Structural Benchmark 1: Linear Elastic Bracket Under Variable Pressure Loading

Problem Overview

This benchmark defines a 3D linear elasticity problem for a mechanical bracket subjected to variable pressure loading. The goal is to develop a data-driven surrogate model that can predict the structural response (stress fields) for any pressure value within a specified range.

Boundary Conditions

Boundary Type	Location	Description
Fixed Support	Four upper cylindrical holes	Fully constrained (u = v = w = 0) - prevents all translation
Pressure Load	Bottom cylindrical hole surface	Applied pressure (parameter to be varied)

Design Parameter

Parameter	Symbol	Range	Units
Pressure	P	10,000 - 20,000	Pa

The pressure is applied as a pressure load on the inner cylindrical surface of the bottom mounting hole.

Physics: 3D Linear Elasticity

Governing Equations

The problem is governed by the equilibrium equations of linear elasticity:

∇ · σ + f = 0

where: σ is the Cauchy stress tensor, f is the body force vector (typically zero for this problem)

Constitutive Relation (Hooke's Law)

σ = C : ε

For isotropic materials:

σ_ij = λε_kkδ_ij + 2με_ij

where: λ, μ are Lamé parameters, ε_ij is the strain tensor

Strain-Displacement Relation

ε = ½(∇u + ∇u^T)

where u = (u, v, w) is the displacement vector.

Material Properties

Standard structural material properties should be assumed (e.g., steel or aluminum). Typical values:

Property	Symbol	Typical Value (Steel)	Units
Young's Modulus	E	200	GPa
Poisson's Ratio	ν	0.3	-
Density	ρ	7850	kg/m³

Mesh

The finite element mesh consists of:

Element Type: Roughly 70K 3D tetrahedral (quadratic, 10 nodes) elements
Mesh Refinement: Higher density near stress concentration areas (holes, fillets)

Output Quantities of Interest

The data-driven model should predict:

1. Von Mises Stress

σ_VM = √[½((σ₁-σ₂)² + (σ₂-σ₃)² + (σ₃-σ₁)²)]

Data-Driven Approach

Data Generation

Sample 35 pressure values uniformly or via Latin Hypercube Sampling in range [10,000, 20,000] Pa
Run FEA simulations for each pressure value
Extract nodal stresses
Split 35 simulation results into 30 for train and 5 for test dataset

Surrogate Model Requirements

Input: Total 4D: 3D Space (x,y,z) + 1D Parameter (Pressure value P ∈ [10,000, 20,000] Pa)
Output: Total 1D: Von Mises stress
Target Accuracy: Less than 1% RMSE test error

Problem Summary

Aspect	Specification
Problem Type	3D Linear Elasticity
Geometry	Mechanical Bracket
Parameter	Pressure (10,000 - 20,000 Pa)
Boundary Conditions	Fixed supports, Pressure load
Output	Von Mises stress field

Benchmark Results

Benchmark Comparison

Models Compared

Model	Full Name	Description
INN	Interpolating Neural Network	Neural network with interpolation-based architecture
DNN	Deep Neural Network	Standard fully-connected deep neural network
KAN	Kolmogorov-Arnold Network	Network based on Kolmogorov-Arnold representation theorem
FNO	Fourier Neural Operator	Neural operator learning mappings in Fourier space

Results Summary

Model	Final RMSE	Training Time	Target Met (<1%)
INN	~0.005	~50-100 sec	Yes
FNO	~0.005-0.008	~500-2000 sec	Yes
DNN	~0.009	~300-3000 sec	Yes
KAN	~0.010-0.025	~1000-3000 sec	Marginal

Key Observations

INN achieves fastest convergence - Reaches target accuracy (~0.5% RMSE) in under 100 seconds
FNO provides stable low error - Consistent ~0.5-0.8% RMSE with moderate training time
DNN shows steady improvement - Converges to target but requires longer training
KAN exhibits high variance - Oscillating error during training, slowest to stabilize

Hardware Configuration

Component	Specification
GPU	NVIDIA GeForce RTX 5080
VRAM	16 GB

Conclusions

Best Performance: INN demonstrates superior efficiency with fastest convergence and lowest error
All models except KAN reliably achieve the <1% RMSE target accuracy
Trade-off: INN offers 10-20x speedup compared to other methods for this benchmark