The aim of this study is to obtain numerical solutions of the modified Fornberg Whitham equation via collocation finite element method combined with operator splitting method. The splitting method is used to convert the original equation into two sub equations including linear and nonlinear part of the equation as a slight modification of splitting idea. After splitting progress, collocation method is used to reduce the sub equations into algebraic equation systems. For this purpose, quintic B-spline base functions are used as a polynomial approximation for the solution. The effectiveness and efficiency of the method and accuracy of the results are measured with the error norms L 2 and L ∞. The presentations of the numerical results are shown by graphics as well.

A new underground powerhouse that combines a run-of river scheme with a pumped storage scheme is planned in course of the Renewal of Forbach HPP in Germany. The powerhouse cavern is designed to have a maximum span and height of 18.9 and 42 m, and a total length of 123 m. The rock mass conditions were found favourable in general. However, comprehensive investigations revealed weakness planes crossing the cavern. The support system was determined by joint evaluation of empirical methods, wedge analysis and the 2-D finite element analyses. Due to the complicated geometry of the cavern and the orientation of the weakness planes, the effect of weakness zones on the stability and deformations could not be evaluated by only 2-D analyses. Implementing an additional 3-D analysis and comparison of the results with the common 2-D approach allowed a more realistic assessment of the rock behaviour, especially under the existence of the weakness zones through the cavern. A very detailed 3-D finite element model for the whole cavern is still not practical, so a relatively larger mesh was adopted. Nevertheless, a reduced model for a limited part of the cavern was also analysed by using a finer mesh. Obtained results were similar indicating the suitability of the adopted numerical models for the rock support assessment. The deformations and the size of the plastic zones were identified as significantly larger for weakness zones passing through the cavern. For these intersection areas additional rock support measures including pre-stressed anchors were therefore planned.

No abstract