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各种问题中。然而，由于应力与应变之间通常存在非光滑的本构关系，弹塑性问题的求解一直颇具挑战。处理这类问题的新颖之处在于引入虚拟有限差分散点，构建一种混合径向基函数生成的有限差分（RBF-FD）方法，该方法首次通过这一原创途径用于求解小应变von Mises弹塑性问题。本文进一步将新方法与两种传统RBF-FD替代方案进行对比，后者在应用于此类问题时失效。这三种方法的差异在于对平衡方程中作用于非光滑应力场的散度算子的离散化方式。此外，采用了一种创新的稳定化技术来处理边界条件，并证明该技术对于任何方法成功收敛都至关重要。这些方法在弹性和弹塑性基准测试中进行了评估，研究了新引入自由参数的可行范围在稳定性、精度和收敛速率方面的表现。