P-refinement

Definition

P-refinement is a method used in finite element analysis to enhance the accuracy of numerical solutions by increasing the polynomial degree of the shape functions within the elements. This technique allows for better approximation of the solution without necessarily increasing the number of elements, making it a computationally efficient approach to improve solution fidelity in simulations.

5 Must Know Facts For Your Next Test

1. P-refinement allows for local enhancement of solution accuracy by increasing polynomial degrees only in regions where higher precision is needed, such as areas with high gradients or singularities.
2. This technique contrasts with h-refinement, where the focus is on reducing element size to achieve more accurate results.
3. In p-refinement, elements can maintain their original shape while improving performance through higher-order polynomials, which can reduce overall computational costs.
4. The choice between p-refinement and h-refinement depends on the problem characteristics and the desired balance between accuracy and computational efficiency.
5. P-refinement can be particularly advantageous in problems with smooth solutions, as higher polynomial orders can approximate these solutions more effectively than just refining the mesh.

Review Questions

• How does p-refinement improve the accuracy of finite element analysis compared to other refinement techniques?
  • P-refinement enhances accuracy by increasing the polynomial degree of shape functions within existing elements instead of refining the mesh. This allows for a more precise approximation of solutions, especially in areas where there are steep gradients or complex behavior. By focusing on adjusting polynomial orders rather than just reducing element sizes, p-refinement can achieve better accuracy while maintaining computational efficiency.
• In what scenarios would you prefer using p-refinement over h-refinement in finite element analysis?
  • P-refinement is preferable when dealing with problems that have smooth solutions or where high accuracy is needed only in localized regions of the domain. If a simulation has areas with strong gradients or singularities, increasing the polynomial order can provide better approximations without requiring a denser mesh, which can be computationally expensive. Conversely, h-refinement may be more beneficial in situations with complex geometries or discontinuities where local mesh density is essential.
• Evaluate how p-refinement contributes to computational efficiency and solution accuracy in large-scale finite element simulations.
  • P-refinement improves computational efficiency by allowing for enhanced accuracy through higher-order polynomials without significantly increasing the number of elements, which helps manage computational resources better. This means that larger-scale simulations can achieve desired accuracy levels with fewer overall computations. Furthermore, by selectively applying p-refinement only where necessary, it helps focus computational efforts on critical areas while keeping overall resource use lower, thus optimizing both speed and precision in large-scale analyses.

Related terms

Finite Element Method (FEM): A numerical technique for finding approximate solutions to boundary value problems for partial differential equations, by dividing a large problem into smaller, simpler parts called finite elements.

h-refinement:

A technique in finite element analysis that involves decreasing the size of elements to increase the mesh density and improve the solution accuracy.

Convergence:

The process by which a numerical solution approaches the exact solution as the mesh is refined or as the polynomial order is increased.