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Abstract A differential quadrature hierarchical finite element method (DQHFEM) using Fekete points was formulated for triangles and tetrahedrons and applied to structural vibration analyses. First, orthogonal polynomials on triangles and tetrahedrons that can be used as bases of the hierarchical finite element method (HFEM) were derived and simple formulas of transforming one dimensional non-uniform nodes to simplexes were presented. Then the non-uniform nodes were used as initial guesses to solve the Fekete points on simplexes through Newton–Raphson's method together with the orthogonal polynomials. New differential quadrature (DQ) rules on simplexes were formulated using the HFEM bases and the Fekete points. The numbers of nodes or bases on different edges and faces and inside the body of the new DQ elements do not relate with each other like the HFEM that can freely assign different numbers of bases on different edges and faces and inside the body. So the new DQ method was named as a differential quadrature hierarchical (DQH) method that uses either interpolation functions or orthogonal polynomials as bases inside the element. Its weak form was named as the DQHFEM. Besides the DQH method and its weak form, a simple method of generating high quality linear and high order triangular and tetrahedral meshes from a single NURBS patch was presented. Numerical tests of the DQHFEM through structural vibration analyses showed that high accuracy results can be obtained using only a few nodes even on curvilinear domains using the DQH bases on both physical and geometric fields. It was concluded that wide applications of the DQH method and the DQHFEM to science and engineering are possible and commercial codes based on them are deserved to be developed. [ABSTRACT FROM AUTHOR]

In this article, based on the most general form of strain gradient theory (MGSGT), a novel extended triangular Mindlin plate element is proposed. To accomplish this aim, first, the quadratic form of energy functional is obtained by vectorizing the higher-order tensors of energy pairs, from which the stiffness and mass matrices of the element are readily derived. In comparison with the standard Mindlin plate element, the new element needs three additional nodal degrees of freedom (DOF) including derivatives of lateral deflection and rotations, which means a total of nine DOFs per node. Also, as compared to the standard Mindlin plate element which requires only C 0 shape functions, the present one requires C 1 continuous smooth shape functions due to second derivatives of deflection and rotations. Hence, cubic polynomials are used to interpolate the displacement components. The new element can be reduced to that based on the modified strain gradient theory (MSGT) and the modified couple stress theory (MCST). Moreover, the standard Mindlin plate element is recovered when the gradient-based material parameters tend to zero. The Mindlin microplates with different boundary conditions are considered as the problem under study whose free vibration and bending are analyzed. The results are compared with the exact solutions and excellent agreement is achieved. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a meshfree-enriched finite element formulation for triangular and tetrahedral elements in the analysis of two and three-dimensional compressible and nearly incompressible solids. The new formulation is first established in two-dimensional case by introducing a meshfree approximation into a linear triangular finite element with an enriched node. The interpolation functions of the four-noded triangular element are constructed by the meshfree convex approximations and are completed to a polynomial of degree one. The reference mapping using the constructed interpolation functions is shown to be invertible everywhere in the element and the global element area is proven to be conserved under a standard three-point integration rule. The triangular element formulation is extendable to the tetrahedral element in three-dimensional case. To provide a locking-free analysis for the nearly incompressible materials, an area-weighted strain smoothing is developed in conjunction with the enriched interpolation functions to yield a discrete divergence-free property at the integration point. The resultant element formulation with strain smoothing is shown to pass the patch test. To introduce the smoothed strain into Galerkin formulation, a modified Hu–Washizu variational principle is adopted to formulate the discrete equations. Since the Kronecker-delta property in element interpolation is held along the element boundary using meshfree convex approximation, boundary conditions can be treated in a standard way. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method. [Copyright &y& Elsevier]

Abstract: We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but some results are also given for the non-conforming case. We obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuška–Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation , so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions. [Copyright &y& Elsevier]

Developability conditions are presented for prestressed curved surfaces to be reduced to plane sheets by relaxing the stresses. Those conditions are applied to an optimization problem for minimizing the deviation of stresses from the target values. Two formulations based on the displacements defined by the local and global coordinates of finite elements are presented. It is shown that the numbers of constraints by those formulations are same if triangular elements are used, and there is an upper bound in the number of the elements in a cutting sheet. Performances of the two formulations are compared in the numerical examples, and the local formulation is shown to lead to more accurate estimation of stresses that are actually generated by connecting and stretching the plane sheets. [Copyright &y& Elsevier]