Professor! Today's topic is the Heisler chart, right? What exactly is it?

It's a classical method that charts one-dimensional unsteady conduction in slabs, cylinders, and spheres using dimensionless parameters. It's used for initial design estimates.

What kind of equation is involved in the Heisler chart?

The Heisler chart is a graphical solution to the transient heat conduction equation for these simple geometries.

How is this equation actually solved using a computer?

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

What specifically is meant by a matrix solution algorithm?

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

High-Throughput Chart

Category: 熱解析 | Integrated 2026-04-06

CAE visualization for heisler charts theory - technical simulation diagram

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Theory and Physics

Overview

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Teacher! Today's topic is about Heisler charts, right? What are they?

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It's a classical method that charts one-dimensional unsteady conduction for slabs, cylinders, and spheres using dimensionless parameters. Used for initial design estimation.

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So, if you cut corners on the slab/cylinder/sphere part, you'll pay for it later. I'll keep that in mind!

Governing Equations

$$ \theta^* = \frac{T - T_\infty}{T_i - T_\infty} = f(Bi, Fo) $$

$$ \theta^*_0 = C_1 e^{-\zeta_1^2 Fo} $$

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 a 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 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). Preconditioned iterative methods are effective for large-scale problems.

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 work with Heisler charts?

Tool Name	Developer/Current	Main File Formats
Ansys Mechanical (formerly ANSYS Structural)	Ansys Inc.	.cdb, .rst, .db, .ans, .mac
Abaqus FEA (SIMULIA)	Dassault Systèmes SIMULIA	.inp, .odb, .cae, .sta, .msg
COMSOL Multiphysics	COMSOL AB	.mph
Simcenter STAR-CCM+	Siemens Digital Industries Software	.sim, .java, .csv

Vendor History and Product Integration Background

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Are the origins of each software quite dramatic?

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). APDL (Ansys Parametric Design Language) based.

Current affiliation: Ansys Inc.

Abaqus FEA (SIMULIA)

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What exactly is Abaqus FEA?

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Developed in 1978 by HKS (Hibbitt, Karlsson & Sorensen). Acquired by Dassault Systèmes in 2005 and integrated into the SIMULIA brand.

Current affiliation: Dassault Systèmes SIMULIA

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Now 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 COMSOL. Strong in multiphysics.

Current affiliation: COMSOL AB

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Wow, the story of development is super 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	3D CAD data exchange format compliant with ISO 10303. 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 must be paid to the correspondence of element types, compatibility of material models, and differences in the representation of loads and boundary conditions. Particularly, 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 deep, aren't they?

Practical Considerations

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