In this paper, we propose a numerical method to solve isotropic elliptic equations on point cloud by generalizing the point integral method. The idea of the point integral method is to approximate the differential operators by integral operators and discretize the corresponding integral equation on point cloud. The key step is to get the integral approximation. In this paper, with proper kernel function, we get an integral approximation for the elliptic operators with isotropic coefficients. Moreover, the integral approximation has been proved to keep the coercivity of the original elliptic operator. The convergence of the point integral method is also proved.

For cable bridges, the cable tension force plays a crucial role in their construction, assessment and long-term structural health monitoring. Cable tension forces vary in real time with the change of the moving vehicle loads and environmental effects, and this continual variation in tension force may cause fatigue damage of a cable. Traditional vibration-based cable tension force estimation methods can only obtain the time-averaged cable tension force and not the instantaneous force. This paper proposes a new approach to identify the time-varying cable tension forces of bridges based on an adaptive sparse time-frequency analysis method. This is a recently developed method to estimate the instantaneous frequency by looking for the sparsest time-frequency representation of the signal within the largest possible time-frequency dictionary (i.e. set of expansion functions). In the proposed approach, first, the time-varying modal frequencies are identified from acceleration measurements on the cable, then, the time-varying cable tension is obtained from the relation between this force and the identified frequencies. By considering the integer ratios of the different modal frequencies to the fundamental frequency of the cable, the proposed algorithm is further improved to increase its robustness to measurement noise. A cable experiment is implemented to illustrate the validity of the proposed method. For comparison, the Hilbert–Huang transform is also employed to identify the time-varying frequencies, which are then used to calculate the time-varying cable-tension force. The results show that the adaptive sparse time-frequency analysis method produces more accurate estimates of the time-varying cable tension forces than the Hilbert–Huang transform method.

In this paper, we consider multiple signals sharing same instantaneous frequencies. This kind of data is very common in scientic and engineering problems. To take ad- vantage of this special structure, we modify our data-driven time-frequency analysis by updating the instantaneous frequencies simultaneously. Moreover, based on the simul- taneously sparsity approximation and fast Fourier transform, some ecient algorithms is developed. Since the information of multiple signals is used, this method is very ro- bust to the perturbation of noise. And it is applicable to the general nonperiodic signals even with missing samples or outliers. Several synthetic and real signals are used to test this method. The performances of this method are very promising.