Advanced numerical simulation methods : from CAD data by Gernot Beer

By Gernot Beer

This publication is an interesting, effortless to learn creation 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, specifically one with no the requirement of mesh iteration, that takes facts at once from CAD courses. in this trip readers will observe the great thing about Non-uniform Rational B-Splines or NURBS and notice how good they're fitted to describing geometry, realize how CAD courses paintings and the way their info can be utilized for simulation. The implementation of NURBS-based simulation is mentioned utilizing Finite aspect and Boundary point expertise. this can be a hands-on e-book with emphasis on implementation and examples of programming in the next 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 know-how and insist growth from a a little conservative undefined, for software program builders, to allow them to see that this can be a expertise with a major destiny and for researchers, within the wish that it'll allure extra humans to paintings during this interesting new box.

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Example text

The princes, he told his correspondent, were always making discoveries, by accidents and sagacity, of things which they were not in quest of. Stage 1: Basis functions 21 Figure 4 Hermite polynomials and their derivatives. The second derivative is given by ⎛ ⎞⎞ ⎛ j=i n=j,n=i m=j,m=i,m=n ∂2 Ni (ξ) 1 1 ξ − ξ m ⎠⎠ ⎝ = ·⎝ · ∂ξ 2 ξi − ξ j ξi − ξ n ξi − ξ m j=1:I n=1:I (12) m=1:I For a quadratic Serendipity function we have ∂ 2 N1 = 1; ∂ξ 2 ∂ 2 N2 = 1; ∂ξ 2 ∂ 2 N3 = −2 ∂ξ 2 (13) Figure 1 shows Lagrange functions of order 1 to 4.

The functions were subsequently called Serendipity functions2 . e. for Serendipity functions the edge nodes are numbered first, for reasons to be revealed later). 5; ∂ξ ∂N3 (ξ) = −2 · ξ ∂ξ (11) Serendipity means a “pleasant surprise’’. It was first coined by Horace Walpole in 1754. In a letter he wrote to a friend Walpole explained an unexpected discovery he had made by reference to a Persian fairy tale, The Three Princes of Serendip. The princes, he told his correspondent, were always making discoveries, by accidents and sagacity, of things which they were not in quest of.

The only difference to B-splines is that the amplitude of the basis functions is determined by the weights. The derivative to u for example is given by: p,q N wi,j − W (u) · Ri,j (u, v) d p,q Ri,j (u, v) = du W 2 (u) (62) Stage 1: Basis functions 39 Figure 23 B-spline functions of order p = q = 2 and with Knot vectors u = (0, 0, 0, 1, 1, 1) and v = (0, 0, 0, 1, 1, 1) with associated anchor locations in the u, v coordinate system (some anchors are hidden from view). 40 Advanced numerical simulation methods Figure 24 Figure 23 continued.

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