Solutions of 2D and 3D non-homogeneous potential problems by using a boundary element-collocation method

Qu, Wenzhen; Chen, Wen; Fu, Zhuojia

doi:10.1016/j.enganabound.2015.04.018

Cited by 19 publications

(7 citation statements)

References 29 publications

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“…Due to the inherent mathematical difficulties, closed-form solutions exist in a few simple cases. Some traditional powerful methods, such as the finite element method(FEM) and the boundary element method(BEM), are used to solve the engineering problems by meshing [2]. In recent years the MFS-DRM approach has become popular, because it has lead to a mesh-free scheme, But only with a certain mathematical skill this approach can be achieved [3].…”

Section: Introductionmentioning

confidence: 99%

Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis

Guo,

Zhuang,

Chen

et al. 2020

Preprint

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A deep learning based collocation method is presented in this paper to solve the three dimensional potential problems in non-homogeneous media. Based on the universal approximation theorem, the neural network can be utilized to approximate solutions for different PDEs in different geometries. The performance of deep learning based method depends on the configurations of the network and other hyper-parameter settings. This makes the choice of neural network configurations extremely important. The configuration of this deep collocation method is setup by comparing different schemes of smooth activation functions, sampling methods for collocation points generation, combined optimizers. Besides, a convergence proof of this deep collocation method in solving non-homogeneous potential problems is performed. Then the deep collocation method is applied to the analysis of different material variations, and it can be concluded that the deep collocation method predicts the temperature and flux accurately for different material variations, especially the exponential material variations. As a result, the deep learning based collocation method shows a great potential in approximating solutions to PDEs.

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Section: Introductionmentioning

confidence: 99%

Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis

Guo,

Zhuang,

Chen

et al. 2020

Preprint

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“…they are characteristic functions of Γ v . SLAE for finding the vector of unknowns (s = 1, 2) is obtained by substituting ( 11)-( 15) in (10) and adding a condition similar to (5):…”

Section: Mathematical Model For a Piecewise Homogeneous Isotropic Hal...mentioning

confidence: 99%

“…Fig. 5 presents the errors of the apparent resistivity calculated by the formula (16) for problem (7)- (10) for electrical profiling and VES over the homogeneous half-plane. Fig.…”

Section: Practical Implementation Of the Proposed Approaches In Model...mentioning

confidence: 99%

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Using of partly-boundary elements as a version of the indirect near-boundary element method for potential field modeling

Zhuravchak¹,

Zabrodska²

2020

Math. Model. Comput.

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In this paper, the partly-boundary elements as a version of the indirect near-boundary element method has been considered. Accuracy and effectiveness of their using for 2D problems of potential theory have been investigated. It is shown that using of partly-boundary elements for objects of canonical shape (circle, square, rectangle, ellipse) and arbitrary polygons allows us to achieve the solution accuracy, which is comparable with the accuracy of the indirect near-boundary element method, and its order of magnitude is higher than in the indirect boundary element method. In this case, the computation time is reduced by 2–2.5 times than in the near-boundary element method case. The software of the proposed approach has been implemented in Python. Practical testing was carried out for the tasks of electrical profiling and vertical electrical sounding in the half-plane with inclusion as a polygon. The recommendations for application of the partly-boundary elements in geophysical practice have been given.

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“…A comparison of the SBM and the indirect boundary integral equation (IBIE) with a one‐point (IBIE‐1P) and eight‐point (IBIE‐8P) integration rules is provided. For the IBIE, the boundary is discretized by exact elements , and the boundary quantity in each element is a constant. With an increasing number of boundary nodes or elements, Figure plots the variation of the numerical errors of the acoustic pressure at the point (6, 0, 0) by using the SBM, the IBIE‐1P, and the IBIE‐8P.…”

Section: Numerical Examplesmentioning

confidence: 99%

Diagonal form fast multipole singular boundary method applied to the solution of high‐frequency acoustic radiation and scattering

Chen

Zheng

2017

Numerical Meth Engineering

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Summary The singular boundary method (SBM) is a recent strong‐form boundary collocation method that uses a linear combination of the fundamental solution of the governing equation to approximate the field variables. Because of its full interpolation matrix, the SBM solution encounters the high computational complexity and storage requirement that limit its applications to large‐scale engineering problems. This paper presents a way to overcome this drawback by introducing the diagonal form fast multipole method. A diagonal form fast multipole singular boundary method is then developed to reduce the computational operations of the SBM with direct solvers from O(N3) to O(NlogN), where N is the number of unknowns. The proposed method works well for acoustic radiation and scattering with nondimensional wave number kD<110 (k is the wave number and D the maximum diameter of the computational domain). Several numerical experiments are provided to show the efficiency and accuracy of the present strategy, including the radiation from a nuclear submarine and the scattering from an airplane. The numerical results clearly demonstrate that the present diagonal form fast multipole singular boundary method breaks the limitations of the SBM and successfully simulates the large‐scale acoustic problems with more than one million unknowns on a desktop computer. Copyright © 2016 John Wiley & Sons, Ltd.

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Solutions of 2D and 3D non-homogeneous potential problems by using a boundary element-collocation method

Cited by 19 publications

References 29 publications

Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis

Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis

Using of partly-boundary elements as a version of the indirect near-boundary element method for potential field modeling

Diagonal form fast multipole singular boundary method applied to the solution of high‐frequency acoustic radiation and scattering

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