Browsing by Author "Sherar, P. A."

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    A novel shape optimization method using knot insertion algorithm in B-spline and its application to transonic airfoil design
    (Academic Journals, 2011-11-16T00:00:00Z) Sherar, P. A.; Thompson, Christopher P.; Xu, B.; Zhong, B.
    A new method using the cubic B-spline curves with nominal uniform knot set to parameterize the geometry is proposed to deal with shape optimization problems. In the method, the control points of the B-spline curves are set to be the design variables in the optimization scheme. A knot insertion algorithm has been introduced in order to keep the geometry unchanged whilst increasing the number of control points at the final optimization stage. The super-reduced idea and the mesh refinement are also employed to deal with the equality constraint and speed up the optimization process. The method is applied to two problems. The first is a 2-dimensional Poisson problem, and the second is an airfoil design problem. In both applications, the results show that the new method is much more efficient when compared with the traditional methods. In the airfoil design problem, the drag of the airfoil has been reduced significantly with much less function calls.
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    Variational based analysis and modelling using B-splines
    (Cranfield University, 2004) Sherar, P. A.; Thompson, Chris
    The use of energy methods and variational principles is widespread in many fields of engineering of which structural mechanics and curve and surface design are two prominent examples. In principle many different types of function can be used as possible trial solutions to a given variational problem but where piecewise polynomial behaviour and user controlled cross segment continuity is either required or desirable, B-splines serve as a natural choice. Although there are many examples of the use of B-splines in such situations there is no common thread running through existing formulations that generalises from the one dimensional case through to two and three dimensions. We develop a unified approach to the representation of the minimisation equations for B-spline based functionals in tensor product form and apply these results to solving specific problems in geometric smoothing and finite element analysis using the Rayleigh-Ritz method. We focus on the development of algorithms for the exact computation of the minimisation matrices generated by finding stationary values of functionals involving integrals of squares and products of derivatives, and then use these to seek new variational based solutions to problems in the above fields. By using tensor notation we are able to generalise the methods and the algorithms from curves through to surfaces and volumes. The algorithms developed can be applied to other fields where a variational form of the problem exists and where such tensor product B-spline functions can be specified as potential solutions.