By Youping Chen

This e-book covers the basics of continuum mechanics, the crucial formula tools of continuum difficulties, the elemental techniques of finite point tools, and the methodologies, formulations, tactics, and purposes of assorted meshless tools. It additionally presents basic and specified systems of meshless research on elastostatics, elastodynamics, non-local continuum mechanics and plasticity with lots of numerical examples. a few easy and demanding mathematical tools are incorporated within the Appendixes. For readers who are looking to achieve wisdom via hands-on event, the meshless courses for elastostatics and elastodynamics are supplied on an incorporated disc.

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Extra info for Meshless Methods in Solid Mechanics

Sample text

M). The number of points is larger than the number of coefﬁcients ai , (i = 1, 2, 3, . . , m ≥ n. We have k−1 I = m [R D (a, x j )]2 + α [R B (a, x j )]2 . 19) for i = 1, 2, 3, . . n. 20) yields n equations for ai , even for m > n. The method is also called point least squares or overdetermined collocation. If m = n, the method becomes simple point collocation. Galerkin In this technique, the coefﬁcients ai are determined from the n equations of weighted residuals Ri = Wi (x)R D (a, x) dV = 0 for i = 1, 2, 3, .

Using four-node two-dimensional element as an example, ﬁnd that u(0, 0) = (U1 + U2 + U3 + U4 )/4, u(ξ, 1) = {(1 + ξ )U3 + (1 − ξ )U4 }/2, u(1, η) = {(1 − η)U2 + (1 + η)U3 }/2, which demonstrate the basic ideas of shape functions. Is the function u(ξ, η) continuous within the element? Is u continuous when it crosses the boundaries η = 1 and ξ = 1? 14. Using four-node two-dimensional element as an example, ﬁnd that u ,ξ (ξ, 1) = (U3 − U4 )/2, u ,η (ξ, 1) = {(1 − ξ )(U4 − U1 ) + (1 + ξ )(U3 − U2 )}/4, which demonstrate the basic ideas of the derivatives of shape functions.

R D (a, xi ) = 0 for i = 1, 2, 3, . . , j, R B (a, xi ) = 0 for i = j + 1, j + 2, . . , n. 14) Subdomain Collocation The complete domain of solution is subdivided into n subdomains. Over n different regions i and i , the integral of the residual is set to zero to obtain n equations for the coefﬁcients ai . R D (a, x) dV = 0 for i = 1, 2, 3, . . 15) for i = J + 1, J + 2, . . , n. 16) i R B (a, x) dS = 0 i Weighted Residual Methods 35 Continuous Least Squares The ai s are chosen to minimize a function I: ∂I =0 ∂ai for i = 1, 2, 3, .