The Improved Element-Free Galerkin Method for Anisotropic Steady-State Heat Conduction Problems

Cheng, Heng; Xing, Zebin; Peng, Miaojuan

doi:10.32604/cmes.2022.020755

Cited by 2 publications

(2 citation statements)

References 40 publications

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“…In order to get rid of the complexity of mesh generation and reduce the time of preprocessing, various meshless methods have devoted considerable attention. These approaches include the elementfree Galerkin method [8][9][10][11], the reproducing kernel particle method [12][13][14][15], the meshless local Petrov-Galerkin method [16,17], the radial basis function collocation method (RBFCM) [18,19], the generalized finite difference method (GFDM) [20][21][22][23], the singular boundary method (SBM) [24,25], the method of fundamental solutions (MFS) [26,27] and the boundary knot method (BKM) [28,29], etc. The successful application of these meshless methods fully demonstrates their development prospect.…”

Section: Introductionmentioning

confidence: 99%

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

Zhang¹,

Wang²,

Chen³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method (LKM) with the dual reciprocity method (DRM). Firstly, the temporal derivative is discretized by a finite difference scheme, and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation. Secondly, the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution. And then, the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation, respectively. The LKM is a recently proposed local radial basis function collocation method with the merits of being simple, accurate, and free of mesh and integration. Compared with the traditional domain-type and boundary-type schemes, the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains. Numerical experiments, including two-and three-dimensional heat transfer models, demonstrated the effectiveness and accuracy of the new methodology.

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Section: Introductionmentioning

confidence: 99%

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

Zhang¹,

Wang²,

Chen³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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“…Zhang et al [35] studied the partial differential equations for 3D transient heat conduction using the IEFG method. Cheng et al [36] studied the IEFG method to solve two-dimensional anisotropic heat conduction problems. The numerical solutions show that the IEFG method has a quicker computational speed.…”

Section: Introductionmentioning

confidence: 99%

Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems

Xing,

Cheng,

Cheng

2023

Mathematics

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This paper uses the physical information neural network (PINN) model to solve a 3D anisotropic steady-state heat conduction problem based on deep learning techniques. The model embeds the problem’s governing equations and boundary conditions into the neural network and treats the neural network’s output as the numerical solution of the partial differential equation. Then, the network is trained using the Adam optimizer on the training set. The output progressively converges toward the accurate solution of the equation. In the first numerical example, we demonstrate the convergence of the PINN by discussing the effect of the neural network’s number of layers, each hidden layer’s number of neurons, the initial learning rate and decay rate, the size of the training set, the mini-batch size, the amount of training points on the boundary, and the training steps on the relative error of the numerical solution, respectively. The numerical solutions are presented for three different examples. Thus, the effectiveness of the method is verified.

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The Improved Element-Free Galerkin Method for Anisotropic Steady-State Heat Conduction Problems

Cited by 2 publications

References 40 publications

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems

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