Topology Optimization of a Bounded Space for a Vibroacoustical Problem in a Low Frequency

摘要

The article deals with the problem of a space with vibroacoustical source generating inside specific conditions, which form a field of some values. All applies to an acoustic field in particular, characterized by an acoustic pressure or field of displacements in vibration problems. In general the field is described by specific dependent variable w(t) in all points of space, whose location are defined by coordinates r. The first aspect of this work relates to the modeling of an induced field, which can be alternative to finite element method (FEM) or boundary element method (BEM) in a low range of frequency. The second aspect is connected with minimization of some significant factor level related to dependent variable w(t) and subsequently to control field inside bounded space in order to get required state. In this work, it is assumed that the field distribution can be described by using modal analysis assumptions. Therefore, the dependent variable w(t) is defined by a modal expansion i.e. the sum over a set of a space's eigenfunctions Ψ(r) (normal modes, characteristic functions) and time components (modal amplitudes, generalized coordinates). If the assumption of the highest values of acoustical or mechanical impedance of the space boundaries (damping properties of boundaries) is made, the modal coupling can be neglected. Such approach results in the vibroacoustical model being faster than alternative FEM or BEM models and suitable for the optimization procedure by using genetic algorithms, in the case when a computational cost is high. Thereafter, the topology optimization problem is formulated, where the influence of boundaries, represented by their impedance and the shape of space, represented by eigenfunctions are considered as the design variables. The genetic algorithm method is applied in order to find a minimum of objective function. In this case the function returns some functional value. As the result of the optimization a topology of the investigated space is obtained in the form of its shape and simultaneously configuration of damping properties of the space's boundaries.