Design Optimization

Topology optimization consists of determining the best arrangement of material within a given domain. In our current study the method is used to optimize the performance of an acoustic horn with respect to efficiency and far-field properties. Here we start with a regular 0.5m long funnel-shaped horn and allow material to be placed inside the horn in order to reach our objective.

The figures above show the shape of the initial horn and its reflection spectrum, that is, the reflection coefficient as a function of frequency. The reflection coefficient R is defined as the quotient between the amplitude of the incoming and the outgoing wave in the waveguide. The horn below is optimized with aim to minimize the value of the reflection coefficient in the range 400-500Hz.

In the far field the acoustic field is essentially the product of a function depending on the distance from the horn and a function depending on the direction. The directivity function is denoted the far-field pattern.

The figure to the left shows a horn optimized with aim to minimize the reflection coefficient at 1200Hz. The figure in the middle shows a horn optimized with aim to minimize the reflection coefficient as well as the value of the far-field pattern straight in front of the horn at 1200Hz. The figure at the right shows the far-field patterns for the horn optimized only with respect to efficiency (dotted blue line), the horn optimized with respect to both efficiency and far-field behavior (solid black line) as well as the initial funnel-shaped horn (dashed red line).

Further work include application of the method for fluid problems.

Status: Completed Ph.D. project

Contact: Eddie Wadbro, Martin Berggren

Horns are interfacial devices that connect a waveguide with the surroundings. They are used as loudspeakers (particularly outdoors), as microwave transducers, and horn-like structures can also be found in optical devices. The two main qualities of a horn are its transmission efficiency and the far-field directivity properties. Within this project, we presently study the trade-off of these properties.

Status: Completed Ph.D. project

Contact: Rajitha Udawalpola, Martin Berggren

The project concerns the designs of aircraft wings and airfoil using gradient-based optimization algorithms within the context of the industrial-style Computational Fluid Dynamics (CFD) code Edge.

The optimization algorithms in order require gradients of the aerodynamics coefficients (drag, lift, moments) with respect to the chosen parameterization of the design. In order to obtain the best computational efficiency, the so called adjoint method is applied here on the complete mapping, from the parameters of design to the values of the cost function. Here, the mapping includes the Euler equations for compressible flow discretized on unstructured meshes by a median-dual finite-volume scheme, the primal-to-dual mesh transformation, the mesh deformation, and the parameterization.

The figures above show the results of an optimization at Mach 0.8395 and angle of attack 3.06 degrees starting with geometry of the ONERA M6 wing. Upper left: pressure contours on the ONERA M6 wing. Bottom left: pressure contours on the final design. Bottom right: superposed geometry of the ONERA M6 wing (grey) and of the final design (yellow). The pressure drag, calculated based on the Euler equations, is reduced by about 30 drag counts and the lift and pitch coefficients are maintained at their initial values.

The method of adjoint derived here has also been applied when coupling the Euler equations with the boundary-layer and parabolized stability equations, with the aim to delay the laminar-to-turbulent transition of the flow. A method often called Natural Laminar Flow design. The delay of transition is an efficient way to reduce the drag due to viscosity at high Reynolds numbers.

The figures below show some result of Natural Laminar Flow design starting with the RAE 2822 airfoil.
Left: the geometry of the RAE (solid) and of the optimized airfoil (dashed). Right: the envelope of envelope of N-factor curves (EoE) of the RAE based on Euler pressure distribution (solid) and on RANS pressure distribution for the same lift (dash-dot), of the optimized geometry based on Euler pressure distribution (dashed) and on RANS pressure distribution for the same lift (dotted). The EoE curves indicate that if transition from laminar to turbulent occurs for example at N=7, the location of the transition would at s/c=0.15 on the initial design (RAE), and at s/c>0.5 on the optimized design.

Status: Completed Ph.D. project

Contact: Olivier.Amoignon@it.uu.se, Martin Berggren

1. Optimization of a variable mouth acoustic horn. Rajitha Udawalpola, Eddie Wadbro, and Martin Berggren. In International Journal for Numerical Methods in Engineering, volume 85, pp 591-606, 2011. (DOI).
2. A hybrid scheme for bore design optimization of a brass instrument. Daniel Noreland, Rajitha Udawalpola, and Martin Berggren. In Journal of the Acoustical Society of America, volume 128, pp 1391-1400, 2010. (DOI).
3. High contrast microwave tomography using topology optimization techniques. Eddie Wadbro and Martin Berggren. In Journal of Computational and Applied Mathematics, volume 234, pp 1773-1780, 2010. (DOI).
4. Shape and topology optimization of an acoustic horn-lens combination. Eddie Wadbro, Rajitha Udawalpola, and Martin Berggren. In Journal of Computational and Applied Mathematics, volume 234, pp 1781-1787, 2010. (DOI).
5. Microwave tomographic imaging as a sequence of topology optimization problems. Eddie Wadbro. In Optimization and Engineering, volume 11, pp 597-610, 2010. (DOI).
6. Megapixel topology optimization on a graphics processing unit. Eddie Wadbro and Martin Berggren. In SIAM Review, volume 51, pp 707-721, 2009. (DOI).
7. Optimization of an acoustic horn with respect to efficiency and directivity. Rajitha Udawalpola and Martin Berggren. In International Journal for Numerical Methods in Engineering, volume 73, pp 1571-1606, 2008. (DOI).
8. Microwave tomography using topology optimization techniques. Eddie Wadbro and Martin Berggren. In SIAM Journal on Scientific Computing, volume 30, pp 1613-1633, 2008. (DOI).
9. Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization. Stefan Jakobsson and Olivier Amoignon. In Computers & Fluids, volume 36, pp 1119-1136, 2007. (DOI).
10. Topology optimization of an acoustic horn. Eddie Wadbro and Martin Berggren. In Computer Methods in Applied Mechanics and Engineering, volume 196, pp 420-436, 2006. (DOI).
11. Topology optimization of wave transducers. Eddie Wadbro and Martin Berggren. In Topological Design Optimization of Structures, Machines and Materials: Status and Perspectives, volume 137 of Solid Mechanics and its Applications, pp 301-310, Springer, Dordrecht, The Netherlands, 2006. (DOI).
12. Topology optimization of mass distribution problems in Stokes flow. Allan Gersborg-Hansen, Martin Berggren, and Bernd Dammann. In Topological Design Optimization of Structures, Machines and Materials: Status and Perspectives, volume 137 of Solid Mechanics and its Applications, pp 365-374, Springer, Dordrecht, The Netherlands, 2006. (DOI).
13. Shape optimization for delay of laminar-turbulent transition. Olivier Amoignon, Jan Pralits, Ardeshir Hanifi, Martin Berggren, and Dan Henningson. In AIAA Journal, volume 44, pp 1009-1024, 2006. (DOI).
14. Shape optimization of an acoustic horn. Erik Bängtsson, Daniel Noreland, and Martin Berggren. In Computer Methods in Applied Mechanics and Engineering, volume 192, pp 1533-1571, 2003. (DOI).
15. Multifrequency shape optimization of an acoustic horn. Martin Berggren, Erik Bängtsson, and Daniel Noreland. In Computational Fluid and Solid Mechanics: 2003, pp 2204-2207, Elsevier Science, 2003. (DOI).
16. Discrete adjoint-based shape optimization for an edge-based finite-volume solver. Olivier Amoignon and Martin Berggren. In Computational Fluid and Solid Mechanics: 2003, pp 2190-2193, Elsevier Science, 2003. (DOI).