Minimization of sound radiation from vibrating bi-material structures using topology optimization

Up to now, work on topological design optimization of vibrating structures against noise radiation has mainly addressed the maximization of eigenfrequencies and gaps between consecutive eigenfrequencies of free vibration, and minimization of the dynamic compliance subject to harmonic loading on the structure. In this paper, we deal with topology optimization problems formulated directly with the design objective of minimizing the sound power radiated from the structural surface(s) into a surrounding acoustic medium. Bi-material elastic continuum structures without material damping are considered. The structural vibrations are excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. It is assumed that air is the acoustic medium and that a feedback coupling to the structure can be neglected. Certain conditions are assumed that imply that the sound power emission from the structural surface can be obtained in a simpler way than by solving Helmholz’ integral equation. Hereby, the computational cost of the structural-acoustical analysis is substantially reduced. Several numerical results are presented and discussed for plate- and pipe-like structures with different sets of boundary and loading conditions.

[1]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[2]  N. Kikuchi,et al.  Topological design for vibrating structures , 1995 .

[3]  Niels Olhoff,et al.  The energy generation and transmission in compound elastic cylindrical shells with heavy internal fluid loading—from parametric studies to optimization , 2006 .

[4]  A. Haddow,et al.  Design of band-gap grid structures , 2005 .

[5]  M. Zhou,et al.  Generalized shape optimization without homogenization , 1992 .

[6]  Niels Olhoff,et al.  A method of design against vibration resonance of beams and shafts , 1985 .

[7]  Niels Olhoff,et al.  Designing vibrating beams and rotating shafts for maximum difference between adjacent natural frequencies , 1984 .

[8]  Jakob S. Jensen,et al.  On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases , 2006 .

[9]  Pauli Pedersen,et al.  Design with Several Eigenvalue Constraints by Finite Elements and Linear Programming , 1982 .

[10]  C. S. Jog.  Reducing Radiated Sound Power by Minimizing the Dynamic Compliance , 2002 .

[11]  Pauli Pedersen,et al.  The Integrated Approach of FEM-SLP for Solving Problems of Optimal Design , 1981 .

[12]  Joachim Bös,et al.  Numerical optimization of the thickness distribution of three-dimensional structures with respect to their structural acoustic properties , 2006 .

[13]  Ole Sigmund,et al.  Systematic design of phononic band–gap materials and structures by topology optimization , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[14]  M. Zhou,et al.  Applications of the COC Algorithm in Layout Optimization , 1991 .

[15]  Edward J. Haug,et al.  Optimization of distributed parameter structures , 1981 .

[16]  N. L. Pedersen.  Maximization of eigenvalues using topology optimization , 2000 .

[17]  D. Tortorelli,et al.  Design sensitivity analysis: Overview and review , 1994 .

[18]  Niels Olhoff.  Topology Optimization of Structures Against Vibration and Noise Emission , 2005 .

[19]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[20]  M. Lax,et al.  On the Radiation Problem at High Frequencies , 1947 .

[21]  N. Olhoff,et al.  Topology Optimization of Vibrating Bi-Material Structures with Respect to Sound Radiation , 2006 .

[22]  Søren Tørholm Christensen,et al.  On analysis and optimization in structural acoustics — Part I: Problem formulation and solution techniques , 1998 .

[23]  M. Bendsøe.  Optimal shape design as a material distribution problem , 1989 .

[24]  Noboru Kikuchi,et al.  Optimal topology design of structures under dynamic loads , 1999 .

[25]  C. Jog.  Topology design of structures subjected to periodic loading , 2002 .

[26]  Jakob S. Jensen,et al.  Maximizing band gaps in plate structures , 2006 .

[27]  Niels Olhoff,et al.  Topology optimization of continuum structures: A review* , 2001 .

[28]  Ole Sigmund,et al.  IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials , 2006 .

[29]  Jakob Søndergaard Jensen.  Efficient Optimization of Dynamic Systems using Padé Approximants , 2006 .

[30]  T. Wu,et al.  A New Look at the High Frequency Boundary Element and Rayleigh Integral Approximations , 2003 .

[31]  Jakob Søndergaard Jensen,et al.  Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures , 2003 .

[32]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[33]  M. P. Bendsøe,et al.  Optimization of material properties for improved frequency response , 1994 .

[34]  Jakob S. Jensen,et al.  Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide , 2005 .

[35]  N. Kikuchi,et al.  Solutions to shape and topology eigenvalue optimization problems using a homogenization method , 1992 .

[36]  C. Swan,et al.  A SYMMETRY REDUCTION METHOD FOR CONTINUUM STRUCTURAL TOPOLOGY OPTIMIZATION , 1999 .

[37]  N. Olhoff,et al.  Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps , 2007 .

[38]  Niels Olhoff,et al.  Regularized formulation for optimal design of axisymmetric plates , 1982 .

[39]  Dmitri Tcherniak,et al.  Topology optimization of resonating structures using SIMP method , 2002 .

[40]  N. Olhoff,et al.  Optimum topology and reinforcement design of disk and plate structures with multiple stiffness and eigenfrequency objectives , 1999 .

[41]  Niels Olhoff,et al.  On analysis and optimization in structural acoustics — Part II: Exemplifications for axisymmetric structures , 1998 .

[42]  Gary H. Koopmann,et al.  Designing Quiet Structures: A Sound Power Minimization Approach , 1997 .

[43]  Ole Sigmund.  Microstructural design of elastic band gap structures , 2001 .

[44]  Ole Sigmund,et al.  On the Design of Compliant Mechanisms Using Topology Optimization , 1997 .

[45]  J. Petersson,et al.  Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima , 1998 .