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A superconvergent point interpolation method (SC-PIM) with piecewise linear strain field using triangular mesh. (English) Zbl 1156.74394
Summary: A superconvergent point interpolation method (SC-PIM) is developed for mechanics problems by combining techniques of finite element method (FEM) and linearly conforming point interpolation method (LC-PIM) using triangular mesh. In the SC-PIM, point interpolation methods (PIM) are used for shape functions construction; and a strain field with a parameter $$\alpha$$ is assumed to be a linear combination of compatible stains and smoothed strains from LC-PIM. We prove theoretically that SC-PIM has a very nice bound property: the strain energy obtained from the SC-PIM solution lies in between those from the compatible FEM solution and the LC-PIM solution when the same mesh is used. We further provide a criterion for SC-PIM to obtain upper and lower bound solutions. Intensive numerical studies are conducted to verify these theoretical results and show that (1) the upper and lower bound solutions can always be obtained using the present SC-PIM; (2) there exists an $$\alpha_{\text{exact}}\in (0, 1)$$ at which the SC-PIM can produce the exact solution in the energy norm; (3) for any $$\alpha \in (0, 1)$$ the SC-PIM solution is of superconvergence, and $$\alpha =0$$ is an easy way to obtain a very accurate and superconvergent solution in both energy and displacement norms; (4) a procedure is devised to find a $$\alpha _{\text{prefer}}\in (0, 1)$$ that produces a solution very close to the exact solution.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74B05 Classical linear elasticity
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