##  [Isogeometric Analysis](/isogeometric-analysis-0) 

 Definition

A numerical simulation approach that uses the same spline-based functions (e.g., NURBS, B-splines) for both exact geometry representation and field approximation, thereby integrating CAD geometry and analysis bases.

 

 

 

 

 

 





## Principle

Principle

Unify geometric representation and solution spaces so that the geometry is represented exactly (or with the CAD parametrization) and approximation properties of splines yield smooth, higher-continuity basis functions for analysis.

 

 

 

 

 





## Demonstration

Demonstration

In structural analysis of a curved shell described by NURBS, use the same NURBS basis for the mid-surface geometry and for displacement approximation; this avoids geometry approximation error and simplifies mesh refinement by knot insertion.

 

 

 

 

## Misapplication

Misapplication

Treating arbitrary low-regularity polygonal meshes as if spline geometry continuity holds, or using incompatible spline spaces for geometry and solution, breaks the isogeometric premise and introduces geometric or approximation inconsistencies.

 

 

 

 

 





## Consequence

Consequence

Proper application yields exact CAD-consistent geometry, higher-order continuity across elements, improved accuracy per degree of freedom for smooth problems, and straightforward h/p/k refinement strategies aligned with CAD operations.

 

 

 

 

## Reversal

Reversal

The opposite is traditional finite element analysis: geometry is approximated by piecewise polynomials on a mesh independent of CAD, producing geometric error and lower inter-element continuity unless special elements are used.

 

 

 

 

 





## Boundary

Boundary

Best suited for problems where CAD geometry is available and spline bases are practical; less advantageous for problems requiring unstructured simplicial meshes, discontinuous solutions, or where CAD parametrizations are unavailable or too complex to integrate.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Competes with classical FEM and CAD-to-mesh pipelines: it reduces geometry error and improves continuity but requires spline technology and can complicate local adaptivity compared to simplicial finite elements.

 

 

 

 

 





## Synthesis

Synthesis

Isogeometric Analysis: an approach that merges CAD parametrizations and analysis by employing spline-based bases for both exact geometry and solution approximation, delivering smoother bases and CAD-consistent discretizations.