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<p>In this paper, a generalized biharmonic equation has been numerically solved using two Fragile Points Methods (FPM). Following up on previous studies, we consider the trial function in the form of local, simple, and discontinuous polynomials. Another method uses the Heaviside step function as the test function, and we call it the Heaviside FPM and the results of the two methods have been compared. In continuation, numerical flux correction was used to correct the inconsistency due to discontinuous trial functions. The Voronoi diagram is applied to divide domains into subdomains, and for this reason, the method has good accuracy on irregular domains. Some numerical examples are presented to evaluate the efficiency, accuracy, and speed of calculations. Numerical integrations are used to solve these numerical examples in each subdomain, which only require one integration point in the Gaussian quadrature scheme, which is an advantage compared to other meshless methods. Additionally, FPM applies to all kinds of boundary conditions, such as Dirichlet, Neumann, and a combination of these two. These examples are presented on regular and irregular domains and show that the domain of the problem does not create a limit for implementation.</p>

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Application of the fragile points method for two-dimensional generalized biharmonic equation on arbitrary domains

  • 期刊论文

  • 发表日期：

    2025年4月3日

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  • Donya Haghighi,
  • Saeid Abbasbandy,
  • Yue Guan,
  • Elyas Shivanian

摘要

In this paper, a generalized biharmonic equation has been numerically solved using two Fragile Points Methods (FPM). Following up on previous studies, we consider the trial function in the form of local, simple, and discontinuous polynomials. Another method uses the Heaviside step function as the test function, and we call it the Heaviside FPM and the results of the two methods have been compared. In continuation, numerical flux correction was used to correct the inconsistency due to discontinuous trial functions. The Voronoi diagram is applied to divide domains into subdomains, and for this reason, the method has good accuracy on irregular domains. Some numerical examples are presented to evaluate the efficiency, accuracy, and speed of calculations. Numerical integrations are used to solve these numerical examples in each subdomain, which only require one integration point in the Gaussian quadrature scheme, which is an advantage compared to other meshless methods. Additionally, FPM applies to all kinds of boundary conditions, such as Dirichlet, Neumann, and a combination of these two. These examples are presented on regular and irregular domains and show that the domain of the problem does not create a limit for implementation.