Bone Growth Analogy for Steel Truss Structures

1.   Literature Review

1.1. Structural Optimization

The optimisation of structures is well established within the field of structural design, however due to the mathematical and computational complexity it remained a field of more academic interest until the last 30-40 years (Querin, et al., 2000) The rise of computer processing power and speeds since has led to the creation of numerous powerful FEA programmes and optimisation algorithms.

In order to understand just how inefficient a simple structural design can be, and why optimization is necessary, take the example of a simply-supported beam. With a seemingly ‘maximum stressed’ cross-section, where the stress in the outer fibre at midspan due to bending is equal to the yield strength of the material, the bulk of the material is under-stressed. In fact, 97% of the material is stressed to less than 75% of the maximum (Xie & Steven, 1997), and yet this is the design method practiced the world over.

Figure 1 – Contours of ‘iso-stress’ with respect to maximum bending stress in beam. Adapted from (Xie & Steven, 1997).

Generating a structural form that fully utilizes the materials contained within defines the concept of ‘fully stressed design’, where, ideally, all the material experiences the same stress. This theorem of fully stressed design has been developed for over a century, notably in classical papers by Michell (1904) and later summarized by Hemp (1973), as cited by Xie & Steven (1997). The concept of optimization in this regard aims for this stress level to be as close as practically possible to the maximum, which should inherently produce a structure minimized with respect to weight, cost, embodied carbon, etc.

The word ‘optimization’ is used liberally, so it is important to define exactly what is meant by it. For a real-life structure designed to withstand numerous different loads and support conditions, it could be said that an optimum solution would be one that best satisfies the various constraints. However, for each load or support case, for example, the solution is ‘sub-optimal’, and so what is really designed is an ‘optimum compromise’ (Querin, et al., 2000). However, in theory, for a single load case under certain circumstances there can be an absolute optimum structure meaning a fully-stressed design (Querin, et al., 2000), although that is not always the case.

1.1.1.   Classes of Optimization

1.1.1.1.  Topology vs Shape vs Size Optimization

The three main types of optimization with respect to what is being optimized, are ‘topology’, ‘shape’, and ‘size’

Topology optimization refers to the configuration of the internal members within a structure (Mortazavi & Togan, 2016), or rather the selection of nodes and their connectivity (Kalyanmoy, et al., 2015) within a set design space.

Shape optimization refers to the geometric layout (Muller & van der Klashorst, 2017), optimizing the coordinates of existing nodes (Kalyanmoy, et al., 2015) within a defined design space.

Size optimization refers to the physical size or properties of members within a structure (Muller & van der Klashorst, 2017), most commonly the cross-sectional areas used.

1.1.1.2.  Truss vs Continuum Optimization

Generally, most optimization methods fit within one of two categories for how the structure will behave during the optimization; ‘truss’ or ‘continuum’.

Truss optimization involves defining a pattern of nodes and elements within a design space, with all possible connections predefined by the designer. A potentially major drawback with truss optimization is that the designer must define every potential truss member in the first place, and so the designer must include the optimum solution in the initial design space. For cases of classic truss optimization, this solution is known, but it might not always be obvious for new or complex problems. Further, the optimum found can not be guaranteed as the absolute optimum (with respect to the variables and criteria defined), but rather the optimum solution contained within the starting geometry.

Continuum optimization treats the design space as a finely meshed plate, rather than a truss. This ensures that every possible solution that fits within the design space can be found, and does not rely on any predefined connections by the designer. It does mean that the designer must carefully choose their design space and mesh density to ensure accuracy- in fact for continuum analysis the quality of the final optimum solution is largely dependent on the fineness of the mesh used.

1.1.2.   Optimization Criteria

In order to ‘optimize’ a structure, one must first define what the design is being optimized with relation to, which is almost always the weight of the structure or some function of that weight. One must also define what criteria is to be used for the evolution of the structure. Typically, this criterion is based around the potential for structural failure (as the basis of structural design is one which is both safe and efficient), which means using some measure of stress or strain. As the material is typically divided into many small elements, combining the average of the stress components felt on each small element seems sensible.

In ductile isotropic materials, such as steel, the typical criterion used is von Mises stress, ${\sigma }^{\mathit{vm}}$

, which for plane stress problems is defined as: ${\sigma }^{\mathit{vm}}=\sqrt{{\sigma }_x^2+{\sigma }_y^2–{\sigma }^x{\sigma }^y+3{\tau }_{\mathit{xy}}^2}\mathrm{ }$

$\text{w}\text{here}{\text{σ}}^{\text{x}}\text{and}{\text{σ}}^{\text{y}}\text{are normal stresses in}\text{ x }\text{and}\text{ y }\text{directions respectively}\text{, }\text{and}{\text{τ}}_{\text{xy}}\text{is the shear stress}$

(Xie & Steven, 1997).

For plane strain problems, the von Mises stress includes the additional stress in the out of plane dimension:

${\sigma }^{\mathit{vm}}=\sqrt{{({\sigma }^x–{\sigma }^y)}^2+{({\sigma }^y–{\sigma }^z)}^2+{({\sigma }^z–{\sigma }^x)}^2+6({\tau }_{\mathit{xy}}^2+{\tau }_{\mathit{yz}}^2+{\tau }_{\mathit{zx}}^2)}$

(Atkins & Escudier, 2014).

A material can be considered to be at its elastic limit, or at the onset of yielding, when its von Mises stress reaches its yield strength (Engineers Edge, 2018), as a result of reaching a critical value of distortional energy stored in the material (Christensen, 2018).

1.2. Existing Optimization Methods

1.2.1.   Evolutionary Structural Optimization (ESO)

One of the first optimisation algorithms to be developed was the ‘Evolutionary Structural Optimisation’ (ESO) method, a form of shape/topology optimization developed by Xie & Steven (1992). The algorithm starts from a full design space with elements covering all possible geometries, and under-stressed elements are removed from the structure iteratively. By slowly removing the inefficient material, the structure will naturally evolve towards a desired optimum (Xie & Steven, 1997). Ideally this optimum is something approaching a fully stressed design, but in practice this is only possible in a few special cases and a lower efficiency criterion is used, for example all elements greater than 25% of the maximum stress (Xie & Steven, 1997). Due to its simplicity- the ESO process is no more sophisticated than running FEA, then deleting one or more elements, then running the FEA again- it has been used as a benchmark for further development of algorithms, and itself has been under continuous development since its introduction.

Some issues with ESO that have been widely discussed include that the algorithm is not able to detect if it is finding a ‘global’ or ‘local’ solution (Kalyanmoy, et al., 2015). The algorithm can become stuck in local solutions whereby the order in which the elements have been removed prevents the global optimum being found. The method can only remove material to eliminate under-stressed elements but cannot add material in order to alleviate over-stressed elements (Querin, et al., 1998). If the global optimum solution contains an element which has been removed in an early iteration of the ESO, the solution can never be found. 

1.2.1.1.  Additive Evolutionary Structural Optimization (AESO)

The Additive ESO algorithm (AESO), developed in 1997, sought to use the principles outlined in ESO, but instead start from an empty design space with just the minimum possible structural form connecting loads to supports as outlined in Figure 2 (Querin, et al., 2000). In effect this is the opposite evolutionary process to the classic ESO i.e. a structure growing to minimize the stresses experienced rather than shrinking to improve efficiency.

A

B

C

D

Figure 2 – AESO method visualised showing definition of design domain, loading and supports (A), discretisation of finite element mesh covering entire design domain (B), creation of minimum structural form connecting loads and supports (C), final mesh representing initial topology for optimisation (D). Adapted from (Querin, et al., 2000).

However, the authors discovered that while the algorithm successfully evolved the structure into one with expected outer optimal shapes of structures, there are typically regions within these boundaries that are under-stressed and the algorithm is not able to remove these areas (Querin, et al., 2000).

1.2.1.2.  Bi-Directional Evolutionary Structural Optimization (BESO)

The Bi-Directional ESO (BESO) combines the processes of the ESO and AESO in order to produce an algorithm capable of both removing and adding material in order to minimise the relative weaknesses of both the parent algorithms (Querin, et al., 1998). The developed BESO algorithm was fully evolutionary, capable of reducing maximum stresses and removing inefficient material. This was an example structure that can react to changes in the environment (loading, supports, material properties, etc.), producing what appears to be a ‘self-designing structure’. Further, the iterative (or evolutionary) nature of the algorithm means that the designer can watch it take shape and stop the algorithm at any point to find a solution more optimal than the last. 

The BESO algorithm has four control parameters to be tuned by the user (Kalyanmoy, et al., 2015). This means that in order to use the algorithm, one must ideally have carried out the analysis and know what is to be expected beforehand. While this is useful in verifying the accuracy of the algorithm, it is not useful in applying the method to a new problem. Further, it makes the algorithm more sensitive to user input and requires a working knowledge of the mathematical side of the algorithm. These inputs are ad hoc which makes the algorithm difficult to apply universally (Kalyanmoy, et al., 2015). A further limitation for the BESO is that it focuses purely on topology and shape optimization, in that any member within the design space can be ‘on’ or ‘off’, and does not allow for size optimization.

1.2.2.   Ant Colony Optimization (ACO)

The behavior of ant colonies was first introduced into computational mathematics in 1992 by Dorigo. Ants, though almost fully blind, can work as a colony to successfully find the shortest paths between their nest and food sources (Camp & Bichon, 2004). When a colony of ants look to find the shortest route between two points, each ant will use pheromones to communicate with the others, by signifying the route that it took (Kaveh, et al., 2008). In the Ant Colony Optimization (ACO) algorithm, the paths represent a possible solution, and the ants represent an iteration. Once each ant has completed a path, the paths are evaluated as to their quality with respect to some criteria and the pheromone intensity of the ants that found a good solution, i.e. travelling on a more optimal path, is increased (Luh & Lin, 2009). This means that the ants will tend to converge on the most optimum solution, although a key part of the algorithm is the decreasing strength of the pheromones over time, so that each ant is inclined to explore new paths to ensure every possible option has been explored (Camp & Bichon, 2004).

With respect to structures, one application of the ACO is to let each path represent a solution of different cross-sectional areas for the various members in a truss system as shown in Figure 3. When each ant has finished each path (i.e. a possible solution of cross-sectional areas has been found), the weight of the system is calculated and the system is also checked for its structural stability. The randomness of each further iteration of produced solutions centered around the tendency to favour previous good solutions due to communication between the colony ensures a global optimum is found (Camp & Bichon, 2004).

Figure 3 – ACO visualised showing different paths (cross-sectional areas) for each element 1,2,…,n. Taken from (Camp & Bichon, 2004).

1.2.3.   Particle Swarm Optimization (PS)

Particle Swarm (PS) is a computational method developed by Kennedy and Eberhart (1995) which has since been applied to numerous optimization tasks. The algorithm is based on the social behavior of animals in a flock, whereby the flock uses ‘social sharing’ of data among members to gain an evolutionary advantage (Perez & Behdinan, 2007). In this method, a set of ‘particles’ within a swarm are randomly generated which then move around the domain of the problem. At each iteration or timestep, the ‘quality’ of the location of each particle is evaluated and compared to the locations of all the other particles within the swarm. In this way each particle can ‘know’ the quality of its own position by communicating with the rest of the swarm (Mortazavi & Togan, 2016). The resultant movement of each particle for the next timestep is dependent on the respective qualities of its own position and that of the swarm’s best location obtained thus far (Kennedy & Eberhart, 1995).

Particle Swarm Optimization (PSO) applied this method to structural design, where the particles can represent, for example, nodes of a truss, and the quality of each position can relate to some criteria (e.g. stress levels) which the designer is seeking to optimize. The PSO method has been shown to be a reliable form of topology optimization with results matching those of traditional benchmark problems (Perez & Behdinan, 2007). More recently, an integrated PSO method (iPSO) has been generated which seeks to increase the communication between particles in order to improve the navigation to the global optimum (Mortazavi & Togan, 2016).     

1.2.4.   Big Bang-Big Crunch Optimization (BB-BC)

The Big Bang-Big Crunch method (BB-BC) was developed by Erol & Eksin (2006) consists of two steps; a Big Bang whereby a set of possible solutions are randomly generated within the search space, and a Big Crunch where the initial random solutions are contracted around what is called a ‘center of mass’, or the weighted average of all the generated solutions with relation to an optimization criterion (Kaveh & Talatahari, 2009). Running a size optimization on a truss, the initial Big Bang phase would involve generating a random set of cross-sections for each member, with each design evaluated to determine the feasibility and quality. The Big Crunch phase calculates the centre of mass of the design set, as well as the best global solution. Subsequent iterations are run, with each Big Bang normally distributed with respect to the previous centre of mass and global best solution, until equilibrium is reached (Camp, 2007). At this point, the search space is redefined as the area around the global best solution to encourage further exploration of the optimum solution (Camp, 2007).

Figure 4 visualizes the algorithm, showing the initial Big Bang phase (left) of randomly generated data points with defined criteria (X₁, X₂) such as structure weight and maximum member stress. The centre of mass for the set is shown in red. Over time, the method will converge upon an optimum solution (right), although the generated designs will be random and so each iteration a small number will be located elsewhere within the criteria map to encourage exploration and ensure the global optimum is found rather than local optimums.    

Figure 4 – BB-BC algorithm visualised showing initial randomly generated Big Bang phase data set (left) and n^th Big Bang phase at equilibrium. Adapted from (Erol & Eksin, 2006).

The BB-BC method performs very well in the latter phases of the optimization in fine tuning the maximum around a local optimum solution. However, issues can arise in the initial global investigation, as the first set of random data can be skewed to a specific part of the search space essentially ‘trapping’ the algorithm in that area (Kaveh & Talatahari, 2009). Increasing the number of data points could reduce the chances of this occurring, but this increases both the running time and costs of the method.

1.2.4.1.  Hybrid Big Bang-Big Crunch Optimization (HBB-BC)

The Hybrid BB-BC (HBB-BC) was later developed in order to improve the initial exploration of the design space to find a global maximum (Kaveh & Talatahari, 2009). The modified algorithm seeks to utilize the PSO to improve the initial exploration in the BB-BC method. HBB-BC utilizes an additional component in the Big Crunch phase, being the local best of each element so far, and places more emphasis on the local and global bests than the centre of mass when defining the centre of the next Big Bang distribution (Kaveh & Talatahari, 2009).

This refined algorithm outperformed the classic BB-BC for finding optimums due to the improved exploration of the search space in the initial stages (Kaveh & Talatahari, 2009). However, the algorithm is increasingly complex due to the combination of the two methods.  

1.3. Potential Developments in Optimization Methods

Currently, methods that seek to solve the TSS optimization are simple size optimization algorithms for trusses that allow a ‘zero value’ for are/stiffness, inherently allowing some sort of topology optimization (as elements can therefore be entirely removed) (Achtziger, 2007). Recent developments in this field by Ali, Ali & Kalyanmoy (2015) and Mortazavi & Togan (2015) sought to integrate traditional evolutionary optimization methods with newer techniques (Mortazavi & Torgan developed an integrated PSO method to simultaneously solve TSS problems). These newer algorithms produced ‘competitive’ results and further application and development is ongoing to apply them to more complex problems with more refined techniques (Kalyanmoy, et al., 2015).

On the other hands, methods that use continuum optimization are typically simple topology methods due to the complexity involved with the FEA required for each iteration. The methods proposed by Xie & Steven (and later Querin) between 1993-2000 with the ESO, AESO, and BESO algorithms have not been significantly developed since. This is perhaps due to the fact that the design of a truss using a continuum design space could be seen as flawed: the structure can not be built entirely as designed, and the optimum solution cannot be directly transferred to a working design. However, with the rapid development of 3D printing, it is not inconceivable that future trusses could take the exact shape of the optimum continuum solution and rapid prototyping could allow for almost instant design verification (Xie & Steven, 1993).

A potential development for a new optimization method is a simplified approach to simultaneous topology, shape, and size (TSS) optimization for a continuum structure. This approach can use the simplicity and evolutionary nature of the BESO method (and its parent algorithms) while introducing some of the more refined techniques identified in the modern algorithms- e.g. ACO, PSO, HBB-BC. 

1.4. Bone Growth Analogy

The human body is one of the most fundamental examples of a ‘self-designing structure’ in the natural world. On a cellular level, the body quite literally changes to accommodate its environment, by strengthening or weakening as a response to the applied external forces (Glass , et al., 2018). These adaptational processes help to reduce injury risk and maintain efficiency with the body functions (Krahl, et al., 1995).

The skeletal system and surrounding soft tissue can experience both ‘atrophy’ (shrinking) and ‘hypertrophy’ (growth). These processes can affect the mass, density, and strength of the bone or muscle through a change in cross-sectional area (Boonyarom & Inui, 2006)- and this is continuous throughout life of a human depending on the stresses experienced. In a single state of external forces, the body will tend to maintain homeostasis, where a period of equilibrium is reached with the cellular structure (Boonyarom & Inui, 2006).

This analogy of bone growth seems perfect to combine with the identified potential developments in the field of structural optimization. The atrophy and hypertrophy of the skeletal system are analogous with the removal and addition of elements in the BESO method, while the real-life change in cross-sectional area on a cellular level applies itself well to creating a TSS optimization method. By hybridizing these concepts into a new algorithm, it is possible that a full TSS optimization method for continuum structures could be developed. This algorithm could mimic perfectly how the structure of bones grow and shrink in response to changing conditions, and the designer could see how the structure forms at each evolutionary state. It is proposed that this method will be called Bone Growth Analogy Optimization (BGA).

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Appendix A