弹塑性不可压问题的平衡有限元法研究

Incompressibility is frequently encountered in the elastic or plastic analysis of solids and fluids. Generally, when dealing with incompressible problems, the conventional displacement finite element method often suffers from the volumetric locking, which will lead to unacceptable deviation between the numerical solution and the exact solution. To avoid volumetric locking for elastoplastic incompressible problems, this program is devoted to develop a new complementary energy based equilibrium finite element method; the motivation is simple: complementary energy can keep the same form for both compressible and incompressible problems and the incompressible condition tailored for displacements is not involved herein, therefore, the equilibrium finite element method is reasonably volumetric locking-free. To this end, the complementary energy principle is firstly proposed for elastic and plastic incompressible problems and the well-posedness is analyzed thereafter; next, an equilibrium element with no redundancies is to be developed for effectiveness purpose; at last, combining the complementary energy principle with the equilibrium element, the equilibrium finite element method for elastic and plastic incompressible problems is established and numerical examples will be studied to testify the feasibility and convergence of the proposed method.

不可压问题广泛存在于固体与流体、弹性与塑性等分析之中。常规的位移有限元法在处理不可压问题时会出现体积闭锁现象，导致数值解与真实解出现较大的偏差。本项目拟采用基于余能原理的平衡有限元法分析弹塑性不可压问题，由于余能原理可以同样的方式处理可压和不可压问题而不涉及位移的不可压条件，平衡有限元法将不存在体积闭锁现象。为此，本项目首先拟建立弹塑性不可压问题的余能原理，分析其适定性；接着，为提高计算效率，提出一种无冗余自由度的平衡单元；最后，结合余能原理和平衡单元，分别建立弹性和塑性问题的平衡有限元法的计算格式，并利用数值算例验证方法的可行性和收敛性。

数值仿真计算已经成为航空航天、土木、机械、电子等领域涉及与分析不可或缺的工具。传统的有限元法在处理不可压、薄板等问题时会出现诸如体积闭锁、剪切闭锁等现象，使得计算结果与真实解存在较大误差。本项目主要从余能原理出发，针对弹性、塑性、动力、Stokes流体等问题发展了新的平衡元以克服体积闭锁、剪切闭锁等难点问题。不仅如此，平衡元与有限元互为补充，可通过并行计算，得到计算误差的严格估计，结合对偶理论，还能得到如点位移、点应力等变量的严格上下界，为大规模结构自适应计算提供依据，并确保计算结果的可靠性。平衡元与以往位移型单元不同，它以应力作为基本变量，需要满足应力在法线方向上的连续性，即面力连续，因此通常做法是选取面力作为基本自由度。本项目还提出了一种基于结点自由度的平衡单元，该单元形成的自由度数目、带宽、条件数变化趋势与有限元法一致，因此，可以认为平衡元相较于常规的有限元法是一种具有竞争力的方法。大量的数值算例验证了所提方法的有效性，突出其固体流体求解一致性和无剪切闭锁的特性。