Cenk Feridunoglu, PhD, Director of Engineering, Empower Operations Corp.
Engineering design plays an important role in achieving fast transition from a product idea to commercialization or improving an existing product. Design process differs among companies where their accumulated expertise and exposure to modern software tools dictate their success.
Design optimization is a concept that more and more engineers start to realize its power. The increase of computing power and new advanced optimization algorithms make optimization practical and useful for iterative search of better designs, based on state-of-the-art simulation tools (FEA, CFD, multi-physics simulation, etc.).
When simulation is completed on a part design, the weakness and strength of the part will be made visible. The engineer or designer looking at the result plots will intrinsically think about some “what if” scenarios to tweak the design for better simulation results.
The “what if” thinking is a natural engineering instinct that is a driving force for design improvement. The number of variables in a design optimization problem reflects its complexity. Experience might be helpful in finding an optimal solution for three to five variable problems without any optimization tool. When the number of variables exceeds five, however, it becomes nearly impossible to find the optimal design by experience or intuition. Optimization tools are thus necessary for efficient and systematic search.
Design optimization tools like OASIS offer a way of maximizing the utility of computing power and simulation software on hand to find better designs faster, pushing the design goals to the extreme while satisfying design constraints. The value gain from any design optimization begins with the problem definition, i.e., the optimization model.
Optimization Model
Objective in a simulation-based optimization is typically selected from the results of simulation, which could include stress, displacement, and drag, and could also be from a CAD model such as volume/mass, center of mass, moment inertia, etc. When an optimization is run, the optimization algorithms search within the given range of input variables for the minimum or maximum value of the objective while satisfying all constraints.
Input variables include selected model properties of a design, e.g., dimensions, angles, etc. from CAD geometry; other parameters can be variables as well, such as loading conditions, material properties, and operating parameters that can vary in the design and operation of a part. A good optimization model will consist of the minimum possible number of design variables that is required to meet the design objectives and satisfy constraints.
Input variable bounds define the search space where the minimum or maximum values of an objective lies in. It is always a good practice to select input bounds that respect the CAD geometric constraints rather than picking arbitrary values.
Constraints are barriers/limits that are applied to the search space which an optimizer treats as binding agreements that make an optimal design feasible. Constraints are categorized as being cheap or expensive. Cheap constraints define relationships between input variables mathematically or can be computed as quickly as a mathematical equation. Any constraint (i.e., stress, displacement, drag, etc.) is considered expensive when simulation is required to evaluate.
Engineers Play an Important Role in Defining an Optimization Model
Experience and engineering intuition play an important role in defining an optimization model. Knowledge incorporated into an optimization model is critical for faster convergence with better results. Unnecessary input variables and larger bounds could make an optimization problem immensely complex and hard to converge. A quick-and-dirty approach would be utilizing all dimensions in a model while applying generous input variable bounds and expecting to find a better design. Might work but not recommended.
For good optimization model development, a variety of info are necessary: known objectives to satisfy project goals, selected important inputs based on engineering judgement and simulation data, reasonable input bounds, and constraints that have to be satisfied. Take the static structural simulation as an example, the objective is generally weight/volume, stress levels, or displacement. The design engineer should pick the most important simulation output depending on the project goal which could be weight reduction, design improvement to overcome stress concentrations, failure avoidance, etc. Also, an initial/reference design would be very helpful to establish a comparison and guide the search. Before building the optimization model, simulations are usually required to identify how features and their dimension’s variability affect the chosen simulation outputs, and to define input bounds that capture the optimal design. The more thoughts that engineers put in at this stage, the more likely it is to arrive at a superior optimal design in a fast manner.
With intelligent optimization tools such as OASIS, engineers can be freed from repetitive trial-and-error, instead can concentrate on the design problem itself by exploring different model formulations and design scenarios. In this way, engineers are empowered to do more.
Changes in the Design Process
While using optimization in design, the design process will have subtle changes. The new process is referred as optimization-drive design, in contrast to experience-driven design when optimization was not applied.
As seen in Figure 1, the top row shows in color the exact same design steps in a typical product development process. The difference between the experience-driven design and optimization-driven design lie on how design revisions are made. Table 1 lists their differences.
Figure 1 Conventional experience-driven design (upper) vs. optimization-driven design (lower).
As shown in Table 1, optimization-driven design enforces clearly defined objective(s), as well as technical specifications that a design needs to satisfy. The optimization algorithm automatically computes how variables affect the design outcomes, drives the design change, and systematically searches for the optimal. This allows engineers explore more design possibilities in a given time frame.
Table 1 Comparison between experience-driven design and optimization-driven design.
Below we will show how one would approach a flywheel and a bracket design following the optimization-driven design process.
Optimization of a Flywheel
Figure 2 shows a basic flywheel and an optimization model. The flywheel is simulated at an angular velocity of 1000 rpm and the Von Mises stress is calculated with a finite element solver. The goal of the optimization is to find the combination of slot dimensions, spacing to center, and number of slots, which will result in minimum mass of the flywheel while keeping the maximum Von Mises stress below a certain level.
The optimization model has an expensive constraint and two cheap constraints. The constraint concerning stress is expensive because it will be evaluated at the end of every simulation call. Cheap constraints are required to maintain the model integrity and are defined by mathematical relationships between input variables (dimensions). The two math equations in Figure 2, along with bounds of the variables, limit the variables from being unrealistic. Figure 3 (a) and (b) show possible invalid model geometries that can be generated without cheap constraints. Geometric values to derive the cheap constraints are indicated in Figure 3 (c).
Figure 2 Basic flywheel and optimization model (unit: mm, kg).
Figure 3 (a) and (b) invalid designs; (c) geometric values to derive cheap constraints.
An engineer needs to first perform analyses to pick the best bounds and define math constraints to ensure the majority of possible designs generated from combinations of variables within the bounds are valid. This will prevent or reduce the geometry and meshing failures for FEA. Then an engineer can specify any expenive constraint from outputs of simulation calls. For constrained problems, OASIS redirects its search towards the feasible area before going full force searching for the optimum of the objective.
Then how to use optimization in a design journey? Let’s take a look at a bracket design example as follows.
Bracket Design and Optimization
Figure 4 shows the initial design of a simple bracket, where a downward force of 300 N is applied at the circular feature.
Figure 4 Initial bracket design with loading conditions.
The design specification of the bracket is dictated by the back plate and circular feature of the design. The critical design parameters, in terms of load bearing (based on simulation results) and total mass, are identified as plate thickness (p), rib thickness (r) and rib length (l).
The initial design satisfies the strength requirement as the maximum Von Mises stress is less than the yield strength of 2.757e7 N/m2 (see Figure 5) with a mass about 132 grams. In most similar cases in practice, engineers should be satisfied with the design and the design task is considered completed.
Figure 5 The maximum Von Mises stress plot for the initial design: Stress (Max) = 1.809e7 N/m^2.
With the optimization, the engineers can continue to reduce the mass of the bracket while keeping the stress at the same level. Hence the engineer defines the objective to minimize the bracket mass, subject to the current maximum Von Mises stress of 1.8e7 N/m2. The optimization problem thus becomes:
For demonstration purpose only, SolidWorks simulation is used for stress analysis and OASIS is used to run the optimization model. This setup leads to the optimal design shown in Figure 6:
Figure 6 The optimal design from the single objective set-up.
This solution may surprise the design engineer because the optimal suggests that the middle rib can be eliminated while still satisfying the constraint. The mass has been reduced to 118.56 g, a drop of 10%.
Well, the design improvement has been great. The question then becomes: what if we want the bracket to be safer? In this case, how much will we sacrifice on the mass? This question in fact brings the need for a multiobjective solution.
A multiobjective optimization model is then formed to explore the design space and find a better design which is lighter and stronger. Below is the optimization model formed for the bracket:
The multiobjective optimization result is a list of trade-off points called Pareto set. Depending on objective importance and decision maker’s preference, a point is selected as the best design. In this case, we are looking for a design that has less mass and less stress (max) than the initial design. Figure 7 is the Pareto set generated from OASIS at the end of an optimization run. There are quite a number of solutions that an engineer can explore and compare.
Figure 7 Pareto set solutions from multiobjective optimization (screenshot from OASIS).
The Pareto set characterizes the design space in terms of mass vs. (maximum Von Mises) stress trade-off points. The Pareto set points are passed into the OASIS Decision Making module for screening and post processing. Since the goal of the optimization is to find better designs, the initial design mass and stress values are applied as filters to the Pareto set (Figure 8).
Figure 8 Decision making with OASIS, Pareto set screening.
The remaining points in the Pareto set are normalized and passed into post-processing where a parallel coordinates plot displays the solution fitness based on the equal importance (weights) of the objectives. Figure 9 shows Designs 67, 64 and 49 ranked higher than the other.
Figure 9 OASIS Pareto set post-processing.
From a closer look at the objective values at the bottom left corner of the screenshot, one can see Design 49 has a better stress value with a higher mass. Designs 64 and 67 have very similar mass values and one can discard Design 64 due to its slightly higher stress value.
After this process, it is the time for the engineer/decision maker to pick the design from the optimization study. Figure 10 plots the optimal designs in the performance space. One can see that the initial design is worse than the single objective optimal and the multiobjective optimal designs. Then the engineer can bring all the design candidates to the decision maker to pick the design, according to their preference.
Figure 10 Performance comparison of obtained optimal designs.
Table 2 lists the initial design, single objective optimum, and multi-objective screened Pareto frontier designs for a more comprehensive reference.
Table 2 Comparison of obtained optimal designs.
Final Remarks
Design optimization is a structured study of searching for better designs with given measurable criteria. Thus, no need for trial-and-error of designs where it gets exponentially harder with an increasing number of input variables. Any simulation output or CAD property can be utilized as an objective or constraint, and any model dimension, material property, loading condition, operation parameter, etc. can be an input in an optimization model.
With a good optimization tool, an engineer can explore multiple design scenarios and be sure for each scenario the best has been found. With a Pareto set, you can also tell your supervisor or manager, with confidence, if you want to save x% cost, you have to sacrifice y% performance. Without such optimization, an engineer may have a sense of failure or guilt that they cannot satisfy the demand from their superiors. Optimization-driven design will facilitate more effective and efficient communication between the management and engineers and avoid wasting time to search for unachievable designs.
As one can see from the above optimization-driven design process, there are changes in one’s approach and thinking. But all one has to do is to focus on the design problem at hand without worrying about optimization itself. A user only needs to focus on how to define the optimization problem, explore different design scenarios, and provide with decision makers all the smart plots and solutions, which are outcome of your knowledge, creativity, and use of intelligent optimization tools.