OPTEC - K.U.Leuven Center of Excellence: Optimization in Engineering

Computer-aided engineering (CAE) tools are widely used to analyse, simulate and design structures in civil and mechanical engineering. The integration of optimization in CAE allows for a design optimization, aimed at finding the best compromise between cost and performance. The optimization of the lay-out or the geometry of a structure or mechanism can be formulated as a topology or shape optimization problem [1]. In the former the optimal distribution of material within the design domain of the structure or mechanism is searched, while in the latter the type of structure (e.g. truss or shell structures) is chosen first and its shape is optimized.

The parametrization of the shape or topology is crucial, as it determines the design space and therefore the variety of structural geometries considered in the optimization. When the aim is to minimize material use, the total volume of material or the strain energy is often considered in the objective function. The objective function is accompanied by a number of constraints in terms of stresses, displacements or natural frequencies ensuring an adequate performance of the structure or mechanism. When the aim is to maximize performance under given physical or cost related constraints, the performance can be part of the objective function. Such an approach has been followed recently to minimize forces transmitted to the supports of a moving mechanism while satisfying constraints imposing limits on input torque and bearing life [2]. The evaluation of the objective function and constraints typically involves simulations of rigid body kinematics, static or dynamic structural analyses, with possibly geometrical and material non-linearities, or vibro-acoustic simulations.
The optimization problem considered is not trivial as most problems in topology and shape optimization are not convex [1] and even do not have a unique solution. This is particularly true for large-scale problems with a large number of design variables, where regularization methods [3] may be required. In some cases, however, the problem can be (partially) reformulated as a convex optimization problem [2] or be approximated by a sequence of convex problems, leading to efficient and numerically tractable solutions. Another major difficulty is the strong dependence of the optimal solution on the data (loads, boundary conditions, material characteristics). It is therefore important to take data uncertainties into account by either robust design optimization or reliability based design optimization.

The aim is to develop methodologies for shape and topology optimization applicable to real-world applications, exploiting the (approximate) convex character whenever possible, and taking into account data uncertainties.

The research forcus at OPTEC is:

• Convex reformulations for shape and topology optimization: Most problems in shape and topology optimization are not convex. A convex reformulation of the problem is only available for a limited class of problems. It will therefore be investigated how (approximate) convex reformulations can be obtained for a wider range of (non-linear) problems.
• Efficient algorithms for robust and reliability-based design optimization: The optimal geometry can be very sensitive to data uncertainties or variations of the parameters in the objective function or constraints. When these uncertainties are accounted for in the optimization, a robust optimal design can be obtained that is insensitive to data uncertainties. Alternatively, data uncertainties may lead to the formulation of reliability-based design optimization problems
• Exploring the optimality achieved within design spaces: The success of the optimization is largely dependent on the choice of the parametrization of the shape or topology, which defines the design space. Whereas for small-scale problems the choice can be based on engineering judgement, this is no longer the case for large-scale problems. It is therefore desirable to develop methods that allow for a quantitative assessment of the optimality achieved within different design spaces. For problems with a large number of design variables, the solution may become nonunique and the set of (near-)optimal solutions needs to be identified. This enables the choice between equivalent design alternatives.
• Reduced order models for shape and topology optimization: In structural dynamics and vibro-acoustics, a model reduction is often applied to accelerate the computations. When a reduced order model is used in the design optimization, it is very important to ensure that the model remains valid for the range of the design variables.