Introduction_FEM_2

Introduction_FEM_2 - Introduction to FEM Lecture 2 Prof...

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Unformatted text preview: Introduction to FEM: Lecture 2 Prof. Jin-Fa Lee http://esl.eng.ohio-state.edu/~csg 0 1 1/2 x Application of FEM Trial Function Space = Testing Function Space = Λ h = Span α , α 1 , , α m − 1 { } φ FEM h = φ α + φ 1 α 1 + + φ m − 1 α m − 1 v = v α + v 1 α 1 + + v m − 1 α m − 1 Note: α i x j ( ) = δ ij , and α i x ( ) ≠ 0 iff x i − 1 ≤ x ≤ x i + 1 Galerkin Statement Find φ FEM h ∈Λ h such that dv dx ε r d φ FEM h dx dx 1 ∫ = vdx 1 ∫ for every v ∈Λ h φ FEM h = φ α + φ 1 α 1 + + φ m − 1 α m − 1 = φ φ 1 φ m − 1 ⎡ ⎣ ⎤ ⎦ α α 1 α m − 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = α α 1 α m ⎡ ⎣ ⎤ ⎦ φ φ 1 φ m ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ Let φ = φ φ 1 φ m − 1 ⎡ ⎣ ⎤ ⎦ , and α = α α 1 α m − 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ Then φ FEM h = φ α = α φ Galerkin Statement Find φ such that v ε r d α dx d α 1 dx d α m − 1 dx ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ d α dx d α 1 dx d α m − 1 dx ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ dx φ = 1 ∫ v α α 1 α m − 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ dx 1 ∫ for every v ε r d α dx d α 1 dx d α m − 1 dx ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ d α dx d α 1 dx d α m − 1 dx ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ dx φ = 1 ∫ α α 1 α m − 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ dx 1 ∫ ⇒ ε r d α dx d α dx dx 1 ∫ ε r d α dx d α 1 dx dx 1 ∫ ε r d α dx d α m − 1 dx dx 1 ∫ ε r d α 1 dx d α dx dx 1 ∫ ε r d α 1 dx d α 1 dx dx 1 ∫ ε r d α 1 dx d α m − 1 dx dx 1 ∫ ε r d α m dx d α dx dx 1 ∫ ε r d α m − 1 dx d α 1 dx dx 1 ∫ ε r d α m − 1 dx d α m − 1 dx dx 1 ∫ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ φ φ 1 φ m ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = α dx 1 ∫ α 1 dx 1 ∫ α m − 1 dx 1 ∫ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ A φ = f A ij = ε r d α i dx d α j dx dx 1 ∫ f i = α i dx 1 ∫ ε r d α dx d α dx dK K i ∫ ε r d α dx d α 1 dx dK K i ∫...
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Introduction_FEM_2 - Introduction to FEM Lecture 2 Prof...

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