Composite finite elements and stabilization of meshless methods

Composite triangular and tetrahedral elements are developed; the composite element with constant volumetric and linear deviatoric strain is the most robust. The stabilities of meshless methods are analyzed and a stress point quadrature with Lagrangian kernels is recommended.; Two types of composite elements are constructed. In the first, a triangle or tetrahedron is divided into subelements. The displacements in subelements are linear. The stress and strain are assumed over the assemblage and a compatibility condition is utilized to derive the strain-displacement relation. The original composite elements by Camacho, and Ortiz, in which the stress and strain are linear (CTLS elements), fail the Babuska-Brezzi test and have poor accuracy and convergence rate in pressure for incompressible materials. In the new composite elements, the stress and strain consist of a constant volumetric part and a linear deviatoric part, (CTCV elements). They satisfy the BB condition, maintain convergence rates higher than the optimal values for linear-displacement elements, and have comparable or better accuracy than CTLS elements. Another kind of composite elements, in which a triangle is divided into three quadrilaterals, are also studied. Their performance is not as good as that of the CT-type elements.; A unified stability analysis of meshless methods is presented. The stability of Eulerian and Lagrangian kernels under different quadrature schemes, and the least square stabilization are investigated using Fourier analysis in one and two dimensions. Three types of instabilities are identified: an instability due to rank deficiency, a tensile instability and an instability under compressive stress; the latter is also found in continua. The tensile instability can be circumvented with Lagrangian kernels and the spurious modes due to rank deficiency can be suppressed by stress points. In two dimensions, the stabilization of stress points depends on their locations. Stress points at the center of quadrilaterals are insufficient for stability; a denser arrangement based on triangles is proved stable. The best approach to stable particle discretizations is to use Lagrangian kernels with stress points. The stable time step for explicit dynamics is also studied.