scholarly journals Direct construction of optimized stellarator shapes. Part 3. Omnigenity near the magnetic axis

2019 ◽  
Vol 85 (6) ◽  
Author(s):  
Gabriel G. Plunk ◽  
Matt Landreman ◽  
Per Helander
Keyword(s):  
Numerical Method ◽  
Plasma Phys ◽  
Magnetic Axis ◽  
Direct Construction ◽  
Numerical Construction ◽  
First Order

The condition of omnigenity is investigated, and applied to the near-axis expansion of Garren & Boozer (Phys. Fluids B, vol. 3 (10), 1991a, pp. 2805–2821). Due in part to the particular analyticity requirements of the near-axis expansion, we find that, excluding quasi-symmetric solutions, only one type of omnigenity, namely quasi-isodynamicity, can be satisfied at first order in the distance from the magnetic axis. Our construction provides a parameterization of the space of such solutions, and the cylindrical reformulation and numerical method of Landreman & Sengupta (J. Plasma Phys., vol. 84 (6), 2018, 905840616); Landreman et al. (J. Plasma Phys., vol. 85 (1), 2019, 905850103), enables their efficient numerical construction.

scholarly journals Figures of merit for stellarators near the magnetic axis

2021 ◽  
Vol 87 (1) ◽  
Author(s):  
Matt Landreman
Keyword(s):  
Magnetic Field ◽  
Field Strength ◽  
Magnetic Axis ◽  
Configuration Design ◽  
New Paradigm ◽  
First Order ◽  
Figures Of Merit ◽  
Minor Radius ◽  
Error Field ◽  
Quantities Of Interest

A new paradigm for rapid stellarator configuration design has been recently demonstrated, in which the shapes of quasisymmetric or omnigenous flux surfaces are computed directly using an expansion in small distance from the magnetic axis. To further develop this approach, here we derive several other quantities of interest that can be rapidly computed from this near-axis expansion. First, the $\boldsymbol {\nabla }\boldsymbol {B}$ and $\boldsymbol {\nabla }\boldsymbol {\nabla }\boldsymbol {B}$ tensors are computed, which can be used for direct derivative-based optimization of electromagnetic coil shapes to achieve the desired magnetic configuration. Moreover, if the norm of these tensors is large compared with the field strength for a given magnetic field, the field must have a short length scale, suggesting it may be hard to produce with coils that are suitably far away. Second, we evaluate the minor radius at which the flux surface shapes would become singular, providing a lower bound on the achievable aspect ratio. This bound is also shown to be related to an equilibrium beta limit. Finally, for configurations that are constructed to achieve a desired magnetic field strength to first order in the expansion, we compute the error field that arises due to second-order terms.

A Difference Method for Plane Problems in Magnetoelastodynamics

1972 ◽  
Vol 39 (3) ◽  
pp. 689-695 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Difference Method ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Symmetric Hyperbolic System ◽  
Von Neumann ◽  
First Order ◽  
Independent Variables

The two-dimensional equations of magnetoelastodynamics are considered as a symmetric hyperbolic system of linear first-order partial-differential equations in three independent variables. The characteristic properties of the system are determined and a numerical method for obtaining the solution to mixed initial and boundary-value problems in plane magnetoelastodynamics is presented. Results on the von Neumann necessary condition are presented. Application of the method to a problem which has a known solution provides further numerical evidence of the convergence and stability of the method.

scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

scholarly journals Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis

2020 ◽  
Vol 86 (1) ◽  
Author(s):  
R. Jorge ◽  
W. Sengupta ◽  
M. Landreman
Keyword(s):  
Magnetic Field ◽  
Aspect Ratio ◽  
Magnetic Flux ◽  
Arbitrary Order ◽  
Analytical Framework ◽  
Field Vector ◽  
Magnetic Axis ◽  
Direct Construction ◽  
Geometrical Properties ◽  
Rotational Transform

A direct construction of equilibrium magnetic fields with toroidal topology at arbitrary order in the distance from the magnetic axis is carried out, yielding an analytical framework able to explore the landscape of possible magnetic flux surfaces in the vicinity of the axis. This framework can provide meaningful analytical insight into the character of high-aspect-ratio stellarator shapes, such as the dependence of the rotational transform and the plasma beta limit on geometrical properties of the resulting flux surfaces. The approach developed here is based on an asymptotic expansion on the inverse aspect ratio of the ideal magnetohydrodynamics equation. The analysis is simplified by using an orthogonal coordinate system relative to the Frenet–Serret frame at the magnetic axis. The magnetic field vector, the toroidal magnetic flux, the current density, the field line label and the rotational transform are derived at arbitrary order in the expansion parameter. Moreover, a comparison with a near-axis expansion formalism employing an inverse coordinate method based on Boozer coordinates (the so-called Garren–Boozer construction) is made, where both methods are shown to agree at lowest order. Finally, as a practical example, a numerical solution using a W7-X equilibrium is presented, and a comparison between the lowest-order solution and the W7-X magnetic field is performed.

FIRST ORDER PHASE TRANSITIONS IN THE BINARY PERCEPTRON

1992 ◽  
Vol 03 (05) ◽  
pp. 1019-1023
Author(s):  
STAM NICOLIS
Keyword(s):  
Phase Transitions ◽  
Numerical Method ◽  
First Order ◽  
The Stability ◽  
First Order Phase ◽  
Open Issues

We present a numerical method for the study of the stability-capacity diagram of the Ising perceptron that is readily scalable, thus providing the opportunity to resolve some open issues on this problem. We also discuss the possibility of studying multi-layer systems.

scholarly journals A Numerical Method for Fuzzy Differential Equations and Hybrid Fuzzy Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
K. Ivaz ◽  
A. Khastan ◽  
Juan J. Nieto
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Sufficient Conditions ◽  
Numerical Algorithms ◽  
Stability And Convergence ◽  
First Order

Numerical algorithms for solving first-order fuzzy differential equations and hybrid fuzzy differential equations have been investigated. Sufficient conditions for stability and convergence of the proposed algorithms are given, and their applicability is illustrated with some examples.

Single-Phase Transient in Single-Elbow Pipe

2006 ◽  
Author(s):  
Janez Gale ◽  
Iztok Tiselj
Keyword(s):  
Numerical Method ◽  
Single Phase ◽  
Accurate Method ◽  
Equation Model ◽  
One Dimensional ◽  
Structure Interaction ◽  
First Order ◽  
Applied Model ◽  
Beam Equations ◽  
Phase Fluid

Mathematical and numerical model needed for description of single-elbow pipe movement in horizontal Z-Y plane coupled with one-dimensional single-phase fluid dynamics is described and discussed. The governing phenomenon is also known as two-way Fluid-Structure Interaction. Standard Skalak’s four-equation model was improved with additional four Timoshenko’s beam equations for description of flexural displacements and rotations. The applied model was solved with improved second-order accurate numerical method that is based on Godounov’s upwind first-order accurate method. The model was successfully used for simulation of the rod impact induced transient and conventional instantaneous valve closure induced transient in the tank-pipe-valve system. Special attention was made to the applicability of the applied numerical method.

scholarly journals An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs)

2020 ◽  
Vol 08 (09) ◽  
pp. 1729-1735
Author(s):  
Omolara Fatimah Bakre ◽  
Ashiribo Senapon Wusu ◽  
Moses Adebowale Akanbi
Keyword(s):  
Numerical Method ◽  
Initial Value Problems ◽  
Single Step ◽  
Initial Value ◽  
First Order
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