Closed-form formula of magnetic gradient tensor for a homogeneous polyhedral magnetic target: A tetrahedral grid example

Geophysics ◽  
2017 ◽  
Vol 82 (6) ◽  
pp. WB21-WB28 ◽  
Author(s):  
Zhengyong Ren ◽  
Chaojian Chen ◽  
Jingtian Tang ◽  
Huang Chen ◽  
Shuanggui Hu ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Closed Form ◽  
Magnetization Vector ◽  
Gradient Tensor ◽  
Magnetic Gradient ◽  
Closed Form Solutions ◽  
Tetrahedral Grid ◽  
Pipeline Model ◽  
The Magnetic Field ◽  
Closed Form Formula

A closed-form formula is developed for the full magnetic gradient tensor of a polyhedral body with a homogeneous magnetization vector. It is based on the direct derivative technique on the closed form of the magnetic field. These analytical expressions are implemented into an easy-to-use C++ package which simultaneously calculates the magnetic potential, the magnetic field, and the full magnetic gradient tensor for magnetic targets. Modern unstructured tetrahedral grids are adopted to represent the polyhedral body so that our code can deal with arbitrarily complicated magnetic targets. A prismatic body is tested to verify the accuracies of our closed-form formula. Excellent agreements are obtained between our closed-form solutions and solutions of a prismatic magnetic body with differences up to machine precision. A pipeline model is used to demonstrate its capability to deal with complicated magnetic targets. This C++ code is freely available to the magnetic exploration community.

New analytical expression of the magnetic gradient tensor for homogeneous polyhedrons

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. A31-A35 ◽  
Author(s):  
Zhengyong Ren ◽  
Huang Chen ◽  
Chaojian Chen ◽  
Yiyuan Zhong ◽  
Jingtian Tang
Keyword(s):  
Magnetic Field ◽  
Unexploded Ordnance ◽  
Gradient Tensor ◽  
Magnetic Gradient ◽  
Analytical Expression ◽  
Closed Form Solutions ◽  
Tensor Data ◽  
Analytical Expressions ◽  
The Magnetic Field ◽  
Surface Integrals

We have developed a new analytical expression for the magnetic-gradient tensor for polyhedrons with homogeneous magnetization vectors. Instead of performing the direct derivative on the closed-form solutions of the magnetic field, it is obtained by first transforming the volume integrals of the magnetic-field tensor into surface integrals over polyhedral facets, in terms of the gradient theorem. Second, the surface divergence theorem transforms the surface integrals over polyhedral facets into edge integrals and structure-simplified surface integrals. Third, we develop analytical expressions for these edge integrals and simplified surface integrals. We use a synthetic prismatic target to verify the accuracies of the new analytical expression. Excellent agreements are obtained between our results and those calculated by other published formulas. The new analytical expression of the magnetic-gradient tensor can play a fundamental role in advancing magnetic mineral explorations, environmental surveys, unexploded ordnance and submarine detection, aeromagnetic and marine magnetic surveys because more and more magnetic tensor data have been collected by magnetic-tensor gradiometry instruments.

Fast 3D forward modeling of the magnetic field and gradient tensor on an undulated surface

2018 ◽  
Vol 15 (3-4) ◽  
pp. 500-512
Author(s):  
Kun Li ◽  
Long-Wei Chen ◽  
Qing-Rui Chen ◽  
Shi-Kun Dai ◽  
Qian-Jiang Zhang ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Forward Modeling ◽  
Gradient Tensor ◽  
The Magnetic Field ◽  
3D Forward Modeling

A NOVEL ANALYTICAL APPROACH FOR PRICING DISCRETELY SAMPLED GAMMA SWAPS IN THE HESTON MODEL

2016 ◽  
Vol 57 (3) ◽  
pp. 244-268
Author(s):  
SANAE RUJIVAN
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Analytical Approach ◽  
Solution Procedure ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Variance Swaps ◽  
Closed Form Solutions ◽  
Volatility Model ◽  
Closed Form Formula

The main purpose of this paper is to present a novel analytical approach for pricing discretely sampled gamma swaps, defined in terms of weighted variance swaps of the underlying asset, based on Heston’s two-factor stochastic volatility model. The closed-form formula obtained in this paper is in a much simpler form than those proposed in the literature, which substantially reduces the computational burden and can be implemented efficiently. The solution procedure presented in this paper can be adopted to derive closed-form solutions for pricing various types of weighted variance swaps, such as self-quantoed variance and entropy swaps. Most interestingly, we discuss the validity of the current solutions in the parameter space, and provide market practitioners with some remarks for trading these types of weighted variance swaps.

On Geometrical Invariants of the Magnetic Field Gradient Tensor in Turbulent Space Plasmas: Scale Variability in the Inertial Range

2019 ◽  
Vol 878 (2) ◽  
pp. 124 ◽  
Author(s):  
Virgilio Quattrociocchi ◽  
Giuseppe Consolini ◽  
Maria Federica Marcucci ◽  
Massimo Materassi
Keyword(s):  
Magnetic Field ◽  
Field Gradient ◽  
Magnetic Field Gradient ◽  
Inertial Range ◽  
Space Plasmas ◽  
Gradient Tensor ◽  
The Magnetic Field

RESPONSE OF DYKE TO OSCILLATING DIPOLE

Geophysics ◽  
1958 ◽  
Vol 23 (1) ◽  
pp. 128-133 ◽  
Author(s):  
James Paul Wesley
Keyword(s):  
Magnetic Field ◽  
Closed Form ◽  
Scalar Potential ◽  
Three Dimensions ◽  
Electromagnetic Response ◽  
Dipole Source ◽  
First Approximation ◽  
Sulfide Ore ◽  
Infinite Conductivity ◽  
The Magnetic Field

A dyke of sulfide ore may be geophysically prospected by observing its electromagnetic response to a slowly oscillating magnetic dipole source. An excellent first approximation of the fields generated is obtained by considering the idealized case of a dyke of infinite conductivity and vanishing thickness in a vacuum. Surprisingly, this idealized problem can be solved exactly in terms of a newly discovered Green’s function for Laplace’s equation (in three dimensions) which is simply expressed in closed form. The magnetic scalar potential and the magnetic field are given for final results.

Closed Form Solutions For Mixed Convection With Magnetohydrodynamic Effect in a Vertical Porous Annulus Surrounding an Electric Cable

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
A. Barletta ◽  
E. Magyari ◽  
S. Lazzari ◽  
I. Pop
Keyword(s):  
Mixed Convection ◽  
Closed Form ◽  
Heat Generation ◽  
Internal Heat ◽  
Boussinesq Approximation ◽  
Closed Form Solutions ◽  
Electric Cable ◽  
Special Cases ◽  
Magnetohydrodynamic Effect ◽  
The Magnetic Field

Mixed convection Darcy flow in a vertical porous annulus around a straight electric cable is investigated. It is assumed that the flow is fully developed and parallel. Moreover, the Boussinesq approximation is used. The magnetic field with a steady electric current in the cable is radially varying according to the Biot–Savart law. Two flow regimes are investigated. The first is mixed convection with negligible effects of internal heat generation due to Joule heating and viscous dissipation. The second is forced convection with important effects of heat generation. In these two special cases, closed form expressions of the velocity profile and of the temperature profile, as well as of the flow rate and the Nusselt number, are obtained. The main features of these solutions are discussed.

scholarly journals Comparison of closed-form solutions to experimental magnetic force between two cylindrical magnets

2021 ◽  
pp. 141-146
Author(s):  
Sampart Cheedket ◽  
Chitnarong Sirisathitkul
Keyword(s):  
Magnetic Field ◽  
Closed Form ◽  
Magnetic Force ◽  
Permanent Magnets ◽  
Magnetic Levitation ◽  
Closed Form Solution ◽  
Vertical Tube ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Set Up

The force between permanent magnets implemented in many engineering devices remains an intriguing problem in basic physics. The variation of magnetic force with the distance x between a pair of magnets cannot usually be approximated as x-4 because of the dipole nature and geometry of magnets. In this work, the force between two identical cylindrical magnets is accurately described by a closed-form solution. The analytical model assumes that the magnets are uniformly magnetized along their length. The calculation, based on the magnetic field exerted by one magnet on the other along the direction of their orientation, shows a reduction in the magnetic force with the distance x and a dependence on the size parameters of magnets. To verify the equation, the experiment was set up by placing two cylindrical neodymium iron boron type magnets in a vertical tube. The repulsive force between the identical upper and lower magnets of 2.5 cm in diameter and 7.5 cm in length was measured from the weight on the top of the upper magnet. The resulting separation between the magnets was recorded as x. The forces measured at x=0.004-0.037 m differ from the values calculated using the analytic solution by -0.55 % to -13.60 %. The calculation also gives rise to a practical remnant magnetic field of 1.206 T. When x is much large than the equation of force is approximated as a simple form proportional to 1/x-4. The finding can be directly used in magnetic levitation as well as applied in calculating magnetic fields and forces in other systems incorporating permanent magnets.

Sign in / Sign up

Export Citation Format

Share Document