The Use of Dual-Number-Automatic-Differentiation With Sensitivity Analysis to Investigate Physical Models

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Malcolm J. Andrews
Keyword(s):  
Automatic Differentiation ◽  
Liquid Droplet ◽  
Free Fall ◽  
Physical Models ◽  
Test Problems ◽  
One Dimensional ◽  
Physical Systems ◽  
Dual Number ◽  
Functional Forms ◽  
Sensitivity Studies

Local sensitivities are explored using dual-number-automatic-differentiation (DNAD) across three mathematical models of physical systems that have increasing complexity. The models are: (1) a model for the approach of a sphere to free fall; (2) the Taylor-analogy-breakup (TAB) model for liquid droplet atomization; and, (3) an evaluation of the BHR model of turbulence for the development of one-dimensional Rayleigh–Taylor driven material mixing. Sensitivity and functional shape parameters are developed that permit a relative study to be quickly performed for each model. Furthermore, compensating errors, measurement parameter sensitivity, and feature sensitivities are investigated. The test problems consider transient (initial condition effects), steady state (final functional forms), and measures of functional shape. Reduced model forms are explored and selected according to sensitivity. Aside from the local sensitivity studies of the models and associated results, DNAD is shown to be one of several useful, quickly implemented tools to investigate a variety of sensitivity effects in models and together with the present results may serve as a means to simplify a model or focus future model developments and associated experiments.

Investigation of two-dimensional nonstationary processes of outflow a gas-saturated liquid from axisymmetric vessels

2012 ◽  
Vol 9 (1) ◽  
pp. 47-52
Author(s):  
R.Kh. Bolotnova ◽  
V.A. Buzina
Keyword(s):  
Comparative Analysis ◽  
Numerical Model ◽  
Liquid Mixture ◽  
Phase Model ◽  
Test Problems ◽  
Two Dimensional ◽  
Nonstationary Processes ◽  
Saturated Liquid ◽  
Two Phase ◽  
One Dimensional

The two-dimensional and two-phase model of the gas-liquid mixture is constructed. The validity of numerical model realization is justified by using a comparative analysis of test problems solution with one-dimensional calculations. The regularities of gas-saturated liquid outflow from axisymmetric vessels for different geometries are established.

Super Accelerated Flow in Diverging Conical Pipes

2011 ◽  
Vol 110-116 ◽  
pp. 880-885
Author(s):  
G.J. Gutierrez ◽  
A. López Villa ◽  
A. Torres ◽  
S. Peralta ◽  
C. A. Vargas
Keyword(s):  
Free Surface ◽  
Free Fall ◽  
Liquid Column ◽  
Dimensional Model ◽  
One Dimensional ◽  
Low Viscosity ◽  
Accelerated Flow ◽  
Theoretical Results

The motion of the upper free surface of a liquid column released from rest in a vertical, conical container is analyzed theoretically and experimentally. An inviscid, one-dimensional model, for a slightly expanding pipe's radius, describes how the recently reported super free fall of liquids occurs in liquids of very low viscosity. Experiments agree with the theoretical results.

Implications of Using Dual Number Derivatives With a Numerical Solution

2013 ◽  
Author(s):  
Suryanarayana R. Pakalapati ◽  
Hayri Sezer ◽  
Ismail B. Celik
Keyword(s):  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Computer Code ◽  
Model Parameters ◽  
Model Parameter ◽  
Systematic Assessment ◽  
Dual Number ◽  
Advection Diffusion Equation ◽  
Model Equations ◽  
Derivatives Of

Dual number arithmetic is a well-known strategy for automatic differentiation of computer codes which gives exact derivatives, to the machine accuracy, of the computed quantities with respect to any of the involved variables. A common application of this concept in Computational Fluid Dynamics, or numerical modeling in general, is to assess the sensitivity of mathematical models to the model parameters. However, dual number arithmetic, in theory, finds the derivatives of the actual mathematical expressions evaluated by the computer code. Thus the sensitivity to a model parameter found by dual number automatic differentiation is essentially that of the combination of the actual mathematical equations, the numerical scheme and the grid used to solve the equations not just that of the model equations alone as implied by some studies. This aspect of the sensitivity analysis of numerical simulations using dual number auto derivation is explored in the current study. A simple one-dimensional advection diffusion equation is discretized using different schemes of finite volume method and the resulting systems of equations are solved numerically. Derivatives of the numerical solutions with respect to parameters are evaluated automatically using dual number automatic differentiation. In addition the derivatives are also estimated using finite differencing for comparison. The analytical solution was also found for the original PDE and derivatives of this solution are also computed analytically. It is shown that a mathematical model could potentially show different sensitivity to a model parameter depending on the numerical method employed to solve the equations and the grid resolution used. This distinction is important since such inter-dependence needs to be carefully addressed to avoid confusion when reporting the sensitivity of predictions to a model parameter using a computer code. A systematic assessment of numerical uncertainty in the sensitivities computed using automatic differentiation is presented.

Enhanced Verification for RELAP5-3D Parameter and Sensitivity Studies

2016 ◽  
Author(s):  
George L. Mesina ◽  
Nolan Anderson
Keyword(s):  
Nuclear Power ◽  
Power Plants ◽  
User Interaction ◽  
Computer Code ◽  
Unintended Consequences ◽  
Physical Models ◽  
Case Processing ◽  
Time Step ◽  
Sensitivity Studies ◽  
Code Changes

The RELAP5-3D1 program solves a complex system of governing, closure and special process equations to model the underlying physics of nuclear power plants. For SQA (software quality assurance), the code must be verified and validated (V&V) to ensure proper performance before release to users. The physical models are validated against data from experiments and plants and verified against specifications for the computer code. In addition to physics, programs such as RELAP5-3D perform numerous other functions and processes that should also be checked to guarantee correct results. Functions include input, output, data management, and user interaction, while processes include restart, time-step backup, code coupling, and multi-case processing. Previous articles have covered the verification of the physical models, restart, and backup through extremely accurate and automated sequential verification applied on a comprehensive suite of test cases to ensure that code changes produced no unintended consequences. New developments have enabled the verification of multi-case and multi-deck processing. These features are frequently used in parameter and code sensitivity studies and therefore must be verified as working correctly. Both theory and application are presented.

scholarly journals A Numerical Approach to Solve Volume-Based Batch Crystallization Model with Fines Dissolution Unit

Processes ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 453 ◽  
Author(s):  
Mukhtar ◽  
Sohaib ◽  
Ahmad
Keyword(s):  
Numerical Approximation ◽  
Chemical Engineering ◽  
Numerical Study ◽  
Moment Generating Function ◽  
Numerical Approach ◽  
Test Problems ◽  
Batch Crystallization ◽  
One Dimensional ◽  
Engineering Sciences ◽  
The Moment

In this article, a numerical study of a one-dimensional, volume-based batch crystallization model (PBM) is presented that is used in numerous industries and chemical engineering sciences. A numerical approximation of the underlying model is discussed by using an alternative Quadrature Method of Moments (QMOM). Fines dissolution term is also incorporated in the governing equation for improvement of product quality and removal of undesirable particles. The moment-generating function is introduced in order to apply the QMOM. To find the quadrature abscissas, an orthogonal polynomial of degree three is derived. To verify the efficiency and accuracy of the proposed technique, two test problems are discussed. The numerical results obtained by the proposed scheme are plotted versus the analytical solutions. Thus, these findings line up well with the analytical findings.

An Avant-Garde Handling of Temporal-Spatial Fractional Physical Models

2020 ◽  
Vol 21 (2) ◽  
pp. 183-194 ◽  
Author(s):  
Imad Jaradat ◽  
Marwan Alquran ◽  
Qutaibeh Katatbeh ◽  
Feras Yousef ◽  
Shaher Momani ◽  
...  
Keyword(s):  
Fractional Derivatives ◽  
Dimensional Space ◽  
Physical Models ◽  
Mathematical Framework ◽  
Analytical Treatment ◽  
Caputo Fractional Derivatives ◽  
Avant Garde ◽  
Differential Transform ◽  
Functional Forms ◽  
Physical Case

AbstractIn the present study, we dilate the differential transform scheme to develop a reliable scheme for studying analytically the mutual impact of temporal and spatial fractional derivatives in Caputo’s sense. We also provide a mathematical framework for the transformed equations of some fundamental functional forms in fractal 2-dimensional space. To demonstrate the effectiveness of our proposed scheme, we first provide an elegant scheme to estimate the (mixed-higher) Caputo-fractional derivatives, and then we give an analytical treatment for several (non)linear physical case studies in fractal 2-dimensional space. The study concluded that the proposed scheme is very efficacious and convenient in extracting solutions for wide physical applications endowed with two different memory parameters as well as in approximating fractional derivatives.

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