Professor, today's topic is about casting solidification simulation, correct? What exactly is it?

It's the solidification analysis of the casting process. It's used for predicting shrinkage cavities, optimizing directional solidification, and designing the gating and riser systems.

How do we actually solve these equations on a computer?

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

What specifically do you mean by matrix solution algorithms?

We solve the system of equations 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.

Solidification Simulation in Casting

Q: What kind of equations are involved in casting solidification simulation?

Casting solidification simulation

Category: 熱解析 | Integrated 2026-04-06

CAE visualization for casting solidification theory - technical simulation diagram

鋳造凝固シミュレーション

Theory and Physics

Overview

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

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Solidification analysis of the casting process. Used for shrinkage cavity prediction, optimization of directional solidification, and gating/riser design.

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Wait, wait, so you mean the solidification solution for the casting process... can it also be used in cases like this?

Governing Equations

$$ \frac{\partial T}{\partial t} = \frac{k}{\rho c_p}\nabla^2 T + \frac{L_f}{c_p}\frac{\partial f_s}{\partial t} $$

$$ t_s = C_m\left(\frac{V}{A}\right)^2 $$

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I see... casting solidification simulation 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 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. higher-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 be used to do casting 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 History and Product Integration Background

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Is the origin of each software quite dramatic?

Ansys Fluent

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

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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 content?

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Developed by CD-adapco. Acquired by Siemens in 2016 and integrated into the Simcenter brand. Polyhedral mesh is a feature.

Current affiliation: Siemens Digital Industries Software

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

COMSOL Multiphysics

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Please 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 super interesting! Please 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	3D CAD data exchange format compliant with ISO 10303. 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 to the correspondence of element types, compatibility of material models, and differences in the representation of loads/boundary conditions. Especially, higher-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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