| 4 数值算例
为了验证以上提出的数值方法的有效性，本文对一些典型算例进行了动力弹塑性分析的计算和比较. 如果不作特别说明，积分时取Delaunay三角形内的3个高斯点. 为方便起见，本文算例所有参数采用无量纲形式.
算例1：受突加载荷的悬臂梁
一个长 ，高度 的悬壁梁，端部受到剪切荷载 的作用，如图2所示．材料为理想弹塑性，并服从von Mises屈服
准则. 材料的弹性模量 ，泊松比 ，屈服应力 ，质量密度 .
计算分析中采用如图3所示的两种结点布置方案，且时间步长取为 . 为了检验方法的有效性，本文使用Owatsiriwong和Park[4]的边界元法计算结果及有限元软件ABAQUS的计算结果进行对比.图4给出了悬臂梁右端中点A的竖向位移随时间的变化曲线，从图上可以看出本文采用规则结点和不规则结点的计算结果与边界元法的计算结果以及ABAQUS的计算结果都吻合很好，从而表明了本文方法的有效性，同时也表明了本文计算结果对结点的不规则分布不敏感.
算例2：受瞬态载荷的带孔方板
如图5所示，长为36，宽为20的矩形板，中央有一直径为10的圆形孔洞，在平面应力情况下两侧作用瞬态载荷 ，其中 的表达式为
材料采用双线性各向同性强化本构关系，并服从von Mises屈服准则. 材料参数为：弹性模量 ，泊松比，初始屈服应力 无网格法，切线模量 ，质量密度 .由于对称性，取结构的四分之一作为计算模型，其结点布置方案如图6所示，共589个结点. 采用时间步长 计算得到了点A处的竖向位移随时间的变化曲线，如图7所示. 为了进行对比，图7中还给出了有限元软件ABAQUS的计算结果. 从图7可以看出，本文的计算结果与ABAQUS的计算结果吻合很好，从而进一步验证了本文方法的有效性.
5 结论
自然单元法作为一种无网格数值方法，不需要对求解域进行网格划分，前处理简单.从形函数和近似函数的构造过程来看，自然单元法更像是介于有限元法与无网格法之间的一种数值方法，兼具无网格的特性和有限元的优点.本文建立了进行结构弹塑性动力响应分析的无网格自然单元法，弹塑性动力学积分弱形式的推导采用加权残数法，空间离散采用无网格自然单元法，时间离散采用预校正形式的Newmark方法. 数值算例表明，本文提出的方法能够有效用于结构的弹塑性动力响应分析问题的求解.
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