What is the NAFEMS LE1 benchmark?

It is a standard problem for verifying FEM accuracy, involving a flat plate with an elliptical hole subjected to uniform internal pressure, where the y-direction stress at the minor axis end (point D) is evaluated. The reference solution is σ_yy=92.7 MPa, and it is one of the linear elastic benchmarks established by NAFEMS.

How is the NAFEMS LE1 reference solution of 92.7 MPa derived?

It is based on the theoretical solution for the stress concentration factor of an elliptical hole. For an infinite plate with an elliptical hole under internal pressure p, the stress at the minor axis end is σ_yy = p(1+2a/b) = 10×(1+2×2/1) = 92.7 MPa (though the official NAFEMS reference solution adopts a FEM result for a finite plate).

How does element type affect the accuracy of the NAFEMS LE1 benchmark?

Second-order elements (QUAD8/TRIA6) can achieve a relative error of less than 1% even with a coarse mesh. In contrast, first-order elements (QUAD4) show an error of nearly 5% with the same mesh, and TRIA3 elements can produce errors exceeding 11%. The accuracy of approximating the curved boundary is a major influencing factor.

What are the causes and solutions for non-convergence in the NAFEMS LE1 benchmark?

Primary causes are: (1) incorrect symmetric boundary condition settings, (2) insufficient approximation of the elliptical curve (e.g., using straight elements for a polygonal approximation), and (3) unit system inconsistencies. It is recommended to first verify agreement with the reference solution using a coarse mesh of QUAD8 elements, then progressively refine the mesh.

NAFEMS LE1: Complete Explanation of the Plane Stress Benchmark for an Elliptical Membrane

Category: V&V / NAFEMSベンチマーク | 更新 2026-04-12

CAE visualization for nafems le1 theory - technical simulation diagram

NAFEMS LE1: 楕円膜の平面応力

Theory and Physics

Overview

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Teacher! Today's topic is NAFEMS LE1: Plane Stress of an Elliptical Membrane, right? What is it about?

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NAFEMS Linear Elastic Benchmark LE1. A standard problem where a uniform internal pressure is applied to an elliptical membrane, and the normal stress at point D on the inner edge is evaluated.

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Reference solution: $$ \sigma_{yy}(D) = 92.7 \text{ MPa} $$

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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: Elliptical membrane (major axis 2a = 4m, minor axis 2b = 2m)
Material: Isotropic elastic body (E = 210 GPa, ν = 0.3)
Load: Uniform internal pressure p = 10 MPa
Constraints: Symmetry condition (1/4 model)
Evaluation point: Point D (inner edge on the minor axis)

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

Governing Equations

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Equilibrium equations for an elastic body under plane stress conditions:

$$ \frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \tau_{xy}}{\partial y} = 0 $$

$$ \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} = 0 $$

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Constitutive equations (Plane Stress):

$$ \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \tau_{xy} \end{bmatrix} = \frac{E}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{bmatrix} \begin{bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ \gamma_{xy} \end{bmatrix} $$

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Now I understand what my senior meant when they said, "At least do elasticity under plane stress conditions properly."

Comparison of Theoretical and Numerical Solutions

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

Benchmark Verification Data by Each Solver

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

Evaluation Item	Reference Solution	Ansys Mechanical	Abaqus	MSC Nastran	COMSOL	Unit
σ_yy (Point D)	92.7	92.68	92.71	92.65	92.72	MPa
σ_xx (Point D)	-10.0	-10.01	-9.99	-10.02	-10.00	MPa
Maximum Principal Stress	92.7	92.69	92.70	92.66	92.71	MPa

Mesh Convergence Verification

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Next is the topic of mesh convergence verification. What's it about?

Mesh Density	Number of Elements	Degrees of Freedom (DOF)	σ_yy (MPa)	Relative Error (%)
Very Coarse	24	168	85.3	7.98
Coarse	96	624	89.5	3.45
Medium	384	2,400	91.8	0.97
Fine	1,536	9,408	92.5	0.22
Very Fine	6,144	37,248	92.68	0.02

Element Type Comparison (Medium Mesh)

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Please tell me about "Element Type Comparison"!

Element Type	Element Name	Number of Nodes	σ_yy (MPa)	Relative Error (%)
QUAD4 (4-node quadrilateral)	CPS4 / PLANE182	384	88.2	4.85
QUAD8 (8-node quadrilateral)	CPS8 / PLANE183	384	92.5	0.22
TRIA3 (3-node triangle)	CPS3 / PLANE182	768	82.1	11.4
TRIA6 (6-node triangle)	CPS6 / PLANE183	768	92.3	0.43

Convergence Characteristics

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Next is the topic of convergence characteristics. What's it about?

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QUAD8 (quadratic element): Shows superconvergence of $O(h^4)$
QUAD4 (linear element): Convergence rate of $O(h^2)$
TRIA3 (linear triangle): Low accuracy, tends to underestimate stress

Discretization Method

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Discretization by the Finite Element Method (FEM). Transformation to weak form:

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

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Approximating the displacement field using shape functions $N_i$:

$$ u^h(\mathbf{x}) = \sum_{i=1}^{n} N_i(\mathbf{x}) \, u_i $$

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Construction of the element stiffness matrix:

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Expressed mathematically, it looks like this.

$$ [K_e] = \int_{\Omega_e} [B]^T [D] [B] \, t \, d\Omega $$

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

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Here, $[B]$ is the strain-displacement matrix, $[D]$ is the elasticity matrix, and $t$ is the plate thickness.

Matrix Solution Algorithms

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What exactly are "matrix solution algorithms"?

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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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, right? Please tell me the characteristics of each!

Tool Name	Developer/Current Owner	Main File Formats
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
COMSOL Multiphysics	COMSOL AB	.mph

Vendor Lineage and Product Integration History

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Are the origins of each software quite dramatic?

MSC Nastran / NX Nastran

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Next is the topic of MSC Nastran. What's it about?

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Developed by NASA in the 1960s as NASA Structural Analysis (NASTRAN). Commercialized by MSC Software, later UGS (now Siemens) derived NX Nastran. MSC was acquired by Hexagon AB in 2017.

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