hyperworks非线性分析理论详解

批注 [张剑龙3]: 非线性解法 基本的牛顿法用于解决非线性问题。 该方法的原理如下图所示，用于一维 问题，可以表达如下：
考虑一个非线性问题：
Consider a nonlinear problem:
Where, u is the displacement vector, P is the global load vector, and L(u) is the nonlinear response of the system (nodal reactions). Note that for a linear problem, L(u) would simply be Ku (as described in the Linear Static Analysis section). Application of Newton's method to this equation leads to an iterative solution procedure:
selected on the NLPARM bulk data card. The basic principle in assessing nonlinear convergence is to compare an error measure of the solution with a pre-determined tolerance level. When the error falls below the prescribed tolerance, the problem is considered converged. In a case of multiple, simultaneous convergence criteria, all criteria need to be satisfied for the solution to be converged.
stiffness matrix
and the vector norm II. II is calculated as:
批注 [张剑龙6]: 位移的相对误差（作为 EUI 汇总在汇 总表中表示）计算为：
批注 [张剑龙4]: 增量加载 对于满足一定稳定性和平滑度条件 的大类问题，假设初始猜测足够接近 真实的力 - 位移路径 L（u），牛顿迭 代法是收敛的。因此，为了提高强非 线性问题的收敛性，总负载 P 通常以 较小的增量应用，如下图所示。在每 个中间负载 P1，P2 等，执行标准牛 顿迭代。
被称为增量加载的这个过程有助于 使连续的迭代保持更接近于真实的 负载路径，从而提高获得最终收敛解 的机会（尽管通常以增加的迭代次数 为代价）。
这种形式很容易通过在牛顿方程的 两边添加 Kn un 而产生，并且在实际 实现中具有一定的优点。
Note that the above scheme is somewhat modified to an equivalent format wherein,
instead of calculating
HyperWorks Solvers
Nonlinear Analysis
OptiStruct > User's Guide > Structural Analysis:
Nonlinear Analysis
Nonlinear Analysis
Nonlinear Quasi-Static Analysis Large Displacement Nonlinear Static Analysis Geometric Nonlinear Analysis HyperWorks Solvers
Nonlinear Solution Method
The basic Newton method is used for the solution of nonlinear problems. The principle of this method is illustrated for a one-dimensional problem in the figure below and can be formulated as follows:
批注 [张剑龙5]: 非线性收敛准则 为了评估非线性过程是否收敛，可以 使用多个收敛准则。可以在 NLPARM 批量数据卡上选择这些标准和相应 的公差。评估非线性收敛的基本原理 是将解决方案的误差测量与预定的 公差水平进行比较。当误差低于规定 的公差时，该问题被认为是收敛的。 在多个同时收敛标准的情况下，需要 满足要收敛的解的所有标准。
其中，u 是位移矢量，P 是全局负载 矢量，L（u）是系统的非线性响应（节 点反应）。 注意，对于线性问题，L （u）将简单地是 Ku（如线性静态分 析部分所述）。 牛顿方法在这个方程 中的应用导致迭代解法：
其中，
在上述公式中，Kn 表示“斜率”矩 阵，定义为在点 un 处的 L（u）曲线 的切线，Rn 是非线性残差。 在一定 的收敛条件下迭代地重复这个过程， 导致残差 Rn 的系统减少，从而导致 收敛。 注意，上述方案被稍微修改为等价格 式，其中，代替计算，直接获得新解 un + 1：
Nonlinear Quasi-Static Analysis
This solution sequence performs quasi-static nonlinear analysis. Presently, the sources of nonlinearity include CONTACT interfaces, GAP elements, and MATS1 elastic-plastic material.
, the new solution un+1 is directly obtained:
This form is readily produced by adding Kn un to both sides of Newton's equation, and has certain advantages in practical implementations.
This procedure, known as incremental loading, helps to keep the consecutive iterations closer to the true load path, thereby improving the chances of obtaining a final, converged solution (though usually at the expense of an increased total number of iterations).
Incremental Loading
For a large class of problems satisfying certain stability and smoothness conditions, the Newton's iterative method is proven to converge, provided that the initial guess is sufficiently close to the true force-displacement path L(u). Hence, to improve convergence for strongly nonlinear problems, the total loading P is often applied in smaller increments, as shown in the figure below. At each of the intermediate loads, P1, P2, etc., the standard Newton iterations are performed.
Nonlinear Quasi-Static Analysis
OptiStruct > User's Guide > Structural Analysis > Nonlinear Analysis:
Nonlinear Quasi-Static Analysis
批注 [张剑龙1]: 非线性准静态分析 大位移非线性静力分析 几何非线性分析
Nonlinear Convergence Criteria
In order to assess whether the nonlinear process has converged, a number of convergence criteria are available. These criteria and respective tolerances can be
Small deformation theory is used in the solution of nonlinear problems, similar to the way it is used with Linear Static Analysis. Inertia relief is also possible. Small deformation theory means that strains should be within linear elasticity range (some 5 percent strain), and rotations within small rotation range (some 5 degrees rotation). This also means that there is no update of gap/contact element locations or orientation due to the deformations – they remain the same throughout the nonlinear computations. The orientation may change, however, due to geometry changes in optimization runs.
The relative error in displacements (printed in the convergence summary as EUI) is calculated as:
Here, A is a normalizing vector consisting of square roots of diagonal elements of
批注 [张剑龙2]: 非线性准静态分析 该解决方案序列执行准静态非线性 分析。目前，非线性的来源包括 CONTACT 接触，GAP 单元和 MATS1 弹性塑料材料。 小变形理论被用于解决非线性问题， 类似于线性静态分析的方法。惯性救 济也是可能的。小变形理论意味着应 变应在线弹性范围内（约 5％应变）， 并在小旋转范围（约 5 度转角）内旋 转。这也意味着由于变形而不存在间 隙/接触单元位置百度文库方向的更新 - 它 们在整个非线性计算中保持不变。然 而，由于优化运行中的几何变化，方 向可能会改变。
Where,
In the above formulas, Kn represents a "slope" matrix, defined as a tangent to the L(u) curve at a point un , and Rn is the nonlinear residual. Repeating this procedure iteratively, under certain convergence conditions, leads to systematic reduction of residual Rn and hence, convergence.