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首页> 外文期刊>Engineering analysis with boundary elements >A new Fragile Points Method (FPM) in computational mechanics, based on the concepts of Point Stiffnesses and Numerical Flux Corrections
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A new Fragile Points Method (FPM) in computational mechanics, based on the concepts of Point Stiffnesses and Numerical Flux Corrections

机译：基于点刚度和数值通量校正的概念的一种新的计算力学脆弱点方法（FPM）

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In this paper, a new method, named the Fragile Points Method (FPM), is developed for computer modeling in engineering and sciences. In the FPM, simple, local, polynomial, discontinuous and Point-based trial and test functions are proposed based on randomly scattered points in the problem domain. The local discontinuous polynomial trial and test functions are postulated by using the Generalized Finite Difference method. These functions are only piece-wise continuous over the global domain. By implementing the Point-based trial and test functions into the Galerkin weak form, we define the concept of Point Stiffnesses as the contribution of each Point in the problem domain to the global stiffness matrix. However, due to the discontinuity of trial and test functions in the domain, directly using the Galerkin weak form leads to inconsistency. To resolve this, Numerical Flux Corrections, which are frequently used in Discontinuous Galerkin methods are further employed in the FPM. The resulting global stiffness matrix is symmetric and sparse, which is advantageous for large-scale engineering computations. Several numerical examples of 1D and 2D Poisson equations are given in this paper to demonstrate the high accuracy, robustness and convergence of the FPM. Because of the locality and discontinuity of the Point-based trial and test functions, this method can be easily extended to model extreme problems in mechanics, such as fragility, rupture, fracture, damage, and fragmentation. These extreme problems will be discussed in our future studies.
机译：在本文中，为工程和科学领域的计算机建模开发了一种称为脆弱点方法（FPM）的新方法。在FPM中，基于问题域中的随机分散点，提出了简单，局部，多项式，不连续和基于点的试验功能。通过使用广义有限差分法来假定局部不连续多项式试验和测试函数。这些功能在全球范围内只是分段连续的。通过将基于点的试验和测试功能实现为Galerkin弱形式，我们将点刚度的概念定义为问题域中每个点对整体刚度矩阵的贡献。但是，由于域中试验和测试功能的不连续，直接使用Galerkin弱形式会导致不一致。为了解决这个问题，在FPM中进一步使用了不连续Galerkin方法中经常使用的数值通量校正。所得的整体刚度矩阵是对称且稀疏的，这对于大规模工程计算是有利的。本文给出了1D和2D泊松方程的几个数值示例，以证明FPM的高精度，鲁棒性和收敛性。由于基于点的试验功能的局部性和不连续性，因此可以轻松地将此方法扩展为对力学中的极端问题进行建模，例如易碎性，破裂，断裂，损坏和破碎。这些极端问题将在我们的未来研究中进行讨论。

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