Patch Test in FEM | Shape Functions, Completeness & Serendipity vs Lagrange Elements

Choosing the right shape functions is one of the most important steps in the Finite Element Method (FEM) — and this is exactly where many errors begin. In this video, I explain the Patch Test in a clear and intuitive way and show how it is used to verify whether a finite element formulation is consistent, complete, and convergent. Using 2D triangular and quadrilateral elements, we cover: Why shape functions matter in FEM Rigid body motion and constant strain requirements What the Patch Test actually checks Why T3, T6, Q4, Q8, and Q9 elements pass the patch test The role of Pascal’s triangle in completeness The difference between Serendipity and Lagrangian (tensor-product) elements This video is especially useful for graduate students, researchers, and engineers working with FEM software such as ABAQUS, ANSYS, PLAXIS, COMSOL, or writing their own finite element codes. 📌 Understanding the patch test helps you avoid spurious stresses, locking, and non-convergent elements. 🔔 If you are learning FEM theory, numerical methods, or computational mechanics, consider subscribing for more in-depth explanations. #FiniteElementMethod #PatchTest #FEM #ShapeFunctions #ComputationalMechanics #NumericalMethods #EngineeringEducation #StructuralEngineering #GeotechnicalEngineering #CivilEngineering