By Anh-Vu Vuong

Isogeometric finite parts mix the numerical answer of partial differential equations and the outline of the computational area given by way of rational splines from machine aided geometric layout. This paintings supplies a well-founded advent to this subject after which extends isogeometric finite parts by way of a neighborhood refinement approach, that is crucial for an effective adaptive simulation. Thereby a hierarchical process is tailored to the numerical necessities and the suitable theoretical homes of the root are ensured. The computational effects recommend the elevated potency and the potential for this neighborhood refinement method.

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We have seen in Sec. 4 that NURBS can be expressed as B-splines in the projective space and therefore all the reﬁnement algorithms are also applicable for NURBS. As seen in Sec. 2 splines are characterized by their knots and their degree. Both aspects can be used for reﬁnement and will be discussed in the following. ,n+p+1 to the new basis Bi,p . ,n+p+2 = (u0 , . . , uk , u, uk+1 , . . , un+p+1 ) that results from inserting u ∈ [uk , uk+1 ) into U . Then it must hold n n Bi,p (u)P i . 40) i=1 By inserting a knot, p + 1 of the initial functions are changed.

2) If this is combined with the conservation of ﬂux ∇ · fd = g with a source term g this results in −κΔϕ = g. 3) In potential ﬂow we use a potential Φ to describe the velocity v = ∇Φ. Under the assumption of incompressibility ∇ · v = 0 this leads to the Laplace problem ΔΦ = 0. 1. The Poisson problem will serve as a model problem for local adaptive reﬁnement in Sec. 5. 2 Continuum Mechanics Another ﬁeld we want to investigate are problems from continuum mechanics. It is based upon the assumption that it is feasible to describe a material body as a continuum.

Ciarlet) Let • T ⊆ R be a bounded closed set with nonempty interior and Lipschitz-continuous boundary (element domain), • P be a ﬁnite-dimensional space of functions on K (space of shape functions) • Σ = p1 , p2 , . . , pk be a basis for P (the set of nodal variables or degrees of freedom). Then (T, P, Σ) is called a ﬁnite element. Typically the shape T of the element domain is triangular or quadrilateral (or tetrahedral or hexagonal in three dimensions) and for P a polynomial space is chosen.