Metamodel-based approaches are very often used for nonlinear optimization. The Successive Response Surface Method (SRSM) (Roux et al. in Int J Numer Methods Eng 42:517–534, 1998; Stander and Craig in Eng Comput 19:431–450, 2002) is one example for the application of such an approach. SRSM computes a smoothing approximation for each system response in each iteration based on a set of sample points generated in a defined domain around the current optimum. This domain is shifted and contracted in each iteration according to the resulting current best point. By default, a linear polynomial is used as the approximation function. A new Linear Interpolation Approach (LInA) has been developed to reduce the number of nonlinear analyses required during the optimization. In contrast to SRSM, LInA uses interpolating linear polynomials rather than regressions. Linear approximation is used to exclude artificial local minima between sample points. Due to the fact that no oversampling is required anymore, interpolation results in a lower number of nonlinear analyses without losing accuracy. A second reduction of the required analyses is achieved by avoiding a complete resampling in each iteration. It is assumed that the vectors defined from the current best point to the remaining sample points should be as orthogonal as possible. Therefore, LInA starts with initial sample points from a Koshal design of experiments (DOE) locally around the initial design point. After having computed the new best point based on the linear approximation, only a few existing sample points are deleted in each iteration and replaced by new ones. Complete QR decomposition is used to generate the new sample points in an orthogonal complementary subspace. First studies with two nonlinear structural optimization problems have shown that the LInA approach leads to a noteworthy reduction of the number of required nonlinear analyses in comparison to SRSM. Furthermore, LInA is compared to two kriging approaches using a subdomain similar as LInA does. LInA shows results comparable to kriging with respect to the number of analysis runs.

基于元模型的方法通常用于非线性优化。连续响应面法（SRSM）（Roux等，Int J Numer Methods Eng 42：517-534，1998; Stander and Craig in Eng Comput 19：431-450，2002）是应用这种方法的一个例子。方法。SRSM基于在围绕当前最佳值的定义域中生成的一组采样点，为每次迭代中的每个系统响应计算平滑近似值。该域在每次迭代中根据生成的当前最佳点进行移动和收缩。默认情况下，线性多项式用作近似函数。已经开发出一种新的线性插值方法（LInA），以减少优化过程中所需的非线性分析次数。与SRSM相反，LInA使用插值线性多项式而不是回归。线性逼近用于排除采样点之间的人工局部最小值。由于不再需要过采样这一事实，因此插值可以减少非线性分析的数量，而不会降低精度。通过避免每次迭代中进行完整的重采样，可以减少所需分析的第二次。假定从当前最佳点到其余采样点定义的向量应尽可能正交。因此，LInA从Koshal实验设计（DOE）的初始样本点开始，局部围绕初始设计点。在基于线性近似计算出新的最佳点之后，每次迭代中仅删除几个现有的采样点，并用新的采样点替换。完全QR分解用于在正交互补子空间中生成新的样本点。对两个非线性结构优化问题的初步研究表明，与SRSM相比，LInA方法显着减少了所需非线性分析的数量。此外，使用与LInA类似的子域，将LInA与两种克里金法进行了比较。LInA的分析次数显示了与克里金法相当的结果。