The Topography Optimization method uses morphing optimization technology to process shell elements by varying the node locations.
Shape Description
For the Topography Optimization method, the shape, 
, is represented by a mesh, 
, and composed of 
shell elements 
. This method is essentially equivalent to the Shape Optimization
method for surface bodies.
| Conformal Shell Mesh |
|
|
Shape Evaluation
For this method, no special treatment is required to perform shape evaluation.
Note: An acceptable mesh quality is expected to be retained during the optimization process. However, excessive deformation of the shape may lead to large approximation errors in the finite element analysis.
Shape Derivative
The application computes the shape derivative using the continuous formalism
defined by Hadamard (see [AJT2004]). That is, given a shape perturbation
, the asymptotic expansion reads:
Where:
 is current shape. |
 is the shape perturbation. |
 is the new shape. |
The shape derivative usually admits the following form:
For the form, the integrand, 
, depends on the criterion, 
, through both the solution state of the mechanical problem and
some corresponding adjoint-state.
Shape Update
The application creates the new shape by updating the position of the mesh
vertices using the shape perturbation 
, as:
Where:
 is the position vector of vertex  at the new shape. |
 is the position vector of vertex  at the current shape. |
 is the shape perturbation. |
Specific programming is in place to preserve mesh-quality.
Summary
Degrees of freedom for this method are based on node location.
- Strengths
This method accurately computes any state variable given proper mesh quality.
Compared to the topology optimization methods, no numerical trickery occurs to evaluate the shape.
This method is specific to local modifications of the shape but also manages large shape changes without remeshing.
- Place in Design Stage
Used in the final stage of the design process for local shape adjustments.
- Limitations
Finite element approximation errors could occur due to poor-quality mesh regions or if large modifications are made.
Because shape optimization does not manage topology changes, additional programming is in place to preserve mesh-quality. This factor can sometimes lead to additional computational requirements.
- Tips
Use a uniform mesh to equally capture geometric details.
References
[9] G. Allaire, M. Schoenauer, Conception optimale de structures, Springer, 2007.
[10] M. Shimoda, Y. Liu, A non-parametric free-form optimization method for shell structures, Structural and Multidisciplinary Optimization, 2014.