The paper introduces isogeometric analysis, a method that uses NURBS (Non-Uniform Rational B-Splines) to construct exact geometric models for finite element analysis. The key features of isogeometric analysis include: 1. **Exact Geometry**: NURBS basis functions are used to represent the exact geometry, eliminating the need for approximate geometric representations. 2. **Mesh Refinement**: The basis functions can be refined (e.g., through knot insertion or order elevation) without changing the geometry, maintaining exact geometry at all levels. 3. **k-Refinement**: A new higher-order refinement strategy, k-refinement, is introduced, which is more efficient and robust than traditional p-refinement. 4. **Isoparametric Concept**: The solution space for dependent variables is represented by the same functions as the geometry, ensuring consistency. 5. **Application in Structural Mechanics**: The method is shown to be effective for linear elasticity problems, thin shell analysis, and fluid dynamics, achieving optimal convergence rates and accurate solutions. The paper also discusses the historical context of finite element analysis and CAD, highlighting the limitations of geometric approximations in finite element analysis and the advantages of isogeometric analysis. Examples and numerical results are provided to demonstrate the effectiveness of the method in various engineering applications.The paper introduces isogeometric analysis, a method that uses NURBS (Non-Uniform Rational B-Splines) to construct exact geometric models for finite element analysis. The key features of isogeometric analysis include: 1. **Exact Geometry**: NURBS basis functions are used to represent the exact geometry, eliminating the need for approximate geometric representations. 2. **Mesh Refinement**: The basis functions can be refined (e.g., through knot insertion or order elevation) without changing the geometry, maintaining exact geometry at all levels. 3. **k-Refinement**: A new higher-order refinement strategy, k-refinement, is introduced, which is more efficient and robust than traditional p-refinement. 4. **Isoparametric Concept**: The solution space for dependent variables is represented by the same functions as the geometry, ensuring consistency. 5. **Application in Structural Mechanics**: The method is shown to be effective for linear elasticity problems, thin shell analysis, and fluid dynamics, achieving optimal convergence rates and accurate solutions. The paper also discusses the historical context of finite element analysis and CAD, highlighting the limitations of geometric approximations in finite element analysis and the advantages of isogeometric analysis. Examples and numerical results are provided to demonstrate the effectiveness of the method in various engineering applications.