Download Meshless Methods in Solid Mechanics by Youping Chen PDF
By Youping Chen
This booklet covers the basics of continuum mechanics, the indispensable formula tools of continuum difficulties, the fundamental ideas of finite point equipment, and the methodologies, formulations, systems, and purposes of varied meshless tools. It additionally presents basic and specific strategies of meshless research on elastostatics, elastodynamics, non-local continuum mechanics and plasticity with plenty of numerical examples. a few uncomplicated and significant mathematical equipment are incorporated within the Appendixes. For readers who are looking to achieve wisdom via hands-on adventure, the meshless courses for elastostatics and elastodynamics are supplied on an integrated disc.
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Use the result det A det B = det (AT B) to show that εi jk εlmn = δil (δ jm δkn − δ jn δkm ) + δim (δ jn δkl − δ jl δkn ) + δin (δ jl δkm − δ jm δkl ) and consequently εi jk εlmk = δil δ jm − δim δ jl , εi jk εl jk = 2δil . 3. Prove the following vector identities involving ∇: ∇ · (∇ F) = ∇ 2 F, ∇(F G) = F∇G + G∇ F, ∇ 2 (F G) = F∇ 2 G + 2(∇ F) · (∇G) + G∇ 2 F, ∇ · (Fv) = (∇ F) · v + F∇ · v, ∇ · (F∇G) = F∇ 2 G + ∇ F · ∇G, ∇ × (∇ F) = 0, ∇ · (∇ × v) = 0, ∇ · (a × b) = (∇ × a) · b − a · (∇ × b), ∇ × (Fv) = ∇ F × v + F∇ × v, ∇ × (∇ × v) = ∇(∇ · v) − ∇ 2 v.
105) t ¨ and U, ˙ Eq. 101) is then For static problems, ignoring the dynamic effect of U reduced to KU = F. 106) This simple form is the ﬁnite element equations of elastostatics. Numerical Integration in Finite Element Method Numerical integration is a very important step in ﬁnite element methods. The integrals, such as those in Eqs. 105), that we obtained previously can all be expressed in the form of F(ξ, η) dξ dη, F(ξ, η, ς) dξ dη dς for two- and three-dimensional problems, respectively. 108) i, j F(ξ, η, ς) dξ dη dς = i, j,k where the summations are extended over all i, j, and k as speciﬁed, the αi j and αi jk are weighting factors, F(ξi , η j ) and F(ξi , η j , ςk ) are the functions of F(ξ, η) and F(ξ, η, ς) evaluated at the points speciﬁed in the arguments, respectively.
3 Fundamentals of Finite Element Method The fundamentals of continuum mechanics were discussed in Chapter 2. In this chapter, we ﬁrst present the integral formulation of continuum problems, and then move on to present the essentials of ﬁnite element methods. The development is necessary as a prerequisite to the subsequent chapters in meshless methods. Integral Formulation of Continuum Problems Physical problem arising in engineering and science can often be represented by a set of differential equations and boundary conditions.