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Abstract
Topology optimization is a design tool which is used in numerous
fields. It can be used whenever the design is driven by weight and
strength considerations. The basic concept of topology optimization
is the interpretation of partial differential equation coefficients
as effective material properties and designing through changing
these coefficients. For example, consider a continuous structure.
Then the basic concept is to represent this structure by small
pieces of material that are coinciding with the elements of a finite
element model of the structure.
This thesis treats stress constrained structural topology
optimization problems. For such problems a stress constraint for an
element should only be present in the optimization problem when the
structural design variable corresponding to this element has a value
greater than zero. We model the stress constrained topology
optimization problem using both discrete and continuous design
variables. Using discrete design variables is the natural modeling
frame. However, we cannot solve realsize problems with the
technological limits of today. Using continuous design variables
makes it possible to also study topology optimization problems of
large scale.
We find the global optimal solution to the stress constrained
topology optimization problem using discrete design variables. The
problem is solved using a parallel cutandbranch method. The cuts
include information about the mathematical structure of our problems
and also their physics. The method shows particularly good speedup
because of the added cuts.
The study of stress constrained topology optimization problem using
continuous design variables constitute the main part of this thesis.
Primarily we study the problem reformulated into standard form via
the Mathematical Program with Equilibrium Constraints (MPEC)
formulations. We look at the two variations: Mathematical Program
with Complementarity Constraints and Mathematical Program with
Vanishing Constraints. These problem formulations are compared to a
restricted problem formulation. The restricted problem include
stress constraints for all elements independently of the values of
the design variables. The investigations include validating
constraint qualifications, attacking the problem formulations
directly, and bounding the objective function value.
We consider different constraint qualifications and whether they
hold for the MPEC formulations of some truss topology optimization
problems. We provide examples in which none of the considered
constraint qualifications hold at the optimal solutions. This occurs
when the upper limits of the design variables become active and
there are nodal displacements that are nonunique. Note that this
situation is generally the case at an optimal solution. However, the
numerical experiments show that the MPEC formulations are not less
robust than the restricted problem formulation. This indicates that
the inherent lack of constraint qualifications is not the main
numerical obstacle.
We further observe that a general nonlinear interiorpoint algorithm
applied to the MPEC formulations outperforms a general nonlinear
activeset sequential quadratic programming method. Inspired by
this, we implement an interiorpoint algorithm such that we have
full control of all aspects of the code.
Solving the stress constrained structural topology optimization
problem is computationally challenging. We therefore present a
technique that decides whether it may payoff to actually treat the
stress constrained problem. The technique finds lower and upper
bounds on the objective function value of the stress constrained
topology optimization problem. It further produces a feasible
design. If the upper and lower bounds are far apart, then one should
invest in attacking the stress constrained structural topology
optimization problem. Otherwise one can use the obtained feasible
design.
Original language  English 

Place of Publication  Kgs. Lyngby, Denmark 

Publisher  Technical University of Denmark 
Number of pages  213 
Publication status  Published  Jun 2010 
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Topology Optimization Problems with DesignDependent Sets of Constraints
Schou, M. H., Stolpe, M., Evgrafov, A., Sigmund, O., Jørgensen, J. B., Svanberg, K. & Kocvara, M.
01/01/2006 → 30/06/2010
Project: PhD