By Anh-Vu Vuong

ISBN-10: 3834824445

ISBN-13: 9783834824448

ISBN-10: 3834824453

ISBN-13: 9783834824455

Isogeometric finite parts mix the numerical resolution of partial differential equations and the outline of the computational area given by means of rational splines from desktop aided geometric layout. This paintings provides a well-founded advent to this subject after which extends isogeometric finite components via an area refinement method, that's crucial for a good adaptive simulation. Thereby a hierarchical strategy is tailored to the numerical requisites and the correct theoretical houses of the foundation are ensured. The computational effects recommend the elevated potency and the possibility of this neighborhood refinement method.

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N. 3: Support of ﬁnite element basis functions Then an isoparameteric element (T, P, Σ) is given by (also see Fig. 4) ⎧ ˆ ⎪ ⎨ T = G(T ), P = {p : T → R : p = pˆ ◦ G−1 , pˆ ∈ Pˆ }, ⎪ ⎩ Σ = {p(G(ˆ ai )), i = 1 . . N }. 79b) Note that Eq. 79a) demands that the mapping is from the same ﬁnite element space on the element, which explains the name “iso-parametric”. Again we make use of the reference element concept (see Sec. 3) but here we do not restrict ourselves to aﬃne mappings only. For example we can now use the isoparametric quadrilateral element of degree one that is given by the position of its four corners.

28) 33 = ν−1 By inserting this equation again into Eq. 22) the stress-strain relation can be simpliﬁed to ⎛ ⎞ ⎛ ⎞⎛ ⎞ σ11 1 ν 0 11 E ⎝σ22 ⎠ = ⎝ν 1 0 ⎠ ⎝ 22 ⎠ . 29) 2 1−ν 0 0 1−ν σ12 12 2 This model will be used later on in Sec. 7. 2 Variational Formulation In order to employ the ﬁnite element method it is now necessary to reformulate the problem into a variational or weak formulation. This step will be the same as for isogeometric analysis because it is also based on variational formulations. The formulations introduced in the previous section lead to classical solutions in the space of continuous functions.

With some calculations Alg. 2 for the computation of the control points Pi can be derived. 3b shows the control points after some knot insertion into the curve in Fig. 3a. There are also special algorithms for inserting several knots at once (see [76]). The extension to multivariate splines is straightforward: Alg. 2 is applied in each parameter direction and therefore computes new rows or columns of control points. For eﬃciency the values αi may be precomputed. 2 Knot insertion algorithm for a B-spline curve of degree p Find index k such that u ∈ [uk , uk+1 ) .

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