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l45 - Lecture 45 Sensitivity Analysis 1 Last Time Completed...

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1 Lecture 45: Sensitivity Analysis

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2 Last Time … Completed consideration of multigrid methods Solved an example problem
3 This Time … We will: Look at emerging methods to perform sensitivity analysis using CFD

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What inputs yield desired outputs ? Adjoint Problem: How is a specific output affected by all inputs? Tangent Problem: How does a specific input affect all outputs? Determine effect of uncertainties and tolerances Plan experiments better Improve single-point simulations Mesh sensitivity Sensitivity to iteration parameters Why Sensitivity and Adjoints? Can gain a lot more information than conventional single- point simulations!
Approaches Continuous approach » Derive sensitivity/adjoint versions of pde » Discretize and solve » Need to repeat for each physical model » Generally more complex Discrete approach » Start with discrete pde and add sensitivity/adjoints » Easier to automate » Adjoints more difficult 5 x m n mn y x    J = m outputs n inputs Jacobian

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6 Continuous Approach: Sensitivity Equation Governing equations are differentiated with respect to each parameter of interest New PDEs are derived for the sensitivity coefficients, as well as appropriate boundary conditions These are solved using numerical methods similar (but not identical) to the original PDE CPU times scale sub-linearly with the number of parameters
7 Example: Transient Heat Conduction in a 1D Slab L x   44 0 | s r x q T T     | xL h T T    0 0 0 0 | | | | | | ( ,0) Parameters: C, k, q , , , , , x x s r x x L x L x L i s r i T q T C q k t x x T q k q T T x T q k h T T x T x T T h T T                           Courtesy K. Dowding, Sandia

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8 Sensitivity Equation for Heat Capacity Differentiate energy equation wrt C: + + 0 Multiply by nominal value of C: +C + 0 Scaled sensitivity of q wrt C: C T q T T q CC C t x t C t x C T T q C C C t C t x C q qC C                                                              PDE for new variable : C C C T k C k T x C x T TT T Ck t x C t                         Courtesy K. Dowding, Sandia
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l45 - Lecture 45 Sensitivity Analysis 1 Last Time Completed...

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