Development of a 4‐node hybrid stress tetrahedral element using a node‐based smoothed finite element method

Abstract

Summary In this paper, a new 4‐node hybrid stress element is proposed using a node‐based smoothing technique of tetrahedral mesh. The conditions for hybrid stress field required are summarized, and the field should be continuous for better performance of a constant‐strain tetrahedral element. Nodal stress is approximated by the node‐based smoothing technique, and the stress field is interpolated with standard shape functions. This stress field is linear within each element and continuous across elements. The stress field is expressed by nodal displacements and no additional variables. The element stiffness matrix is calculated using the Hellinger‐Reissner functional, which guarantees the strain field from displacement field to be equal to that from the stress field in a weak sense. The performance of the proposed element is verified by through several numerical examples.

Rotation‐free triangular shell element using node‐based smoothed finite element method

“…The shell element was formulated by combining the plate element and the membrane element using the coordinate transformation. For a membrane element that fit well with the proposed plate element, a method that had been previously developed by the current authors was used to construct the membrane element. The penalty method was used for the rotation boundary conditions.…”

“…However, since the CST element exhibits a very stiff behavior, it is difficult to obtain a solution appropriately for the membrane dominant problem. In the proposed shell element, a membrane element developed by the current authors is used. This subsection summarizes the formulation of the membrane element.…”

“…The first term in Equation is identical to the stiffness matrix of NS‐FEM, which has a smaller bandwidth than the proposed element . For computational efficiency, the first integral term in Equation is used in the same way as NS‐FEM and the second integral is calculated by three‐point numerical integration.…”

“…The membrane behavior of the proposed element was formulated using the approach proposed by the current authors . Since there was no verification of the triangular element for linear elasticity, the cantilever and the Cook's skew beam problem were chosen as the benchmark problem to confirm the plane stress behavior of the membrane element.…”

A 3‐node C0 triangular element for the modified couple stress theory based on the smoothed finite element method

“…However, the stiffness of the constant strain triangular element tends to be very stiff. Therefore, a hybrid stress element proposed by the current authors is used for approximating the strain energy related to the first‐order derivative term. In the original paper, a tetrahedral element was proposed, and a triangular element is formulated using a similar method.…”