形状记忆合金(shape memory alloys,简称SMA)具有复杂的热力本构关系,为了模拟SMA 及其组合结构复杂的受力和变形行为,在数值模拟中需要采用可靠且高效的应力点积分算法. 隐式应力点回映算法已经成功应用于形状记忆合金的数值模拟,但在复杂加载条件下,荷载增量较大时有可能导致整体非线性迭代求解不收敛. 推广了局部误差控制的显式子步积分算法,首次将其应用于形状记忆合金及其组合结构这类热力相变问题的应力点积分,并通过数值算例对所提算法和隐式应力点回映算法进行了比较. 数值结果表明:对于大规模数值模拟和计算,整体子步步数决定着总体计算时间;所提出的修正Euler 自动子步方案可以有效减少整体子步步数,在保证相同计算精度的前提下能够大幅提高有限元计算效率,因而更适合大规模形状记忆合金智能结构的数值模拟.

Biot’s symmetric indefinite linear systems of equations are commonly encountered in finite-element computations of geotechnical problems. The development of efficient solution methods for Biot’s linear systems of equations is of practical importance to geotechnical software packages. In conjunction with the Krylov-subspace iterative method symmetric quasi-minimal residual (SQMR), some zero-level fill-in incomplete factorization preconditioning techniques including a symmetric successive overrelaxation (SSOR) type method and several zero-level incomplete LU [ILU(0)] methods are investigated and compared for Biot’s symmetric indefinite linear systems of equations. Numerical experiments are carried out based on three practical geotechnical problems. Numerical results indicate that ILUð0Þ preconditioners are classical and generally efficient when adequately stabilized. However, the tunnel problem provides a counterexample demonstrating that ILU(0) preconditioners cannot be fully stabilized by preliminary scaling, reordering, making use of perturbed matrices, or dynamically selecting pivots. Compared with the investigated ILUð0Þ preconditioners, the recently proposed modified SSOR preconditioner is less efficient but is robust over the range of problems studied.

Purpose – The initial stiffness method has been extensively adopted for elasto-plastic finite element analysis. The main problem associated with the initial stiffness method, however, is its slow convergence, even when it is used in conjunction with acceleration techniques. The Newton-Raphson method has a rapid convergence rate, but its implementation resorts to non-symmetric linear solvers, and hence the memory requirement may be high. The purpose of this paper is to develop more advanced solution techniques which may overcome the above problems associated with the initial stiffness method and the Newton-Raphson method. Design/methodology/approach – In this work, the accelerated symmetric stiffness matrix methods, which cover the accelerated initial stiffness methods as special cases, are proposed for non-associated plasticity. Within the computational framework for the accelerated symmetric stiffness matrix techniques, some symmetric stiffness matrix candidates are investigated and evaluated. Findings – Numerical results indicate that for the accelerated symmetric stiffness methods, the elasto-plastic constitutive matrix, which is constructed by mapping the yield surface of the equivalent material to the plastic potential surface, appears to be appealing. Even when combined with the Krylov iterative solver using a loose convergence criterion, they may still provide good nonlinear convergence rates. Originality/value – Compared to the work by Sloan et al., the novelty of this study is that a symmetric stiffness matrix is proposed to be used in conjunction with acceleration schemes and it is shown to be more appealing; it is assembled from the elasto-plastic constitutive matrix by mapping the yield surface of the equivalent material to the plastic potential surface. The advantage of combining the proposed accelerated symmetric stiffness techniques with the Krylov subspace iterative methods for large-scale applications is also emphasized.