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Discrete Adjoint derivative computation - help understanding Finite Difference |
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topheruk
Join Date: Mar 2018
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I am looking for some guidance on the implementation of the Discrete Adjoint method, specifically the computation of the partial derivatives via Finite Differencing.
source(s): DAFoam: An Open-Source Adjoint Framework for Multidisciplinary Design Optimization with OpenFOAM https://arc.aiaa.org/doi/10.2514/1.J058853 The first major step in solving the Discrete Adjoint is computing the partial derivatives of the vector of flow residuals ( ![]() ![]() Set ![]() ![]() ![]() where the subscripts ![]() ![]() ![]() ![]() ![]() ![]() ![]() In the DAFoam implementation, a graph-colouring method is used to accelerate the computation of the partial derivatives via Finite Difference. This works by exploiting the sparsity of a Jacobian matrix to simultaneously perturb sets of columns that influence independent rows. An example is given for a ![]() ![]() Assumptions: 1. For an unperturbed and converged flow solution, each cell contains a ![]() ![]() Questions: 1. when the source(s) discusses graph-colouring, what matrix are they referring to? Is the ![]() ![]() 2. when perturbing a flow variable e.g. the u-component of velocity, is ![]() 3. how does perturbing a flow variable in a cell influence other rows in the matrix? Appreciate any input, Chris Last edited by topheruk29; July 21, 2021 at 06:57. Reason: missed out vector definitions |
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adjoint, mathematics, optimization |
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