Auto­mated Opti­miza­tion using Adjoint Flow Solvers

Here at FRIEND­SHIP SYSTEMS, we recently carried out a case study for an auto­mated opti­miza­tion process based on the shape sen­si­tiv­i­ties computed by an adjoint CFD solver. The open-source opti­miza­tion toolkit Dakota by Sandia National Labs, that is inte­grated in CAESES® through a direct inter­face, provided an opti­miza­tion strategy that can directly receive the gradient infor­ma­tion obtained from coupling adjoint shape sen­si­tiv­i­ties to CAD model para­me­ters as input (check out this recent article). Based on this infor­ma­tion, the algo­rithm selects the para­me­ter com­bi­na­tion for the next variant that is then created by CAESES® and analyzed by the adjoint flow solver.

The geometry con­sid­ered in this study was a simple fic­tional duct with a 90-degree bend that could resemble a com­po­nent taken from an internal com­bus­tion engine. Inlet and outlet were fixed in position, shape and ori­en­ta­tion, but every­thing in between was free to change. The geometry was modeled using an intel­li­gent variable surface design in stream­wise direc­tion and 13 defining para­me­ters that control the shape of the duct’s path and cross-section. The geometry was trans­ferred using a ​“colored” STEP format that allows trans­fer­ring patch names.

The flow solver STAR-CCM+ was used to solve the primal and adjoint equa­tions for every variant. The fluid was air with an inlet velocity of 50m/​s, which cor­re­sponds to a common gas velocity in engine parts. The overall com­pu­ta­tion time, includ­ing the meshing with about 33.000 cells, was about 6 minutes. The surface shape sen­si­tiv­i­ties were cal­cu­lated by STAR-CCM+ after the adjoint sim­u­la­tion was com­pleted and exported in Ensight Gold format, to be able to import this infor­ma­tion to CAESES®. From within CAESES®, the STAR-CCM+ cal­cu­la­tions were trig­gered and con­trolled with JAVA macros. The overall pressure drop was used as the objec­tive function.

13 geometry variants were analyzed in the course of the auto­mated opti­miza­tion, whereby a local minimum with a 16% improve­ment of the objec­tive function was already found after a very quick descent at the 5th variant, after which no sig­nif­i­cant improve­ment happened any more. A con­ven­tional opti­miza­tion with a standard deter­min­is­tic algo­rithm was carried out for com­par­i­son purposes. Although this lead to a slightly better local minimum (approx­i­mately 20% improve­ment), it required a sig­nif­i­cantly higher number of variants, and there­fore function eval­u­a­tions (67 new designs). Espe­cially apparent are the small steps during the explo­ration phase of the algo­rithm that are required for the numer­i­cal deter­mi­na­tion of the local gradient direc­tion. Obvi­ously, this step is omitted when using the gradient infor­ma­tion from the adjoint analysis, which is crucial for the poten­tial time saving of this opti­miza­tion approach.

The con­ver­gence of the design can also be very well observed in the dif­fer­ence of the shape sen­si­tiv­ity plots between initial and opti­mized design. The shape sen­si­tiv­ity in most parts of the geometry has been reduced to almost 0, except for the areas where the mod­i­fi­ca­tion of the geometry was con­strained by the fixed inlet and outlet geometries. 

Summary

Using adjoint shape sen­si­tiv­i­ties and coupling them to the CAD model para­me­ters in the context of opti­miza­tion allows for a sig­nif­i­cant speed up of the con­ver­gence process and an opti­mized geometry that can directly be fed into the down­stream CAD design process. Addi­tion­ally, since the expenses do not scale with the number of para­me­ters, all form para­me­ters of the model can be involved into an opti­miza­tion, without having to do a pre-selec­tion. One must consider, however, that the pre­dic­tions based on the adjoint sen­si­tiv­i­ties are only valid for small (in strict math­e­mat­i­cal sense infin­i­tes­i­mal) changes to the geometry and that using this infor­ma­tion for opti­miza­tion is a local pro­ce­dure. If a global optimum is searched, one might still have to precede the actual opti­miza­tion by a Design of Exper­i­ments (e.g. using random sampling algo­rithms such as the Latin Hyper­cube Sampling) to scan the wider design space. 

More Infor­ma­tion

If you are inter­ested in fast and auto­mated shape opti­miza­tion using adjoint infor­ma­tion, then simply get in touch with us. We look forward to dis­cussing it with you in the context of your indi­vid­ual engi­neer­ing appli­ca­tion. More infor­ma­tion about using Dakota together with CAESES® can be found in this recent blog post about response surface methods. 

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