Strong-form meshless methods received much attention in recent years and are being extensively researched and applied to a wide range of problems in science and engineering. However, the solution of elasto-plastic problems has proven to be elusive because of often non-smooth constitutive relations between stress and strain. The novelty in tackling them is the introduction of virtual finite difference stencils to formulate a hybrid radial basis function generated finite difference (RBF-FD) method, which is used to solve smallstrain von Mises elasto-plasticity for the first time by this original approach. The paper further contrasts the new method to two alternative legacy RBF-FD approaches, which fail when applied to this class of problems. The three approaches differ in the discretization of the divergence operator found in the balance equation that acts on the non-smooth stress field. Additionally, an innovative stabilization technique is employed to stabilize boundary conditions and is shown to be essential for any of the approaches to converge successfully. Approaches are assessed on elastic and elasto-plastic benchmarks where admissible ranges of newly introduced free parameters are studied regarding stability, accuracy, and convergence rate.

翻译：强形式无网格方法近年来受到广泛关注，并被深入研究和应用于科学及工程领域的众多问题。然而，由于应力与应变之间常存在非光滑的本构关系，弹塑性问题的求解一直具有挑战性。本文的创新之处在于引入虚拟有限差分模板，构建一种混合径向基函数生成有限差分（RBF-FD）方法，并首次通过这种原始方法求解小应变von Mises弹塑性问题。论文进一步将新方法与两种无法适用于此类问题的传统RBF-FD方法进行对比。这三种方法在平衡方程中作用于非光滑应力场的散度算子离散化方式上有所不同。此外，采用了一种创新的稳定技术来稳定边界条件，并证明该技术对于任何方法成功收敛都是至关重要的。通过弹性与弹塑性基准算例对这些方法进行评估，研究新引入自由参数的稳定范围、精度及收敛速率。