Topology Optimisation
- Sumit Basu
- May 12
- 9 min read
Traditional heuristic design methodologies can be augmented by computational tools that can decide how material must be distributed over a domain in space so that a well-formulated and mathematically well-posed design goal can be met. But conjuring up a workable design demands robust software tools and computational power. In a discussion moderated by Prof Anupam Saxena (IIT Kanpur), Profs G. K. Ananthasuresh and C.S Jog (IISc, Bangalore), V. S. Nagendra Reddy B (NIT Kuruskshetra), Prabhat Kumar (IIT Hyderabad) and Rajit Ranjan (IIT Jodhpur) listed the amazing breakthroughs made and challenges ahead.
If you are a designer in a modern car manufacturing company tasked with designing a new model, your starting point will, most certainly, be an existing successful design that looks close to your goal and not a blank drawing sheet. After you have your starting point, the process will be incremental. You will bring your experience to bear on the process, evaluate your manufacturing and packaging constraints, look at regulatory requirements that must be satisfied in your intended market, estimate loads that your design will have to bear and carefully decide all the other things that text-book design methods demand. But it is unlikely that you will deviate much from what you know as established practise. The longitudinal rails, roof pillars
and floor cross-beams will remain. In fact, you have a very strong reason to include them --- the basic
design of a car is so evolved now that questioning established design norms is legitimately considered foolhardy.
But suppose you start, not exactly with a blank drawing sheet, but with something a bit more abstract --- just the definition of an empty space into which your design must fit. Then you set your performance goals, go for a coffee, and let mechanics figure out how best to fill that space with material so that your goals are met. Of course the design will depend on what goals and constraints are specified. The resulting design may or may not be manufacturable, but such designs sometimes open doors to ideas that were hidden by our faithful adherence to existing design wisdom. Also, with modern flexible manufacturing techniques, even `crazy' designs can be built.
Optimising structures intuitively
The first glimpses of topology optimisation are seen in the early graphical methods for determining member loads in trusses. Maxwell's construction for trusses uses the fact that a truss has axial members only. So, given the distribution of externally applied loads and support reactions, we can create ingenuous force diagrams to find internal forces in the members. In 1904, Michell used this idea to claim that given a set of prescribed loads, a class of minimum weight trusses can be designed where all members are stressed to the extent that they can bear.
Michell's idea was revolutionary for its time. In modern parlance, his method involves replacing the loci of the maximum and minimum principal stress directions in any continuum elasticity problem with small truss members. As these loci are locally perpendicular to each other, the result is a set of curves, connecting the load points to the supports, perpendicular to each other and approximated with small straight lines. Each truss member must be appropriately sized so that it can carry the maximum allowable load. If we can do this consistently, we will arrive at the structure that carries the prescribed loads only through tension and compression. For instance, a simple cantilever subjected to a tip load and its Michell truss are shown in the Fig 1.
Modern Tools of Topology Optimisation
Mathematically, modern topology optimisation sets out to perform a seemingly impossible task. Given a volume inside which the designed product must fit, the loads that it will be subjected to and the properties of the material that will be deployed, decide which points in the volume will have material and which will not, so that the compliance of the designed structure is the minimum possible.
The homogenisation method
In the 1980s, homogenisation theory, which was developed to determine upper and lower bounds on the overall properties of heterogeneous materials like composites and porous solids, emerged as a potent tool for accomplishing this task. The central idea of this approach was to determine the spatial distribution of elastic stiffness in a structure to achieve the objective function.
Any point in a large engineering structure has a hidden microstructure that determines the elastic properties at that point. There is a natural separation in length scales between the macro- and the micro domains, i.e. the details of the microstructure manifest on a scale much smaller than the one in which the engineering structure exists. The scale separation mathematically allows us to imagine that the elastic stiffness at each point in the big structure originates from an invisible space tiled with small but identical `unit cells', each just large enough to capture the statistical heterogenity of the microstructure.
The displacement varies smoothly and slowly on the macrostructure. However, on the scale of the microstructure, the variation is made rougher by the heterogeneities present. So, in the end, the displacement has two parts. It is a smooth function at the macroscale with superposed small but significant noise at the microscale. Consequently, strains, which are the spatial derivatives of the displacement, also have a smooth part called the macrostrain and a noisy part that manifests at the microscale. The
engineering structure feels the macrostrain. Our larger goal then becomes optimising the microstructure in such a way that the objective of minimising the compliance of the engineering structure is met.
Note an important distinction that will become important a bit later. In this method, the engineering structure topology is not an ensemble of points with and without materials. The stiffness in the macroscale varies continuously depending on the microstructure underneath. The design is allowed to retain ` grey' points., not merely black and white ones.
This ansatz allows us to come up with a method to generate the effective stiffness at every element of the structure, given the macrostrain that the point is experiencing and the properties of the constituents making up the microstructure. For example, if the invisible microstructure is a solid with spherical pores, the porosity becomes a design variable that governs the effective stiffness.
This is an expensive process because for each element in the structure you will need to solve a Finite Element problem over a microscopic unit cell, to determine the noisy strain. In a general 3 dimensional case, this will require you to apply 6 unit macrostrains one after the other. The effective stiffness at that point will
be the result of this computation. A further FE computation on the engineering structure will be needed to determine if the distribution of point-wise varying stiffness is good enough to achieve our objective.
If we do not achieve minimum compliance, we need to tweak the microstructure to change the distribution of effective stiffness over the engineering structure. This will need another cycle of two scale FE computations.
This also leads us to another important question: how we tweak the microstructure to achieve our goal? Tweaking the microstructure typically means determining, for every element, a new set of parameters that characterise the microstructure. The determination should be made such that the compliance of the engineering structure decreases with the new choices. This means that we need to have a measure of how `sensitive' the global compliance is to changes in each of the microstructural parameters. Directing the optimisation process towards its stated objective is another challenge that needs to be overcome. We will talk about it a bit later.
The SIMP magic
The homogenisation method, despite standing on firm mathematical grounds, is computationally expensive. This is because it requires two layers of computations as described above. Also,
computation of the sensitivities that are so important for driving the solution towards its stated objective is also a bit cumbersome.
This is where the ``Solid Isotropic Material with Penalisation" (SIMP) method brings in massive levels of simplification. The objective remains the same -- given a fixed design domain, applied loads, supports and a limited amount of material, we want our topology optimisation algorithm to tell us where material should exist so that the structure is as stiff as possible. Ideally, the FE mesh that describes the domain should have elements that contain either material or are voids. Thus, with every element, we can associate a switch which we still call the density and which is either 0 or 1.
For a mesh with millions of elements, attaching 0 or 1 to each in order to attain the maximum possible stiffness, is again an intractable problem. Instead, SIMP allows the density to take any value between 0 and 1 but penalises intermediate densities heavily, so the optimiser naturally avoids them. The trick it uses to achieve this is simple to the point of being magical. If the isotropic base solid material has a modulus E₀, SIMP assumes that an element with density ρ has a modulus ρᵖ E₀. The exponent p is generally close to 3 and that small step itself makes the optimiser abhor intermediate densities and prefers 0 or 1 instead.
Within a FE framework, SIMP also provides a simple method of determining sensitivities. In this case, the state variables are the densities in every element. We need to know how the overall compliance changes with changes in these densities. A simple calculation shows that the rate of change in overall compliance with the density of an element is proportional to the element strain energy which is routinely computed by FE codes.
Efficient methods to move towards a more favourable density distribution using the computed sensitivities have also been developed. For example, the Method of Moving Asymptotes (MMA) has emerged as the dominant non-linear optimisation algorithm for solving the SIMP problem.
With SIMP+MMA we have a scalable, easy to implement computational tool fully compatible with standard FEM. Often SIMP does produce what are known as checkerboard patterns. It is also known to suffer from sensitivity to the FE mesh. However, both these problems are known and have been studied in great detail. For instance, density and sensitivity filters can be designed that alleviate these problems.
The way ahead
Topology optimisation techniques have attained a level of generality that makes venturing beyond simple compliance minimisation eminently possible. Optimising the topology of a structure so that the smallest fundamental frequency of vibration or critical buckling load is maximised is possible within the described framework when we redefine the objective appropriately. So are problems involving geometric nonlinearities.
Handling of geometric nonlinearities assume importance in `compliant mechanisms' which attain their mobility from the flexibility of their constituents rather than hinges, pins and sliders that the rigid body counterparts depend on. In compliant mechanisms, on the other hand, flexibility is distributed over the structure. The motion of a snake through continuous body deformation and active actuation
is a great biomechanical demonstration of a compliant mechanism.
At the simplest level, the optimisation problem for a compliant mechanism can simply be aimed at distributing material in the most efficient way to maximise the displacement at a specified output point in the mechanism. This needs to be done while honouring the equilibrium equations at every step and confining the structure within the design space. A more ambitious goal might be to achieve a target displacement output such that the output point adheres to a specified path with a high degree of precision.
An important application is soft grippers that are capable of handling delicate payloads. These grippers and other devices using flexible kinematics are extremely important for designing soft robots that can crawl into crevices, navigate difficult terrains and possibly, climb down precipices.
The field of topology optimisation continues to see flux of new ideas. One of them is that of contact-aided compliant mechanisms (CACMs), an important extension of compliant mechanisms where contact is no longer treated as a numerical nuisance. Contact can stabilise bucklings and snap throughs, act as mechanical stops and in general allow a vastly larger number of design options to be explored.
They are an important extension of ordinary compliant mechanisms in topology optimisation because contact enormously expands the kinds of motion and functionality that can be achieved. Adding the possibility of contact between parts introduces severe nonlinearities as now the contact constraints (also called impenetrability constraints) have to be included in the problem definition. This is a sure source of non-smooth behaviour in the optimisation process. But despite the added difficulties, contact-aided mechanisms can be used to maximise gripping forces in soft grippers. In fact, it has been used to control the sequence in which contact events occur so as to produce a pre-defined gripping behaviour. They have also been used to produce bistable devices.
Topology optimisation techniques derived for structures are also finding non-structural applications. The development of additive manufacturing techniques makes the prototyping of these optimised devices possible. Large computational resources and especially GPU computing enable efficient simulations.
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