Topology optimization (TO) is a mathematical method that optimizes material layout within a given design space, for a given set of loads, boundary conditions and constraints with the goal of maximising the performance of the system. TO is different from shape optimization in the sense that the design can attain any shape within the design space, instead of dealing with predefined configurations.
Contents
- Problem statement
- Implementation methodologies
- Discrete
- Solving the problem with continuous variables
- Topology Optimization for stiff structures
- References
The conventional TO formulation uses a finite element method[FEM] to evaluate the design performance. The design is optimised using nonlinear programming techniques such as the optimality criteria algorithm, the method of moving asymptotes and genetic algorithms.
Currently, engineers mostly use TO at the concept level of a design process. Due to the free forms that naturally occur, the result is often difficult to manufacture. For that reason the result emerging from TO is often fine-tuned for manufacturability. Adding constraints to the formulation in order to increase the manufacturability is an active field of research. In some cases results from topology optimization can be directly manufactured using additive manufacturing.
Problem statement
A topology optimisation problem can be written in the general form of an optimization problem as
subject to
The problem statement includes the following:
Evaluating
Implementation methodologies
There are various implementation methodologies that have been used to solve TO problems.
Discrete
Solving TO problems in a discrete sense is done by discretizing the design domain into finite elements. The material densities inside these elements are then treated as the the problem variables. In this case material density of one indicates the presence of material, while zero indicates an absence of material. Due to the attainable topological complexity of the design being dependent of the amount of elements, a large amount is preferred. Large amount of finite elements increase the attainable topological complexity, but come at a cost. Firstly, solving the FEM system becomes more expensive. Secondly, algorithms that can handle a large amount (several thousands of elements is not uncommon) of discrete variables with multiple constraints are unavailable. Moreover, they are unpractically sensitive to parameter variations. In literature problems with up to 30000 variables have been reported
Solving the problem with continuous variables
The earlier stated complexities with solving TO problems using binary variables has caused the community to search for other options. One is the modelling of the densities with continuous variables. The material densities can now also attain values between zero and one. Gradient based algorithms that handle large amounts of continuous variables and multiple constraints are available. But the material properties have to be modelled in a continuous setting. This is done through interpolation. One of the most implemented interpolation methodologies is the SIMP method (Simplified Isotropic Material with Penalisation ). This interpolation is essentially a power law
Topology Optimization for stiff structures
A stiff structure is one that has the least possible displacement when given certain set of boundary conditions. A global measure of the displacements is the strain energy (also called compliance) of the structure under the prescribed boundary conditions. The lower the strain energy the higher the stiffness of the structure. So, the problem statement involves the objective functional of the strain energy which has to be minimized.
On a broad level, one can visualize that more the material, lesser will be the deflection as there is more material to resist the loads. So, the optimization requires an opposing constraint, the volume constraint . This is in reality a cost factor, as we would not want to spend a lot of money on the material. To obtain the total material utilized, an integration of the selection field over the volume can be done.
Finally the elasticity governing differential equations are plugged in so as to get the final problem statement.
subject to:
But, a straightforward implementation in the Finite Element Framework of such a problem is still infeasible owing to issues such as:
- Mesh dependency—Mesh Dependency means that the design obtained on one mesh is not the one that will be obtained on another mesh. The features of the design become more intricate as the mesh gets refined.
- Numerical instabilities—The selection of region in the form of a chess board.
Some techniques such as Filtering based on Image Processing are currently being used to alleviate some of these issues.
Nowadays, the programs can run 3D topology optimization problems.