Next is 'License and Terms of Use.' What does this cover?

Depending on the type of open-source license (e.g., GPL, LGPL, Apache, BSD), obligations for publishing modified code and restrictions on commercial use vary. It is recommended to review the license terms before using it in a project and consult with your company's legal department in advance. Also consider the handling of derivative works and the possibility of dual licensing.

Could you explain the concept behind this equation?

The displacement field $\mathbf{u}$ that renders the functional $\Pi$ stationary is the equilibrium solution. Solvers like CalculiX and Code_Aster implement Galerkin methods based on this variational principle.

Um... what physical phenomenon does each term represent?

By applying this integral form to each control volume and numerically evaluating the flux across the surfaces, the discrete equations are obtained.

Gmsh Mesh Generation

Q: Professor! Today's topic is about Gmsh mesh generation, right? What is it?

Gmsh is an open-source CAD and mesh generation tool. It enables fast tetrahedral/hexahedral mesh generation using Delaunay, Frontal, and HXT parallel algorithms. It is fully scriptable via GEO, Python, and Julia APIs.

Category: 解析 | Integrated 2026-04-06

CAE visualization for gmsh meshing theory - technical simulation diagram

Gmshメッシュ生成

Theory and Physics

Overview

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Teacher! Today's topic is about Gmsh mesh generation, right? What is it like?

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Gmsh is an open-source CAD/mesh generation tool. It enables fast tetra/hexa mesh generation via Delaunay, Frontal, and HXT parallel algorithms. It is fully scriptable via GEO/Python/Julia API.

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Wait, wait, it's open-source, so does that mean it can be used in cases like this too?

Governing Equations

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Expressing this with an equation, it looks like this.

$$\text{Delaunay condition}: \text{No other points inside the circumcircle}$$

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Hmm, just the equation doesn't really click for me... What does it represent?

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Element size field:

$$h(\mathbf{x}) = \min_i \left(h_{min,i} + \frac{h_{max,i}-h_{min,i}}{1+\exp(-k(d_i-d_0))}\right)$$

Theoretical Foundation

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I've heard of "theoretical foundation," but I might not fully understand it...

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The numerical solution methods for Gmsh mesh generation are based on the Finite Volume Method (FVM) or the Finite Element Method (FEM). Being open-source, its greatest advantage is the ability to check and modify algorithm details at the source code level. Discretization schemes and convergence criteria logic, which are black boxes in commercial solvers, can be directly verified, making it particularly suitable for academic research and method development. Continuous improvement and bug fixes by the community ensure its quality.

Licensing and Terms of Use

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Next is "Licensing and Terms of Use"! What is this about?

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Depending on the type of open-source license (GPL, LGPL, Apache, BSD, etc.), obligations for publishing modified code and restrictions on commercial use differ. It is recommended to check the license terms before using it in a project and to consult with the internal legal department beforehand. Also consider the handling of derivative works and the possibility of dual licensing.

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Wow~, the talk about open-source licenses is super interesting! Tell me more.

Theoretical Background of Numerical Methods

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Next is "Theoretical Background of Numerical Methods"! What is this about?

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Explains the theoretical foundation of the numerical methods implemented by open-source CAE tools.

Variational Principle of the Finite Element Method (FEM)

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Please teach me about the "Finite Element Method"!

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The principle of minimum potential energy, which is the foundation of structural analysis:

$$ \Pi(\mathbf{u}) = \frac{1}{2} \int_{\Omega} \boldsymbol{\sigma} : \boldsymbol{\varepsilon} \, d\Omega - \int_{\Omega} \mathbf{f} \cdot \mathbf{u} \, d\Omega - \int_{\Gamma_t} \mathbf{t} \cdot \mathbf{u} \, d\Gamma $$

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The displacement field $\mathbf{u}$ that makes $\Pi$ stationary is the equilibrium solution. CalculiX and Code_Aster implement the Galerkin method based on this variational principle.

Conservation Law of the Finite Volume Method (FVM)

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Please teach me about the "Finite Volume Method"!

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The FVM adopted by OpenFOAM is based on the integral conservation law for a control volume:

$$ \frac{\partial}{\partial t} \int_{V} \rho \phi \, dV + \oint_{S} \rho \phi \mathbf{u} \cdot d\mathbf{S} = \oint_{S} \Gamma \nabla \phi \cdot d\mathbf{S} + \int_{V} S_\phi \, dV $$

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Discrete equations are obtained by applying this integral form to each control volume and numerically evaluating the fluxes on the faces.

Licensing and Quality Assurance

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Please teach me about "Licensing and Quality Assurance"!

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Because the source code is public, open-source CAE allows third-party verification of algorithms. On the other hand, there is no vendor support like with commercial tools, so information sharing within user communities and forums is important.

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Wow~, the talk about open-source is super interesting! Tell me more.

Application Conditions and Precautions

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I've heard of "Application Conditions and Precautions," but I might not fully understand it...

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Results from OSS tools should always be verified with known benchmark problems.
Be aware of incompatibilities between versions (especially differences between OpenFOAM forks).
It is recommended to confirm the accuracy of OSS by comparing results with commercial tools.
When documentation is insufficient, direct reference to the source code may be necessary.

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Wait, wait, "tool results" means it can be used in cases like this too?

Dimensionless Parameters and Dominant Scales

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I've heard of "Dimensionless Parameters and Dominant Scales," but I might not fully understand it...

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Understanding the dimensionless parameters governing the physical phenomenon being analyzed is fundamental to appropriate model selection and parameter setting.

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Peclet Number Pe: Relative importance of convection and diffusion. Pe >> 1 indicates convection dominance (stabilization methods required).
Reynolds Number Re: Ratio of inertial forces to viscous forces. A fundamental parameter for fluid problems.
Biot Number Bi: Ratio of internal conduction to surface convection. For Bi < 0.1, the lumped capacitance method can be applied.
Courant Number CFL: Indicator of numerical stability. For explicit methods, CFL ≤ 1 is required.

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Ah, I see! So that's how the mechanism of "the physical phenomenon being analyzed" works.

Verification via Dimensional Analysis

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Please teach me about "Verification via Dimensional Analysis"!

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Dimensional analysis based on Buckingham's Π theorem is effective for order-of-magnitude estimation of analysis results. Using characteristic length $L$, characteristic velocity $U$, and characteristic time $T = L/U$, estimate the order of each physical quantity beforehand to confirm the validity of the analysis results.

Classification and Mathematical Characteristics of Boundary Conditions

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I've heard that if you get the boundary conditions wrong, everything fails...

Type	Mathematical Expression	Physical Meaning	Example
Dirichlet Condition	$u = u_0$ on $\Gamma_D$	Specification of variable value	Fixed wall, specified temperature
Neumann Condition	$\partial u/\partial n = g$ on $\Gamma_N$	Specification of gradient (flux)	Heat flux, force
Robin Condition	$\alpha u + \beta \partial u/\partial n = h$	Linear combination of variable and gradient	Convective heat transfer
Periodic Boundary Condition	$u(x) = u(x+L)$	Spatial periodicity	Unit cell analysis

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Choosing appropriate boundary conditions is directly linked to solution uniqueness and physical validity. Insufficient boundary conditions lead to an ill-posed problem, while excessive ones cause contradictions.

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I've grasped the overall picture of Gmsh mesh generation! I'll try to be mindful of it in my practical work from tomorrow.

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Yeah, you're doing great! Actually getting your hands dirty is the best way to learn. If you don't understand something, feel free to ask anytime.

Coffee Break Yomoyama Talk

The Delaunay Theory that Gave Birth to Gmsh—A Gift from Geometers Across a Century

At the core of Gmsh lies "Delaunay triangulation," a geometry theory proposed in 1934. It is a partition satisfying the condition that "no other points lie inside the circumcircle of any triangle," and this provides the theoretical basis for automatically generating the "least distorted" mesh. Christophe Geuzaine and Jean-François Remacle of Gmsh extended this algorithm to three dimensions and added a method to accurately follow boundaries using Bézier curves. When the first version was released in 1998, it surprised the research community with the thought, "Can a personally developed tool do this much?"

Physical Meaning of Each Term

Time Variation Term of Conserved Quantity: Represents the rate of change over time of the physical quantity in question. Becomes zero for steady-state problems. 【Image】When filling a bathtub with hot water, the water level rises over time—this "rate of change per time" is the time variation term. The state where the valve is closed and the water level is constant is "steady," and the time variation term is zero.
Flux Term (Flow Term): Describes the spatial transport/diffusion of a physical quantity. Broadly classified into convection and diffusion. 【Image】Convection is like "a river's current carrying a boat," where things are carried along by the flow. Diffusion is like "ink naturally spreading in still water," where things move due to concentration differences. The competition between these two transport mechanisms governs many physical phenomena.
Source Term (Generation/Destruction Term): Represents the local generation or destruction of a physical quantity, such as external forces or reaction terms. 【Image】Turning on a heater in a room "generates" thermal energy at that location. When fuel is consumed in a chemical reaction, mass is "destroyed." It is the term representing physical quantities injected into the system from the outside.

Assumptions and Applicability Limits

The continuum assumption holds for the spatial scale.
The constitutive laws of materials/fluids (stress-strain relation, Newtonian fluid law, etc.) are within the applicable range.
Boundary conditions are physically valid and mathematically well-defined.

Dimensional Analysis and Unit Systems

Variable	SI Unit	Notes / Conversion Memo
Characteristic Length $L$	m	Must match the unit system of the CAD model.
Characteristic Time $t$	s	For transient analysis, time step should consider CFL condition and physical time constants.

Numerical Methods and Implementation

Details of Numerical Methods

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Specifically, what kind of algorithm solves Gmsh mesh generation?

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Explains the key points of the numerical methods and implementation for Gmsh mesh generation.

Compilation and Build

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I've heard of "Compilation and Build," but I might not fully understand it...

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Building from source code uses CMake or dedicated build systems (like wmake for OpenFOAM). Proper version management of dependency libraries (MPI, PETSc, BLAS/LAPACK, etc.) is important. Linux environment is recommended, but it can also be built on Windows using WSL2 or Docker containers.

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