Abstract
The concept of isogeometric analysis is proposed. Basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element h- and p-refinement schemes are presented and a new, more efficient, higher-order concept, k-refinement, is introduced. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. A k-refinement strategy is shown to converge toward monotone solutions for advection–diffusion processes with sharp internal and boundary layers, a very surprising result. It is argued that isogeometric analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses several advantages.
Highlights
In this paper we introduce a new method for the analysis of problems governed by partial differential equations such as, for example, solids, structures and fluids
The approach we have developed is based on NURBS (Non-Uniform Rational B-Splines), a standard technology employed in Computer Aided Design (CAD) systems
Infinite plate with circular hole under constant in-plane tension in the x-direction. In this two-dimensional example, we present in some detail the NURBS analysis of a problem in solid mechanics having an exact solution
Summary
In this paper we introduce a new method for the analysis of problems governed by partial differential equations such as, for example, solids, structures and fluids. Smooth geometry completely eliminated the entropy layers even when the flow fields were approximated by linear elements on the curved geometry; see Fig. 2 This result explains why methods which employ smooth geometric mappings are widely used in airfoil analysis (see [3]). The construction of finite element geometry (i.e., the mesh) is costly, time consuming and creates inaccuracies It is clear from the smaller size of the CAE industry compared with the CAD industry that the most fruitful direction would be to attempt to change, or replace, finite element analysis with something more CAD-like. By employing highorder, k-refinement strategies, convergence toward monotone solutions is obtained This surprising result seems to contradict numerical analysis intuitions and suggests the possibility of linear difference methods that are simultaneously robust and highly accurate.
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