MIT2_092F09_lec20 - 2.092/2.093 Finite Element Analysis of...

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2.092/2.093 Finite Element Analysis of Solids & Fluids I Fall ‘09 Lecture 20 - Wave Propagation Response Prof. K. J. Bathe MIT OpenCourseWare Quiz #2: Closed book, 6 pages of notes, no calculators. Covers all materials including this week’s lectures. L w C = t w (C depends on material properties) For this system, C is the wave speed (given), L w is the critical wavelength to be represented, t w is the total time for this wave to travel past a point, L e is the “effective length” of a finite element, and is equivalent to L n w (given). To solve, we should use Δ t = L C e . Mesh L e should be smaller than the shortest wave length we want to pick up. To establish a mesh, we use low-order elements (4-node elements in 2D, 8-node elements in 3D). We use the central difference method, which requires stability. We need to ensure that Δ t Δ t cr = ω 2 n . Recall that in nonlinear analysis, the wave speed changes. We know that ω n max ω n ( m ) m where ω n is the largest frequency of an assembled finite element mesh and max ω n ( m ) is the largest element m frequency of all elements in the mesh. Then we can use 2 Δ t = ( m ) max ω n m and conservatively, we use a slightly smaller value. 1
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Lecture 20 Wave Propagation Response 2.092/2.093, Fall ‘09 ( m ) How to Find ω n For lower-order elements, we have bounds for ω n ( m ) . (See
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