Structural optimization methods are applied in virtual product development since several years. In general, these methods are separated in Sizing, Shape, and Topology Optimization methods, respectively.


The application of the different methods as well as their potential are strongly related to the available design freedom. Thus, structural optimization should start in the virtual design process as early as possible. Due to our fundamental knowledge and extensive experience the FEMopt Studios GmbH is the best partner to support your development process with the right optimization tool.

Sizing optimization is applied when structural shape and structural topology are already fixed. A large application field is the design of composite structures, e.g. carbon fibre or glass fibre reinforced laminates. In the design process of composites the sizing optimization is used to determine the optimal stacking sequence which consists of the optimal fibre orientation and the optimal layer thickness. The performance of composite structures can be significantly improved by optimal cross sections. This may result in improved structural stiffness or lower structural weight.

Fiber optimization

Fibre optimization of quadratic plate with corner support and central loading


Topology optimization is extensively used in industrial optimization tasks. Among other things this is caused by the simple formulation of topology optimization problems. It requires the common definitions of objective and constraints and the specification of the allowable design space but no extra parametrization. A parametrization is not necessary because topology optimization does not distinguish between geometry and analysis model. All definitions are based on the analysis model. The final result of a topology optimization reflect the optimal material distribution in the design space. These general approach offers the possibility to end up with completely new structural designs. In any case, these results have to transferred by a postprocessing step into a CAD model again.

2-D Topologieoptimierung

Optimal material distribution of a cantilever under tip load


Shape Optimization has the goal to change the shape of a structure such that the optimization problem is solved in the best possible way. Famous optimization goals are minimizing weight, maximizing stiffness, modification of eigenfrequencies, or minimizing stresses. The actual available optimization tools often require a separate parametrization step where the optimization parameters are specified on an extra geometry model, e.g. a CAD model (ANSYS) or morphing boxes (OptiStruct). This time consuming and complex parametrization step requires the knowledge of the optimal geometry to ensure that the parametrization is able to reflect this geometry. Usually the optimal geometry is a priori not known. Thus, several reparametrization steps are necessary in order to improve the parametrization until the optimal geometry is converged. Due to decreasing development times and the necessary effort these reparametrization steps are usually not executed. Hence, optimization potential and structural performance is lost. The high parametrization effort is the main reason that shape optimization is rarely applied compared to topology optimization. But it is well known that shape optimization results give valuable support in complex design processes. The following sections introduce an innovative approach that simplifies the parametrization tremendously and make the extra geometry model unnecessary.

Formoptimierung Emmrich Haube

Shape optimization of a clamped cowling subjected to tip load


New Methods of numerical shape optimization

FE-based parametrization allows for a direct formulation of the common optimization problem:

  • Minimize the objective such that
  • all constraints are feasible and
  • no variable bounds are violated.


In order to achieve this goal the shape of the structure can be modified. Usually the shape modifications are restricted to a specified domain the so called design domain. Applying FE-based parametrization allows for a definition of this design domain by a part ID, a property ID, a node set, or an element set. By this approach the FE-nodes in the design domain can be identified. These nodes are called design nodes because the coordinates of the nodes serve as optimization variables. This simple and robust definition ensures a large design space with a high flexibility. The large number of optimization variables require gradient based optimization strategies and adjoint sensitivity analysis to ensure efficient algorithms.


  • K.-U. Bletzinger, M. Firl, F. Daoud, Approximation of derivatives in semi-analytical structural optimization, Computers and Structures, 86, 2008
  • H. Masching, M. Fischer, M. Firl, K.-U. Bletzinger, Finite Element Based Structural Optimization in Object-Oriented Parallel Programming, PARENG 2011, 12.-15. April, 2011, Corsica, Italy
  • M. Firl, M. Fischer,  F. Daoud,  K.-U. Bletzinger, Structural optimisation and nonlinear simulation based on object-oriented and parallel programming, 2nd Aircraft Structural Design Conference, October 26-28, 2010, London, UK.


Regularization techniques

It is well known that such a large number of optimization variables may yield to undesired geometries, e.g.

  • mesh dependent solutions
  • solutions with large mesh deformations,
  • non-aesthetic solutions
  • non-manufacturable designs.


This class of undesired solutions can be eliminated by suitable regularization techniques. These robust and simple techniques control

  • surface curvature
  • and mesh quality.


Gradient filter

The gradient filter is applied to control the surface curvature. Such filter strategies are also applied in topology optimization in order to specify the minimal allowable size of structural members. The filter methods are based on well known mathematical theory of mollifier functions that are extensively applied in signal theory. The filters eliminate responses with high frequencies whereby responses with lower frequencies are preserved. The properties of the filter method are basically controlled by the radius of the filter functions. This ensures robust optimization results and allows for easy analysis of parameter studies. A very important outcome of the filter method is the mesh independence. Thus, the optimal geometry is independent from the applied discretization.


Mesh regularization

Controlling mesh quality during shape optimization is very important because accuracy of FE-results strongly depends on the mesh quality. Meshes with large element defomation generate numerical stiffness which disturbes the design improvement by the shape optimization. During renalysis or physical tests these numerical stiffness is usually not present. Thus the optimization results cannot be verified. In the context of mesh quality the available optimization methods show serious differences. The optimization results computed by our software Carat++ show a very high mesh quality. Thus, the models ensure very accurate analysis results. A small selection of application examples is presented here.


  • M. Firl, R. Wüchner, K.-U. Bletzinger, Regularization of Shape Optimization Problems using FE-based Parametrization, submitted to Structural and Multidisciplinary Optimization