4. Optimum design 4.1. Introduction to design optimisation The design process The design of many engineering systems can be a complex process. Assumptions must be made to develop realistic models that can be subjected to mathematical analysis by available methods, and the models must be verified by experiments. Many possibilities and factors must be considered during the problem formulation. In most situations, the economic considerations play a very important role in designing cost-effective systems. The design of a system is an iterative process, which includes the analysis of trial designs one after another until an acceptable design is obtained. In the design process, the designer estimates the trial designs of the system based on simple to complex mathematical analyses, intuition, and experience. The trial design is then analysed to determine if it is acceptable. If it is then the design process is terminated; if not, new trial designs will be generated and analysed until an acceptable design is obtained. Shown in Figure 4-1 is a five-step “system evolution model” which can be used to keep the design process well organised (Arora, 2102). In this model, the first step is to precisely define the specifications for the system; this step involves the interactions between the designer and the sponsor of the project in order to quantify the system specifications. The preliminary design of the system takes place in the second step. Various design concepts will be considered; simplified models are often used at this stage. The third step is the detailed design. The fourth and fifth steps may or may not be necessary for all systems. These stages involve the fabrication of a prototype system and testing. Note that during the design process, modifications to the system specifications may take place hence the presence of feedback loops. Figure 4-1: System evolution model (Arora, 2102) Conventional versus optimum design process Shown in Figure 4-2 are the conventional design approach and the optimum design method. Note that both methods are iterative and share some similar calculation processes. The key features of the two processes are: - The optimum design method (Figure 4-2b) has block 0, in which the problem is formulated and an objective function that measure the merit of different designs is defined - Both methods require data to describe the system in block 1 - Both methods require an initial design estimate in block 2 - Both methods require analysis of the system in block 3 - In block 4, the conventional design method (Figure 4-2a) checks to ensure that the performance criteria are met, while the optimum design method (Figure 4-2b) checks for satisfaction of all of the constraints formulated in block 0 - In block 5, stopping criteria for the two methods are checked in order to stop the iteration - In block 6, the conventional design method the design is updated based on the designer’s experience and instinct and other information gathered from one or more trial designs; the optimum design method uses optimisation concepts and procedures to update the current design. Figure 4-2: Comparison of (a) conventional design and (b) optimum design methods (Arora, 2102) 4.2. Optimum design problem formulation According to Arora (2102), it is generally accepted that the proper definition and formulation of a problem take approximately 50 percent of the total effort needed to solve it. The importance of properly formulating a design optimisation problem must be stresses because the optimum solution will be only as good as the formulation. For example, if one forgets to include a critical constraint in the formulation, most likely the optimum solution will violate such constraint. Therefore, it is critical to follow well-defined procedures to formulate the design optimisation problems. In this section, the five-step formulation procedure - Project/problem description: The designer should ask the question “Are the project goals clear?” - Data and information collection: “Is all the information available for the problem to be solved?” - Definition of design variables: “What are the variables? How do I identify them?” - Optimisation criterion: “How do I know that my design I is the best?” - Formulation of constraints: “What restrictions do I have on my design?” will be introduced along with some examples. The problem formulation process The optimum design problem formulation involves the translation of descriptive statements into well-defined mathematical statements. As mentioned above, the optimisation methods are iterative in which the process of finding solutions started by selecting a set of trial designs. The trial designs are then analysed, evaluated, and compared with the design constraints. If the trial designs do not meet the requirements, then new set of trial design will be generated and the process is continued until an optimum solution is reached. Below is the design and optimise procedure of a cantilever beam Project/problem description The formulation process begins with the development of descriptive statement for the problem (usually by the project’s owner or sponsor). The stamen describes the overall objectives of the project and the requirements to be met. This is also called the statement of work, which is shown in the following paragraph and in Figure 4-3. Figure 4-3: Cantilever beam of a hollow square cross-section “Consider the design of a hollow square-cross-section cantilever beam to support a load of 20 at its end as shown in Figure 4-3. The beam is made of steel of length 2 . The failure conditions for the beam are as follows: - The material should not fail under the action of the load - The deflection of the free end should not be more than 1 - The width-to-thickness ratio should be no more than 8 - A minimum mass beam is desired - The width, , and thickness, , of the beam must be: 60 300 and 10 40 ” Data and information collection To develop a mathematical formulation for the problem, we need to gather information on material properties, performance requirements, resource limits, and cost of raw materials to name only a few. In addition, most problems require the capability to analyse trial designs. Therefore, analysis procedures and analysis tools must be identified at this stage. In many cases, the problem statement is vague, hence, assumptions need to be made in order for the problem to be formulated and solved. The information needed for the cantilever beam design are the expression for bending and shear stress, and the deflection of the free end. Such information can be found by using the equations below: - Cross-sectional area of the beam 2 - Moment of inertia 2 - Moment about the neutral axis of the area above the neutral axis - Bending moment - Shear force - Bending stress - Shear stress - Deflection If the type of material is known, then the Young modulus, shear modulus, allowable bending stress, and allowable shear stress can be assumed to be: Modulus of elasticity 21 10 Shear modulus 8 10 Allowable bending stress 165 Allowable shear stress 90 Defining of design variables The next step in the formulation process is to identify a set of variables that describe the system, which is referred to as the design variables. These variables are also called optimisation variables or free variables because the designer need to assign values to them. Different values for the variables will produce different designs. The number of independent variables are the design degrees of freedom of a problem. In general, the following considerations should be given in identifying the design variables for a problem: - Design variables should be independent from each other. If they are not, there should be some equality constraints between them; - A minimum number of design variables required to properly formulate a design problem must exist; - At the problem formulation phase, the designer should identify as many design variables as possible. During the optimisation process, some of the variables can be assigned fixed values. - A numerical value should be given to each identified design variable to determine whether or not a trial design of the system can be specified. Applying the above points to the cantilever beam problem, one should notice that it has been required that the mass of the beam must be minimum; the beam length, , and the density of the material, , are constant, hence, the variable is the cross-sectional area, , of the beam. This cross-sectional area, in turn, depends on the two design variables beam width, , and wall thickness, . Objective function (optimisation criterion) There are many designs or solutions available for one particular problem. The question that a designer should ask is how to compare designs and select a best design. For this we must have a criterion or an objective function from which the merit of each design can be specified. The objective function must be a scalar function and it must be a function of the design variables. This objective function can be minimised or maximised depends on the requirements of the problem. The selection of a proper objective function is an important decision in the design process. Some examples of the objective functions are cost (to be minimised), profit (to be maximised), weight (to be minimised), ride quality of a vehicle (to be maximised). In many actual situations, two or more objective functions may be identified i.e. a designer may want to minimise the weight of the beam at the same time minimise the deflection or stress at certain point of the beam. These are called multi-objective design optimisation problems, with such optimisation problems weighting factors are often introduced by the designer to highlight the importance of a certain objective function. The objective function for the cantilever beam design problem can be formulated from the equation to calculate the cross-sectional area as , 2 4 4 [4-1] Formulation of constraints All restrictions placed on the design are collectively called constraints. The final step in the formulation process is to identify all constraints (some of the constraints are described in the problem description) and develop expressions for them. Most of the realistic systems must be designed and fabricated with the given resources and must meet the performance requirements. For example, structural members should not fail under normal operating loads; the vibration of the system has to be different from the natural frequency. These constraints must depend on the design variables, hence, their value will change with different designs. Meaningful constraints must be functions of at least one of the design variables. Below are several types of constraints - Linear and nonlinear constraints - Equality and inequality constraints: i.e. , or - Explicit and implicit constraints: explicit – clearly, precisely, and directly stated; implicit – implied though not directly expressed The cantilever beam design constraints are as follows - Bending stress constraint ⇒ 0 ⇒ 0 ⇒ 0 - Shear stress constraint ⇒ 0 ⇒ 0 ⇒ 0 - Deflection constraint ⇒ 0 ⇒ 0 - Width-to-thickness constraint 8 ⇒ 8 0 - Dimension restrictions 60 300 ⇒ 60 0 and 10 40 ⇒ 10 0 300 0 40 0 The optimisation problem is to find values of and that minimise the cross-sectional area or minimise the material cost. 4.3. Monte Carlo and/or Excel Solver (gradient based method) Will be demonstrated during the lecture 4.4. Examples - Design a can to hold at least 500 of liquid as shown in Figure 4-4. The diameter of the can should be no more that 8cm and no less that 3.5 cm; the height should be no more than 18 cm and no less than 8cm. The can is produced in billions, therefore it is desirable to minimise the material cost. - Design a minimum mass tubular column, shown in Figure 4-5, with length, , that can support a load, , without buckling or overstressing. The column is Figure 4-4: Design fixed at the base and free at the top. It is known that the buckling load or of a can critical load for the given column and the moment of inertial about the vertical axis are calculated as and , respectively. - A company owns two sawmills and two forests. The capacity of each sawmill (logs/day) and the distance between the forest and the sawmill (km) are shown in the following table. Each forest can yield up to 200 logs/day for the duration of the project, and the cost to Figure 4-5: (a) Column design, (b) R-t transport the logs is estimated at $10/km/log. At least formulation, (c) Ri-Ro formulation 300 logs are needed daily. Minimise the total daily cost of transporting the logs. Data for sawmills Mill Distance from mill A Distance from mill B Mill capacity per day (km) (km) (logs/day) A 24.0 20.5 240 B 17.2 18.0 300