book

Magnetic Materials and 3D Finite Element Modeling

by João A. Bastos, Nelson Sadowski

April 2017

Intermediate to advanced

402 pages

9h 2m

English

CRC Press

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Preface
Authors
Chapter 1 Statics and Quasistatics EIectromagnetics Brief Presentation
1.1 Introduction1.2 Maxwell’s Equations1.3 Maxwell’s Equations: Local Form1.4 Maxwell’s Equations: Iintegral Form1.5 Maxwell’s Equations In Low Frequency1.6 Electrostatics1.6.1 Refraction of the Electric Field1.6.2 Laplace’s and Poisson’s equations of the electric field for dielectric media1.6.3 Laplace’s Equation of The Electric Field FR Conductive Media1.7 Magnetostatic Fields1.7.1 Equation rot H = J1.7.2 Equation div B = 01.7.3 Equation rot E = 01.7.4 Biot-Savart Law1.7.5 Magnetic Field Refraction1.7.6 Energy in The Magnetic Field1.8 Magnetic Materials1.8.1 Diamagnetic Materials1.8.2 Paramagnetic Materials1.8.3 Ferromagnetic Materials1.8.3.1 General Presentation1.8.3.2 Influence of Iron on Magnetic Circuits1.8.4 Permanent Magnets1.8.4.1 General Presentation1.8.4.2 Principal Types of Permanent Magnets1.8.4.3 Dynamic 0peration of Permanent Magnets1.9 Inductance and Mutual Inductance1.9.1 Definition of Inductance1.9.2 Energy in a Linear System1.10 Magnetodynamic Fields1.10.1 Maxwell’s Equations for the Magnetodynamic Field1.10.2 Penetration of Time-Dependent Fields in Conducting Materials1.10.2.1 Equation for H1.10.2.2 Equation for B1.10.2.3 Equation for E1.10.2.4 Equation for I1.10.2.5 SoIution of the Equations1.11 Fields defined by potentials1.11.1 Electric Scalar Potential1.11.2 Magnetic scalar potential1.11.3 Magnetic vector potential1.11.4 Electric vector potential1.12 Final considerations
References
Chapter 2 Ferromagnetic Materials and lron Losses
2.1 Introduction2.2 Basic Concepts2.3 Loss Components2.4 Iron Losses Under Alternating, Rotating, and DC‐Biased Inductions2.4.1 Epstein’s Frame And Workbench2.4.1.1 Methodology for Iron Loss Separation2.4.1.2 Results for Two Different Iron Sheets2.4.1.3 Considering Eddy Current in Epstein’s Frame Corners2.4.1.4 Improved Model for the Eddy Current Losses2.4.1.5 Results Verification by 3D FE Modeling2.4.2 Single Sheet Tester2.4.3 Rotational Single Sheet Tester2.4.4 DC-Biased Induction2.5 Final ConsiderationsReferences
Chapter 3 Scalar Hysteresis Modeling
3.1 Introduction3.2 Preisach’s Scalar Model3.2.1 Magnetization ln Terms of Everett’s Function3.2.2 Identification of Everett’s Function3.2.3 Results Obtained With Preisach’s Scalar Model3.3 Jiles‐Atherton Scalar Model3.3.1 Original (Direct) Jiles–Atherton Model3.3.2 Inverse Jiles–Atherton Model3.3.3 Jiles–Atherton Model Parameter Determ ination3.3.4 Results Obtained with the Jiles–Atherton Model3.3.5 Modified Jiles–Atherton Hysteresis Model3.3.6 Determ ination of Parameter R in the Modified Jiles–Atherton Model3.3.7 Results of the Modified Jiles–Atherton Model3.4 Final ConsiderationsReferences
Chapter 4 Vector Hysteresis Modeling
4.1 Introduction4.2 Vector Model Obtained With the Superposition of Scalar Models4.2.1 Model Principle4.2.2 Identification of the Parameters of the Model4.2.3 Results of the Vector Model4.3 Vector Generalization of the Jiles‐Atherton Scalar Models4.3.1 Vector Generalization of the Original Jiles–Atherton Model4.3.2 Vector Generalization of the Inverse Jiles–Atherton Model4.3.3 Some Aspects of the Jiles–Atherton Vector Model and Results4.4 Remarks Concerning the Vector Behavior of Hysteresis4.5 Final ConsiderationsReferences
Chapter 5 Finite Element Method Brief Presentation
5.1 Introduction5.2 Galerkin Method: Basic Concepts Using Real Coordinates5.2.1 Equations and Numerical Implementation5.2.2 Boundary Conditions5.2.2.1 Dirichlet Boundary Condition: Imposed Potential5.2.2.2 Neumann Condition: Unknown Nodal Values on the Boundary5.2.3 First Order 2D Finite Element Program5.2.4 Example for the Finite Element Program5.3 Generalization of the Fem: Using Reference Coordinates5.3.1 High-Order Finite elements: general5.3.2 High-Order Finite elements: notation5.3.3 High-Order Finite Elements: Implementation5.3.4 Continuity of finite elements5.3.5 Polynomial Basis5.3.6 Transformation of Quantities: Jacobian5.3.7 Evaluation of the integrals
5.4 Numerical Integration
5.5 Some Finite Elements5.5.1 First-Order triangular element5.5.2 Second-order triangular element5.5.3 First-order tetrahedral element5.5.4 Implementation Aspects5.6 Using Edge Elements5.6.1 Magnetostatic equation using the vector potential5.6.2 Brief explanation of edge shape functions5.6.3 Applying the Edge Element Shape Functions5.6.4 Implementing the first-order tetrahedron edge element shape functions5.6.5 Applying the galerkin method5.6.6 Coding Tetrahedral Edge Elements5.7 Final ConsiderationsReferences
Chapter 6 Using Nodal EIements with Magnetic Vector Potential
6.1 Introduction6.2 Main Equations6.2.1 Magnetostatic Governing Equation6.2.2 Defining Some Operations6.3 Applying the Galerkin Method6.4 Uniqueness of the Solution: Coulomb’s Gauge6.5 Implementation6.6 Example and Comparisons6.7 Final ConsiderationsReferences

Chapter 7 Source-Field Method for 3D Magnetostatic Fields
7.1 Introduction7.2 Magnetostatic Case: Scalar Potential7.2.1 Main Equations7.2.2 Hs Calculation: Edge Tree7.2.3 Facet Tree7.2.4 Applying the Galerkin Method7.2.5 Elemental Matrices: Evaluation, Notation, and Array Dimensions7.2.6 Considering Permanent Magnets7.2.7 Boundary Conditions7.3 Magnetostatic Case: Vector Potential7.3.1 Main Equations7.4 Implementation aspects and Conventions7.4.1 Building the Facets7.4.2 Building the Edges7.4.3 Building the Edge Tree7.4.4 Building the Conductor Facet Tree and Calculating the Flux of J7.4.5 Calculating Hs7.4.6 Applying the Boundary Conditions7.5 Computational implementation7.5.1 Main Subroutines for the Scalar Potential Formulation7.5.2 Main Subroutines for the Vector Potential Formulation7.6 Example and Results7.7 Final ConsiderationsReferences
Chapter 8 Source-Field Method for 3D Magnetodynamic Fields
8.1 Introduction8.2 Formulation Considering Eddy Currents: Time Stepping8.2.1 Governing Equations8.3 Formulation considering eddy currents: complex formulation8.4 Field-Circuit Coupling8.4.1 Basic Equations8.4.2 Applying the Galerkin Method8.4.3 Formulation Considering Eddy currents and Electric Circuit Coupling8.5 Computational Implementation8.6 Differential Permeability Method8.6.1 Nonlinear Cases8.6.2 Anisotropic Cases8.7 Example and Results8.7.1 Eddy Currents, Circuit Coupling, Regular Permeability8.7.2 Example of an Isotropic Nonlinear Case with differential permeability8.7.3 Anisotropic Magnetic Circuit8.7.4 Scalar Hysteresis: A didactical case8.7.5 Vector Hysteresis Anisotropic Case: Team Workshop Problem 328.8 Final ConsiderationsReferences
Chapter 9 Matrix-Free Iterative Solution Procedure for Finite Element Problems
9.1 Introduction9.2 Classical Fem: T-Scheme9.3 Proposed Technique: N-Scheme9.4 Implementation9.5 Convergence9.6 Implementation of N-Scheme with SOR9.7 Applying the N-Scheme in Nonstationary Solvers9.8 CC Algorithm Implementation9.9 Examples and Results9.9.1 Two-Dimensional Electrostatic Problem9.9.2 Three-Dimensional Nonlinear Case Using SOR Technique9.9.3 Example with a large number of unknowns9.10 Results and Discussion9.11 Final ConsiderationsReferences
Index

Content preview from Magnetic Materials and 3D Finite Element Modeling

5.4 NUMERICAL INTEGRATION

When calculating, for example, the first term of Equation 5.69, it is necessary to evaluate the following integral:

ρ D \int_{0}^{1} \int_{0}^{1 - v} (1 - u - v) d u d v = ρ D \int_{0}^{1} {(u - \frac{u^{2}}{2} - u v)}_{0}^{1 - v} d v

= ρ D \int_{0}^{1} (1 - v - \frac{{(1 - v)}^{2}}{2} - v (1 - v)) d v = \frac{p D}{6}

which is the result shown in Equation 5.69.

It is clear from the example earlier that even for the very simple 2D element, the integration requires much work. This is in spite of the fact that the $det J$ is a constant. For second‐order elements, the shape functions are much more complex and $det J$ is not necessarily constant. Therefore, in general, it is practically impossible to evaluate ...

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Magnetic Materials and 3D Finite Element Modeling

by João A. Bastos, Nelson Sadowski

5.4 NUMERICAL INTEGRATION

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