What kind of equations are involved in melting and solidification simulation?

The governing equations for melting and solidification simulation.

What specifically does the matrix solver algorithm entail?

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

Melting and Solidification Simulation

Q: Professor! Today's topic is melting and solidification simulation, right? What is it?

It's a coupled fluid flow and heat transfer analysis involving phase change. The solid-liquid coexistence region is modeled using the enthalpy-porosity method.

Q: How do you actually solve these equations on a computer?

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

Category: 熱解析 | Integrated 2026-04-06

CAE visualization for melting solidification theory - technical simulation diagram

融解・凝固シミュレーション

Theory and Physics

Overview

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Teacher! Today's topic is about melting/solidification simulation, right? What is it like?

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Coupled flow and heat transfer analysis involving phase change. The solid-liquid coexistence region is modeled using the enthalpy-porosity method.

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So, if you cut corners on the flow and heat transfer part involving phase change, you'll face serious consequences later. I'll keep that in mind!

Governing Equations

$$ \frac{\partial}{\partial t}(\rho H) + \nabla\cdot(\rho\mathbf{u}H) = \nabla\cdot(k\nabla T) $$

$$ S_i = -\frac{C(1-f_l)^2}{f_l^3+\epsilon}u_i $$

Discretization Method

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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 matrix 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 are 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 face serious consequences later. I'll keep that in mind!

Implementation in Commercial Tools

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So, what software can be used for melting/solidification simulation?

Tool Name	Developer/Current	Main File Formats
Ansys Fluent	Ansys Inc.	.cas, .dat, .msh, .jou
Simcenter STAR-CCM+	Siemens Digital Industries Software	.sim, .java, .csv
COMSOL Multiphysics	COMSOL AB	.mph
Abaqus FEA (SIMULIA)	Dassault Systèmes SIMULIA	.inp, .odb, .cae, .sta, .msg

Vendor Lineage and Product Integration History

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Does each software have a dramatic origin story?

Ansys Fluent

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Next is about Ansys Fluent. What's the story?

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Developed by Fluent Inc. Acquired by Ansys in 2006. A general-purpose CFD solver based on unstructured grids.

Current affiliation: Ansys Inc.

Simcenter STAR-CCM+

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Next is about Simcenter STAR. What's the story?

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Developed by CD-adapco. Acquired by Siemens in 2016 and integrated into the Simcenter brand. Features polyhedral meshes.

Current affiliation: Siemens Digital Industries Software

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After hearing this, I finally understand why development is so important!

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

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Wow, the story of development is incredibly interesting! Tell me more.

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.
CGNS	.cgns	CFD Data	CFD General Notation System. Standard exchange format for CFD results.
VTK	.vtk/.vtu	Visualization	Visualization Toolkit format. Used by ParaView, etc.

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When converting models between different solvers, attention is needed regarding 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 are actually very profound.

Practical Considerations

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Are there things like "field wisdom" that aren't in textbooks?

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