This paper develops least-squares methods for the solution of linear elastic problems in both two and three dimensions. Our main approach is defined by simply applying the L2 norm least-squares principle to a stress-displacement system: the constitutive and the equilibrium equations. It is shown that the homogeneous least-squares functional is elliptic and continuous in the H(div; Ω)d × H1(Ω)d ...

In the classical approach to elasticity problems, the components of the displacement field are the primary unknowns. In an “intrinsic” approach, new unknowns with more physical or geometrical meanings, such as a strain tensor field or a rotation field for instance, are instead taken as the primary unknowns. We survey here recent progress about the mathematical analysis of such methods applied t...

We seldom measure H from the center of earth. Therefore what we can easily measure is the change of the potential energy ∆W = (mg)(H −Ho) (8.2) The energy is always positive. It can e zero but it cannot be negative. The gravity force F = mg is directed towards the center of earth. Therefore there is a need for the negative sign in Eq. (8.2)). In the coordinate system of Fig. (8.1), the gravity ...

Conditions for Stable Approximation Schemes Basic idea: Mimic structure of continuous problem. To establish stability of continuous problem, only used last two spaces in top sequence and last three spaces in bottom sequence. Λn−1(K) dn−1 −−−→ Λn(K)→ 0 ↗ Sn−2 ↗ Sn−1 Λn−2(V) dn−2 −−−→ Λn−1(V) dn−1 −−−→ Λn(V)→ 0. Thus, look for five finite dimensional spaces connected by a similar structure, i.e.,...

I systems and the Internet have facilitated the creation of used-product markets that feature a dramatically wider selection, lower search costs, and lower prices than their brick-and-mortar counterparts do. The increased viability of these used-product markets has caused concern among content creators and distributors, notably the Association of American Publishers and Author’s Guild, who beli...

Online retailers have maintained a price advantage over brick-and-mortar retailers since they were not required to collect sales tax. Recently, several states have required that the online retailer Amazon collect sales tax during checkout. Using transaction-level data, we document that households living in these states reduce Amazon purchases by 8% after sales taxes were implemented, implying a...

We present two kinds of lowest-order virtual element methods for planar linear elasticity problems. For the first one we use nonconforming method with a stabilizing term. It can be interpreted as modification Crouzeix–Raviart finite suggested in Hansbo and Larson (2003) to method. second conforming component displacement vector other. This approach seen an extension idea Kouhia Stenberg (1995) ...