math.DSApr 8, 2026

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

Authors: Iswari Sahu, Ramanath Garai, S. Chakraverty

Organizations: Department of Mathematics, National Institute of Technology Rourkela, India

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Abstract

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 (αα), number of rows of holes (NN), tapering parameters (φφ and ψψ), and exponential loading parameter (γγ), 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.

Figures & tables

Figure 1: Geometric model of a tapered perforated beam
Figure 2: Schematic diagram of PINN
Figure 3: Schematic diagram of Domain mapped FL-TFC
Critical DeflectionKp~\widetilde{K_{p}}DFL-TFCGalerkinExisting [ 20 ]Existing [ 32 ]
W~p(0.5)\widetilde{W}_{p}(0.5)001.30211.30211.30211.3023
10100.64480.64480.64470.6448
25250.36610.36610.36610.3661
Table 1: Comparison of maximum bending deflection of a solid beam with uniform load q0~=1\widetilde{q_{0}}=1 corresponding to α=1,γ=0,ϕ=0,ψ=0,N=0\alpha=1,\gamma=0,\phi=0,\psi=0,N=0
W~p(X)\widetilde{W}_{p}(X)S-SC-S
DFL-TFCPINNGalerkinDFL-TFCPINNGalerkin
W~p(0.1)\widetilde{W}_{p}(0.1)4.42814.42804.42811.05521.10401.0552
W~p(0.5)\widetilde{W}_{p}(0.5)14.642514.642514.64259.59969.61239.5996
W~p(0.9)\widetilde{W}_{p}(0.9)5.02265.02275.02263.88353.84433.8835
Table 2: Comparison of bending deflection of a tapered perforated beam for both S-S and C-S boundary conditions corresponding to α=0.3,N=4,γ=1,ϕ=0,ψ=0,Kp~=10,and q0~=10\alpha=0.3,N=4,\gamma=1,\phi=0,\psi=0,\widetilde{K_{p}}=10,\text{and}\ \widetilde{q_{0}}=10
W~p(X)\widetilde{W}_{p}(X)S-SC-S
DFL-TFCPINNGalerkinDFL-TFCPINNGalerkin
W~p(0.1)\widetilde{W}_{p}(0.1)0.79800.80050.79800.17870.20900.1787
W~p(0.5)\widetilde{W}_{p}(0.5)2.10032.10012.10031.17141.18161.1714
W~p(0.9)\widetilde{W}_{p}(0.9)0.56100.55740.56100.35310.33130.3531
Table 3: Comparison of bending deflection of a tapered perforated beam for both S-S and C-S boundary conditions corresponding to α=0.5,N=2,γ=0,ϕ=0.5,ψ=0.5,Kp~=10,and q0~=5\alpha=0.5,N=2,\gamma=0,\phi=0.5,\psi=0.5,\widetilde{K_{p}}=10,\text{and}\ \widetilde{q_{0}}=5
Figure 4: DFL-TFC and PINN solution for α=0.8,γ=5,ϕ=0.5,ψ=0.5,N=3,Kp~=10,q0~=1\alpha=0.8,\gamma=5,\phi=0.5,\psi=0.5,N=3,\widetilde{K_{p}}=10,\widetilde{q_{0}}=1
α\alphaNNS-SC-S
W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)
0.114.496614.86985.11111.08819.83933.9801
45.534818.24726.57451.819214.12315.7176
0.313.546211.71043.93490.71346.87602.7865
44.428114.64255.02261.05529.59963.8835
0.513.341611.03033.69300.65006.32602.5647
Table 4: Bending deflection of a tapered perforated beam for C-S boundary conditions with varying α\alpha and NN corresponding to γ=1,ϕ=0,ψ=0,Kp~=10,and q0~=10\gamma=1,\phi=0,\psi=0,\widetilde{K_{p}}=10,\text{and}\ \widetilde{q_{0}}=10
ϕ\phiψ\psiS-SC-S
  W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)
0.10.11.04503.15880.95870.21351.78050.6363
0.50.96522.73680.76620.20151.52750.4918
0.90.89592.39480.63260.19091.32510.3954
0.50.10.86122.40300.68110.18911.34940.4392
0.50.79802.10030.56100.17871.17140.3531
Table 5: Bending deflection of a tapered perforated beam for S-S and C-S boundary conditions with varying ϕ\phi and ψ\psi corresponding to γ=0,α=0.5,N=2,Kp~=10,and q0~=5\gamma=0,\alpha=0.5,N=2,\widetilde{K_{p}}=10,\text{and}\ \widetilde{q_{0}}=5
γ\gammaKp~\widetilde{K_{p}}S-SC-S
  W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)
110.32350.86990.23420.05920.42550.1322
50.27410.74420.20230.05520.39270.1225
100.22980.63070.17340.05090.35820.1122
310.95752.72920.77300.17890.17890.4723
50.80692.33910.67290.16581.30980.4390
Table 6: Bending deflection of a tapered perforated beam for S-S and C-S boundary conditions with varying γ\gamma and Kp~\widetilde{K_{p}} corresponding to α=0.8,N=3,ϕ=ψ=0.5,and q0~=1\alpha=0.8,N=3,\phi=\psi=0.5,\text{and}\ \widetilde{q_{0}}=1
Figure 5: Variation of bending deflection with respect to filling ratio α\alpha for a tapered perforated beam subjected to an exponential load
Figure 6: Variation of bending deflection with respect to exponential load parameter γ\gamma for a tapered perforated beam resting on foundation
Figure 7: Effect of tapering parameter ϕ\phi on the bending behaviour of a tapered perforated beam resting on a foundation
Figure 8: Effect of tapering parameter ψ\psi on the bending behaviour of a tapered perforated beam resting on a foundation
Figure 9: Effect of elastic foundation parameter Kp~\widetilde{K_{p}} on the bending behaviour of a tapered perforated beam with exponential load
Figure 10: Loss figures of PINN and DFL-TFC frameworks at different parameters for S-S boundary conditions
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
14.0750×10−64.0750\times 10^{-6}23.86 s133.1803×10−83.1803\times 10^{-8}4.94 s
25.3076×10−65.3076\times 10^{-6}39.43 s141.8148×10−101.8148\times 10^{-10}5.80 s
34.7255×10−64.7255\times 10^{-6}47.43 s154.8919×10−104.8919\times 10^{-10}7.08 s
Table 7: Comparison of PINN and DFL-TFC for parameters α=0.7\alpha=0.7 , γ=3\gamma=3 , ϕ=ψ=0.5\phi=\psi=0.5 , N=1N=1 , Kp~=10\widetilde{K_{p}}=10 , and q0~=2\widetilde{q_{0}}=2 (S-S boundary conditions)
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
15.3951×10−55.3951\times 10^{-5}15.60 s133.4304×10−103.4304\times 10^{-10}5.84 s
24.1046×10−74.1046\times 10^{-7}34.86 s148.7719×10−118.7719\times 10^{-11}6.16 s
32.7828×10−62.7828\times 10^{-6}43.26 s159.9721×10−119.9721\times 10^{-11}8.21 s
Table 8: Comparison of PINN and DFL-TFC for parameters α=0.8\alpha=0.8 , γ=4\gamma=4 , ϕ=ψ=0.3\phi=\psi=0.3 , N=2N=2 , Kp~=5\widetilde{K_{p}}=5 , and q0~=1\widetilde{q_{0}}=1 (S-S boundary conditions)
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
17.9079×10−57.9079\times 10^{-5}36.55 s131.5806×10−81.5806\times 10^{-8}5.14 s
22.2729×10−32.2729\times 10^{-3}57.14 s143.9759×10−113.9759\times 10^{-11}6.72 s
31.4391×10−41.4391\times 10^{-4}58.31 s156.8713×10−136.8713\times 10^{-13}8.90 s
Table 9: Comparison of PINN and DFL-TFC for parameters α=0.2\alpha=0.2 , γ=5\gamma=5 , ϕ=ψ=0.2\phi=\psi=0.2 , N=1N=1 , Kp~=8\widetilde{K_{p}}=8 , and q0~=4\widetilde{q_{0}}=4 (S-S boundary conditions)
Figure 11: Loss figures of PINN and DFL-TFC frameworks at different parameters for C-S boundary conditions
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
12.6243×10−62.6243\times 10^{-6}35.78 s131.3258×10−71.3258\times 10^{-7}4.80 s
25.8697×10−55.8697\times 10^{-5}56.07 s145.5581×10−105.5581\times 10^{-10}7.11 s
31.2247×10−61.2247\times 10^{-6}49.03 s154.3449×10−114.3449\times 10^{-11}9.44 s
Table 10: Comparison of PINN and Domain mapped FL-TFC for parameters α=0.7\alpha=0.7 , γ=3\gamma=3 , ϕ=ψ=0.5\phi=\psi=0.5 , N=1N=1 , Kp~=10\widetilde{K_{p}}=10 , and q0~=2\widetilde{q_{0}}=2 (C-S boundary conditions)
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
11.6465×10−61.6465\times 10^{-6}27.36 s136.6098×10−106.6098\times 10^{-10}6.39 s
22.7963×10−52.7963\times 10^{-5}45.26 s141.9484×10−101.9484\times 10^{-10}8.09 s
31.1195×10−51.1195\times 10^{-5}41.58 s151.7089×10−131.7089\times 10^{-13}9.66 s
Table 11: Comparison of PINN and Domain mapped FL-TFC for parameters α=0.8\alpha=0.8 , γ=4\gamma=4 , ϕ=ψ=0.3\phi=\psi=0.3 , N=2N=2 , Kp~=5\widetilde{K_{p}}=5 , and q0~=1\widetilde{q_{0}}=1 (C-S boundary conditions)
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
18.8678×10−48.8678\times 10^{-4}35.76 s134.9020×10−84.9020\times 10^{-8}6.71 s
22.7195×10−42.7195\times 10^{-4}58.58 s142.7857×10−112.7857\times 10^{-11}8.23 s
33.7222×10−13.7222\times 10^{-1}71.00 s158.0515×10−128.0515\times 10^{-12}10.34 s
Table 12: Comparison of PINN and Domain mapped FL-TFC for parameters α=0.2\alpha=0.2 , γ=5\gamma=5 , ϕ=ψ=0.2\phi=\psi=0.2 , N=1N=1 , Kp~=8\widetilde{K_{p}}=8 , and q0~=4\widetilde{q_{0}}=4 (C-S boundary conditions)
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