Algorithms For Topology Optimization Problems Of Elastic Vibration Structure With Certain And Stochastic Coefficients

The optimization problems of elastic vibration structure exist in various research fields of engineering design.The optimization problems can be divided into three types: size,shape and topology optimization.Among them,the topology optimization has more advantages,since it can deal with complicated topology changes.In this thesis,the topology optimizations of elastic vibration structure with certain coefficients and with stochasic coefficients are studied.The maximization of the smallest eigenfrequency of structural vibration is investigated firstly in this thesis.By introducing the ersatz material,the linearized elasticity system is extended into a fixed background domain.To represent the two different material regions,the piecewise constant level set(PCLS)method is applied.By sensitivity analysis,the functional derivative of the smallest eigenfrequency with respect to PCLS function is derived.A restarted conjugate gradient penalty algorithm is proposed.When the multiplicity of eigenfrequency happens,the eigenmode closed to the one used in the last iteration is selected.Numerical experiments of 2D and 3D cantilevers are conducted to illustrate the effectiveness of the proposed algorithm.The maximization of the smallest eigenvalue with stochastic coefficients is also investigated in this thesis.Using the series expansion of the stochastic coefficients,the infinite stochastic coefficients are truncated into the finite dimensional ones.The Rank-1Lattices methods is used to generate the random points.The minimization of the mean value of the smallest eigenvalue is set as the objective function.The PCLS method is applied to represent the real material domain and ersatz material domain.By sensitivity analysis,the functional derivative of the objective function with respect to PCLS function is derived.A restarted conjugate gradient penalty algorithm is proposed.When the multiplicity of eigenvalue happens,the eigenmode closed to the one used in the last iteration is selected.A 2D numerical example is conducted to show the differences between the numerical results with certain coefficients and with stochastic coefficients.