Shape control of beams under general loading conditions is implemented using piezoceramic actuators to provide the control forces. The objective of the shape-control is to minimize the maximum deflection of the beam to obtain a min-max deflection configuration with respect to loading and piezo-actuators. In practice, the loading on a beam is a variable quantity with respect to its magnitude, and this aspect can be handled easily by optimizing the magnitude of the applied voltage to achieve the min-max deflection. This property of the smart materials technology overcomes the problem of one-off conventional optimal designs which become suboptimal when the loading magnitude changes.In addition to the magnitude of the applied voltage, the optimal values for the locations and the lengths of the piezo-actuators are determined to achieve the min-max deflection. Due to the complexity of the governing equations involving finite length piezo patches, the numerical results are obtained by the finite-difference method. The analysis of the problem shows the effect of the actuator locations, lengths and the applied voltage on the maximum deflection. The optimal values for the actuator locations and the voltage are determined as functions of the load locations and load magnitudes, respectively. The effect of the actuator length on the min-max deflection is investigated and it is observed that the optimal length depends on the applied voltage. Finally, it is shown that using multiple actuators are more effective than a single actuator in the cases of complicated loading.

A study is made using numerical experiments to see the effect of the parameters in the explicit Euler-discretized form of a one-dimensional. nonlinear. reaction-diffusion equation. Based on a series of these experiments, one of the main results obtained is that diffusion. which is usually perceived as having a stabilizing effect. is able to produce chaotic as well as divergent numerical solutions. Furthermore. the discretization parameters are also able to produce chaotic results. From the results presented herein. i t is shown that varying the parameters can produce solutions that are single numbers. periodic. aperiodic (chaos), or divergznt.

SUMMARYA finite element weighted residual process has been used to solve transient linear and non-linear two-dimensional heat conduction problems. Rectangular prisms in a space-time domain were used as the finite elements. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in tables with analytical, as well as other numerical data. The finite element method compared favourably with these results. It was found to be stable, convergent to the exact solution, easily programmed, and computationally fast. Finally, the method does not require constant parameters over the entire solution domain.

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SUMMARYThe method of 'least squares', which falls under the category of weighted residual processes, is applied as a time-stepping algorithm to one-dimensional transient problems including the heat conduction equation, diffusion-convection equation, and a non-linear unsaturated flow equation. Comparison is made with other time-stepping algorithms, and the least squares method is seen to offer definite advantages.