Narrow Results

This paper presents a generalized gradient smoothing technique, the corresponding smoothed bilinear forms, and the smoothed Galerkin weakform that is applicable to create a wide class of efficient numerical methods with special properties including the upper bound properties. A generalized gradient smoothing technique is first presented for computing the smoothed strain fields of displacement functions with discontinuous line segments, by "rudely" enforcing the Green's theorem over the smoothing domain containing these discontinuous segments. A smoothed bilinear form is then introduced for Galerkin formulation using the generalized gradient smoothing technique and smoothing domains constructed in various ways. The numerical methods developed based on this smoothed bilinear form will be spatially stable and convergent and possess three major important properties: (1) it is variationally consistent, if the solution is sought in a Hilbert space; (2) the stiffness of the discretized model will be reduced compared to the model of the finite element method (FEM) and often the exact model, which allows us to obtain upper bound solutions with respect to both the FEM solution and the exact solution; (3) the solution of the numerical method developed using the smoothed bilinear form is less insensitive to the quality of the mesh, and triangular meshes can be used perfectly without any problems. These properties have been proved, examined, and confirmed by the numerical examples. The smoothed bilinear form establishes a unified theoretical foundation for a class of smoothed Galerkin methods to analyze solid mechanics problems for solutions of special and unique properties: the node-based smoothed point interpolation method (NS-PIM), smoothed finite element method (SFEM), node-based smoothed finite element method (N-SFEM), edge-based smoothed finite element method (E-SFEM), cell-based smoothed point interpolation method (CS-PIM), etc.

This paper presents an examinations of the G space theory that was established recently for a unified formulation of compatible and incompatible models for mechanics problems using the finite element and meshfree settings. Using the generalized gradient smoothing technique, we first give a general definition for G spaces with more details on the G¹ space containing both continuous and discontinuous functions. The physical meanings and implications of various numerical treatments used in the G space theory are discussed in detail. Both normed and un-normed G spaces are discussed with emphases on the normed G spaces. Some important properties and a set of useful inequalities for the normed G spaces are proven in theory and analyzed in detail. Because discontinuous functions are allowed in a G space, much more types of function approximation methods and techniques can be used to create shape functions for numerical models. These models can be compatible and incomputable but are all stable and converge to the exact solution to the corresponding strong formulation as long as it is well-posed, based on the normed G space theory. Methods based on normed G space theory does not use the derivatives of the displacement functions in the formulation and is known as the weakened weak (W²) formulation that has a number of attractive properties such as conformability, softness, upper/lower bound, superconvergence, and ultra accuracy.

Recently, the edge-based and node-based smoothed finite element method (ES-FEM and NS-FEM) has been proposed for Reissner–Mindlin plate problems. In this work, in order to utilize the numerical advantages of both ES-FEM and NS-FEM for static and vibration analysis, a hybrid smoothing technique based beta FEM ($\beta$FEM) is presented for Reissner–Mindlin plate problems. A tunable parameter $\beta$ is introduced to tune the proportion of smoothing domains calculated by ES-FEM or NS-FEM, which controlled the accuracy of the results. Numerical illustrations in both static and free vibration analysis are conducted. The shear locking free property, converge property and dynamic stability are carefully examined via several well-known benchmark examples. Moreover, an experimental test is carefully designed and conducted for validations, in which the mode values and shape of a rectangular steel plate is tested. Numerical examples demonstrate the advantages of $\beta$FEM, in comparison with the standard FEM, ES-FEM and NS-FEM using the same meshes. The numerical and experimental results are in good agreement with each other and the $\beta$FEM achieves the best accuracy among all the methods for the static or free vibration analysis of plates.

In this work, a three-dimensional (3D) nonlinear smoothed finite element method (S-FEM) solver is developed for large deformation problems. Node-based and face-based S-FEM using automatically generable four-noded tetrahedral elements (NS-FEM-Te4 and FS-FEM-Te4) are adopted to find the solution bounds in strain energy. The lower bound solutions are obtained using FEM-Te4 and FS-FEM-Te4, while the upper bound solutions are obtained using NS-FEM-Te4. A combined $\alpha$S-FEM-Te4 with a scaling factor $\alpha$ that controls the combination is constructed to find nearly exact solutions for the nonlinear solids mechanics problems through adjusting $\alpha$. This is achieved using the property that a successive change of scaling factor $\alpha$ can make the model transform from “overly-stiff” to “overly-soft”. Considering the properties of FS-FEM and NS-FEM, a selective FS/NS-FEM-TE4 is also used to solve 3D nonlinear large deformation problems, which produces a lower bound in strain energy. Hyperelastic Mooney–Rivlin and Ogden materials are both used in this study. Numerical examples reveal that S-FEM-Te4 is an effective method for obtaining solution bounds together with the standard FEM, and the FS-FEM-Te4, NS-FEM-Te4 and selective FS/NS-FEM-TE4 are robust with the high accuracy and computational efficiency for large deformation nonlinear problems.