## Book description

Magnetic Materials and 3D Finite Element Modeling explores material characterization and finite element modeling (FEM) applications. This book relates to electromagnetic analysis based on Maxwell’s equations and application of the finite element (FE) method to low frequency devices. A great source for senior undergraduate and graduate students in electromagnetics, it also supports industry professionals working in magnetics, electromagnetics, ferromagnetic materials science and electrical engineering.

The authors present current concepts on ferromagnetic material characterizations and losses. They provide introductory material; highlight basic electromagnetics, present experimental and numerical modeling related to losses and focus on FEM applied to 3D applications. They also explain various formulations, and discuss numerical codes.

• Furnishes algorithms in computational language

• Summarizes concepts related to the FE method

• Uses classical algebra to present the method, making it easily accessible to engineers

Written in an easy-to-understand tutorial format, the text begins with a short presentation of Maxwell’s equations, discusses the generation mechanism of iron losses, and introduces their static and dynamic components. It then demonstrates simplified models for the hysteresis phenomena under alternating magnetic fields. The book also focuses on the Preisach and Jiles–Atherton models, discusses vector hysterisis modeling, introduces the FE technique, and presents nodal and edge elements applied to 3D FE formulation connected to the hysteretic phenomena.

The book discusses the concept of source-field for magnetostatic cases, magnetodynamic fields, eddy currents, and anisotropy. It also explores the need for more sophisticated coding, and presents techniques for solving linear systems generated by the FE cases while considering advantages and drawbacks.

## Table of contents

- Preface
- Authors
- Chapter 1 Statics and Quasistatics EIectromagnetics Brief Presentation
- 1.1 Introduction
- 1.2 Maxwellâs Equations
- 1.3 Maxwellâs Equations: Local Form
- 1.4 Maxwellâs Equations: Iintegral Form
- 1.5 Maxwellâs Equations In Low Frequency
- 1.6 Electrostatics
- 1.6.1 Refraction of the Electric Field
- 1.6.2 Laplaceâs and Poissonâs equations of the electric field for dielectric media
- 1.6.3 Laplaceâs Equation of The Electric Field FR Conductive Media
- 1.7 Magnetostatic Fields
- 1.7.1 Equation rot H = J
- 1.7.2 Equation div B = 0
- 1.7.3 Equation rot E = 0
- 1.7.4 Biot-Savart Law
- 1.7.5 Magnetic Field Refraction
- 1.7.6 Energy in The Magnetic Field
- 1.8 Magnetic Materials
- 1.8.1 Diamagnetic Materials
- 1.8.2 Paramagnetic Materials
- 1.8.3 Ferromagnetic Materials
- 1.8.3.1 General Presentation
- 1.8.3.2 Influence of Iron on Magnetic Circuits
- 1.8.4 Permanent Magnets
- 1.8.4.1 General Presentation
- 1.8.4.2 Principal Types of Permanent Magnets
- 1.8.4.3 Dynamic 0peration of Permanent Magnets
- 1.9 Inductance and Mutual Inductance
- 1.9.1 Definition of Inductance
- 1.9.2 Energy in a Linear System
- 1.10 Magnetodynamic Fields
- 1.10.1 Maxwellâs Equations for the Magnetodynamic Field
- 1.10.2 Penetration of Time-Dependent Fields in Conducting Materials
- 1.10.2.1 Equation for H
- 1.10.2.2 Equation for B
- 1.10.2.3 Equation for E
- 1.10.2.4 Equation for I
- 1.10.2.5 SoIution of the Equations
- 1.11 Fields defined by potentials
- 1.11.1 Electric Scalar Potential
- 1.11.2 Magnetic scalar potential
- 1.11.3 Magnetic vector potential
- 1.11.4 Electric vector potential
- 1.12 Final considerations
- References
- Chapter 2 Ferromagnetic Materials and lron Losses
- 2.1 Introduction
- 2.2 Basic Concepts
- 2.3 Loss Components
- 2.4 Iron Losses Under Alternating, Rotating, and DCâBiased Inductions
- 2.4.1 Epsteinâs Frame And Workbench
- 2.4.1.1 Methodology for Iron Loss Separation
- 2.4.1.2 Results for Two Different Iron Sheets
- 2.4.1.3 Considering Eddy Current in Epsteinâs Frame Corners
- 2.4.1.4 Improved Model for the Eddy Current Losses
- 2.4.1.5 Results Verification by 3D FE Modeling
- 2.4.2 Single Sheet Tester
- 2.4.3 Rotational Single Sheet Tester
- 2.4.4 DC-Biased Induction
- 2.5 Final Considerations
- References
- Chapter 3 Scalar Hysteresis Modeling
- 3.1 Introduction
- 3.2 Preisachâs Scalar Model
- 3.2.1 Magnetization ln Terms of Everettâs Function
- 3.2.2 Identification of Everettâs Function
- 3.2.3 Results Obtained With Preisachâs Scalar Model
- 3.3 JilesâAtherton Scalar Model
- 3.3.1 Original (Direct) JilesâAtherton Model
- 3.3.2 Inverse JilesâAtherton Model
- 3.3.3 JilesâAtherton Model Parameter Determ ination
- 3.3.4 Results Obtained with the JilesâAtherton Model
- 3.3.5 Modified JilesâAtherton Hysteresis Model
- 3.3.6 Determ ination of Parameter R in the Modified JilesâAtherton Model
- 3.3.7 Results of the Modified JilesâAtherton Model
- 3.4 Final Considerations
- References
- Chapter 4 Vector Hysteresis Modeling
- 4.1 Introduction
- 4.2 Vector Model Obtained With the Superposition of Scalar Models
- 4.2.1 Model Principle
- 4.2.2 Identification of the Parameters of the Model
- 4.2.3 Results of the Vector Model
- 4.3 Vector Generalization of the JilesâAtherton Scalar Models
- 4.3.1 Vector Generalization of the Original JilesâAtherton Model
- 4.3.2 Vector Generalization of the Inverse JilesâAtherton Model
- 4.3.3 Some Aspects of the JilesâAtherton Vector Model and Results
- 4.4 Remarks Concerning the Vector Behavior of Hysteresis
- 4.5 Final Considerations
- References
- Chapter 5 Finite Element Method Brief Presentation
- 5.1 Introduction
- 5.2 Galerkin Method: Basic Concepts Using Real Coordinates
- 5.2.1 Equations and Numerical Implementation
- 5.2.2 Boundary Conditions
- 5.2.2.1 Dirichlet Boundary Condition: Imposed Potential
- 5.2.2.2 Neumann Condition: Unknown Nodal Values on the Boundary
- 5.2.3 First Order 2D Finite Element Program
- 5.2.4 Example for the Finite Element Program
- 5.3 Generalization of the Fem: Using Reference Coordinates
- 5.3.1 High-Order Finite elements: general
- 5.3.2 High-Order Finite elements: notation
- 5.3.3 High-Order Finite Elements: Implementation
- 5.3.4 Continuity of finite elements
- 5.3.5 Polynomial Basis
- 5.3.6 Transformation of Quantities: Jacobian
- 5.3.7 Evaluation of the integrals
- 5.4 Numerical Integration
- 5.5 Some Finite Elements
- 5.5.1 First-Order triangular element
- 5.5.2 Second-order triangular element
- 5.5.3 First-order tetrahedral element
- 5.5.4 Implementation Aspects
- 5.6 Using Edge Elements
- 5.6.1 Magnetostatic equation using the vector potential
- 5.6.2 Brief explanation of edge shape functions
- 5.6.3 Applying the Edge Element Shape Functions
- 5.6.4 Implementing the first-order tetrahedron edge element shape functions
- 5.6.5 Applying the galerkin method
- 5.6.6 Coding Tetrahedral Edge Elements
- 5.7 Final Considerations
- References
- Chapter 6 Using Nodal EIements with Magnetic Vector Potential
- 6.1 Introduction
- 6.2 Main Equations
- 6.2.1 Magnetostatic Governing Equation
- 6.2.2 Defining Some Operations
- 6.3 Applying the Galerkin Method
- 6.4 Uniqueness of the Solution: Coulombâs Gauge
- 6.5 Implementation
- 6.6 Example and Comparisons
- 6.7 Final Considerations
- References
- Chapter 7 Source-Field Method for 3D Magnetostatic Fields
- 7.1 Introduction
- 7.2 Magnetostatic Case: Scalar Potential
- 7.2.1 Main Equations
- 7.2.2 Hs Calculation: Edge Tree
- 7.2.3 Facet Tree
- 7.2.4 Applying the Galerkin Method
- 7.2.5 Elemental Matrices: Evaluation, Notation, and Array Dimensions
- 7.2.6 Considering Permanent Magnets
- 7.2.7 Boundary Conditions
- 7.3 Magnetostatic Case: Vector Potential
- 7.3.1 Main Equations
- 7.4 Implementation aspects and Conventions
- 7.4.1 Building the Facets
- 7.4.2 Building the Edges
- 7.4.3 Building the Edge Tree
- 7.4.4 Building the Conductor Facet Tree and Calculating the Flux of J
- 7.4.5 Calculating Hs
- 7.4.6 Applying the Boundary Conditions
- 7.5 Computational implementation
- 7.5.1 Main Subroutines for the Scalar Potential Formulation
- 7.5.2 Main Subroutines for the Vector Potential Formulation
- 7.6 Example and Results
- 7.7 Final Considerations
- References
- Chapter 8 Source-Field Method for 3D Magnetodynamic Fields
- 8.1 Introduction
- 8.2 Formulation Considering Eddy Currents: Time Stepping
- 8.2.1 Governing Equations
- 8.3 Formulation considering eddy currents: complex formulation
- 8.4 Field-Circuit Coupling
- 8.4.1 Basic Equations
- 8.4.2 Applying the Galerkin Method
- 8.4.3 Formulation Considering Eddy currents and Electric Circuit Coupling
- 8.5 Computational Implementation
- 8.6 Differential Permeability Method
- 8.6.1 Nonlinear Cases
- 8.6.2 Anisotropic Cases
- 8.7 Example and Results
- 8.7.1 Eddy Currents, Circuit Coupling, Regular Permeability
- 8.7.2 Example of an Isotropic Nonlinear Case with differential permeability
- 8.7.3 Anisotropic Magnetic Circuit
- 8.7.4 Scalar Hysteresis: A didactical case
- 8.7.5 Vector Hysteresis Anisotropic Case: Team Workshop Problem 32
- 8.8 Final Considerations
- References
- Chapter 9 Matrix-Free Iterative Solution Procedure for Finite Element Problems
- 9.1 Introduction
- 9.2 Classical Fem: T-Scheme
- 9.3 Proposed Technique: N-Scheme
- 9.4 Implementation
- 9.5 Convergence
- 9.6 Implementation of N-Scheme with SOR
- 9.7 Applying the N-Scheme in Nonstationary Solvers
- 9.8 CC Algorithm Implementation
- 9.9 Examples and Results
- 9.9.1 Two-Dimensional Electrostatic Problem
- 9.9.2 Three-Dimensional Nonlinear Case Using SOR Technique
- 9.9.3 Example with a large number of unknowns
- 9.10 Results and Discussion
- 9.11 Final Considerations
- References
- Index

## Product information

- Title: Magnetic Materials and 3D Finite Element Modeling
- Author(s):
- Release date: April 2017
- Publisher(s): CRC Press
- ISBN: 9781351831512

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