A Substructuring Method to Apply Topology Optimization to System-Component Problems

A Substructuring Method to Apply Topology Optimization to System-Component Problems

Stolk, M.

Langelaar, M. (mentor)
Van Keulen, A. (mentor)

Mechanical, Maritime and Materials Engineering

Precision and Microsystems Engineering

Engineering Mechanics

2017-01-16

Since its introduction, Topology Optimization (TO) has been applied to a broad range of design cases. TO is a material distribution method which finds the optimal material lay-out for a given design space while upholding given constraints and boundary conditions. This has made the method popular in structural mechanics where it is used to find concept designs for a wide range of structural problems. In this thesis TO will be applied to a structural case inspired by a problem found in ship design: the connections between the hull and the decks as well as the connections between decks. These connections have a significant influence on the behaviour of the ship as they determine the stiffness of the ship. With TO it is possible to find new designs for these connections. Applying TO on these connections is not straightforward as they are a part of a large system, and the interaction with the system must be taken into account for realistic results. This complicates the transformation of the ship response into boundary conditions for the TO model. In this thesis this problem is translated into a general problem where the design region and the global model will be modelled as separate substructures.The Lagrange multiplier method is introduced in this thesis to couple the substructures with non-conforming meshes. Tests in this thesis show the advantages of the method, it is able to couple non-conforming substructures and the error between a coupled highly nonconforming structure (up to 6 times smaller elements between the substructures) and a single structure remains below 1%. However, the Lagrange multiplier method is not free of disadvantages: it introduces additional unknowns, it makes the system indefinite and gaps and penetrations can be present on the interface due to the weak compatibility, changing the stiffness of the interface. The optimization process is only applied to one substructure. The test results show that the coupling has an effect on the stiffness of the interface, resulting in a different design when the non-conformity between substructures increases. A buffer-zone is introduced in order to exclude the interface elements from the optimization process and this will prevent the influence of coupling on the TO. The designs almost match perfectly with this buffer zone and the differences between the displacements and the compliance of the single structure case is _ 0.3%. Using static condensation it is possible to reduce the size of the system resulting in a reduction of the computational cost. The Degrees of Freedom (DoFs) of the global structures, and the Lagrange multipliers are removed, leaving only the DoFs of the substructure that is optimized. This will lead to a system that has the size of the model in the design region and which is unaffected by the size of the systems connected to it. The designs obtained using the condensed systems do not differ from the design obtained using the complete system, while the computational cost per iteration is greatly reduced and made constant. New steps are introduced that will add to the total computational time, such as constructing the condensed stiffness matrices. But these steps only need to be done once and this increase of computational cost is lower than the total reduction obtained. The method has been tested on a problem consisting of 4 substructures with non-conforming element sizes. With the correct buffer-zone the designs found by TO are almost equal and the difference in compliance is low (_ 2.3%). The power of static condensation is also shown in this test case, the calculation time of the optimization process is reduced by 40 to 80 %. Concluding that a design region can be coupled as a substructure to an existing model and optimized without substantially increasing the computational cost.

Topology optimization
Substructures
Lagrange multiplier method
non-conforming meshes
System-component problem

http://resolver.tudelft.nl/uuid:ac8d7262-a19a-4a85-bb11-e1cdfb4ab411

2017-01-16

Student theses

master thesis

(c) Delft University of Technology