English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In industry, shape optimization problems are of utter importance whendesigning structures such as aircraft, automobiles and turbines. For many ofthese applications, the structure changes over time, with a prescribed ornon-prescribed movement. Therefore, it is important to capture these featuresin simulations when optimizing the design of the structure. Using gradientbased algorithms, deriving the shape derivative manually can become verycomplex and error prone, especially in the case of time-dependent non-linearpartial differential equations. To ease this burden, we present a high-levelalgorithmic differentiation tool that automatically computes first and secondorder shape derivatives for partial differential equations posed in the finiteelement frameworks FEniCS and Firedrake. The first order shape derivatives arecomputed using the adjoint method, while the second order shape derivatives arecomputed using a combination of the tangent linear method and the adjointmethod. The adjoint and tangent linear equations are symbolically derived forany sequence of variational forms. As a consequence our methodology works for awide range of PDE problems and is discretely consistent. We illustrate thegenerality of our framework by presenting several examples, spanning the rangeof linear, non-linear and time-dependent PDEs for both stationary and transientdomains.

Automatic shape derivatives for transient PDEs in FEniCS and Firedrake