Shape Optimisation with PDE and Geometric Constraints


Shape optimisation is concerned with design problems where the geometry of an object is to be determined that minimises or maximises some objective. The relationship between the geometry and the objective is typically governed by a PDE. Examples include: determining the optimal shape of a wing that minimises the drag coefficient, while preserving its lift; determining the shape of a bridge of a given mass that best supports its load; determining the bedrock shape beneath a glacier that explains its observed motion. While shape optimisation has gained immense popularity in academia, there are several key issues that we will address in this project in order to bring the methodology to industrial relevance.

We use a technique known as shape calculus in order to find deformation vector fields that morph a given initial shape into an optimal shape.


Mesh deformations

We create a mesh of the domain in order to solve the PDE that governs the relationship between the geometry and the objective. Optimising for the shape of the domain then translates to moving the nodes of the mesh; here one needs to be careful that this deformation does not lead to overlapping or strong distortion of the mesh. Stretched triangles as see in the left of the picture below can lead to numerical inaccuracy.

We have formulated new mesh deformation methods based on conformal mappings so that the mesh stays well-behaved and we can avoid costly remeshing [1].


Left: Deformed geometry using classical deformation methods. We can see strongly stretched triangles in the area of high curvature. This leads to numerical inaccuracy in computations on this mesh. Right: Using nearly conformal mappings the mesh quality remains good throughout the optimisation.

Furthermore, in [2] we have explored the use of high-order methods to discretise deformation vector fields. High-order methods offer higher accuracy and are desirable when shapes of higher regularity (e.g. without kinks) are required.

Geometric constraints

An important class of constraints are geometric constraints on the shape itself. Two classical and well-understood examples for this are constraints on the volume or the location of the barycentre of the shape. Another requirement is that often the shape needs to be contained in a feasible region. This can be due to the existence of other parts in the design or because of regulations. For example, in Formula 1 the design of the each component of the car is strongly regulated by the FIA. Though this kind of constraint is crucial when trying to solve real-world problems, it is usually only considered in the parametric case, e.g. by bounding the location of nodes of free form deformation boxes, and there is surprisingly little literature treating its inclusion in a continuous, non-parametric setting.

We have formulated an augmented Lagrangian method to incorporate such a constraint when the feasible region is given by a box. and the method can be extended to general convex domains.

Above (click to see animation): We consider an airfoil and solve the Navier-Stokes equations. The objective to be maximised is the aerodynamic efficiency: the ratio of lift and drag. We furthermore impose the additional geometric constraint that the shape needs to be contained in the black box - this is similar to the bounding boxes that the wings of Formula 1 cars are subject to.

Future Work

Second order optimisation methods

At the moment the most common optimisation methods used in shape optimisation are gradient-based methods like steepest descent or L-BFGS. Our goal is to apply a semi-smooth Newton method to the full system of optimality conditions. This has two advantages: Firstly, solving for the full system means that we can solve for the shape and the PDE at the same time, thus making the method more efficient as we don't have to solve the PDE completely on every intermediary domain. Secondly, Newton converges superlinearly close to the optimal shape and hence fewer iterations are necessary.

Navier-Stokes with Turbulence

To model the flow around a car or airplane accurately, much higher Reynolds numbers of order 100 000 to 1 000 000 are necessary. At this speed the flow will be turbulent and we cannot expect to be able to solve the steady Navier-Stokes equations numerically. Instead, the typical approach is to consider the Reynolds-averaged Navier-Stokes (RANS) equations: here one averages over the time-dependent velocity and pressure field and then closes the equations by prescribing a model for the Reynolds stress. We expect several difficulties when extending our approach to the RANS equations: first, solving the RANS equations itself is difficult and good implementations are usually commercial and add several opaque stabilisation methods in order to ensure robustness of the solver. Second, the turbulence models often contain non-differentiable as well as mesh-dependent quantities - this means that it is not clear what the underlying continuous problem is, which we need to know in order to calculate the adjoint and shape derivative.


[1] José A. Iglesias, Kevin Sturm, and Florian Wechsung. Shape optimisation with nearly conformal transformations. arXiv preprint arXiv:1710.06496 (2017).

[2] Alberto Paganini, Florian Wechsung, and Patrick E. Farrell. Higher-order moving mesh methods for PDE-constrained shape optimizationarXiv preprint arXiv:1706.03117 (2017).