scholarly journals Application of Generalized (Hyper-) Dual Numbers in Equation of State Modeling

2021 ◽  
Vol 3 ◽  
Author(s):  
Philipp Rehner ◽  
Gernot Bauer
Keyword(s):  
Equation Of State ◽  
Open Source ◽  
Critical Points ◽  
Higher Order ◽  
Dual Numbers ◽  
Science And Engineering ◽  
Thermodynamic Potentials ◽  
Complex Models ◽  
Derivatives Of

The calculation of derivatives is ubiquitous in science and engineering. In thermodynamics, in particular, state properties can be expressed as derivatives of thermodynamic potentials. The manual differentiation of complex models can be tedious and error-prone. In this work, we revisit dual and hyper-dual numbers for the calculation of exact derivatives and show generalizations to higher order derivatives and derivatives with respect to vector quantities. The methods described in this paper are accompanied by an open source Rust implementation with Python bindings. Applications of the generalized (hyper-) dual numbers are given in the context of equation of state modeling and the calculation of critical points.

scholarly journals Convergence Ball and Complex Geometry of an Iteration Function of Higher Order

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 28 ◽  
Author(s):  
Deepak Kumar ◽  
Ioannis Argyros ◽  
Janak Sharma
Keyword(s):  
Complex Geometry ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Order Method ◽  
Eighth Order ◽  
Science And Engineering ◽  
Convergence Ball ◽  
Convergence Results ◽  
Iteration Function ◽  
Derivatives Of

Higher-order derivatives are used to determine the convergence order of iterative methods. However, such derivatives are not present in the formulas. Therefore, the assumptions on the higher-order derivatives of the function restrict the applicability of methods. Our convergence analysis of an eighth-order method uses only the derivative of order one. The convergence results so obtained are applied to some real problems, which arise in science and engineering. Finally, stability of the method is checked through complex geometry shown by drawing basins of attraction of the solutions.

scholarly journals Higher Order Automatic Differentiation with Dual Numbers

2020 ◽  
Author(s):  
László Szirmay-Kalos
Keyword(s):  
Gaussian Curvature ◽  
Automatic Differentiation ◽  
Arbitrary Order ◽  
Higher Order ◽  
Dual Numbers ◽  
Computer Implementation ◽  
Particular Solution ◽  
Derivatives Of ◽  
Curve Fairing ◽  
Curvature Computation

In engineering applications, we often need the derivatives of functions defined by a program. The approach chosen for derivative computation must be algebraic to allow computer implementation. A particular solution to obtain first derivatives is the application of dual numbers. This paper proposes simple and compact generalizations of this idea to obtain derivatives of arbitrary order for single or multi-variate functions and the automatic handling of 0/0 ambiguities in the calculations. We also provide the C++ code that takes advantage of operator overloading and recursion. The method is demonstrated by path animation, Gaussian curvature computation, and curve fairing.

scholarly journals Jerk within the Context of Science and Engineering—A Systematic Review

Vibration ◽  
2020 ◽  
Vol 3 (4) ◽  
pp. 371-409
Author(s):  
Hasti Hayati ◽  
David Eager ◽  
Ann-Marie Pendrill ◽  
Hans Alberg
Keyword(s):  
Systematic Review ◽  
Literature Review ◽  
Fatigue Cracks ◽  
Higher Order ◽  
Historical Background ◽  
Science And Engineering ◽  
Rapid Changes ◽  
Higher Derivatives ◽  
University Science ◽  
Derivatives Of

Rapid changes in forces and the resulting changes in acceleration, jerk and higher order derivatives can have undesired consequences beyond the effect of the forces themselves. Jerk can cause injuries in humans and racing animals and induce fatigue cracks in metals and other materials, which may ultimately lead to structure failures. This is a reason that it is used within standards for limits states. Examples of standards which use jerk include amusement rides and lifts. Despite its use in standards and many science and engineering applications, jerk is rarely discussed in university science and engineering textbooks and it remains a relatively unfamiliar concept even in engineering. This paper presents a literature review of the jerk and higher derivatives of displacement, from terminology and historical background to standards, measurements and current applications.

scholarly journals Traversable wormholes in Einsteinian cubic gravity

2019 ◽  
Vol 35 (06) ◽  
pp. 2050017 ◽  
Author(s):  
Mohammad Reza Mehdizadeh ◽  
Amir Hadi Ziaie
Keyword(s):  
Equation Of State ◽  
Gravitational Lensing ◽  
Energy Condition ◽  
Higher Order ◽  
Energy Conditions ◽  
Geodesic Motion ◽  
De Sitter ◽  
Asymptotically Flat ◽  
Traversable Wormholes ◽  
Standard Energy

In this work, we investigate wormhole configurations described by a constant redshift function in Einstein-Cubic gravity ( ECG ). We derive analytical wormhole geometries by assuming a particular equation of state ( EoS ) and investigate the possibility that these solutions satisfy the standard energy conditions. We introduce exact asymptotically flat and anti-de Sitter (AdS) spacetimes that admit traversable wormholes. These solutions are obtained by imposing suitable values for the parameters of the theory so that the resulted geometries satisfy the weak energy condition ( WEC ) in the vicinity of the throat, due to the presence of higher-order curvature terms. Moreover, we find that AdS solutions satisfy the WEC throughout the spacetime. A description of the geodesic motion of time-like and null particles is presented for the obtained wormhole solutions. Also, using gravitational lensing effects, observational features of the wormhole structure are discussed.

A C0 three-node triangular element based on preprocessing approach for thick sandwich plates

2017 ◽  
Vol 21 (6) ◽  
pp. 1820-1842
Author(s):  
Wu Zhen ◽  
Ma Rui ◽  
Chen Wanji
Keyword(s):  
Transverse Shear ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Higher Order ◽  
Triangular Element ◽  
Shear Stresses ◽  
Sandwich Plates ◽  
Transverse Shear Stress ◽  
Transverse Shear Stresses ◽  
Derivatives Of

This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

Expansions of the mildly relativistic dispersion function for nearly perpendicular electron cyclotron ray tracing

1999 ◽  
Vol 61 (1) ◽  
pp. 121-128 ◽  
Author(s):  
I. P. SHKAROFSKY
Keyword(s):  
Ray Tracing ◽  
Computer Programs ◽  
Higher Order ◽  
Electron Cyclotron ◽  
Dispersion Function ◽  
Relativistic Dispersion ◽  
Plasma Dispersion ◽  
Derivatives Of ◽  
Cyclotron Harmonic ◽  
Experienced Difficulty

To trace rays very close to the nth electron cyclotron harmonic, we need the mildly relativistic plasma dispersion function and its higher-order derivatives. Expressions for these functions have been obtained as an expansion for nearly perpendicular propagation in a region where computer programs have previously experienced difficulty in accuracy, namely when the magnitude of (c/vt)2 (ω−nωc)/ω is between 1 and 10. In this region, the large-argument expansions are not yet valid, but partial cancellations of terms occur. The expansion is expressed as a sum over derivatives of the ordinary dispersion function Z. New expressions are derived to relate higher-order derivatives of Z to Z itself in this region of concern in terms of a finite series.

Open Higher Order Continuous-Time Dynamic Model with Mixed Stock and Flow Data and Derivatives of Exogenous Variables

1991 ◽  
Vol 7 (3) ◽  
pp. 404-408 ◽  
Author(s):  
K. Ben Nowman
Keyword(s):  
Continuous Time ◽  
Higher Order ◽  
Flow Data ◽  
Continuous Time Model ◽  
Exogenous Variables ◽  
Time Model ◽  
Time Dynamic ◽  
Mixed Stock ◽  
Gaussian Estimation ◽  
Derivatives Of

This paper is concerned with deriving formulae for higher order derivatives of exogenous variables for use in estimating the parameters of an open secondorder continuous time model with mixed stock and flow data and first and second order derivatives of exogenous variables which are not observable. This should provide the basis for the future estimation of continuous time models in a range of applied areas using the new Gaussian estimation computer program developed by Nowman [4].

Phase diagrams and higher-order critical points

1975 ◽  
Vol 12 (1) ◽  
pp. 345-355 ◽  
Author(s):  
Robert B. Griffiths
Keyword(s):  
Phase Diagrams ◽  
Critical Points ◽  
Higher Order
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