Size, Shape and Topology Optimization

Structural optimization is determining the best possible material distribution within a physical volume domain to transmit the applied load safely. To achieve this, the constraints imposed by the manufacturer and eventual use must also be considered. These may include increasing stiffness (increasing strain energy or minimizing compliance), minimizing stress levels, minimizing displacement, altering the fundamental natural frequency, and increasing the buckling load.

1. Size Optimization:

In Size optimization, the designer knows what the structure will look like but does not know the size of the components which make up that structure. For example, if a cantilever beam was going to be used, its length and position may be known, but not its cross-sectional dimensions. Another example would be a truss structure where its overall dimensions may be known but not the cross-sectional areas of each truss element (bar). Yet another example would be the thickness distribution of a shell structure. So basically, any feature of a structure where its size is required but where all other aspects of the structure are known.

2. Shape Optimization:

In Shape optimization, the unknown is the form or contour of some part of the boundary of a structural domain. The shape or boundary could either be represented by an unknown equation or by a set of points whose locations are unknown. The shape optimization modifies the design geometry to meet the required objective. If a beam with circular holes were to undergo loading, shape optimization would be able to change the geometry of the holes to withstand the loading as best as possible.

3. Topology Optimization:

Topology optimization is the most general structural optimization, combining size and shape optimization. In discrete cases, such as for truss structures, it is achieved by allowing the design variables, such as the cross-sectional areas of the truss members, to have a value of zero or a minimum gauge size. For continuum-type structures in two dimensions (2D), topology changes can be achieved by allowing the thickness of a sheet to have values of zero at different locations, thereby determining the number and shape of the cavities (holes). The same effect can be achieved for continuum-type structures in three dimensions (3D) by having a density-like variable that can take any value down to zero.

In size and shape optimization, the size and shape of the components of a structure can be manipulated. They can have any value within their limits, but they must always be present. But if the designer/engineer does not know what the shape or size of the structure should be, then topology optimization needs to be used. Topology optimization gives the product’s initial design, while size/shape optimization is used to update the current design. The basic flowchart depicts the process of the Topology optimization is shown in the following figure.

Basic flowchart of Topology Optimization

In size optimization, the design variables are set to determine the size of a component in the structure, such as the cross-sectional area of a truss or the thickness of a plate. With shape optimization, the internal and or external boundary of a predetermined topological layout is modified. In size and shape optimization, assumptions on the final structure are made from an initial design. No such assumptions are made in topology optimization since it starts only from a predetermined design domain with applied load and boundary conditions.

Topology Optimization in Truss:

In a discrete case, such as a truss, topology optimization is achieved by taking the cross-sectional area of truss members as design variables and then allowing these variables to take the value 0, i.e., bars are removed from the truss.

References:

http://carat.st.bv.tum.de/caratuserswiki/index.php/Users:Structural_Optimization/General_Formulation

https://www.mid.t.u-tokyo.ac.jp/en/research/index.html

https://projekter.aau.dk/projekter/files/280752551/DMS4_JH.pdf

http://eprints.utar.edu.my/3404/1/CI-2019-1404301-1.pdf

https://slideplayer.com/slide/10947129/

https://www.intechopen.com/chapters/70489

https://www.tandfonline.com/doi/full/10.1080/10286608.2013.820280