Direct Coupling of Para­met­ric CAD and Adjoint CFD

In the context of dif­fer­ent R&D projects over the last few years, we at FRIEND­SHIP SYSTEMS have devel­oped and imple­mented a new method for our product CAESES® that is based on the direct coupling of our effi­cient CAD para­me­ter­i­za­tion and adjoint CFD com­pu­ta­tions. The main advan­tages of this method are:

• The largest possible design space can be con­sid­ered for opti­miza­tion. All para­me­ters in the model can be involved, without the need for the user to under­stand the effect of each parameter.
• The effort does not scale with the number of parameters!
• No mesh morphing as in other CAD-free opti­miza­tion solu­tions based on adjoint CFD. The CAD model is modified directly.
• The opti­mized geometry can directly be used in down­stream CAD without re-modeling!
• Geometry con­straints can be easily mon­i­tored and fulfilled!
• The local gradient of the objec­tive function is computed directly. It can be fed into the opti­miza­tion algo­rithm for faster convergence.

Intro­duc­tion

Using para­met­ric mod­el­ling in the design process allows for an effi­cient vari­a­tion of the geometry. The total number of shape-defining para­me­ters for a typical para­met­ric model of a flow-exposed geometry (like a ship hull, tur­bo­ma­chin­ery or engine com­po­nent) in CAESES®, however, can typ­i­cally reach the range of 20 to 50. This means that for complex free-form geome­tries the number of degrees of freedom is still so large that, when using a direct opti­miza­tion approach, taking into account all para­me­ters, a high number of function eval­u­a­tions, i.e., CFD com­pu­ta­tions, would be required to identify trends with respect to the nec­es­sary geometry mod­i­fi­ca­tions leading to an improve­ment of the con­sid­ered objec­tive function(s). Because of the long com­pu­ta­tion times, this approach becomes infea­si­ble, espe­cially when dealing with complex flow sit­u­a­tions. In order to adapt the expenses to the avail­able resources, the design engineer would typ­i­cally select a suitable subset of para­me­ters, either based on his expe­ri­ence or his tech­ni­cal intu­ition. Besides the dis­ad­van­tage of reducing the avail­able design space for the opti­miza­tion, the selec­tion of this para­me­ter subset is often dif­fi­cult, since the effect on the objec­tive function is not known a priori. This is espe­cially true when the engineer does not have enough expe­ri­ence to make a mean­ing­ful selec­tion or when the model was produced by someone else, intro­duc­ing the addi­tional uncer­tainty of the specific impact of every para­me­ter on the geometry. 

Adjoint CFD

The decisive infor­ma­tion for the flow opti­miza­tion of a given geometry is the cor­re­la­tion between the objec­tive function J and the form para­me­ters αi. This cor­re­la­tion can be math­e­mat­i­cally expressed through the so-called sen­si­tiv­i­ties – the rate of change (i.e., the gradient) of the objec­tive when varying the form para­me­ters. The indi­vid­ual sen­si­tiv­i­ties can be approx­i­mated by the finite dif­fer­ence quotient, which for n para­me­ters in a clas­si­cal gradient based opti­miza­tion approach would require n+1 CFD com­pu­ta­tions to evaluate the cor­re­spond­ing modified geometry for every form para­me­ter. As men­tioned above, this direct approach is there­fore not really applic­a­ble for a high number of para­me­ters and expen­sive sim­u­la­tions. Of course, intel­li­gent DoE strate­gies can possibly reduce the number of function eval­u­a­tions, but they still sig­nif­i­cantly scale with the number of free parameters.

So-called adjoint methods, however, go the reverse way. Instead of eval­u­at­ing the change in objec­tive function due to a vari­a­tion of para­me­ter value, the required vari­a­tion of the para­me­ter values for a desired change in objec­tive function is computed. Within a single com­pu­ta­tion, the adjoint method will yield the full gradient of the objec­tive function, irre­spec­tively of the number of para­me­ters. The full analysis would require a con­ven­tional (primal) CFD com­pu­ta­tion, followed by one adjoint com­pu­ta­tion for each con­sid­ered objec­tive. In the context of shape opti­miza­tion, the adjoint analysis will provide the so-called shape sen­si­tiv­ity as a result. This is given as field infor­ma­tion on the surface of the model and describes the change of objec­tive function due to normal dis­place­ment of the surface cells (∂J/∂nk). A positive value of the shape sen­si­tiv­ity indi­cates a local dis­place­ment in positive normal direc­tion and vice-versa. The sen­si­tiv­ity is given by the adjoint analysis with respect to the CFD dis­cretiza­tion of the model surface. For industry-relevant cases, the CFD geometry is typ­i­cally described by ten thou­sands of nodes, and there­fore degrees of freedom, so that the obtained sen­si­tiv­ity is described with a very high degree of detail, in a quasi-con­tin­u­ous way. In a CAD-free approach, this infor­ma­tion can be directly used for a dis­place­ment of the grid nodes and, there­fore, a defor­ma­tion of the initial geometry. When using this approach, however, a dis­ad­van­tage is given by the fact that the shape mod­i­fi­ca­tions are dif­fi­cult to feed back into the design process and geo­met­ri­cal con­straints (e.g. due to pro­duc­tion restric­tions) can be violated. This leads to the moti­va­tion of devel­op­ing a method that maps these initial sen­si­tiv­i­ties for the degrees of freedom of the CFD geometry to the cor­re­spond­ing sen­si­tiv­i­ties ∂J/∂αi (rate of change of the objec­tive function due to change of para­me­ter values) of the CAD model parameters. 

Mapping Adjoint Shape Sen­si­tiv­i­ties to the CAD Model Parameters

In order to map the adjoint shape sen­si­tiv­i­ties to the CAD model para­me­ters, a new object – the Sen­si­tiv­ity Com­pu­ta­tion – that takes care of all the nec­es­sary steps, was imple­mented into CAESES®. Inputs to this object are the surfaces and the adjoint CFD results. Here is a screenshot:

Ini­tially, the adjoint sen­si­tiv­i­ties are inter­po­lated from the CFD dis­creti­sa­tion to the surface tes­sel­la­tion of the CAD model. Then, the local normal dis­place­ment of the model surface due to para­me­ter vari­a­tion – ∂nk/∂αi, the so-called design velocity – has to be deter­mined. By tracking the depen­den­cies for all model surfaces selected for this oper­a­tion, all influ­enc­ing para­me­ters are auto­mat­i­cally deter­mined. These are per­turbed by an indi­vid­ual amount related to a spec­i­fied absolute dis­place­ment of the model surface in positive and negative direc­tion and the normal dis­place­ments of the surface tes­sel­la­tion nodes are eval­u­ated. For each element, the gradient ∂nk/∂αi can then be computed from the dis­place­ment due to the per­tur­ba­tion. Like the adjoint shape sen­si­tiv­ity ∂J/∂nk, this data can be visu­al­ized as a plot on the model surface.

By com­par­ing the two plots, one can already get a more or less good visual esti­ma­tion about which para­me­ters influ­ence areas with high shape sen­si­tiv­ity and, there­fore, have a more pro­nounced impact of the objec­tive function. A more precise state­ment can be obtained by creating the product of ∂J/∂nk and ∂nk/∂αi for every tes­sel­la­tion element, weight­ing it with the local area, creating the sum over the surface and thereby com­put­ing the sen­si­tiv­ity ∂J/∂αi for every parameter. 

These scalar values for all involved para­me­ters are col­lected and dis­played in a table. They show the user which of the para­me­ters have the biggest influ­ence on the objec­tive function and in which direc­tion they should be changed in order to have a positive influ­ence on the objec­tive. This gradient infor­ma­tion can be used to select a subset of the most influ­en­tial para­me­ters, as well as for a sub­se­quent auto­mated opti­miza­tion process. In the former case, the para­me­ter values can be manually changed or involved in a con­ven­tional opti­miza­tion. In the latter case, the gradient infor­ma­tion can directly be used by a suitable opti­miza­tion algo­rithm to deter­mine the search direc­tion towards the (nearest local) optimum. This sig­nif­i­cantly speeds up the con­ver­gence, because the gradient does not have to be deter­mined by the opti­miza­tion algo­rithm in a numer­i­cal way. 

More Infor­ma­tion

This text is part of the paper ​“Direct Coupling of Para­met­ric CAD and Adjoint CFD for the Effi­cient Opti­miza­tion of Flow Geome­tries”. It addi­tion­ally includes detailed case studies includ­ing pictures and diagrams for the opti­miza­tion of an auto­mo­tive rear wing, a ship hull and a duct. Gradient-based opti­miza­tion strate­gies are also con­sid­ered for finally having a fully auto­mated design process. Follow-Up Article Check out our follow-up article opti­miza­tion using adjoint flow analysis which describes this automa­tion pro­ce­dure by means of a duct example. If you are inter­ested in the full article, just get in touch with us. 

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