By Gernot Beer
This booklet is an wonderful, effortless to learn advent to complicated numerical modeling. the purpose of the booklet is to guide the reader on a trip in the direction of the ‘holy grail’ of numerical simulation, particularly one with no the requirement of mesh new release, that takes info without delay from CAD courses. in this trip readers will observe the great thing about Non-uniform Rational B-Splines or NURBS and spot how good they're fitted to describing geometry, notice how CAD courses paintings and the way their information can be utilized for simulation. The implementation of NURBS-based simulation is mentioned utilizing Finite aspect and Boundary aspect know-how. it is a hands-on booklet with emphasis on implementation and examples of programming in a better point language are given. it's been written for clients of simulation software program, to allow them to comprehend some great benefits of this new expertise and insist development from a a bit of conservative undefined, for software program builders, to allow them to see that this can be a expertise with a massive destiny and for researchers, within the wish that it'll allure extra humans to paintings during this intriguing new box.
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Extra info for Advanced numerical simulation methods : from CAD data directly to simulation results
Stage 1: Basis functions Figure 8 Pierre Étienne Bézier. Figure 9 B-spline functions of order 0, 1 and 2. 25 26 Advanced numerical simulation methods First we compute the functions for order p = 0 (constant) and for i = 0, . . , N. Ni,0 (u) = 1 if ui u < ui+1 0 otherwise (24) Higher order basis functions are defined by referencing lower order functions: Ni,p (u) = ui+p+1 − u u − ui Ni,p−1 (u) + Ni+1,p−1 (u) ui+p − ui ui+p+1 − ui+1 (25) It can be seen that the higher order functions are a linear combination of the lower order functions.
Although this gives better control over the shape of the curve, it would be also desirable to be able to control the continuity of the basis functions. This can be achieved by B-splines. B-splines The further development of Bernstein polynomials can be attributed to Bezier5 and de Boor. To define B-splines we start with a knot vector. This is a vector containing a series of non-decreasing values of the local coordinate: = (u0 u1 ··· uN ) (23) We define the entries in the vector as knots and the interval between knots as the knot span.
BIBLIOGRAPHY   G. Beer, O. Sigl, and J. Brandl. Recent developments and application of the boundary element method. In Petruzczak and Pande, editors, Numerical models in geomechanics, pages 461–467. Balkema, 1997. H. G. Salamon, and E. Georgiadis. A new digital method for threedimensional stress analysis in elastic media. Rock Mechanics, 5(189–202), 1973. G. J. P. Collins. The failure of an offshore platform. Concrete International, 19(8):28–35, 1997. W. Wappenhaus and J. Richter. Die erste Statik der Welt.