My teacher once asked me how I thought the first engineer designed the very first bridge. While I would require a time machine to be certain, I can logically infer that he or she plagiarized. Not from another engineer because at this point there weren’t any but from the design of nature and learning from our predecessor we are still doing the same.
Then there’s another question, why?
The answer was given by a British guy most of us have probably heard about, named Charles Darwin. Yes, the one who proposed the theory of evolution. Don’t worry I won’t bore you with needless biology just one idea. Evolution is nothing but a large-scale optimization simulation.
Optimization, what exactly is that?
Optimization is the process of finding out the best solution from the series of given constraints. Mathematical optimization deals with the selection of the best criteria from the series of available alternatives. Any optimization problem consists of three main elements namely:
- Optimization variables/Design variables
- Cost function, also known as the objective function, is denoted as f (x). Constraints are expressed as equalities and inequalities, gi(x).
There are various types of optimization problems consisting of one or multiple variables and depending upon the existence of constraints. Similarly, optimization variables can be discrete or continuous, problems can be static or dynamic, systems can be deterministic or stochastic (involving randomness/ probability), and equations can be linear or non-linear.
The simplest type of optimization problem are those involving linear equations and simple constraints. For example: questions involving maximizing/minimizing like:
Maximize P = 50x1 + 80x2
Subject to x1 + 2x2 ≤ 32
3x1 + 4x2 ≤ 84
x1, x2 ≥ 0
Due to this widely efficient approach mathematical optimization has penetrated various fields, one of which is Structural Optimization (SO). Just like any other forms of optimization SO deals with the process of determining the best material distribution but concerns within a physical volume domain, to safely transmit or support the applied loading condition.
It has become a very important tool for engineers and especially toward us civil engineers. During SO the constraints imposed by the manufacture and the usage condition should be taken in consideration. These include increasing the stiffness, reducing stress, displacement and manufacturing with advanced methods.
What are the types of Structural Optimization?
Structural Optimization is then further divided into 3 types:
- Size Optimization
- Shape Optimization
- Topology Optimization
Size Optimization
In this, the engineer/designer knows what the structure looks like but not the size of the component making it. Examples include not knowing the cross-sectional area of each truss element while designing a truss structure.
Shape Optimization
Here, the designer doesn’t know about the contour of some part of the boundary of a structural domain. When solving such optimization problems shape or boundary can be represented by an unknown equation or a set of points whose equations are unknown.
Topology Optimization
Topology optimization is defined as the design optimization formulations that allow for the prediction of the layout of structural or mechanical system. It is the most general form of SO and used where there are moderate amount of design variable. In TO the layout should be the outcome of the procedure.
It should be considered as the companion discipline that provides the user with a new type of design that can either be processed directly or be further refined using shape and size optimization.
While the result of TO should be consistent with the result of shape and size optimization due to the change in the design parameterization direct comparison cannot be carried out easily. Unlike shape and size optimization where they can have any values between their limit but values must always be present such isn’t the case for topology optimization. It is used when the engineer doesn’t know what the shape and size of the structure should be.
The development of mathematical programming and more specifically linear programming techniques and the simplex method has contributed to the stress-constrained minimum weight design of truss structure.
Distinctive features of topology optimization:
- Elastic property of a material as a function of its density can vary over entire design domain.
- Material can be permanently removed form design domain.
Categories of topology optimization:
There are several topology optimization methods which can be grouped into two categories
- Optimality criteria methods
- Heuristic/Intuitive criteria methods
Optimality Criteria
They are the indirect method of topology optimization and satisfy a set of criteria related to the behavior of structure. These are generally more rigorous methods (based on Kuhn-Tucker optimality condition) and are used for cases where there are large number of design variables and few constraints.
This includes the following topology optimization procedures:
- Homogenization
- Solid Isotropic Material with Penalization (SIMP)
- Level set method
- Growth method for Truss structure
Heuristic/Intuitive criteria methods
These method are derived from intuition and observations of engineer working in a field from a long time relating to engineering processes and biological system. They cannot guarantee optimality but can provide viable efficient solutions. They include:
- Fully Stressed Design (FSD)
- Computer Aided Optimization (CAO)
- Soft-Kill Option (SKO)
- Evolutionary Structural Optimization (ESO)
- Bidirectional ESO (BESO)
- Sequential Element Rejection and Admission (SERA)
- Isolines/Isosurface design
STEP 1: A solid block was designed with the shown dimension and extruded length 500mm.
STEP 2: The geometry of the base was fixed as the base of the bridge where the abutment needs to be attached should be solid and not optimized.
STEP 3: To simulate the live load on bridge, 10,000N of load was added on the lower span where the vehicle moves as it is a through type bridge.
STEP 4: We then set the best stiffness to weight ratio at 50% i.e the final optimized product will be 50% of the initial.
STEP 5: Then mesh was created for the made topology for further optimization.
STEP 6: The optimization simulation was run and the final result was obtained.
STEP 7: The optimized topology wasn’t in the best shape due to sharp edges so for easier time during 3D printing the model was made smoother.
So, the question arises, why optimization? In the world where every microsecond, penny and micron matter, optimization helps us to save just that. It is more cost effective than traditional guess and check method and saves a lot of time than just building-and-testing.
Although the conceptual design of a bridge depends on the designer’s intuition and ability to recognize the role of members in transferring weight and loads to the earth, currently, structural optimization is not only an important tool for sizing structure members but also for helping the designer find the most suitable shape of a structure from a structural and an architectural point of view.