The Trefftz methods for 3D biharmonic equation using directors and in-plane biharmonic functions

IF 8.7 2区 工程技术 Q1 Mathematics Engineering with Computers Pub Date : 2024-04-29 DOI:10.1007/s00366-024-01977-1
Chein-Shan Liu, Chung-Lun Kuo
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Abstract

Because the complete set of Trefftz functions for the 3D biharmonic equation is not yet well established, a multiple-direction Trefftz method (MDTM) and an in-plane biharmonic functions method (IPBFM) are deduced in the paper. Inspired by the Trefftz method for the 2D biharmonic equation, a novel MDTM incorporates planar directors into the 2D like Trefftz functions to solve the 3D biharmonic equation. These functions being a series of biharmonic polynomials of different degree, automatically satisfying the 3D biharmonic equation, are taken as the bases to expand the solution. Then, we derive a quite large class solution of the 3D biharmonic equation in terms of 3D harmonic functions, and 2D biharmonic functions in three sub-planes. The 2D biharmonic functions are formulated as the Trefftz functions in terms of the polar coordinates for each sub-plane. Introducing a projective variable, we can obtain the projective type general solution for the 3D Laplace equation, which is used to generate the 3D Trefftz type harmonic functions. Several numerical examples confirm the efficiency and accuracy of the proposed MDTM and IPBFM.

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使用导演和平面内双谐函数的三维双谐方程的特雷弗茨方法
由于三维双谐波方程的全套特雷弗兹函数尚未完全建立,本文推导出一种多方向特雷弗兹方法(MDTM)和一种平面内双谐波函数方法(IPBFM)。受用于二维双谐波方程的 Trefftz 方法的启发,一种新型 MDTM 将平面导向融入类似 Trefftz 函数的二维双谐波方程中,以求解三维双谐波方程。这些函数是一系列不同度数的双谐波多项式,自动满足三维双谐波方程,并以此为基础展开求解。然后,我们用三维谐函数和三个子平面上的二维谐函数推导出了三维双谐方程的大类解。二维双谐函数是以每个子平面的极坐标表示的 Trefftz 函数。通过引入一个投影变量,我们可以得到三维拉普拉斯方程的投影型通解,并用它来生成三维特雷弗兹型谐函数。几个数值示例证实了所提出的 MDTM 和 IPBFM 的效率和准确性。
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来源期刊
Engineering with Computers
Engineering with Computers 工程技术-工程:机械
CiteScore
16.50
自引率
2.30%
发文量
203
审稿时长
9 months
期刊介绍: Engineering with Computers is an international journal dedicated to simulation-based engineering. It features original papers and comprehensive reviews on technologies supporting simulation-based engineering, along with demonstrations of operational simulation-based engineering systems. The journal covers various technical areas such as adaptive simulation techniques, engineering databases, CAD geometry integration, mesh generation, parallel simulation methods, simulation frameworks, user interface technologies, and visualization techniques. It also encompasses a wide range of application areas where engineering technologies are applied, spanning from automotive industry applications to medical device design.
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