By Chang-New Chen
This booklet offers numerical differential quadrature (DQ) - dependent tools lately constructed by way of the writer. Their skill for fixing typical clinical and engineering difficulties is confirmed. those tools are the conventional differential quadrature, the prolonged differential quadrature and the comparable discrete aspect research tools. those novel numerical strategies are either effective and trustworthy. they're compatible for constructing resolution algorithms for varied computational mechanics issues of arbitrarily advanced geometry. this is often proven for numerous entire examples corresponding to bars and beams, trusses, frames, common box difficulties, elasticity difficulties or bending of plates.
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Additional resources for Discrete Element Analysis Methods of Generic Differential Quadratures
0 0 ¯ξ ¯ξ D 1(N −3) D1(N −2) 2 2 ¯ξ ¯ξ D 1(N −3) D1(N −2) 0 0 ... . . ... . 1 0 ... 0 0 ... ¯ ξ ¯ξ DN (N −3) D N (N −2) 2 2 ... ¯ ξ ¯ξ D D ... 14 2 ¯ ξ ... D 14 ... 1 . . 0 0 ¯ξ D N4 2 ¯ξ D N4 N (N −3) N (N −2) 0 0 ¯ξ D ⎤ ⎥ ¯ξ D 1N ⎥ 1(N −1) 2 2 ⎥ ξ ξ ¯ ¯ ⎥ D 1(N −1) D1N ⎥ 0 0 ⎥ ⎥ . ⎥ ⎥ . 115) . ⎥ ⎥ 0 0 ⎥ ⎥ 0 1 ⎥ ⎥ ¯ξ ¯ξ ⎥ D N (N −1) DN N ⎦ 2 2 ¯ξ ¯ξ D D N (N −1) NN Using [T ] in Eq. 113), the weighting coeﬃcients for the C 2 − C 0 − ˆ = N ¯ − 4. For applying the C 2 − C EDQ model can be obtained with N 0 2 C − C EDQ model to the DQEM analysis, the physical ﬁrst and second order derivatives at the two EDQ element boundary nodes are discrete EDQ ¯ − 1)th columns parameters.
1) where w is the transverse displacement, E is Young’s modulus, I is the moment of inertia of the cross-sectional area, p is the distributed load and k is the foundation modulus. 3) V = dx dx respectively. 1 DQEM Formulation For elements having no distributed load, the DOF of the element can be 4. Since Eq. 1) is a fourth order diﬀerential equation, without using a certain technique to calculate the four equivalent nodal loads for the distributed load of a four-DOF cubic element and include them into the natural transition conditions or natural boundary conditions, the order of approximate displacement must at least be four and the element must at least have one discrete point for deﬁning a discrete element equilibrium equation at the discrete point.
0 1 0 ... 117) Using [T ] in Eq. 113), the weighting coeﬃcients for the C 1 −C 0 EDQ model ˆ =N ¯ − 1. For applying the C 1 − C 0 EDQ model to can be obtained with N the transient structural analysis, the physical ﬁrst order derivative at the ﬁrst EDQ element boundary node is a discrete EDQ parameter. Consequently, the elements of the second column of the related EDQ weighting coeﬃcient matrix ξm need to be multiplied by l. 5 The ﬁrst four frequency factors of a clamped square plate solved by the DQEM using the C 1 − C 0 − C 1 EDQ model generated by using the equivalent DQ model with two auxiliary nodes inside the physical EDQ model Order of appr.
Discrete Element Analysis Methods of Generic Differential Quadratures by Chang-New Chen