Comparative Study of Bending Analysis using Physics-Informed Neural Networks and Numerical Dynamic Deflection in Perforated nanobeam

Ramanath Garai; Iswari Sahu; S. Chakraverty

cs.LGApr 15, 2026

Comparative Study of Bending Analysis using Physics-Informed Neural Networks and Numerical Dynamic Deflection in Perforated nanobeam

Authors: Ramanath Garai, Iswari Sahu, S. Chakraverty

Organizations: Department of Mathematics, National Institute of Technology Rourkela, Rourkela-769008, India.

Abstract

In this chapter, we investigate the bending behavior of a perforated nanobeam subjected to sinusoidal loading using an efficient and computationally robust Physics-Informed Functional Link Constrained Framework with Domain Mapping (DFL-TFC) method. Our aim is to determine the relationship between static bending response and dynamic deflection of a perforated nanobeam for various perforation cases. The static bending is obtained using the FL-TFC with Domain mapped method, whereas dynamic deflection is determined using the Galerkin method. The proposed approach employs the theory of functional connections (TFC) to systematically embed governing differential equation constraints into a constrained expression (CE), which exactly satisfies all prescribed initial and boundary conditions (ICs and BCs) and domain of differential equation is mapped to domain of orthogonal polynomials. Within this framework, the free function appearing in the constrained expression is expressed through a functional link neural network (FLNN). The cost is minimized by the mean square residual of DE, allowing training without requiring complex deep network architectures. Relationship between static and dynamic defection of simply-supported (S-S) perforated nanobeams has been investigated here. FL-TFC with Domain mapped method eliminates the need for deep and complex neural network architectures while ensuring accuracy, efficiency, and strict satisfaction of boundary conditions as compared to standard PINN.

Figures & tables

Figure 1: Perforated nanobeam structural model

$n$	$\widetilde{W_{p}}^{d}(0.3)$	$\widetilde{W_{p}}^{d}(0.5)$	$\widetilde{W_{p}}^{d}(0.9)$
8	108.3092	133.8678	41.3674
9	108.3056	133.8730	41.3690
10	108.3056	133.8730	41.3690
11	108.3056	133.8730	41.3690
12	108.3056	133.8730	41.3690
13	108.3056	133.8730	41.3690

Table 1: Convergence table of dynamic deflection for the S-S perforated nanobeam with parameters

N=2

\alpha=0.5

, and

\bar{\alpha}=0.2

Filling Ratio $(\alpha)$	N	Deflection $(\widetilde{W_{p}})$	Static $(\widetilde{W_{p}}^{s})$	Dynamic $(\widetilde{W_{p}}^{d})$	Ratio $=\frac{\widetilde{W_{p}}^{d}}{\widetilde{W_{p}}^{s}}$
0.3	$1$	$\widetilde{W_{p}}(0.3)$	1.3534	122.7182	90.6711
		$\widetilde{W_{p}}(0.5)$	1.6729	151.6880	90.6711
		$\widetilde{W_{p}}(0.9)$	0.5170	46.8742	90.6711
	$2$	$\widetilde{W_{p}}(0.3)$	1.8658	127.8701	68.5328
		$\widetilde{W_{p}}(0.5)$	2.3063	158.0562	68.5328
		$\widetilde{W_{p}}(0.9)$	0.7127	48.8420	68.5328

Table 2: Relationship between the static and dynamic deflection of a simply supported (S-S) perforated nanobeam for different values of

\alpha

and

N

with the parameter

\bar{\alpha}=0.2

Non-local Parameter ( $\bar{\alpha}$ )	N	Deflection $(\widetilde{W_{p}})$	Static $(\widetilde{W_{p}}^{s})$	Dynamic $(\widetilde{W_{p}}^{d})$	Ratio $=\frac{\widetilde{W_{p}}^{d}}{\widetilde{W_{p}}^{s}}$
$0.1$	$1$	$\widetilde{W_{p}}(0.1)$	0.3586	45.6687	127.3527
		$\widetilde{W_{p}}(0.6)$	1.1037	140.5539	127.3527
		$\widetilde{W_{p}}(0.8)$	0.6821	86.8671	127.3527
$0.2$	$2$	$\widetilde{W_{p}}(0.1)$	0.5296	41.3690	78.1092
		$\widetilde{W_{p}}(0.6)$	1.6300	127.3208	78.1092
		$\widetilde{W_{p}}(0.8)$	1.0074	78.6886	78.1092

Table 3: Relationship between the static and dynamic deflection of a simply supported (S-S) perforated nanobeam for different values of

\bar{\alpha}

and

N

with the parameter

{\alpha}=0.5

Panel of Figure 2 — Figure 2: Static and dynamic deflection relationship for a simply supported (S–S) perforated nanobeam with varying filling ratio $\alpha$

Panel of Figure 3 — Figure 3: Comparison of static and dynamic deflection for different numbers of holes $N$ in a simply supported perforated nanobeam with the parameter $\alpha=0.5$ and $\bar{\alpha}=0.2$

Panel of Figure 4 — Figure 4: Relationship between static and dynamic deflection of an S–S perforated nanobeam with variation in non-local parameter $\bar{\alpha}$ for the parameters $\alpha=0.8$ and $N=2$

Structural Mechanics

Structural mechanics studies how structures deform, carry loads, vibrate, buckle, and fail under different materials, geometries, and conditions.

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Physics-Informed Functional Link Constrained Framework with Domain Mapping for Solving Bending Analysis of an Exponentially Loaded Perforated Beam

Iswari Sahu, Ramanath Garai, S. Chakraverty

Department of Mathematics, National Institute of Technology Rourkela, India

Physics-Informed ML Structural Mechanics PDE Solving

Open paper AskPDF arXiv

This article presents a novel and comprehensive approach for analyzing bending behavior of the tapered perforated beam under an exponential load. The governing differential equation includes important factors like filling ratio (

α

), number of rows of holes (

N

), tapering parameters (

φ

and

ψ

), and exponential loading parameter (

γ

), providing a realistic and flexible representation of perforated beam configuration. Main goal of this work is to see how well the Domain mapped physics-informed Functional link Theory of Functional Connection (DFL-TFC) method analyses bending response of perforated beam with square holes under exponential loading. For comparison purposes, a corresponding PINN-based formulation is developed. Outcomes clearly show that the proposed DFL-TFC framework gives better results, including faster convergence, reduced computational cost, and improved solution accuracy when compared to the PINN approach. These findings highlight effectiveness and potential of DFL-TFC method for solving complex engineering problems governed by differential equations. Within this framework, hidden layer is replaced by a functional expansion block that enriches input representation via orthogonal polynomial basis functions, and the domain of DE mapped to corresponding domain of orthogonal polynomials. A Constrained Expression (CE), constructed through the Theory of Functional Connections (TFC) using boundary conditions, ensures that constraints are exactly satisfied. In CE, free function is represented using a Functional Link Neural Network (FLNN), which learns to solve resulting unconstrained optimization problem. The obtained results are further validated through the Galerkin and PINN solutions.

Abstract

Structural Mechanics

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