清华大学有限元分析课件4-1 Strong Form for Multidimensional Scalar Field Problems_85002962.pdfVIP

第4 章 多维标量场问题 4.1 强形式 Strong Form for Multidimensional Scalar Field Problems Finite Element Method 4.1 强形式 Divergence Theorem and Green’s Formula   Vector and matrix notations ⎡ ∂ ⎤ q = q i + q j = ⎡ qx ⎤ ∇ = i ∂ + j ∂ = ⎢ ∂x ⎥ x y ⎢ q ⎥ ∂x ∂y ⎢ ∂ ⎥ ⎣ y ⎦ ⎢ ⎥ ⎡ ∂θ ⎤ ⎣ ∂y ⎦ ⎢ ⎥ ∂q ∂q ∂θ ∂θ ∂x x y T ∇ θ = i + j = ⎢ ⎥ ∇ ⋅q = + ≡ div q = ∇ q ∂x ∂y ⎢ ∂θ ⎥ ∂x ∂y ⎣ ∂y ⎦ The flow leaving a point   Green’s theorem 0 ∫Ω∇ θ dΩ = ∫Γ θn dΓ if θ (x ,y ) ∈C and integrable   Divergence theorem 0 ∫Ω∇ ⋅q dΩ = ∫Γ q ⋅n dΓ if q ∈C and integrable   Green’s formula ∫Ω w∇ ⋅q dΩ = ∫Γ wq ⋅n dΓ − ∫Ω∇ w ⋅q dΩ 2/4 School of Aerospace, Tsinghua University Finite Element Method 4.1 强形式 Heat conduction The balance of heat energy in steady state — The heat flux q flowing out through the boundaries of the control volume equals the heat generated s. ∇ ⋅q − s = 0 The generalized Fourier law ⎡ k k ⎤ q = −D∇ T D = ⎢ kxx kxy ⎥ — the conductivity matrix ⎣ yx yy ⎦ = kI — isotropic materials The prescribed temperature boundary condition T (x ,y ) = T (x ,y ) on ΓT Essential boundary conditions y