A lot of research has been performed on error estimation and adaptive mesh refinement. However, adaptive methods are not yet an industrial tool, partly because the need for a link to traditional CAD-system makes this difficult in industrial practice. Here, the use of an isogeometric analysis framework may facilitate more widespread adoption of this technology in industry, as adaptive mesh refinement does not require any further communication with the CAD system.

Classical Lagrange polynomial finite element shape functions are C0 continuous, and when they are used to represent general non-planar surfaces the resulting discrete geometry becomes non-smooth (i.e. discontinuous surface normal). Such faceted finite elements impose problems for sliding and frictional contact between two and more solid objects. In isogeometric analysis this artifact is removed by using C1-continuous splines.

In many fluid-structure interaction problems the appropriate resolution in the finite element mesh for fluid and structures are very different. Thus, the fluid-structure interface is not consistently represented and this affects the quality of the numerical solution. However, by using splines the interface may be represented exactly even though the parametric representations of the interface are different on the fluid and structure side.

In numerical simulations of rotating wind turbine blades, helicopter rotors, tidal turbines, ships propeller etc. exposed to fluid flow it is a common technique to let the fluid mesh in the vicinity of the rotor rotate with it and then have a fixed surrounding fluid mesh. Use of traditional Lagrange polynomial finite elements introduces inconsistent representation of the circular interface between rotating and fixed fluid mesh. Here, the use of non-uniform rational B-splines (NURBs) eliminates this discrepancy which typically makes the solution prone to instabilities.

Many scientific and engineering problems involve optimization of the geometrical shape. Examples include: Optimal turbine blade, optimal ship hull, optimal automotive engines etc. Traditionally the high quality initial geometry representation is degraded during the optimization cycle by the change in geometry based on the finite element solution using C0-continuous basis functions. However, by using spline finite elements one may keep the quality of the geometry representations as well as coherent transfer of between the geometry generator and the finite element analysis.

Scientific and engineering problems described by a partial differential equation involving a 4th order differential operators should preferably be solved with C1-continuous finite elements. Thin walled structures modeled with plates and shells are a relevant case in this respect. Traditionally this has been handled by introducing rotational degrees of freedom in addition to regular translational degrees of freedoms. Rotations introduce complexity that one wish to avoid if possible. However, by using C1 continuous splines one may use finite elements with translational degrees of freedom only.

Published October 19, 2010

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