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SUMMARYA posteriori error estimates and an adaptive reÿnement scheme of ÿrst-order least-squares meshfree method (LSMFM) are presented. The error indicators are readily computed from the residual. For an elliptic problem, the error indicators are further improved by applying the Aubin-Nitsche method. It is demonstrated, through numerical examples, that the error indicators coherently re ect the actual error. In the proposed reÿnement scheme, Voronoi cells are used for inserting new nodes at appropriate positions. Numerical examples show that the adaptive ÿrst-order LSMFM, which combines the proposed error indicators and nodal reÿnement scheme, is e ectively applied to the localized problems such as the shock formation in uid dynamics.

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SUMMARYA new meshfree method for the analysis of elasto-plastic deformation is presented. The method is based on the proposed first-order least-squares formulation for elasto-plasticity and the moving leastsquares approximation. The least-squares formulation for classical elasto-plasticity and its extension to an incrementally objective formulation for finite deformation are proposed. In the formulation, equilibrium equation and flow rule are enforced in least-squares sense, i.e. their squared residuals are minimized, and hardening law and loading/unloading condition are enforced pointwise at each integration point. The closest point projection method for the integration of rate-form constitutive equation is inherently involved in the formulation, and thus the radial-return mapping algorithm is not performed explicitly. The proposed formulation is a mixed-type method since the residuals are represented in a form of first-order differential system using displacement and stress components as nodal unknowns. Also the penalty schemes for the enforcement of boundary and frictional contact conditions are devised and the reshaping of nodal supports is introduced to avoid the difficulties due to the severe local deformation near contact interface. The proposed method does not employ structure of extrinsic cells for any purpose. Through some numerical examples of metal forming processes, the validity and effectiveness of the method are discussed.