Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC‐PIM)

Abstract

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?

Modeling and simulation of bioheat transfer in the human eye using the 3D alpha finite element method (αFEM)

“…It has been found that the NS-FEM works well for the polygonal elements [24]. Because of its upper bound properties [24], and the ease of generating triangular (2D) and tetrahedron (3D) elements for complicated domains, it is used in this work for formulating our FEM for bioheat transfer problems. A brief introduction for the formulations of the triangular elements in 2D problems and tetrahedron elements in 3D problems is now presented.…”

On computing upper and lower bounds on the outputs of linear elasticity problems approximated by the smoothed finite element method

“…References [14,15] provide the same inequality (19) and proved that the strain energy computed with SFEM is also an upper bound on the exact strain energy, but only under the hypothesis that shape functions exist for exact solutions. Many numerical examples show that SFEM does give upper bounds on the true strain energy, except for some cases where the meshes are too coarse.…”

“…It has been found that in some situations the strain energy computed by SFEM bounds the exact strain energy from above [14][15][16]. This method stems from mesh-free finite element research and was first proposed to develop a stabilized nodal integration scheme for the Galerkin mesh-free method.…”

“…In [15] a proof was given, but only in the limited case that the exact solution is in the subspace spanned by the nodal shape functions. The authors also presented a proof for the softening effect shown by the stiffness matrix of SFEM, while in [18] a theoretical explanation of SFEM was given.…”

A Cell-Based Smoothed Point Interpolation Method for Axisymmetric Problems

“…The smoothed strain field overcomes the problems associated with continuity of the approximation field over the problem domain through elimination of the need for the derivatives of the shape functions. Different smoothing domains can be adopted for the smoothing operation leading to different SPIMs [3][4][5][6]. The simplest SPIM is perhaps the cellbased SPIM (CSPIM) in which the cells of a triangular background mesh are used as the smoothing domains.…”