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The paper proposes a density gradient based approach to topology optimization under design-dependent boundary loading. Since there is no explicit boundary representation in density-based topology optimization, the design-dependent boundary loads are implicitly imposed through a domain integration of spatial gradient of the density field. The density gradient based loading is formally derived and justified based on the smoothed boundary method for numerical solution of PDE. The Heaviside projection, density filtering and local refinement are combined to efficiently control the thickness of the loading interface and the number of elements within it. For problems with both loading and non-loading boundary, an auxiliary density field is introduced to keep track of the loading boundary. Both 2D and 3D heat conduction problems, linear elasticity problems and coupled thermoelastic problems under design-dependent boundary loading are presented to illustrate the effectiveness and efficiency of the method. [ABSTRACT FROM AUTHOR]

• A novel multi-extracting velocity-synchronous chirplet transform (MEVCT) for efficient and effective time–frequency analysis. • An adaptive chirp angle search method for improving computational efficiency and velocity synchronous basis accuracy. • A higher-order iterative extraction operator for eliminating the non-reassigned time–frequency points. • The improved IF accuracy and anti-noise capability of MEVCT provide ideal TFR for synchronous multi-component signals. • Demonstrated advantages of MEVCT by numerical simulations as well as experimental tests. In many engineering applications, the instantaneous frequency (IF) needs to be identified rapidly and precisely, especially in rotating machinery condition monitoring, where quasi-real-time analysis is a favorable option, and any error in shaft speed (identified from the instantaneous frequency) needs to be reduced to prevent it from accumulating with time. The modern time–frequency analysis (TFA) method provides a way for IF identification; however, sometimes, it still suffers from energy smearing and closely spaced frequency interference. In addition, many TFA methods based on the chirplet transform also present the problem of excessive computation when searching for the optimal parameters. To overcome these limitations, we propose the multi-extracting velocity-synchronous chirplet transform (MEVCT), an enhanced TFA method, in this paper. In the MEVCT, a refined chirp parameter search is developed first to reduce the computational cost while increasing time–frequency accuracy. Then, an iterative high-order IF estimator is constructed to allow the IF to achieve or approach the actual values even in highly nonlinear situations. After that, the obtained IF estimate is substituted into the simultaneous extraction operator to reduce energy dispersion further and eliminate non-reassigned points (NRPs). Simulated and experimental signal analyses demonstrate that MEVCT is able to eliminate NRPs completely. Furthermore, when handling synchronous multicomponent signals, the proposed method has higher IF accuracy with less computational cost and better anti-noise capability. [ABSTRACT FROM AUTHOR]