A new numerical method is presented in this paper. This method directly solves partial differential equations in the Cartesian coordinate system. It can be easily applied to solve irregular domain problems without introducing the coordinate transformation technique. The concept of the present method is different from the conventional discretization methods. Unlike the conventional numerical methods where the discrete form of the differential equation only involves mesh points inside the solution domain, the new discretization method reduces the differential equation into a discrete form which may involve some points outside the solution domain. The functional values at these points are computed by the approximate form of the solution along a vertical or horizontal line. This process is called extrapolation. The form of the solution along a line can be approximated by Lagrange interpolated polynomial using all the points on the line or by low order polynomial using 3 local points. In this paper, the proposed new discretization method is ®rst validated by its application to solve sample linear and nonlinear differential equations. It is demonstrated that the present method can easily treat different solution domains without any additional programming work. Then the method is applied to simulate incompressible¯ows in a smooth expansion channel by solving Navier±Stokes equations. The numerical results obtained by the new discretization method agree very well with available data in the literature. All the numerical examples showed that the present method is very ef®cient, which is suitable for solving irregular domain problems.

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This paper presents a new haptics interface device based on cable-driven parallel manipulators and an ultrasonic phased array. Unlike previous studies, our method combines force and tactile feedback at the same time. It consists of two main parts: force feedback generated by cable tension and tactile feedback provided by an ultrasonic phased array. While these parts have no direct interference, combining them can offer a synergistic effect on haptic perception. Through a series of experiments, the tactile rendering algorithm for this device is established. The output pressure of ultrasonic phased array is dependent on input command value, modulation frequency, modulation waveform, and the position of the focal point in the workspace. The results of psychophysical experiments are evaluated to determine the absolute threshold of perceivable ultrasonic tactile feedback when the cable-driven force acts on fingers. Finally, we carry out a test to confirm the accuracy advantage of our system in the virtual environment. The results of our study indicate that this device has a wide range of applications in the field of aerial haptic display.

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