104 research outputs found
B\'ezier curves that are close to elastica
We study the problem of identifying those cubic B\'ezier curves that are
close in the L2 norm to planar elastic curves. The problem arises in design
situations where the manufacturing process produces elastic curves; these are
difficult to work with in a digital environment. We seek a sub-class of special
B\'ezier curves as a proxy. We identify an easily computable quantity, which we
call the lambda-residual, that accurately predicts a small L2 distance. We then
identify geometric criteria on the control polygon that guarantee that a
B\'ezier curve has lambda-residual below 0.4, which effectively implies that
the curve is within 1 percent of its arc-length to an elastic curve in the L2
norm. Finally we give two projection algorithms that take an input B\'ezier
curve and adjust its length and shape, whilst keeping the end-points and
end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure
analytical stiffness matrix for curved metal wires
Abstract The paper presents an analytic stiffness matrix for curved thin metal wires, derived by the application of the second Castigliano's Theorem. The matrix accounts both bending and axial stiffness contributions in plane. The beam geometry is described by a cubic polynomial function of the curvature radius with a monotonical attitude angle as the independent variable. The solution proposed if fully analytical although a consistent number of adding factors appear. Some test cases are discussed and compared with Finite Element solutions, formed by a plentiful assembly of straight beams
Doctor of Philosophy
dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest
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