MIT2_094S11_lec15

# MIT2_094S11_lec15 - 2.094 — Finite Element Analysis of...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2.094 — Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 15 - Field problems Prof. K.J. Bathe MIT OpenCourseWare Heat transfer, incompressible/inviscid/irrotational ﬂow, seepage ﬂow, etc. Reading: Sec. 7.2-7.3 • Diﬀerential formulation • Variational formulation • Incremental formulation • F.E. discretization 15.1 Heat transfer Assume V constant for now: S = Sθ ∪ Sq θ(x, y, z, t) is unkown except θ|Sθ = θpr . In addition, q s |Sq is also prescribed. 15.1.1 Diﬀerential formulation I. Heat ﬂow equilibrium in V and on Sq . ∂θ II. Constitutive laws qx = −k ∂x . ∂θ qy = −k ∂ y ∂θ qz = −k ∂z (15.1) (15.2) III. Compatibility: temperatures need to be continuous and satisfy the boundary conditions. 61 MIT 2.094 15. Field problems Heat ﬂow equilibrium gives � � � � � � ∂ ∂θ ∂ ∂θ ∂ ∂θ k + k + k = −q B ∂x ∂x ∂y ∂y ∂z ∂z (15.3) where q B is the heat generated per unit volume. Recall 1D case: unit cross-section dV = dx · (1) (15.4) q |x − q |x+dx + q B dx = 0 � ∂ qx q |x − q |x + dx + q B dx = 0 ∂x (15.5) � − � ∂ ∂x ∂ ∂x −k � k ∂θ ∂x ∂θ ∂x � � (15.6) B� � dx dx � +q � =0 (15.7) = −q B (15.8) We also need to satisfy ∂θ = qS ∂n k (15.9) on Sq . 15.1.2 � θ Principle of virtual temperatures ∂ ∂x � ∂θ k ∂x � + ··· + q B � =0 (15.10) � ( θ�S = 0 and θ to be continuous.) θ � � θ V ∂ ∂x � � � ∂θ k + · · · + q B dV = 0 ∂x (15.11) 62 MIT 2.094 15. Field problems Transform using divergence theorem (see Ex 4.2, 7.1) � � � �T Sq � B θ kθ θq dV + θ q S dSq ���� dV = V V heat ﬂow ⎛ ∂θ ∂x ⎞ ⎜ ⎜ θ =⎜ ⎜ ⎝ ∂θ ∂y (15.12) Sq ⎟ ⎟ ⎟ ⎟ ⎠ � (15.13) ∂θ ∂z ⎡ k k=⎣ 0 0 0 k 0 ⎤ 0 0⎦ k (15.14) Convection b oundary condition � � qS = h θe − θS (15.15) where θe is the given environmental temperature. Radiation � � �4 � 4 q S = κ∗ (θr ) − θS � � �2 � � r �� � 2 = κ∗ (θr ) + θS θ + θS θr − θS � � = κ θr − θS (15.16) (15.17) (15.18) where κ = κ(θS ) and θr is given temperature of source. At time t + Δt � � � � T t+Δt t+Δt � S t+Δt B θ k θ dV = θ q dV + θ t+Δtq S dSq V V t+Δt θ = tθ + θ Let t+Δt (i) or θ t+Δt (0) with θ = t+Δt (i−1) θ (15.19) Sq (15.20) + Δθ (i) (15.21) t =θ (15.22) From (15.19) � �T (i) θ t+Δtk(i−1) Δθ � dV V � � �T t+Δt � (i−1) = θ t+Δtq B dV − θ t+Δtk(i−1) θ dV V V � � �� � (i−1) (i) S + θ t+Δth(i−1) t+Δtθ e − t+Δtθ S + ΔθS dSq (15.23) Sq where the ΔθS (i) term would be moved to the left-hand side. We considered the convection conditions � � � S θ t+Δth t+Δtθ e − t+Δtθ S dSq (15.24) Sq The radiation conditions would be included similarly. 63 MIT 2.094 15. Field problems F.E. discretization ˆ θ = H1x4 · t+Δtθ4x1 t+Δt � ˆ θ2x1 = B2x4 · t+Δtθ4x1 t+Δt S ˆ θ = H S · t+Δtθ t+Δt for 4-node 2D planar element (15.25) (15.26) (15.27) For (15.23) � ⎛ ⎞ � gives (i) θ t+Δtk(i−1) Δθ � dV =⇒ ⎝ ���� t+Δtk(i−1) �B dV ⎠ Δ �ˆ(i) BT θ ��� � �� � ��� V �T V � 4x2 θ t+Δtq B dV ⇒ V � 2x4 2x2 (15.28) 4x1 H T t+Δtq B dV (15.29) V � θ � T t+Δt (i−1) t+Δt � (i−1) k θ �� dV ⇒ V B T t+Δtk(i−1) B dV � V � θ S T t+Δt (i−1) h � t+Δt e θ− � t+Δt S (i−1) θ t+Δt ˆ(i−1) � + ΔθS (i) �� dSq θ �� known (15.30) � =⇒ Sq ⎛ � Sq 15.2 ⎛ ⎞⎞ (15.31) T ˆ ˆ HS H S � t+Δth(i−1) ���� ⎝ t+Δtθ e − ⎝ t+Δtθ (i−1) +Δ �ˆ(i) ⎠⎠ dSq θ � �� ��� � �� � � �� � 4x1 1x4 4x1 4x1 4x1 Inviscid, incompressible, irrotational ﬂow 2D case: vx , vy are velocities in x and y directions. �·v =0 ∂ vx ∂ vy + =0 ∂x ∂y ∂ vx ∂ vy − =0 ∂y ∂x or Reading: Sec. 7.3.2 (15.32) (incompressible) (15.33) (irrotational) (15.34) Use the potential φ(x, y ), vx = ⇒ ∂φ ∂x vy = ∂φ ∂y (15.35) ∂2φ ∂2φ + 2 = 0 in V ∂ x2 ∂y (15.36) (Same as the heat transfer equation with k = 1, q B = 0) 64 MIT OpenCourseWare http://ocw.mit.edu 2.094 Finite Element Analysis of Solids and Fluids II Spring 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ...
View Full Document

## This note was uploaded on 12/29/2011 for the course ENGINEERIN 2.094 taught by Professor Prof.klaus-jürgenbathe during the Spring '11 term at MIT.

Ask a homework question - tutors are online