The structures of (1−x)PbTiO3–xBiFeO3 (x=0.3 and 0.6) were investigated by means of the neutron powder diffraction. A splitting shift between Fe and Ti atoms was found along the c axis in 0.7PbTiO3–0.3BiFeO3; however, this splitting does not appear in 0.4PbTiO3–0.6BiFeO3. The tetragonal phase of PbTiO3–BiFeO3 exhibits a large spontaneous polarization. The negative thermal expansion of PbTiO3 is significantly enhanced in a wide temperature range by the BiFeO3 substitution. The average bulk thermal expansion coefficient of 0.4PbTiO3–0.6BiFeO3 is a¯v=−3.92×10−5°C−1, which is much strong in the known negative thermal expansion oxides.

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This review paper presents a methodological study on possible and existing meshfree methods for solving the partial differential equations (PDEs) governing solid mechanics problems, based mainly on the research work in the past two decades at the authors group. We start with a discussion on the general steps in a meshfree method based on nodes, with the displacements as the primary variables. We then examine the major techniques used in each of these steps: (1) techniques for displacement function approximations using nodes, (2) approximation of the gradient of the displacements or strains based on nodes and a background T-cells that can be automatically generated and refined, and (3) formulation techniques for producing algebraic equations. The function approximation techniques include node-based interpolation methods, cell-based interpolation methods, function smoothing techniques, and moving least squares approximation techniques. The gradient approximation includes direct differentiation, gradient smoothing, and special strain construction. Formulation techniques include strong-form, weakform, local weakform, weak-strong-form, and weakened weakform (W2). In theory, a meshfree method can be developed using a combination of function approximation, gradient approximation, and formulation techniques, which can lead to matrix of a large number of possible methods. This review attempts to provide an overall methodological review, rather than a usual review of comparing different methods. We hope to show readers the differences between the forests, and just between the trees.

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