Isogeometric analysis

Builds on Non-uniform rational B-spline B-spline Galerkin method Finite element method

Definition

Let $\hat\Omega = (0, 1)^d$ be the parameter domain. Let $R_1, \ldots, R_n \colon \hat\Omega \to \R$ be NURBS basis functions, built from B-splines on given knot vectors with positive weights. Let $P_1, \ldots, P_n \in \R^d$ be control points. Assume the geometry map $\mathbf{F}(\xi) = \sum_{i=1}^{n} R_i(\xi) P_i$ is a bijection from $\hat\Omega$ onto the physical domain $\Omega$ , with a smooth inverse. Isogeometric analysis is the Galerkin method for a boundary value problem on $\Omega$ with the discrete space $V_h = \operatorname{span} \set{ R_i \circ \mathbf{F}^{-1} : 1 \le i \le n }.$ Refinement is done by knot insertion and degree elevation, which leave $\mathbf{F}$ unchanged.

Remarks

The same functions $R_i$ define the geometry and the solution, as in isoparametric finite elements, hence the name. With $p$ the spline degree and knots of multiplicity one, the basis functions are $C^{p-1}$ across knots.

Example: the quarter circle $\set{(x, y) : x^2 + y^2 = 1, \ x, y \ge 0}$ is the quadratic NURBS curve with control points $(1, 0)$ , $(1, 1)$ , $(0, 1)$ and weights $1$ , $\tfrac{\sqrt{2}}{2}$ , $1$ . A quarter annulus is then given exactly by a geometry map that is quadratic in the angular parameter and linear in the radial one.

Non-example: a standard finite element method on a mesh of straight-sided triangles approximates the annulus by a polygon. Its geometry is not given by the solution basis of a NURBS map from $\hat\Omega$ , so it is not isogeometric.

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Tag	`VF17XWCK8Z`
URI	`m2://VF17XWCK8Z`
Kind	definition
Writing	W0 stub
MSC	65N30, 65D07, 74S22
Defines	Isogeometric analysis
Uses	Non-uniform rational B-spline, B-spline, Galerkin method, Finite element method

Isogeometric analysis

Definition

Remarks

Reading path

Further reading

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