By Anh-Vu Vuong

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 offers a well-founded advent to this subject after which extends isogeometric finite components by way of an area refinement method, that is crucial for an effective adaptive simulation. Thereby a hierarchical strategy is tailored to the numerical specifications and the appropriate theoretical houses of the root are ensured. The computational effects recommend the elevated potency and the possibility of this neighborhood refinement method.

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1. We will also see that the bilinear forms fulﬁll the requirements of Th. 5. Laplace Equation By multiplication of Eq. 1) with a test function v, integration over the domain Ω and partial integration we get (∇u, ∇v) dx = − Ω v(∇u, n) dS. 37) Γ For zero Dirichlet boundary conditions only we let V = H01 (Ω) with | · |H 1 as the norm (due to the Poincar´e-Friedrichs inequality), and deﬁne (∇u, ∇v) dΩ. 39) 26 3 Mathematical Models and Finite Element Analysis and a(v, v) = |v|H01 , ∀v ∈ V. 40) In the case of nonzero Dirichlet boundary conditions u = g on Γ it is assumed that g ∈ H 1/2 (Γ).

N + 1 end for for i = 0, . . , , n + 1 do P i = αi P i + (1 − αi )P i−1 end for return P 0 , . . 3: Reﬁned curves. The dashed lines are control polygons. ,n+p+1 to the new basis Bi,p with a degree p > p. The knot vector U is created from the knot vector U by increasing every multiplicity mi , i = 0, . . , n + p + 1 by one. This ensures that Bi,p does not obtain a higher smoothness at these knots so that the initial curve can no longer be represented in this basis. Moreover, all functions are aﬀected and all control points are changed, except those that are interpolatory.

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. This neglects the atomic nature and therefore the body is assumed to be inﬁnitely dividable. We will concentrate on problems from elasticity, which again lead to elliptic partial diﬀerential equations. There are plentiful references that deal with this topic and it is only possible to give an incomplete and subjective selection for continuum mechanics in general [50, 92] or elasticity [72, 31, 75].