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A novel neural network method for solid mechanics problems of heterogeneous bars.

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Title: A novel neural network method for solid mechanics problems of heterogeneous bars.
Authors: Lanka, Mitra Ramakrishna1 (AUTHOR) lankamn@mail.uc.edu, Liu, Gui-Rong1 (AUTHOR)
Source: Engineering Applications of Artificial Intelligence. Sep2025:Part A, Vol. 156, pN.PAG-N.PAG. 1p.
Subjects: Domain decomposition methods, Boundary value problems, Solid mechanics, Finite element method, Inhomogeneous materials
Abstract: This paper introduces a novel neural network (NN) method using Physics-Informed Neural Networks (PINNs) to solve boundary value problems in solid mechanics for heterogeneous materials. Existing domain decomposition methods in PINNs require multiple NNs and indiscriminate data sampling, often with heuristically chosen NN architectures. In contrast, this work proposes a domain variation strategy that solves heterogeneous problems using a single-hidden-layered NN. The strategy uses boundary, interface, and collocation spatial coordinates from all sub-domains to generate variations of the heterogeneous solid, which facilitates the establishment of mechanical connections within the NN. From each variation, a unique dataset is generated and fed into an instance of NN, referred to as sub-NN, which output displacements and their gradients. Outputs from all sub-NNs are utilized to compute boundary losses (via boundary conditions), interface losses (via continuity conditions), and governing equation losses, which are minimized using Adam and Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization methods. The number of data samples and optimal neurons for convergence are determined through theoretical studies. The method's effectiveness is demonstrated on a heterogeneous bar fixed at one end and subjected to a load at the other. Validation against Finite Element Method (FEM) simulations shows mean squared error (MSE) of 0.04, with a mean absolute percentage error (MAPE) of 4.64 % for the 20-material bar problem involving force variation. The NN model effectively captures displacements, stresses, and strain discontinuities at the material interfaces. This approach offers an efficient alternative to domain decomposition methods and a framework for solving multiple problems via surrogate modeling. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract:This paper introduces a novel neural network (NN) method using Physics-Informed Neural Networks (PINNs) to solve boundary value problems in solid mechanics for heterogeneous materials. Existing domain decomposition methods in PINNs require multiple NNs and indiscriminate data sampling, often with heuristically chosen NN architectures. In contrast, this work proposes a domain variation strategy that solves heterogeneous problems using a single-hidden-layered NN. The strategy uses boundary, interface, and collocation spatial coordinates from all sub-domains to generate variations of the heterogeneous solid, which facilitates the establishment of mechanical connections within the NN. From each variation, a unique dataset is generated and fed into an instance of NN, referred to as sub-NN, which output displacements and their gradients. Outputs from all sub-NNs are utilized to compute boundary losses (via boundary conditions), interface losses (via continuity conditions), and governing equation losses, which are minimized using Adam and Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization methods. The number of data samples and optimal neurons for convergence are determined through theoretical studies. The method's effectiveness is demonstrated on a heterogeneous bar fixed at one end and subjected to a load at the other. Validation against Finite Element Method (FEM) simulations shows mean squared error (MSE) of 0.04, with a mean absolute percentage error (MAPE) of 4.64 % for the 20-material bar problem involving force variation. The NN model effectively captures displacements, stresses, and strain discontinuities at the material interfaces. This approach offers an efficient alternative to domain decomposition methods and a framework for solving multiple problems via surrogate modeling. [ABSTRACT FROM AUTHOR]
ISSN:09521976
DOI:10.1016/j.engappai.2025.111084