EMBEDDED STRONG DISCONTINUITY MODEL BASED MULTISCALE FINITE ELEMENT METHOD FOR STRAIN LOCALIZATION ANALYSIS
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摘要: 固体材料的应变局部化行为是导致结构破坏失效的重要因素之一，开展相关数值模拟分析对于结构安全性评估具有重要意义。然而由于材料的非均质和多尺度特性，采用传统数值方法进行求解时通常需要从最小特征尺度离散求解的结构，这将大幅度增加计算规模和成本。针对这一问题，本文提出了一种基于嵌入强间断模型的多尺度有限元方法。该方法从粗细两个尺度离散求解模型，首先在细尺度单元上引入嵌入强间断模型来描述单元间断特性，所附加的跳跃位移自由度则通过凝聚技术进行消除，从而保持细尺度单元刚度阵维度不变。其次，提出了一种增强多节点粗单元技术，其可根据局部化带与粗单元边界相交情况自适应动态地增加粗节点，新构造的增强数值基函数可以捕捉细尺度间断特性，完成物理信息从细单元到粗单元的准确传递以及宏观响应的快速分析；再次，在细尺度解的计算中，将细尺度解分解为降尺度解与单胞局部摄动解，从而消除弹塑性分析时单胞内部的不平衡力。最后，通过两个典型算例分析，并与完全采用细单元的嵌入有限元结果进行对比，验证了所提出算法的正确性与有效性。Abstract: Strain localization is a common factor that may lead to the failure of solid structure and its numerical analysis becomes an important aspect for the structural safety evaluation. Due to the heterogeneity and multiscale nature, however, traditional numerical methods need to resolve the structure at the fine scale to obtain reasonable results, which increases drastically the computational scale and cost. To solve this problem, an embedded strong discontinuity model based multiscale finite element method is proposed here. In this method, both coarse and fine scale elements are used to represent the structure. The embedded strong discontinuity model is first introduced into the fine element to describe the discontinuity and the corresponding additional displacement jump degree of freedom on the elemental level can be eliminated with the condensation technique, which keeps the dimensions of the stiffness matrix unchanged. Then, an enhanced multi-node coarse element technique is proposed, which can adaptively insert coarse nodes according to the intersection between the discontinuity line and coarse element boundary and thus guarantees the proper transformation of information between the fine and coarse elements. The problem can then be effectively solved on the coarse scale level. Moreover, a solution decomposition technique, in which the fine scale solution is decomposed into the downscaling and local perturbation solutions, is adopted to eliminate the unbalance forces within the unit cell in the elasto-plastic analysis. Finally, two representative examples are presented to demonstrate the accuracy and effectiveness of the proposed method through the comparisons with the results of the embedded finite element method.
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图 2 局部化单元内角点集合Se+的选取
Figure 2. Selection for the corner nodes set Se+ in a discontinuous element
图 3 (a)多节点粗单元，(b)~(d)分别为1号、9号和5号粗节点的边界指定位移分布图
Figure 3. (a) a multi-node coarse element, the distributions of the prescribed boundary values for (b) coarse node 1, (c) coarse node 9 and (d) coarse node 5
图 5 不同方法作用反力与加载位移的结果比较
Figure 5. Results of the reaction force versus the imposed displacement with different methods
图 6 不同方法竖直位移的云图
Figure 6. Contours of the vertical displacement with different methods
图 7 基础水平加载算例示意图
Figure 7. Illustration of the horizontal loading test of the foundation
图 8 不同方法作用反力与加载位移的结果比较
Figure 8. Results of the reaction force versus the imposed displacement with different methods
图 10 不同方法水平位移的云图
Figure 10. Contours of the horizontal displacement with different methods
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