- Sensitivity governing equations: X-momentum sensitivity equation: ρ∂x∂uu^​+ρ∂y∂uv^​​+ρ∂z∂uw^​+ρ∂x∂uu^​+ρ∂y∂vu^​+ρ∂z∂wu

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- Sensitivity governing equations: X-momentum sensitivity equation: ρ∂x∂uu^​+ρ∂y∂uv^​​+ρ∂z∂uw^​+ρ∂x∂uu^​+ρ∂y∂vu^​+ρ∂z∂wu^​−μt​(∂x2∂2u^​+∂y2∂2u^​+∂z2∂2u^​)=−∂x∂P^​−μ1​Du^−ρC2​∣u∣u^−21​ρ∣u∣u−2ωv^​ Y-momentum sensitivity equation: ρ∂x∂vu^​+ρ∂y∂vv^​​+ρ∂z∂vw^​+ρ∂x∂uv^​+ρ∂y∂vv^​+ρ∂z∂wv^​−μt​(∂x2∂2v^​+∂y2∂2v^​+∂z2∂2v^​)=−∂y∂P​−μ1​Dv^−ρC2​∣v∣v^−21​ρ∣v∣v+2ωu^​ Z-momentum sensitivity equation: ρ∂x∂wu^​+ρ∂y∂wv^​​+ρ∂z∂ww​+ρ∂x∂wu​+ρ∂y∂wv​+ρ∂z∂ww​−μt​(∂x2∂2w​+∂y2∂2w​+∂z2∂2w​)=−∂z∂P​−μ1​Dw−ρC2​∣w∣w−21​ρ∣w∣w​ Continuity/Poisson sensitivity equation: −∂x2∂2P​−∂y2∂2P​−​∂z2∂2P​=ρ∂x∂​(u^∂x∂u​+v^∂y∂u​+w∂z∂u​)+ρ∂y∂​(u^∂x∂v​+v^∂y∂v​+w^∂z∂v​)+ρ∂z∂​(u^∂x∂w​+v^∂y∂w​+w∂z∂w​)+ρ∂x∂​(u∂x∂u^​+v∂y∂u^​+w∂z∂u^​)+ρ∂y∂​(u∂x∂v^​+v∂y∂v^​+w∂z∂v^​)+ρ∂z∂​(u∂x∂w​+v∂y∂w​+w∂z∂w​)+μ1​D∂x∂u^​+ρC2​∂x∂(∣u∣u^)​+2ρ​∂x∂(u∣u∣)​+2ω∂x∂v^​+μ1​D∂y∂v^​+ρC2​∂y∂(∣v∣v^)​+2ρ​∂y∂(v∣v∣)​−2ω∂y∂u^​+μ1​D∂z∂w​+ρC2​∂z∂(∣w∣w^)​+2ρ​∂z∂(w∣w∣)​​ In the previous equations the sensitivity parameters such as P^ become our variable, while the flow parameters such as P are independent of the sensitivity parameters (constants).
After finite volume discretization, we get a set of linear algebraic equations which are very similar to lincarized Navicr-stokes cquation as: [LK​HF​][V^P^​]=[eg​] Where; [A]=[LK​HF​] are function of flow parameters and independent of the sensitivity parameters (constant). V^=⎣⎡​u^v^w​⎦⎤​ - Linear system solver: We could have a solution with Gauss elimination using low number of cells. But, for large grid number (optimum mesh) direct solution for such system become unacceptable. Instead, one of the iterative methods for solving linear system of equations should be used. Unfortunately, due to the non-orthogonality associated to the untrusted mesh the resultant [A] matrix is not positive definite matrix, off-diagonal dominant matrix, and non-symmetric. We tried to use a lot of iterative methods such as G-S, ILU, and GMRES. GMRES gave me some convergence, but did not reach to that value obtained by Gauss climination, as it showed so much slow convergence. We tried to use Algebraic multi grid "AMG" codes (due to its robustness as it is already used in CFD soft wares) available online, but they did not converge or reach to that value obtained by Gauss elimination We tried to use block iterative such as solving: LV^=e−HP^ Then FP^=g−KV^ But not converged. - Conclusion: So, we seek to solve such equation system AX=b in an iterative way that could fastly and correctly converge to same solution obtained from Gauss-elimination. We believe in AMG solver, but may be the codes available online are not effective for my case.