A meshless method based on boundary integral equations and radial basis functions for biharmonic-type problems

Li, Xin; Zhu, Jialin; Zhang, Shougui

doi:10.1016/j.apm.2010.07.030

Cited by 20 publications

(4 citation statements)

References 35 publications

(37 reference statements)

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“…(15). Furthermore, to estimate the computational accuracy appropriately, the maximum absolute error (E MA ), the root-mean-squared error (E RMS ), and the relative error (E Rel ) are respectively introduced as [20,25,38]                             …”

Section: Verification Of the Present Methods And The Determination Of mentioning

confidence: 99%

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Feng

Yan

Lin

et al. 2020

Appl. Math. Mech.-Engl. Ed.

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Based on the nonlocal theory and Mindlin plate theory, the governing equations (i.e., a system of partial differential equations (PDEs) for bending problem) of magnetoelectroelastic (MEE) nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle. The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions (MPS) to solve the governing equations numerically. It is confirmed that for the present bending model, the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points. Finally, the effects of different boundary conditions, applied loads, and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method. Some important conclusions are drawn, which should be helpful for the design and applications of electromagnetic nanoplate structures.

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Section: Verification Of the Present Methods And The Determination Of mentioning

confidence: 99%

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Feng

Yan

Lin

et al. 2020

Appl. Math. Mech.-Engl. Ed.

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show abstract

“…In the following tests, we obtained the solution of ( 1) by means of DRM, FDRM, CDNN and FCDNN, respectively. Their network size is (120, 60, 50, 50, 40), (60, 60, 50, 50, 40), (100, 50,30,30,20) and (50,50,30,30,20), respectively. It is easy to know that the number of parameters for the methods is 14100, 10800, 14000 and 10000, respectively.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Based on this coupled scheme, the finite difference technique [9,18,19], finite element technique and mixed element technique [20][21][22][23][24] are naturally used to solve the two second-order equations. In addition, the collocation method [25][26][27] and radial basis functions (RBF) method [28][29][30] are also approaches considered to solve (1).…”

Section: Introductionmentioning

confidence: 99%

Solving a Class of High-Order Elliptic PDEs Using Deep Neural Networks Based on Its Coupled Scheme

Li¹,

Zhang

et al. 2022

Mathematics

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Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al. to solve a class of variational problems that generally stem from partial differential equations, we present a coupled deep neural network (CDNN) to solve the fourth-order biharmonic equation by splitting it into two well-posed Poisson’s problems, and then design a hybrid loss function for this method that can make efficiently the optimization of DNN easier and reduce the computer resources. In addition, a new activation function based on Fourier theory is introduced for our CDNN method. This activation function can reduce significantly the approximation error of the DNN. Finally, some numerical experiments are carried out to demonstrate the feasibility and efficiency of the CDNN method for the biharmonic equation in various cases.

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“…From 1990, RBFs have been used as meshless (or meshfree) methods [14][15][16][17] to approximate the solutions of partial differential equations (PDEs). [18][19][20][21][22] The so-called Kansa's method is one of the collocation methods, [18] in which the approximate solution is obtained by collocation with RBFs on nodes.…”

Section: Introductionmentioning

confidence: 99%

Application of radial basis functions to evolution equations arising in image segmentation

Li¹,

Li²

2014

Chinese Phys. B

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In this paper, radial basis functions are used to obtain the solution of evolution equations which appear in variational level set method based image segmentation. In this method, radial basis functions are used to interpolate the implicit level set function of the evolution equation with a high level of accuracy and smoothness. Then, the original initial value problem is discretized into an interpolation problem. Accordingly, the evolution equation is converted into a set of coupled ordinary differential equations, and a smooth evolution can be retained. Compared with finite difference scheme based level set approaches, the complex and costly re-initialization procedure is unnecessary. Numerical examples are also given to show the efficiency of the method.

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A meshless method based on boundary integral equations and radial basis functions for biharmonic-type problems

Cited by 20 publications

References 35 publications

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Solving a Class of High-Order Elliptic PDEs Using Deep Neural Networks Based on Its Coupled Scheme

Application of radial basis functions to evolution equations arising in image segmentation

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