Meshless Investigation for Nonlocal Elasticity: Static and Dynamic
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The numerical treatment of nonlocal problems, which taking into account material microstructures, by means of meshless approaches is promising due to its efficiency in addressing integropartial differential equations. This thesis focuses on the investigation of meshless methods to nonlocal elasticity. Firstly, mathematical constructions of meshless shape functions are introduced and their properties are discussed. Shape functions based upon different radial basis function (RBF) approximations are implemented and solutions are compared. Interpolation errors of different meshless shape functions are examined. Secondly, the Point Collocation Method (PCM), which is a strong-form meshless method, and the Local Integral Equation Method (LIEM) that bases on the weak-form, are presented. RBF approximations are employed both in PCM and LIEM. The influences of support domains, different kinds of RBFs and free parameters are studied in PCM. While in LIEM, analytical forms of integrals, which is new in meshless method, is addressed. And, the number of straight lines that enclose the local integral domain as well as the integral radius are analyzed. Several examples are conducted to demonstrate the accuracy of PCM and LIEM. Besides, comparisons are made with Abaqus solutions. Then, PCM and LIEM are applied to nonlocal elastostatics based on the Eringen’s model. Formulations of both methods are reported in the nonlocal frame. Numerical examples are presented and comparisons between solutions obtained from both methods are made, validating the accuracy and effectiveness of meshless methods for solving static nonlocal problems. Simultaneously, the influence of characteristic length and portion factors are investigated. Finally, LIEM is employed to solve nonlocal elastodynamic problems. The Laplace transform method and the time-domain technique are implemented in LIEM respectively as the time marching schemes. Numerical solutions of both approaches are compared, showing reasonable agreements. The influence of characteristic length and portion factors are investigated in nonlocal dynamic cases as well.
AuthorsHuang Thesis, Xuejiao
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