An adaptive meshless technique for solving bilateral obstacle problems

Abstract Bilateral obstacle problems are central to the study of partial differential equations (PDEs) and variational inequalities, with applications in optimal control, elasticity, and material deformation under constraints. However, solving them numerically presents significant challenges due to complex interactions at the contact boundaries. This study presents a meshless method for solving bilateral obstacle problems, providing an alternative to traditional finite element and finite difference approaches. A key advantage of this method is its ability to adaptively refine the free surfaces between contact regions, which enables more accurate simulations. The modified finite-particle method (MFPM) is applied for discretization, and the Picard iteration technique is used to solve the resulting system. Numerical experiments show that MFPM achieves second-order accuracy, similar to central difference methods, while offering better energy efficiency, particularly when refining surfaces between coincident and non-coincident regions. This approach provides a flexible and computationally efficient tool for tackling complex obstacle problems in various physical and engineering contexts.