| 景钊,王思齐,刘彦杰.基于离散里兹法的任意形状Mindlin板弯曲分析[J].计算力学学报,2025,42(4):597~606 |
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| 基于离散里兹法的任意形状Mindlin板弯曲分析 |
| Discrete Ritz Method for bending analysis of arbitrarily shaped Mindlin plates |
| 投稿时间：2024-02-29 修订日期：2024-04-30 |
| DOI：10.7511/jslx20240229001 |
| 中文关键词: 离散里兹法 弯曲 数值方法 任意形状板 变刚度 |
| 英文关键词:Discrete Ritz Method bending numerical method arbitrarily shaped plate variable stiffness |
| 基金项目:国家自然科学基金(12102352);中央高校基金(D5000230074);秦创原建设两链融合专项(23LLRH0002)资助项目. |
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| 中文摘要: |
| 里兹法需构造满足边界条件的全域试函数并在板域上对应变能积分,求解定义在复杂几何域上的变分问题极为困难。为此,本文提出了一种离散能量的新型无网格数值方法,即离散里兹法(DRM)。该方法将任意几何构型板置入一个可覆盖它的最小标准矩形域,通过在标准矩形域内板的真实几何边界内外开孔模拟其几何构型;之后,基于勒让德多项式构造满足位移边界条件的全域试函数;再建立标准矩形域上的能量泛函并用高斯积分点对板域进行离散,其中任意几何构型板的能量分布可通过标准矩形域开孔内高斯积分点处刚度置零模拟,即通过扩展区间积分、全域高斯积分点离散、全域变刚度表征与全域试函数模拟任意几何构型板的应变能;最后,基于最小势能原理求泛函驻值获得任意几何构型板弯曲问题的数值解。离散里兹法解决了里兹法在复杂几何域上无法求解的问题,且求解公式与计算程序完全标准。基于离散里兹法对复杂几何构型Mindlin板进行弯曲分析,并与已有文献和有限元法(FEM)结果进行对比,验证了该方法的准确性、稳定性和通用性。 |
| 英文摘要: |
| It is difficult for Ritz method to solve variational problems defined in complex geometric domains,as it requires the construction of a global trial function that satisfies boundary conditions and integration of strain energy.A novel meshless numerical method for discretizing energy is therefore proposed in this study,known as the Discrete Ritz Method(DRM).In the DRM,an arbitrarily shaped plate is placed within a minimum standard rectangular domain,where inside or outside cutouts are created according to the real geometric boundary of the plate to simulate its geometry.Next,a global trial function satisfying the boundary condition is constructed based on Legendre polynomials.The energy functional on the standard rectangular domain is then established and the plate domain is discretized by Gauss integration points,in which energy distribution of the arbitrarily shaped plate can be simulated by setting the stiffness of Gauss integration points in the opening of the standard rectangular domain to zero.The strain energy the of the arbitrarily shaped plate can be represented that is,via extended interval integral,global Gaussian points discretization,global variable stiffness characterization,and global admissible function.Finally,based on the minimum potential energy principle,the stationary value of the functional is obtained and the numerical solution for bending of the arbitrarily shaped plate is acquired.The DRM enables Ritz method to be applied in complex geometric domains,and its solution formula and calculation program are completely standard.Based on the DRM,the bending analysis of a Mindlin plate with complex geometry is carried out,and results of the DRM are compared with results in the existing literature and the FEM,verifying the accuracy,stability,and generality of the DRM. |
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