University:
Massachusetts Institute of Technology
Course:
2.094 | Finite Element Analysis of Solids and Fluids II
Academic year:

2022
Views:

305

Pages:

4
Author:

debatehl4dcq

2.094 — Finite Element Analysis of Solids and Fluids Lecture 8 - Convergence of displacement-based FEM (A) Find u ∈ V such that a(u, v) = (f , v) ∀v ∈ V (Mathematical model) (8.1) a(v, v) > 0 ∀v ∈ V, (8.2) v= � 0. where (8.2) implies that structures are supported properly. E.g. (B) F.E. Problem Find uh ∈ Vh such that a(uh , vh ) = (f , vh ) ∀vh ∈ Vh a(vh , vh ) > 0 ∀vh ∈ Vh , vh �= 0 (8.3) (8.4) Properties eh = u − uh (I) a(eh , vh ) = 0 ∀vh ∈ Vh (II) a(uh , uh ) ≤ a(u, u) (8.5) (8.6) � � (C) Assume Mesh� � � e.g. Mesh� h1 h1 � � “is contained in” Mesh� � � not contained in Mesh� h2 h2 We assume (C), but need another property (independent of (C)) (III) a(eh , eh ) ≤ a(u − vh , u − vh ) ∀vh ∈ Vh (8.7) uh minimizes! (Recall eh = u − uh ) Proof: Pick wh ∈ Vh . �0 �� a(eh + wh , eh + wh ) = a(eh , eh ) + � 2a(e + a(wh , wh ) �� h , wh ) � �� (8.8) ≥0 Equality holds for (wh = 0) a(eh , eh ) ≤ a(eh + wh , eh + wh ) = a(u − uh + wh , u − uh + wh ) (8.9) (8.10) Take wh = uh − vh . a(eh , eh ) ≤ a(u − vh , u − vh ) (8.11) Using property (III) and (C), we can say that we will converge monotonically, from below, to a(u, u): Pascal triangle (2D) ⇒ 4-node element is complete to k = 1. ⇒ 9-node element is complete to k = 2. (Ch. 4.3) error in displacement ∼ C · hk+1 (8.12) (C is a constant determined by the exact solution, material property. . . ) error in stresses ∼ C · hk error in strain energy ∼ C · h2k (8.13) (← these C are diﬀerent) (8.14) Hence, E − Eh = C · h2k (roughly equal to) (8.15) By theory, log (E − Eh ) = log C + 2k log h (8.16) By experiment, we can evaluate log(E − Eh ) for diﬀerent meshes and plot log(E − Eh ) vs. log h We need to use graded meshes if we have high stress gradients. Example Consider an almost incompressible material: �V = vol. strain (8.17) � · v → very small or zero (8.18) or We can “see” diﬃculties: p = −κ�V κ = bulk modulus As the material becomes incompressible (ν = 0.3 → 0.4999) � κ→∞ p → ﬁnite number �V → 0 (8.19) (8.20) (Small error in �V results in huge error on pressure as κ → ∞, the constant C in (8.15) can be very large ⇒ locking)

Convergence of Displacement-Based FEM

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