Professor, today's topic is magnetostrictive analysis, right? What exactly is it?

It's the strain deformation of a material due to magnetization. It's a cause of vibration and noise in transformer cores and is used in the design of Terfenol-D actuators.

What kind of equations are involved in magnetostrictive analysis?

The magnetostrictive constitutive equations, which couple the magnetic field and mechanical strain fields.

How do we actually solve these equations computationally?

We use spatial discretization via the Finite Element Method (FEM). This involves assembling element stiffness matrices to construct the global stiffness equation.

What specifically do you mean by matrix solution algorithms?

We solve the system of equations using either direct methods (like LU or Cholesky decomposition) or iterative methods (like the Conjugate Gradient or GMRES method). For large-scale problems, preconditioned iterative methods are particularly effective.

磁歪解析

Category: 解析 | Integrated 2026-04-06

Theory and Physics

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Strain deformation of materials due to magnetization. Cause of vibration and noise in transformer cores. Terfenol-D actuator design.

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I see... The strain of materials due to magnetization seems simple at first glance, but it's actually very profound, isn't it?

Governing Equations

$$ \varepsilon = \frac{3\lambda_s}{2}\left(\cos^2\theta - \frac{1}{3}\right) $$

$$ \boldsymbol{\sigma} = \mathbf{C}:(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{mag}) $$

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I see... The foundation for describing magnetostriction analysis seems simple at first glance, but it's actually very profound, isn't it?

Discretization Methods

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How do you actually solve these equations on a computer?

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We use spatial discretization by the Finite Element Method (FEM). We assemble the element stiffness matrices and construct the global stiffness equation.

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We perform transformation to the weak form (variational form) and use formulation by the Galerkin method using test functions and shape functions. The choice of element type (low-order elements vs. high-order elements, full integration vs. reduced integration) directly affects the trade-off between solution accuracy and computational cost.

Matrix Solution Algorithms

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What exactly do you mean by matrix solution algorithms?

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We solve the simultaneous equations using direct methods (LU decomposition, Cholesky decomposition) or iterative methods (CG method, GMRES method). For large-scale problems, preconditioned iterative methods are effective.

Solver	Classification	Memory Usage	Applicable Scale
LU decomposition	Direct Method	O(n²)	Small to Medium Scale
Cholesky decomposition	Direct Method (Symmetric Positive Definite)	O(n²)	Small to Medium Scale
PCG Method	Iterative Method	O(n)	Large Scale
GMRES method	Iterative Method	O(n·m)	Large Scale / Non-symmetric
AMG Preconditioner	Preprocessing	O(n)	Very Large Scale

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So, if you cut corners on the finite element method part, you'll pay for it later. I'll keep that in mind!

Implementation in Commercial Tools

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So, what software can you use to do magnetostriction analysis?

Tool Name	Developer/Current	Main File Format
COMSOL Multiphysics	COMSOL AB	.mph
JMAG-Designer	JSOL Corporation	.jmag, .jproj
Ansys Mechanical (formerly ANSYS Structural)	Ansys Inc.	.cdb, .rst, .db, .ans, .mac
Abaqus FEA (SIMULIA)	Dassault Systèmes SIMULIA	.inp, .odb, .cae, .sta, .msg

Vendor Lineage and Product Integration History

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Is the background of each software quite dramatic?

COMSOL Multiphysics

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Tell me about "COMSOL Multiphysics"!

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Founded in Sweden in 1986. Started as FEMLAB with MATLAB integration, later renamed to COMSOL. Strong in multiphysics.

Current Affiliation: COMSOL AB

JMAG-Designer

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What exactly is JMAG?

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Developed by Japan's JSOL Corporation. An electromagnetic field analysis tool specialized for electrical equipment design.

Current Affiliation: JSOL Corporation

Ansys Mechanical (formerly ANSYS Structural)

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Tell me about "Ansys Mechanical"!

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Developed in 1970 by Swanson Analysis Systems Inc. (SASI). Based on APDL (Ansys Parametric Design Language).

Current Affiliation: Ansys Inc.

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Ah, I see! Founded in Sweden in 1986, that's how it was set up.

File Formats and Interoperability

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Are there any points to note when transferring data between different software?

Format	Extension	Type	Overview
STEP	.stp/.step	Neutral CAD	ISO 10303 compliant 3D CAD data exchange format. Supports geometry + PMI.
IGES	.igs/.iges	Neutral CAD	Early CAD data exchange standard. Has issues with surface data compatibility. Transition to STEP is progressing.
VTK	.vtk/.vtu	Visualization	Visualization Toolkit format. Used by ParaView, etc.

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When converting models between different solvers, attention is needed to the correspondence of element types, compatibility of material models, and differences in the representation of loads and boundary conditions. Particularly, high-order elements and special elements (cohesive elements, user-defined elements, etc.) often cannot be directly converted between solvers.

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I see... Formats seem simple at first glance, but they're actually very profound, aren't they?

Practical Considerations

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Are there any "