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Real-time, high-fidelity linear elastostatic beam models for engineering education

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Abstract

Euler-Bernoulli beam theory is a fundamental subject in structural engineering education, as it characterizes the quintessence of a bending phenomenon. Since this theory relies on assumptions, we generally use finite element analysis when it comes to predicting the physical behavior of a real beam. However, finite element analysis is cumbersome for engineering education because it is not conducive to real-time or repetitive analyses during a lecture. To address this difficulty, we propose to exploit a certified reduced basis analysis to produce a real-time, high-fidelity beam model for engineering education. For demonstration, we construct two three-dimensional, square cross-sectional beam models with certified reduced basis analysis and compare them to the corresponding finite element beam models in terms of accuracy and efficiency. A comparative study shows that the certified reduced basis beam models permit actual errors and runtimes less than 0.1 % and 0.003 seconds, respectively, on average. As a result, we believe that the certified reduced basis beam models are compelling alternatives to their finite element counterparts and would greatly enhance engineering education.

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Abbreviations

a(·,·):

Bilinear functional

β :

Continuity constraint box

C :

Fourth-order constitutive tensor

\(\overline C \) :

Effective constitutive tensor

C j :

Stability constraint sample

c 12 :

Lamé’s first elastic constant

c 66 :

Lamé’s second elastic constant

D :

Parameter domain

E :

Young’s modulus

e n :

Unit normal vector

e t :

Unit tangent vector

∥ e ∥X :

X-norm error

∥ ê ∥X :

Dual norm of the residual

f(·):

Linear functional

G :

Shear modulus

\({\cal J}\) :

Objective function of a minimization problem

L i :

ei directional beam length in the reference domain

L o i :

ei directional beam length in the original domain

M :

Number of points in CJ closest to a given μ

\({\cal N}\) :

Dimension of a finite element space

N :

Dimension of a reduced basis space

p :

Distributed load

Q :

Number of affinely decomposed terms

s :

Output of interest

T :

Linear map for geometric transformation

u :

Exact solution of a PDE

\({u^{\cal N}}\) :

Finite element solution of a PDE

u N :

Reduced basis solution of a PDE

v :

Test function

X :

Hilbert space

\({X^{\cal N}}\) :

Finite element space

X N :

Reduced basis space

x :

Spatial coordinate in the reference domain

x o :

Spatial coordinate in the original domain

α :

Stability (or coercivity) constant

α LB :

Lower bound of the stability constant

α UB :

Upper bound of the stability constant

ΓD :

Dirichlet boundary

ΓN :

Neumann boundary

Δ x u :

X-norm error bound

ε :

Target tolerance in the SCM

ζ :

Reduced basis function

θ :

Parameter-dependent function

μ :

d-tuple parameter vector

v :

Poisson’s ratio

σ :

Normal stress

σ von :

Von Mises stress

T :

Shear stress

φ :

Finite element basis function

Ω:

Reference domain

Ωo :

Original domain

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Acknowledgments

This research was supported by the National Research Foundation of Korea (NRF) grant and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Korea government (MSIT) (No. 2019R1A5A6099595) and the Ministry of Education (No. NRF-2016R1D1A1B03930126), respectively.

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Authors and Affiliations

  1. Department of Aerospace Engineering, Pusan National University, Busan, 46241, Korea

    Shinseong Kang & Kyunghoon Lee

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  1. Shinseong Kang
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Corresponding author

Correspondence to Kyunghoon Lee.

Additional information

Shinseong Kang received his B.S. and M.S. from the Department of Aerospace Engineering, Pusan National University, Busan, South Korea. He is currently studying for the Doctorate at Aerospace Engineering from Pusan National University. His research interests include the certified reduced basis technique for composite analysis.

Kyunghoon Lee is a Professor at the Department of Aerospace Engineering, Pusan National University, Busan, South Korea. He received his Ph.D. in Aerospace Engineering from Georgia Institute of Technology, Atlanta, GA, USA in 2010. His research interests are design computation under uncertainty via rapid simulation.

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Kang, S., Lee, K. Real-time, high-fidelity linear elastostatic beam models for engineering education. J Mech Sci Technol 35, 3483–3495 (2021). https://doi.org/10.1007/s12206-021-0721-y

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  • DOI: https://doi.org/10.1007/s12206-021-0721-y

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