接触问题的三角形载荷离散FFT加速算法

接触问题控制方程的有效求解，往往涉及到复杂的数学理论知识，而在实际工程应用中，接触应力分布又具有高度随机性。为高效快速求解任意载荷分布下固体的接触响应，基于三角形载荷离散单元，嵌入离散卷积快速傅里叶变换（DC-FFT）算法，提供了一种高精度、高可靠度的计算方法。相比于通常采用的分段均布载荷离散方法，三角形单元的解析求解略显复杂，但能更好地模拟接触载荷任意分布的特性，对于接触边缘处载荷由零递增或递减为零的情况，也可以予以充分考虑。为优化三角形载荷离散单元的求解方法，根据接触影响系数矩阵的“激励-响应”特性，推导了三角形载荷单元和均布载荷单元作用下的应力分量解析解。通过构造包含影响系数矩阵的离散卷积形式应力解，将某一目标节点在所有载荷单元作用下，重复度极高的矩阵运算叠加过程，采用DC-FFT算法来简化加速计算。通过程序编程计算，分析验证了所提出算法的精确度和高效性。

FFT acceleration algorithm for contact problems based on triangular element discretization

Effectively solving the governing equations for contact problems often involves complex mathematical theory, while the distribution of contact stress is highly random in practical engineering applications. This study proposes a novel algorithm based on the triangular load discrete element and the discrete convolution fast Fourier transform (DC-FFT) algorithm. This algorithm provides a high-precision and reliable method for efficiently solving the contact response of a solid under any load distribution. Compared to the commonly used uniform load element discrete method, the analytical solution of the triangular element is more complex. However, it better simulates the characteristics of contact load distribution, accounting for situations where the load at the contact edge increases from zero or decreases to zero. The stress component under the action of the triangular and uniform load elements is derived based on the “excitation-response” characteristics of the contact influence coefficient matrix. This information is used to optimize the solution method of the triangular load discrete element. By constructing the stress solution in the form of a discrete convolution, including the influence coefficient matrix, the stress superposition effect of a target node under the action of all elements can be further simplified and accelerated by using the DC-FFT algorithm for highly repetitive matrix calculations. Programming and calculation analysis show that the proposed algorithm based on the triangular load element is accurate and efficient.