解决‘Snap-through’的快速增量/迭代方法

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"M.A.Crisfield的文章‘A Fast Incremental/Iterative Solution Procedure that Handles "Snap-through"’是关于非线性有限元分析中处理‘Snap-through’问题的一个经典研究。Snap-through是指结构在非线性加载过程中，由于过度变形而突然穿越稳定状态的现象，常见于大位移、大应变的结构分析。文章提出的解决方案是基于Riks的方法，通过添加一个约束方程来固定加载步长，使加载水平成为额外的变量，以克服极限点问题。" 在非线性有限元分析中，解决Snap-through现象是一项挑战。传统的增量迭代方法可能会在结构达到极限状态时遇到收敛困难或停滞不前。Riks的方法引入了一个新的概念，即在标准平衡方程之外添加一个约束方程，这个约束直接控制了加载步长，确保在位移-荷载空间中的增量步不越过极限点。这种方法使得结构能够在不稳定的边缘继续进行迭代，从而有效地处理Snap-through问题。 Crisfield在此基础上对Riks的方法进行了改进，使其适应有限元方法的框架。他将这种改良的Riks方法与修改后的牛顿-拉弗森方法结合，既包括原版也包括加速形式。修改后的牛顿-拉弗森方法是一种强大的非线性求解策略，通过迭代更新节点的残差和刚度矩阵，逐步逼近解。当与Riks的加载步长控制结合时，可以显著提高算法的收敛性能，尤其在处理涉及Snap-through的复杂非线性问题时。文章通过一系列示例验证了该技术的有效性，这些示例可能包括大变形情况，如薄壳结构的弯曲、桥梁的弹性后屈曲等，这些情况在实际工程中都可能出现Snap-through现象。通过这些实例，Crisfield展示了新方法如何不仅能够成功地穿越极限点，而且还能改善传统无约束迭代过程的收敛特性。这篇文章为非线性有限元分析提供了一种快速且有效的解决方案，特别是在处理结构可能出现Snap-through行为的情况下，具有重要的理论和实际意义。这一方法的实施和优化对于提升工程计算的效率和准确性至关重要。

c0mpu1m & SfnlclweJ Vol 13. pp 55-62

Printed in Great Britam All rights remvcd

0043-7949/81/01055-Qaso2.m~

A FAST INCREMENTAL/ITERATIVE SOLUTION

PROCEDURE THAT HANDLES

“SNAP-THROUGH”

M. A. CRISFIELD

Transport and Road Research Laboratory, Department of the Environment,

Crowthornc, Berkshire RGl 1 6AV, England

(Received 25 April 1980)

Abstract- Riks [l) has recently proposed a new solution procedure for overcoming limit points. To this

end, he adds, to the standard equilibrium equations, a constraint equation fixing the length of the incre-

mental load step in load/deflection space. The applied load level becomes an additional variable.

The present paper describes a means of modifying Rik’s approach so that it is suitable for use with the

finite element method. The procedure is applied in conjunction with the modified Newton-Rapbson method

in both its original and accelerated forms. The resulting techniques not only allow limit points to be passed.

but also, improve the convergence characteristics of the unconstrained iterative procedures. Illustrative

examples include the large deflection analysis of shallow elastic shells and the collapse analysis of a stiffened

steel diaphragm from a box-girder bridge.

INTRODUCllON

Snap-through and snap-back buckling phenomena

(Fig. 1) pose some of the most difficult problems in non-

linear struotural analysis. For most practical problems,

it is quite unnecessary to trace such a convulated load/

deflection path as that shown in Fig. 1. Indeed, were

such a path to be traced, most analyses would trace the

static path ABCDEFGHIJ and thus infer the dynamic

“snaps”. Although the analyses are therefore somewhat

artificial, they may be very important.

Dd*n,on. p

Fig. 1. “Snap buckling”

For some problems. all that may appear to be

required is the load level at the first limit point. How-

ever, without analysis techniques that allow the limit

points to be passed. even this information may be un-

available or unreliable. “Collapse loads” are often

associated with a failure to achieve convergence with

the iterative solution procedure. However, it may only

be the iterative solution procedure that has collapsed

(possibly as a consequence of round-off error). For

other problems. the analysis may be performed on an

individual component of a complete structure. In such a

situation, it may be important to obtain information on

the nature of the load shedding following the limit

points, in order to assess the performance of the com-

plete structure.

When analysing relatively simple structures, it is

tempting to try and avoid the full complexities of a

“snap analysis” by applying a simple form of displace-

ment control. Such ‘displacement control” may simul-

ate a physical testing procedure. For instance, referring

to Fig. 1, if the displacement p were to be prescribed,

the limit point B could be passed and the load-shedding

curve BC could be traced. However, a similar procedure

would fail at_ or just before, the limit point G. This fail-

ure might not matter if the analyst could conclude

“Following the (local) maximum at F, there is a very

sharp drop-off in load”. However, the dramatic non-

linear behaviour associated with the limit point G may

induce a failure in the incremental/iterative solution

procedure at point E (Fig. 1). In such a situation, the

analyst is left with no information on the nature of the

failure and may not even be sure that he has a htruc-

tural (rather than numerical) collapse. Consequently,

non-linear finite element computer programs should

be provided with solution procedures that will handle

such %napping phenomena” particularly if this can be

achieved without resorting to many ‘special tech-

niques” [2]. Present procedures generally involve the

use of fictitious springs [3] or adopt some form of

“displacement control” [4, S]. The main disadvantage

of the former method is the trial and error often in-

volved in the selection of the appropriate springs [6].

The disadvantages of the latter method relate to the

selection of the appropriate displacement variable. For

instance, Maewal and Nachbar [7] note the necessity

to change the prescribed displacement variable fol-

lowing slow convergence or divergence of the iterative

solution procedure.

Began et al. [6,8] use the ‘kurrent stiffness param-

eter” [S], to predict the position of the local maximum

or minimum. They then suppress the equilibrium itera-

tions in the ncighbourhood of the extremum (limit)

point and reverse the sign of the load following a change

in the sign of the determinant of the tangent stiffness

matrix. The supression of the equilibrium iterations

dictates the provision of very small load increments

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解决‘Snap-through’的快速增量/迭代方法

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