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Mikhail Shashkov - One of the best experts on this subject based on the ideXlab platform.
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analysis and optimization of inner products for mimetic finite difference methods on a triangular grid
Mathematics and Computers in Simulation, 2004Co-Authors: R Liska, Mikhail Shashkov, Victor G GanzhaAbstract:The Support Operator method designs mimetic finite difference schemes by first constructing a discrete divergence Operator based on the divergence theorem, and then defining the discrete gradient Operator as the adjoint Operator of the divergence based on the Gauss theorem connecting the divergence and gradient Operators, which remains valid also in the discrete case. When evaluating the discrete gradient Operator, one needs to define discrete inner products of two discrete vector fields. The local discrete inner product on a given triangle is defined by a 3 × 3 symmetric positive definite matrix M defined by its six independent elements-parameters. Using the Gauss theorem over our triangle, we evaluate the discrete gradient in the triangle. We require the discrete gradient to be exact for linear functions, which gives us a system of linear equations for elements of the matrix M. This system, together with inequalities which guarantee positive definiteness of the matrix M, results in a one parameter family of inner products which give exact gradients for linear functions. The traditional inner product is a member of this family. The positive free parameter can be used to improve another property of the discrete method. We show that accuracy of the method for quadratic functions improves with decreasing this parameter, however, at the same time, the condition number of the matrix M, which is the local matrix of the linear system for computing the discrete gradient, increases to infinity when the parameter goes to zero, so one needs to choose a compromise between accuracy and solvability of the local system. Our analysis has been performed by computer algebra tools which proved to be essential.
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mimetic finite difference methods for diffusion equations on unstructured triangular grid
2004Co-Authors: Victor G Ganzha, Mikhail Shashkov, R Liska, Christoph ZengerAbstract:A finite difference algorithm for solution of stationary diffusion equation on unstructured triangular grid has been developed earlier by a Support Operator method. The Support Operator method first constructs a discrete divergence Operator from the divergence theorem and then constructs a discrete gradient Operator as the adjoint Operator of the divergence. The adjointness of the Operators is based on the continuum Gauss theorem which remains valid also for discrete Operators. Here we extend the method to general Robin boundary conditions, generalize it to time dependent heat equation and perform the analysis of space discretization. One parameter family of discrete vector inner products, which produce exact gradients for linear functions, is designed. Our method works very well for discontinuous diffusion coefficient and very rough or very distorted grids which appear quite often e.g. in Lagrangian simulations.
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Support Operator method for laplace equation on unstructured triangular grid
Selcuk Journal of Applied Mathematics, 2002Co-Authors: Victor G Ganzha, Mikhail Shashkov, R Liska, Christoph ZengerAbstract:A finite difference algorithm for solution of generalized Laplace equation on unstructured triangular grid is constructed by a Support Operator method. The Support Operator method first constructs discrete divergence Operator from the divergence theorem and then constructs discrete gradient Operator as the adjoint Operator of the divergence. The adjointness of the Operators is based on the continuum Green formulas which remain valid also for discrete Operators. Developed method is exact for linear solution and has second order convergence rate. It is working well for discontinuous diffusion coefficient and very rough or very distorted grids which appear quite often e.~g. in Lagrangian simulations. Being formulated on the unstructured grid the method can be used on the region of arbitrary geometry shape. Numerical results confirm these properties of the developed method.
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adjoint Operators for the natural discretizations of the divergence gradient and curl on logically rectangular grids
Applied Numerical Mathematics, 1997Co-Authors: James M Hyman, Mikhail ShashkovAbstract:Abstract We use the Support-Operator method to derive new discrete approximations of the divergence, gradient, and curl using discrete analogs of the integral identities satisfied by the differential Operators. These new discrete Operators are adjoint to the previously derived natural discrete Operators defined using ‘natural’ coordinate-invariant definitions, such as Gauss' theorem for the divergence. The natural Operators cannot be combined to construct discrete analogs of the second-order Operators div grad, grad div, and curl curl because of incompatibilities in domains and in the ranges of values for the Operators. The same is true for the adjoint Operators. However, the adjoint Operators have complementary domains and ranges of values and the combined set of natural and adjoint Operators allow a consistent formulation for all the compound discrete Operators. We also prove that the Operators satisfy discrete analogs of the major theorems of vector analysis relating the differential Operators, including div A = 0 if and only if A = curl B ; curl A = 0 if and only if A = grad ϕ .
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conservative finite difference methods on general grids
1996Co-Authors: Mikhail ShashkovAbstract:INTRODUCTION Governing Equations Elliptic Equations Heat Equation Equation of Gas Dynamic in Lagrangian Form The Main Ideas of Finite-Difference Algorithms 1-D Case 2-D Case Methods of Solution of Systems of Linear Algebraic Equation Methods of Solution of Systems of Nonlinear Equations METHOD OF Support-OperatorS Main Stages The Elliptic Equations Gas Dynamic Equations System of Consistent Difference Operators in 1-D Inner Product in Spaces of Difference Functions and Properties of Difference Operators System of Consistent Difference Operators in 2-D THE ELLIPTIC EQUATIONS Introduction Continuum Elliptic Problems with Dirichlet Boundary Conditions Continuum Elliptic Problems with Robin Boundary Conditions One-Dimensional Support Operator Algorithms Nodal Discretization of Scalar Functions and Cell-Centered Discretization of Vector Functions Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions Numerical Solution of Test Problems Two-Dimensional Support Operator Algorithms Nodal Discretization of Scalar Functions and Cell-Valued Discretization of Vector Functions Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions Numerical Solution of Test Problems Conclusion Two-Dimensional Support Operator Algorithms Discretization Spaces of Discrete Functions The Prime Operator The Derived Operator Multiplication by a Matrix and the Operator D The Difference Scheme for the Elliptic Operator The Matrix Problem Approximation and Convergence Properties HEAT EQUATION Introduction Finite-Difference Schemes for Heat Equation in 1-D Finite-Difference Schemes for Heat Equation in 2-D LAGRANGIAN GAS DYNAMICS Kinematics of Fluid Motions Integral Form of Gas Dynamics Equations Integral Equations for One Dimensional Case Differential Equations of Gas Dynamics in Lagrangian Form The Differential Equations in 1D. Lagrange Mass Variables The Statements of Gas Dynamics Problems in Lagrange Variables Different Forms of Energy Equation Acoustic Equations Reference Information Characteristic Form of Gas Dynamics Equations Riemann's Invariants Discontinuous Solutions Conservation Laws and Properties of First Order Invariant Operators Finite-Difference Algorithm in 1D Discretization in 1D Discrete Operators in 1D Semi-Discrete Finite-Difference Scheme in 1D Fully Discrete, Explicit, Computational Algorithm Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme Stability Conditions Homogeneous Finite-Difference Schemes. Artificial Viscosity Artificial Viscosity in 1D Numerical Example Finite Difference Algorithm in 2D Discretization in 2D Discrete Operators in 2D Semi-Discrete Finite-Difference Scheme in 2D Stability Conditions Finite-Difference Algorithm in 2D Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme Artificial Viscosity in 2D Numerical Example APPENDIX: FORTRAN CODE DIRECTORY General Description of Structure of Directories on the Disk Programs for Elliptic Equations Programs for 1D Equations Programs for 2D Equations Programs for Heat Equations Programs for 1D Equations Programs for 2D Equations Programs for Gas Dynamics Equations Programs for 1D Equations Programs for 2D Equations Bibliography
Adam Burrows - One of the best experts on this subject based on the ideXlab platform.
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bethe hydro an arbitrary lagrangian eulerian multidimensional hydrodynamics code for astrophysical simulations
Astrophysical Journal Supplement Series, 2008Co-Authors: Jeremiah W Murphy, Adam BurrowsAbstract:In this paper, we describe a new hydrodynamics code for one- and two-dimensional (1D and 2D) astrophysical simulations, BETHE-hydro, that uses time-dependent, arbitrary, unstructured grids. The core of the hydrodynamics algorithm is an arbitrary Lagrangian-Eulerian (ALE) approach, in which the gradient and divergence Operators are made compatible using the Support-Operator method. We present 1D and 2D gravity solvers that are finite differenced using the Support-Operator technique, and the resulting system of linear equations are solved using the tridiagonal method for 1D simulations and an iterative multigrid-preconditioned conjugate-gradient method for 2D simulations. Rotational terms are included for 2D calculations using cylindrical coordinates. We document an incompatibility between a subcell pressure algorithm to suppress hourglass motions, and the subcell remapping algorithm and present a modified subcell pressure scheme that avoids this problem. Strengths of this code include a straightforward structure, enabling simple inclusion of additional physics packages, the ability to use a general equation of state, and most importantly, the ability to solve self-gravitating hydrodynamic flows on time-dependent, arbitrary grids. In what follows, we describe in detail the numerical techniques employed and, with a large suite of tests, demonstrate that BETHE-hydro finds accurate solutions with second-order convergence.
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bethe hydro an arbitrary lagrangian eulerian multi dimensional hydrodynamics code for astrophysical simulations
arXiv: Astrophysics, 2008Co-Authors: Jeremiah W Murphy, Adam BurrowsAbstract:In this paper, we describe a new hydrodynamics code for 1D and 2D astrophysical simulations, BETHE-hydro, that uses time-dependent, arbitrary, unstructured grids. The core of the hydrodynamics algorithm is an arbitrary Lagrangian-Eulerian (ALE) approach, in which the gradient and divergence Operators are made compatible using the Support-Operator method. We present 1D and 2D gravity solvers that are finite differenced using the Support-Operator technique, and the resulting system of linear equations are solved using the tridiagonal method for 1D simulations and an iterative multigrid-preconditioned conjugate-gradient method for 2D simulations. Rotational terms are included for 2D calculations using cylindrical coordinates. We document an incompatibility between a subcell pressure algorithm to suppress hourglass motions and the subcell remapping algorithm and present a modified subcell pressure scheme that avoids this problem. Strengths of this code include a straightforward structure, enabling simple inclusion of additional physics packages, the ability to use a general equation of state, and most importantly, the ability to solve self-gravitating hydrodynamic flows on time-dependent, arbitrary grids. In what follows, we describe in detail the numerical techniques employed and, with a large suite of tests, demonstrate that BETHE-hydro finds accurate solutions with 2$^{nd}$-order convergence.
Jeremiah W Murphy - One of the best experts on this subject based on the ideXlab platform.
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bethe hydro an arbitrary lagrangian eulerian multidimensional hydrodynamics code for astrophysical simulations
Astrophysical Journal Supplement Series, 2008Co-Authors: Jeremiah W Murphy, Adam BurrowsAbstract:In this paper, we describe a new hydrodynamics code for one- and two-dimensional (1D and 2D) astrophysical simulations, BETHE-hydro, that uses time-dependent, arbitrary, unstructured grids. The core of the hydrodynamics algorithm is an arbitrary Lagrangian-Eulerian (ALE) approach, in which the gradient and divergence Operators are made compatible using the Support-Operator method. We present 1D and 2D gravity solvers that are finite differenced using the Support-Operator technique, and the resulting system of linear equations are solved using the tridiagonal method for 1D simulations and an iterative multigrid-preconditioned conjugate-gradient method for 2D simulations. Rotational terms are included for 2D calculations using cylindrical coordinates. We document an incompatibility between a subcell pressure algorithm to suppress hourglass motions, and the subcell remapping algorithm and present a modified subcell pressure scheme that avoids this problem. Strengths of this code include a straightforward structure, enabling simple inclusion of additional physics packages, the ability to use a general equation of state, and most importantly, the ability to solve self-gravitating hydrodynamic flows on time-dependent, arbitrary grids. In what follows, we describe in detail the numerical techniques employed and, with a large suite of tests, demonstrate that BETHE-hydro finds accurate solutions with second-order convergence.
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bethe hydro an arbitrary lagrangian eulerian multi dimensional hydrodynamics code for astrophysical simulations
arXiv: Astrophysics, 2008Co-Authors: Jeremiah W Murphy, Adam BurrowsAbstract:In this paper, we describe a new hydrodynamics code for 1D and 2D astrophysical simulations, BETHE-hydro, that uses time-dependent, arbitrary, unstructured grids. The core of the hydrodynamics algorithm is an arbitrary Lagrangian-Eulerian (ALE) approach, in which the gradient and divergence Operators are made compatible using the Support-Operator method. We present 1D and 2D gravity solvers that are finite differenced using the Support-Operator technique, and the resulting system of linear equations are solved using the tridiagonal method for 1D simulations and an iterative multigrid-preconditioned conjugate-gradient method for 2D simulations. Rotational terms are included for 2D calculations using cylindrical coordinates. We document an incompatibility between a subcell pressure algorithm to suppress hourglass motions and the subcell remapping algorithm and present a modified subcell pressure scheme that avoids this problem. Strengths of this code include a straightforward structure, enabling simple inclusion of additional physics packages, the ability to use a general equation of state, and most importantly, the ability to solve self-gravitating hydrodynamic flows on time-dependent, arbitrary grids. In what follows, we describe in detail the numerical techniques employed and, with a large suite of tests, demonstrate that BETHE-hydro finds accurate solutions with 2$^{nd}$-order convergence.
Kim J. Vicente - One of the best experts on this subject based on the ideXlab platform.
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ecological interface design for petrochemical applications Supporting Operator adaptation continuous learning and distributed collaborative work
Computers & Chemical Engineering, 2001Co-Authors: Kim J. VicenteAbstract:Abstract Future Support systems for Operators of petrochemical refineries will have to Support Operator adaptation to unanticipated events, foster continuous learning, and facilitate distributed, collaborative work. This paper describes Ecological Interface Design, a candidate framework for human–computer interface design that has the potential to fulfill these diverse demands. Support for adaptation and continuous learning is demonstrated though the design of a novel Operator interface for a fluid catalytic cracking unit. While the framework forms a basis upon which a distributed, collaborative Support system may be built, no such design is presented here. The process of the application of the framework is described in detail, including the domain modelling activity and a description of the resulting graphical user interface. Limitations to applying the design approach to operational plants are discussed.
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Supporting Operator problem solving through ecological interface design
Systems Man and Cybernetics, 1995Co-Authors: Kim J. Vicente, Klaus Christoffersen, A PereklitaAbstract:This paper describes two experiments evaluating ecological interface design (EID), a novel theoretical framework for the design of interfaces for complex human-machine systems. According to EID, to properly Support Operator problem solving activities, an interface should display both the physical and functional properties of the work domain in the form of a multilevel representation based on the abstraction hierarchy. To evaluate this claim, two interfaces for a thermal-hydraulic process simulation were developed, one based on a traditional format containing only physical information (P) and another based on EID which also contained information about higher-order functional variables (P+F). The findings of Experiment 1 are consistent with the claim that an interface based on an abstraction hierarchy representation can provide more Support for problem solving than an interface based on physical variables alone, thereby providing some initial Support for the EID framework. There was also some evidence to indicate that theoretical expertise is required to enjoy the full benefits of the P+F interface. The findings of Experiment 2 indicate that subjects who exhibited effective diagnosis performance using the P+F interface tended to start their search at a high level of abstraction and gradually work their way down to more detailed levels, as predicted. Furthermore, previous experience with the DURESS system was found to be the most reliable background variable that predicted performance. >
Gisle Andresen - One of the best experts on this subject based on the ideXlab platform.
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evaluation of ecological interface design for nuclear process control situation awareness effects
Human Factors, 2008Co-Authors: Catherine M. Burns, Jordanna Kwok, Greg A. Jamieson, Robin Welch, Gyrd Skraaning, Nathan Lau, Gisle AndresenAbstract:Objective: We determine whether an ecological interface display for nuclear power plant operations Supports improved situation awareness over traditional and user-centered displays in a realistic environment. Background: Ecological interface design (EID) has not yet been fully evaluated with real Operators facing realistic scenarios. Method: Ecological displays were evaluated alongside traditional and user-centered “advanced” displays in a full-scope nuclear power plant simulation. Licensed plant Operators used the displays in realistic scenarios that either had procedural Support or did not have procedural Support. All three displays were evaluated for their ability to Support Operator situation awareness. Results: A significant three-way interaction effect was observed on two independent measures of situation awareness. For both measures, ecological displays improved situation awareness in scenarios that did not have procedural Support, primarily in the detection phases of those scenarios. No other pron...
