Topology optimization of lightweight structures using a meshless Shepard function approximant

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Conference Proceeding
Advances in Applied Mechanics Research, Conference Proceedings - 7th Australasian Congress on Applied Mechanics, ACAM 2012, 2012, pp. 1022 - 1029
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This paper aims to propose a new structural topology optimization method using a Shepard function approximant and meshless Galerkin weak-forms, totally based on a set of arbitrarily scattered field nodes in the design domain. The moving least square (MLS) is applied to construct the meshless shape functions, in which the meshless shape function with the zero-order consistency will degenerate to a specific family of functions: "Shepard function", while the shape function with the first-order consistency refers to the widely studied "MLS shape function". The Shepard function is utilized to develop a physically meaningful dual-level density approximant due to its non-negative and rangerestricted properties. First, the Shepard function is included into the problem formulation to construct a nodal density-based non-local approximant with enhanced smoothness in terms of the original set of nodal density variables. So the density at any node can be evaluated according to the densities inside the influence domain of the current node. Second, in numerical implementation, the Shepard method with a singular weight function is employed to develop a point-wise density interpolant, in which the density at any computational point is determined by the surrounding nodal density variables within the influence domain of the concerned computational point. As a result, both the topology and the field quantities can be described via the dual-level point-wise density approximation scheme, just in terms of a set of generic design variables scattered at field nodes. MLS shape function is used to implement the meshless approximations of the system equations, while the Lagrangian multiplier method is included to enforce the essential boundary conditions due to the lack of Kronecker delta function property of the MLS shape function. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its applicability in avoiding numerical instabilities.
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