Multiresolution method for bending of plates with complex shapes

doi:10.1007/s10483-023-2972-8

Applied Mathematics and Mechanics (English Edition) ›› 2023, Vol. 44 ›› Issue (4): 561-582.doi: https://doi.org/10.1007/s10483-023-2972-8

Multiresolution method for bending of plates with complex shapes

Jizeng WANG^1,2, Yonggu FENG^1,2, Cong XU^1,2, Xiaojing LIU^1,2, Youhe ZHOU^1,2

1. Key Laboratory of Mechanics on Disaster and Environment in Western China (Lanzhou University), the Ministry of Education, Lanzhou 730000, China;
2. College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China

收稿日期:2022-08-05 修回日期:2023-01-16 发布日期:2023-03-30
通讯作者: Jizeng WANG, E-mail: jzwang@lzu.edu.cn
基金资助:
the National Natural Science Foundation of China (No. 11925204) and the 111 Project (No. B14044)

Multiresolution method for bending of plates with complex shapes

Jizeng WANG^1,2, Yonggu FENG^1,2, Cong XU^1,2, Xiaojing LIU^1,2, Youhe ZHOU^1,2

1. Key Laboratory of Mechanics on Disaster and Environment in Western China (Lanzhou University), the Ministry of Education, Lanzhou 730000, China;
2. College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China

Received:2022-08-05 Revised:2023-01-16 Published:2023-03-30
Contact: Jizeng WANG, E-mail: jzwang@lzu.edu.cn
Supported by:
the National Natural Science Foundation of China (No. 11925204) and the 111 Project (No. B14044)

摘要/Abstract

摘要： A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains. To realize this method, we design a new wavelet basis function, by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains. In the solution of differential equations, various derivatives of the unknown function are denoted as new functions. Then, the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals. Therefore, the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative. During the application of the proposed method, boundary conditions can be automatically included in the integration operations, and relevant matrices can be assured to exhibit perfect sparse patterns. As examples, we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes. By comparing the solutions obtained by the proposed method with the exact solutions, the new multiresolution method is found to have a convergence rate of fifth order. The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method (FEM) with tens of thousands of elements. In addition, because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order, we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.

关键词: multiresolution, generalized Coiflet, wavelet integral collocation method, irregular domain, complex shape, high accuracy, plate bending

Abstract: A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains. To realize this method, we design a new wavelet basis function, by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains. In the solution of differential equations, various derivatives of the unknown function are denoted as new functions. Then, the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals. Therefore, the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative. During the application of the proposed method, boundary conditions can be automatically included in the integration operations, and relevant matrices can be assured to exhibit perfect sparse patterns. As examples, we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes. By comparing the solutions obtained by the proposed method with the exact solutions, the new multiresolution method is found to have a convergence rate of fifth order. The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method (FEM) with tens of thousands of elements. In addition, because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order, we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.

Key words: multiresolution, generalized Coiflet, wavelet integral collocation method, irregular domain, complex shape, high accuracy, plate bending

中图分类号:

65L11
34B15

Jizeng WANG, Yonggu FENG, Cong XU, Xiaojing LIU, Youhe ZHOU. Multiresolution method for bending of plates with complex shapes[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(4): 561-582.

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Multiresolution method for bending of plates with complex shapes

Multiresolution method for bending of plates with complex shapes

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