polyhedral Meshes

World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, Sep 20, 2009

We present new finite element methods for Helmholtz and Maxwell equations for general three-dimen... more We present new finite element methods for Helmholtz and Maxwell equations for general three-dimensional polyhedral meshes, based on domain decomposition with boundary elements on the surfaces of the polyhedral volume elements. The methods use the lowest-order polynomial spaces and produce sparse, symmetric linear systems despite the use of boundary elements. Moreover, piecewise constant coefficients are admissible. The resulting approximation on the element surfaces can be extended throughout the domain via representation formulas. Numerical experiments confirm that the convergence behavior on tetrahedral meshes is comparable to that of standard finite element methods, and equally good performance is attained on more general meshes.

Numerical Modeling of Proppant Transport in Complex Hydraulic Fracture Propagation

Hydraulic fracturing is a stimulation technique used in unconventional reservoirs to generate suf... more Hydraulic fracturing is a stimulation technique used in unconventional reservoirs to generate sufficient reservoir contact for viable production. During stimulation, proppant is pumped into the system to keep fractures open when, at a later stage, injection ceases and fluid is recovered as fractures close. Thus, controlling proppant distribution in the fracture is important to improve well productivity. Several factors influence proppant placement and settling such as proppant size, proppant density and carrier-fluid rheology. However, accurately predicting, and consequently optimizing, proppant placement in complex fracture systems remains an industry challenge. The objective of this study is to use advanced modeling techniques of proppant transport inside complex fracture systems using a 3D simulator that couples geomechan ics, fracture mechanics, fluid behavior and proppant transport to assess the optimization of stimulation treatments. In this study proppant transport models incorporate the interaction between hydraulic fractures (HF) and natural fractures (NF) of various sizes and orientations. Treatments were simulated in systematic variations of proppant size, proppant density and fluid viscosity. Fluid rheology, proppant size and density are all major parameters affecting the proppant settling rate. In more complex systems, proppant placement is strongly influenced by the location, size, and orientation of natural fractures. Based on the size of proppant and viscosity of the injected fluid, proppant may or may not enter the natural fractures.

Variable Precision Computing

This report summarizes the activities and major accomplishments of the Variable Precision Computi... more This report summarizes the activities and major accomplishments of the Variable Precision Computing Strategic Initiative project. The overarching goal of this project was to initiate and promote a new paradigm in High Performance Computing (HPC) that would fundamentally change how we represent and make use of representations of real numbers in finite precision and that would establish Lawrence Livermore National Laboratory (LLNL) as a leader in precision-related research for HPC. We pursued three integrated and concurrent research thrusts: new, more dynamic data representations to address data motion and capacity limitations; improved mixed precision algorithms to accelerate computation and to leverage new hardware; and new tools to help developers reason about precision and to automate code transformations. Within these thrust areas, we made significant advances in the use of compressed array data types in numerical calculations, in the theoretical justification, in improved performance, and in the prospects of hardware implementation; in the development of metrics to significantly reduce the storage needs of molecular dynamics simulations; in the development and analysis of efficient mixed-precision algorithms for time-dependent partial differential equations and for orthogonal factorization; and in the capability of tools that can analyze sensitivity of computed results to choices in precision and that can make automated code transformations based on such analysis. We established that there are definite opportunities to realize significant improvements in HPC above and beyond the 2x limits of traditional mixed precision using the technologies we have developed. The project produced numerous papers, presentations, and posters in important venues as well as open-source software contributions to the community, and many of the new capabilities have initiated or are being leveraged in ongoing projects.

Domain Decomposition Solvers for Frequency-Domain Finite Element Equations

Springer eBooks, Oct 5, 2010

The paper is devoted to fast iterative solvers for frequency-domain finite element equations appr... more The paper is devoted to fast iterative solvers for frequency-domain finite element equations approximating linear and nonlinear parabolic initial boundary value problems with time-harmonic excitations. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple linear elliptic system for the amplitudes belonging to the sine-and to the cosine-excitation or a large nonlinear elliptic system for the Fourier coefficients in the linear and nonlinear case, respectively. The fast solution of the corresponding linear and nonlinear system of finite element equations is crucial for the competitiveness of this method.

Local Lagrangian reduced-order modeling for Rayleigh-Taylor instability by solution manifold decomposition

arXiv (Cornell University), Jan 18, 2022

Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various ... more Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various disciplines of science and engineering, including astrophyics, atmospheric sciences and climate, geophysics, and fusion energy. Analytical methods cannot be applied to explain the long-time behavior of Rayleigh-Taylor instability, and therefore numerical simulation of the full problem is required. However, in order to capture the growth of amplitude of perturbations accurately, both the spatial and temporal discretization need to be extremely fine for traditional numerical methods, and the long-time simulation may become prohibitively expensive. In this paper, we propose efficient reduced order model techniques to accelerate the simulation of Rayleigh-Taylor instability in compressible gas dynamics. We introduce a general framework for decomposing the solution manifold to construct the temporal domain partition and temporally-local reduced order model construction with varying Atwood number. We propose two practical approaches in this framework, namely decomposition by using physical time and by penetration distance respectively. Numerical results are presented to examine the performance of the proposed approaches.

Reduced order models for Lagransian hydrodynamics

arXiv (Cornell University), Apr 23, 2021

As a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial ... more As a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial setting, Lagrangian hydrodynamics is characterized by moving meshes, advection-dominated solutions, and moving shock fronts with sharp gradients. These challenges hinder the existing projection-based model reduction schemes from being practical. We develop several variations of projection-based reduced order model techniques for Lagrangian hydrodynamics by introducing three different reduced bases for position, velocity, and energy fields. A time-windowing approach is also developed to address the challenge imposed by the advection-dominated solutions. Lagrangian hydrodynamics is formulated as a nonlinear problem, which requires a proper hyper-reduction technique. Therefore, we apply the over-sampling DEIM and SNS approaches to reduce the complexity due to the nonlinear terms. Finally, we also present both a posteriori and a priori error bounds associated with our reduced order model. We compare the performance of the spatial and time-windowing reduced order modeling approaches in terms of accuracy and speed-up with respect to the corresponding full order model for several numerical examples, namely Sedov blast, Gresho vortices, Taylor-Green vortices, and triple-point problems.

Local Lagrangian reduced-order modeling for the Rayleigh-Taylor instability by solution manifold decomposition

Journal of Computational Physics, 2023

Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various ... more Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various disciplines of science and engineering, including astrophyics, atmospheric sciences and climate, geophysics, and fusion energy. Analytical methods cannot be applied to explain the long-time behavior of Rayleigh-Taylor instability, and therefore numerical simulation of the full problem is required. However, in order to capture the growth of amplitude of perturbations accurately, both the spatial and temporal discretization need to be extremely fine for traditional numerical methods, and the long-time simulation may become prohibitively expensive. In this paper, we propose efficient reduced order model techniques to accelerate the simulation of Rayleigh-Taylor instability in compressible gas dynamics. We introduce a general framework for decomposing the solution manifold to construct the temporal domain partition and temporally-local reduced order model construction with varying Atwood number. We propose two practical approaches in this framework, namely decomposition by using physical time and by penetration distance respectively. Numerical results are presented to examine the performance of the proposed approaches.

S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Models

arXiv (Cornell University), Mar 29, 2022

While projection-based reduced order models can reduce the dimension of full order solutions, the... more While projection-based reduced order models can reduce the dimension of full order solutions, the resulting reduced models may still contain terms that scale with the full order dimension. Hyper-reduction techniques are sampling-based methods that further reduce this computational complexity by approximating such terms with a much smaller dimension. The goal of this work is to introduce a points selection algorithm developed by Shin and Xiu [SIAM J. Sci. Comput., 38 (2016), pp. A385-A411], as a hyper-reduction method. The selection algorithm is originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity S that measures both the column orthogonality and the determinant, we refer to the algorithm as S-OPT. Numerical examples are provided to demonstrate the performance of S-OPT and to compare its performance with an over-sampled Discrete Empirical Interpolation (DEIM) algorithm. We found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy for a given number of indices.

OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Oct 17, 2019

libROM is a collection of C++ classes that compute reduced order models and hyperreduced order mo... more libROM is a collection of C++ classes that compute reduced order models and hyperreduced order models for systems of ordinary differential equations. libROM includes parallel, adaptive methods for proper orthogonal decomposition, and parallel, non-adaptive methods for hyperreduction using the discrete empirical interpolation method.

Energy conserving quadrature based dimensionality reduction for nonlinear hydrodynamics problems

Expanded Mixed Multiscale Finite Element Methods and Their Applications for Flows in Porous Media

Multiscale Modeling & Simulation, 2012

We develop a family of expanded mixed Multiscale Finite Element Methods (Ms-FEMs) and their hybri... more We develop a family of expanded mixed Multiscale Finite Element Methods (Ms-FEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed Multiscale Finite Element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity, and Lagrange multipliers. We use multiscale basis functions for both the velocity and the gradient of pressure. In the expanded mixed MsFEM framework, we consider both separable and non-separable spatial scales. Specifically, we analyze the methods in three categories: periodic separable scales, G−convergent separable scales, and a continuum of scales. When there is no scale separation, using some global information can significantly improve the accuracy of the expanded mixed MsFEMs. We present a rigorous convergence analysis of these methods that includes both conforming and nonconforming formulations. Numerical results are presented for various multiscale models of flow in porous media with shale barriers that illustrate the efficacy of the proposed family of expanded mixed MsFEMs.

Linearized Coulomb Collision Operator for Simulation of Interpenetrating Plasma Streams

IEEE Transactions on Plasma Science, 2019

We present an extension of a linearized Coulomb collision operator, previously used in several Eu... more We present an extension of a linearized Coulomb collision operator, previously used in several Eulerian kinetic codes for like-species collisions and unlike-species collisions in the case where the backgrounds about which the linearization is made all are in collisional equilibrium, to the situation of interpenetrating plasma streams. In the latter case, the backgrounds can not be taken to be in equilibrium and a significant generalization is required. Our development is targeted towards the Eulerian kinetic plasma code LOKI, which evolves the Vlasov-Poisson or Vlasov-Maxwell system in a Cartesian "2+2dimensional" phase space. The extended operator has been implemented in a test code, and results of both quantitative verification and qualitative "realizability" tests are presented.

Helmholtz and Maxwell equations on general

polyhedral meshes

S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Models

Cornell University - arXiv, Mar 29, 2022

While projection-based reduced order models can reduce the dimension of full order solutions, the... more While projection-based reduced order models can reduce the dimension of full order solutions, the resulting reduced models may still contain terms that scale with the full order dimension. Hyper-reduction techniques are sampling-based methods that further reduce this computational complexity by approximating such terms with a much smaller dimension. The goal of this work is to introduce a points selection algorithm developed by Shin and Xiu [SIAM J. Sci. Comput., 38 (2016), pp. A385-A411], as a hyper-reduction method. The selection algorithm is originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity S that measures both the column orthogonality and the determinant, we refer to the algorithm as S-OPT. Numerical examples are provided to demonstrate the performance of S-OPT and to compare its performance with an over-sampled Discrete Empirical Interpolation (DEIM) algorithm. We found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy for a given number of indices.

$Research paper thumbnail of Simulation de fuite de fluide et reflux dans une région souterraine fracturée$

Simulation de fuite de fluide et reflux dans une région souterraine fracturée

Selon certains aspects, un ensemble d'equations d'ecoulement de regulation peut etre defi... more Selon certains aspects, un ensemble d'equations d'ecoulement de regulation peut etre defini pour un fluide de systeme de puits representant un modele d'ecoulement unidimensionnel dans une region souterraine fracturee. Un premier sous-ensemble des equations d'ecoulement de regulation peut representer un ecoulement dans une fracture, et un second sous-ensemble des equations d'ecoulement de regulation peut representer un ecoulement dans un milieu de reservoir adjacent a la fracture. Un ensemble reduit d'equations d'ecoulement de regulation peut etre genere par l'elimination du second sous-ensemble de l'ensemble d'equations d'ecoulement de regulation sur la base d'un accouplement fluidique entre la fracture et le milieu de reservoir. L'ecoulement de fluide de systeme de puits dans la region souterraine fracturee peut etre simule sur la base de l'ensemble reduit d'equations d'ecoulement de regulation.

libROM is a collection of C++ classes that compute reduced order models and hyperreduced order mo... more libROM is a collection of C++ classes that compute reduced order models and hyperreduced order models for systems of ordinary differential equations. libROM includes parallel, adaptive methods for proper orthogonal decomposition, and parallel, non-adaptive methods for hyperreduction using the discrete empirical interpolation method.