SUMMARYThis paper presents a novel face-based smoothed finite element method (FS-FEM) to improve the accuracy of the finite element method (FEM) for three-dimensional (3D) problems. The FS-FEM uses 4-node tetrahedral elements that can be generated automatically for complicated domains. In the FS-FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS-FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non-linear solid mechanics problems. In addition, a novel domain-based selective scheme is proposed leading to a combined FS/NS-FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS-FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS-FEM is found better than that of the FEM.

A linearly conforming point interpolation method (LC-PIM) is developed for 2D solid problems. In this method, shape functions are generated using the polynomial basis functions and a scheme for the selection of local supporting nodes based on background cells is suggested, which can always ensure the moment matrix is invertible as long as there are no coincide nodes. Galerkin weak form is adopted for creating discretized system equations, and a nodal integration scheme with strain smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method (FEM) using linear triangle elements and the radial point interpolation method (RPIM) using Gauss integration, the LC-PIM can achieve higher convergence rate and better efficiency.

I looked at this paper with interest, as it promised an upper bound solution, which apparently could not be related to some form of complementary potential energy and to statically admissible solutions, but these expectations were reduced, when I verified that in Figure 4, for a force-driven problem, the method could provide 'upper bounds' that are smaller than the exact solution.Nevertheless, in the abstract and at the beginning of Section 5. Theorem 3 proves that there is a bound of the exact solution, but only for a very particular problem: 'the solution of an LC-PIM model, which is obtained using shape functions constructed using bases containing the exact solution'. For this model/mesh the results are valid, but why would anyone look for another solution in this case is a mystery.In the remainder of the paper I could not find the trace of a proof of bounding properties for the case when the exact solution is not contained in the basis of the LC-PIM model, which is certainly the most common situation.As Theorem 2 holds it is reasonable that in many cases, perhaps in most, the strain energy of the LC-PIM solutions is higher than its exact value. But this does not allow for the proclamation of a 'bound' that is a 'unique' property of the method.A bound is a precise term, which would be used appropriately if Theorem 3 could be supported for all LC-PIM solutions or, in practical terms, if situations such as those in Figure 4 could not exist.The property is not unique to this method, because a theoretical background exists [1], which sets a context for the determination of bounds of the strain energy, defined for force-driven (û = 0), displacement-driven (t = 0) as well as for mixed problems. It is used by many to attain the 'dream of many decades' that is mentioned in the Introduction of the paper.These are the fundamental points, but reading the paper also raised the following questions:• The conditions that are stated to prove that the method has no zero-energy modes, at the end of Section 5.1, are necessary, but not sufficient. Isn't it possible to have, for some particular geometries, zero-energy modes?