The Upper Bound Property for Solid Mechanics of the Linearly Conforming Radial Point Interpolation Method (Lc-Rpim)

Abstract

It has been proven by the authors that both the upper and lower bounds in energy norm of the exact solution to elasticity problems can now be obtained by using the fully compatible finite element method (FEM) and linearly conforming point interpolation method (LC-PIM). This paper examines the upper bound property of the linearly conforming radial point interpolation method (LC-RPIM), where the Radial Basis Functions (RBFs) are used to construct shape functions and node-based smoothed strains are used to formulate the discrete system equations. It is found that the LC-RPIM also provides the upper bound of the exact solution in energy norm to elasticity problems, and it is much sharper than that of LC-PIM due to the decrease of stiffening effect. An effective procedure is also proposed to determine both upper and lower bounds for the exact solution without knowing it in advance: using the LC-RPIM to compute the upper bound, using the standard fully compatible FEM to compute the lower bound based on the same mesh for the problem domain. Numerical examples of 1D, 2D and 3D problems are presented to demonstrate these important properties of LC-RPIM.

A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory

“…In such cases, we can still expect Equation (83) holds approximately and hence a convergence rate of 1.5 in G 1 semi-norm. This has been confirmed in many numerical examples presented in [22,26,33], where numerical rates of about 1.4 were often found. We were excited about the higher convergence rate, but could not give a good explanation then.…”

“…To produce a softer model, Liu discovered that we can simply create a model that uses a smaller number (but larger than the minimum number) of smoothing domains, as it is done in the NS-PIM [19,26] and in NS-RPIM [33]. Theoretically, one can make a model as softer as desired and can even make a model that is singular!…”

Certified solutions for hydraulic structures using the node‐based smoothed point interpolation method (NS‐PIM)

“…However, the energy obtained from NS-PIM is an overestimate of the exact energy, and the displacement is always an upper bound of the exact solution in the energy form [17]. For the gravity dam, we also prove the energy bound property as shown in Figure 12.…”

“…It was found that NS-PIM is at least linearly conforming, can provide much better stress solutions, is much more tolerant to mesh distortion, is immune from volumetric locking and works well for linear triangular and tetrahedral background cells [16]. More important, the NS-PIM can obtain an upper bound solution in energy norm for elasticity problems with homogeneous essential boundary conditions [16,17]. Owing to the excellent properties, the NS-PIM (and NS-RPIM [18]) has been further developed for conducing the adaptive analysis [19], the contact problems [20], the heat transfer and the thermoelasticity problems [21,22] and provides upper bound solutions in energy norm as it does for elasticity problems.…”

Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

“…Combining the system in Equation (26) and the system in Equation (13) with a continuous scalar scaling factor ∈[0, 1] leads to the following potential energy functional of -FEM for 2-D problems:ˆ From Equation (27), one can note that…”