Could you explain the 'problem setup'?

Geometry: A thick plate (thickness t = 1.0 m) Material: Isotropic elastic (E = 210 GPa, ν = 0.3) Load: Out-of-plane distributed load Constraints: Fixed boundary conditions on an edge Evaluation Point: Point D (on the top surface of the plate)

I've heard we solve it by breaking down the continuous equations. How is that done specifically?

It involves spatial discretization using the Finite Element Method (FEM), based on the weak form derived via the Galerkin method.

NAFEMS LE10: Thick-Walled L-Shaped Bracket

Q: Professor! Today's topic is NAFEMS LE10: The Thick L-Shaped Bracket, right? What is it about?

It's the NAFEMS linear elastic benchmark LE10. It's a thick plate bending problem. The stress at point D is evaluated.

Q: Um... what physical phenomena do each of these terms represent?

Solution Method 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

Category: 解析 | Integrated 2026-04-06

CAE visualization for nafems le10 theory - technical simulation diagram

NAFEMS LE10: 厚肉L型ブラケット

Theory and Physics

Overview

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Professor! Today's topic is NAFEMS LE10: The Thick L-Shaped Bracket, right? What is it about?

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NAFEMS Linear Elastic Benchmark LE10. A thick plate bending problem. Evaluates stress at point D.

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Reference solution: $$ \sigma_{yy}(D) = 5.38 \text{ MPa} $$

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Now I understand what my senior meant when they said, "At least do the linear elastic benchmark properly."

Problem Setup

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Please tell me about the "Problem Setup"!

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Geometry: Thick plate (thickness t = 1.0 m)
Material: Isotropic elastic body (E = 210 GPa, ν = 0.3)
Load: Out-of-plane distributed load
Constraints: Edge boundary conditions
Evaluation point: Point D (top surface of plate)

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Hearing this, I finally grasp why linear elastic benchmarks are so important!

Governing Equations

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Equilibrium equations for a 3D elastic body:

$$ \nabla \cdot \boldsymbol{\sigma} + \mathbf{b} = \mathbf{0} $$

$$ \sigma_{ij,j} + b_i = 0 \quad (i=1,2,3) $$

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Strain-displacement relation (infinitesimal strain):

$$ \varepsilon_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i}) $$

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Ah, I see! So that's how the equilibrium equations for a 3D elastic body work.

Comparison of Theoretical and Numerical Solutions

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Our budget and time are limited, which one offers the best cost-performance?

Benchmark Verification Data by Each Solver

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What does "benchmark verification by each solver" mean specifically?

Evaluation Item	Reference Solution	Ansys Mechanical	Abaqus	MSC Nastran	NX Nastran	Unit
σ_yy (Point D, Top Surface)	5.38	5.377	5.382	5.375	5.379	MPa
Maximum Deflection	Reference	Matches	Matches	Matches	Matches	mm

Mesh Convergence Verification (20-node Hexahedral Element)

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What does "mesh convergence verification" mean specifically?

Mesh	Number of Elements	Degrees of Freedom (DOF)	σ_yy (MPa)	Relative Error(%)
2×2×1	4	315	4.82	10.4
4×4×2	32	1,815	5.21	3.16
8×8×4	256	11,907	5.35	0.56
16×16×8	2,048	86,427	5.378	0.04

Element Type Comparison

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What does "element type comparison" mean specifically?

Element Type	Nodes/Element	σ_yy (MPa)	Relative Error(%)	Remarks
HEX8 (8-node hexahedron)	8	4.15	22.9	Shear locking occurs
HEX8-RI (Reduced integration)	8	5.12	4.83	Hourglass control required
HEX20 (20-node hexahedron)	20	5.378	0.04	Recommended
TET4 (4-node tetrahedron)	4	3.28	39.0	Not recommended
TET10 (10-node tetrahedron)	10	5.31	1.30	Acceptable range

Discretization Method

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Spatial discretization by the Finite Element Method (FEM). Weak form based on the Galerkin method:

$$ \int_\Omega \delta\varepsilon^T \sigma \, d\Omega = \int_{\Gamma} \delta u^T \bar{t} \, d\Gamma + \int_\Omega \delta u^T b \, d\Omega $$

Matrix Solution Algorithm

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What does "matrix solution algorithm" mean specifically?

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Solver	Classification	Memory Usage	Applicable Scale

LU DecompositionDirect MethodO(n²)Small to Medium Scale Cholesky DecompositionDirect Method (Symmetric Positive Definite)O(n²)Small to Medium Scale PCG MethodIterative MethodO(n)Large Scale

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Ah, I see! So that's how the Finite Element Method works.

Implementation in Commercial Tools

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There are many different software tools, right? Please tell me the characteristics of each!

Tool Name	Developer/Current Owner	Main File Format
MSC Nastran / NX Nastran	MSC Nastran (Hexagon), NX Nastran (Siemens Digital Industries Software)	.bdf, .dat, .f06, .op2, .pch
Abaqus FEA (SIMULIA)	Dassault Systèmes SIMULIA	.inp, .odb, .cae, .sta, .msg
Ansys Mechanical (formerly ANSYS Structural)	Ansys Inc.	.cdb, .rst, .db, .ans, .mac

Vendor Lineage and Product Integration History

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Are the origins of each software tool somewhat dramatic?

MSC Nastran / NX Nastran

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Next is MSC Nastran. What's it about?

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Developed as NASA Structural Analysis (NASTRAN) in the 1960s. Commercialized by MSC Software. MSC was acquired by Hexagon AB in 2017.

Current affiliation: MSC Nastran (Hexagon), NX Nastran (Siemens Digital Industries Software)

Abaqus FEA (SIMULIA)

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What does "Abaqus FEA" mean specifically?

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Developed by HKS (Hibbitt, Karlsson & Sorensen) in 1978. Acquired by Dassault Systèmes in 2005.

Current affiliation: Dassault Systèmes SIMULIA

File Formats and Interoperability

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

Format	Extension	Type	Overview
STEP	.stp/.step	Neutral CAD	3D CAD data exchange compliant with ISO 10303.
IGES	.igs/.iges	Neutral CAD	Early CAD exchange standard.
Nastran Bulk Data	.bdf	Solver-specific	Element definitions like CTETRA, CHEXA.
Abaqus Input	.inp	Solver-specific	Element definitions like C3D20R, C3D8I.

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Risks during conversion: Element type mapping (e.g., CHEXA→C3D20R) can be automated, but integration point locations and node numbering order may differ.

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Professor, your explanation is clear! My confusion about formats has cleared up.

Judgment Criteria

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I've heard of "Judgment Criteria," but I might not fully understand it...