--- title: "Convergence requirements for displacement functions" subject: "Finite Element Method" chapter: 5 key: "semester_8/finite_element_method:convergence-requirements-polynomial-elements" source: https://www.bctengineer.com/q/semester_8/finite_element_method/convergence-requirements-polynomial-elements site: BCT Engineering --- # Convergence requirements for displacement functions Finite Element Method · Chapter 5: Interpolation Functions - Asked in: year not given, 5 marks - Video: [Convergence Requirements for deriving shape function and displacement function in FEA](https://www.youtube.com/watch?v=6l6cqeIJ7i0) — Dr. S P Palekar - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/finite_element_method/convergence-requirements-polynomial-elements - Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-5 ## Question State and explain the convergence requirements for the displacement (interpolation) function of a finite element. Explain how Pascal's triangle helps in choosing polynomial terms. ## Short answer Interpolation functions must be complete (represent rigid-body and constant strain states), compatible (C0 for bars/plane, C1 for beams), contain no spurious rigid-body modes, and be geometrically isotropic. Pascal's triangle supplies symmetric complete polynomial terms so element stiffness is frame-invariant. ## Answer For the FE solution to converge to the exact one as the mesh is refined, the assumed displacement function must satisfy these conditions. #### Requirements 1. **Completeness.** The function must be able to represent rigid-body motion (constant terms) and constant strain states. For example, a bar needs $u = a_1 + a_2x$: $a_1$ is rigid-body translation and $a_2$ is constant strain. As elements shrink, the strain in each becomes nearly constant, so this is essential. 2. **Compatibility (conformity).** Displacements must be continuous across element boundaries so that no gaps or overlaps develop. For second-order problems (bars, plane elasticity) this means $C^0$ continuity; for beams the slope must also be continuous ($C^1$). 3. **Rigid-body modes.** An element should have zero strain energy under rigid-body movement. 4. **Geometric isotropy (invariance).** The function should not prefer any direction, so the result should not change if the local axes are rotated. This needs complete polynomial terms of a given order. Elements that satisfy all the conditions converge monotonically. Elements that satisfy completeness but not compatibility (non-conforming elements) can still converge, provided they pass the patch test. #### Pascal's triangle ```text Order 0: 1 Order 1: x y Order 2: x2 xy y2 Order 3: x3 x2y xy2 y3 Order 4: x4 x3y x2y2 xy3 y4 ``` - Terms are taken row by row so that the polynomial is complete up to a given degree and has geometric isotropy. - The number of terms must equal the number of nodal degrees of freedom: 3 terms (1, x, y) for the 3-node triangle; 4 terms (1, x, y, xy) for the 4-node rectangle, where $xy$ is chosen because it is symmetric. - Terms must be selected symmetrically about the vertical axis of the triangle to maintain isotropy. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For a bar with u = a1 + a2 x, what do a1 and a2 represent? A. a1 is rigid-body translation and a2 is constant strain B. a1 is strain and a2 is translation C. Both are rotation D. Both are curvature Answer: A. Completeness needs rigid-body motion (constant term) and constant strain states. (Key point: Completeness) 2. What continuity is needed across element boundaries for beams? A. C0 only B. C1, with slope also continuous C. None D. C3 Answer: B. Bars and plane elasticity need C0 continuity; beams need slope continuity (C1) too. (Key point: Compatibility) 3. What does geometric isotropy require? A. Only x terms B. Only constant terms C. The function should not prefer any direction, using complete polynomial terms D. Direction-dependent terms Answer: C. Complete polynomials of a given order keep the result unchanged when axes are rotated. (Key point: Isotropy) 4. How many terms and which are used for a 3-node triangle? A. 4: 1, x, y, xy B. 6 C. 2 D. 3: 1, x, y Answer: D. The number of terms must equal the number of nodal DOF; 1, x, y complete the first order. 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