This paper describes a new efficient approach based on the concept of reduced basis for large deformation analysis. The domain problem is discretized using the meshfree particle radial point interpolation method (RPIM), which inherently possesses the Kronecker’s delta property. Meshless numerical integration is evaluated by the Cartesian transformation method (CTM), which enhances the performance of the RPIM. In addition, we also introduce a new approach to further improve the capability of the current CTM in evaluation of numerical integration for problems with complex geometries, i.e., by incorporation of the non-uniform rational B-splines function (NURBS) into the CTM. The emphasis of the paper is on the nonlinear nature of the large deformation problems, which are often solved by an iterative scheme. Conventional Newton–Raphson technique usually requires high cost due to the fact that several load steps are usually performed, and multiple iterations are needed in each load step. This low computational efficiency can be overcome, as proposed in this work, by using the so-called combined approximation, which approximates the full-size solution by a set of reduced bases. In other words, reduction of the problem size can be obtained, leading to reduction of the computational time, while accuracy is almost preserved.

本文基于大变形分析的约简基概念描述了一种新的有效方法。域问题是使用无网格粒子径向点插值方法 (RPIM) 离散化的，该方法固有地具有 Kronecker 的 delta 属性。无网格数值积分通过笛卡尔变换方法 (CTM) 进行评估，从而提高了 RPIM 的性能。此外，我们还引入了一种新方法来进一步提高当前 CTM 在评估复杂几何问题的数值积分方面的能力，即将非均匀有理 B 样条函数 (NURBS) 合并到 CTM 中。论文的重点是大变形问题的非线性特性，这些问题通常通过迭代方案来解决。传统的 Newton-Raphson 技术通常需要高成本，因为通常执行多个载荷步，并且每个载荷步都需要多次迭代。正如本文中提出的那样，可以通过使用所谓的组合近似来克服这种低计算效率，该近似通过一组减少的基数来近似全尺寸解。换句话说，可以减少问题的大小，从而减少计算时间，同时几乎保持准确性。