Lecture 1: Introduction to Design Optimization

Welcome to Design Optimization! In this course, we will explore how to systematically make engineering systems better. This first lecture will lay the groundwork by defining what design optimization is, why it’s so critical in modern engineering, and how it differs from traditional design approaches and other related fields.

What is Design Optimization?

Engineering encompasses a variety of established activities, including analysis, design, fabrication, sales, research, and development of systems. Among these, the design of systems is a fundamental field in the engineering profession. Historically, the process of designing and fabricating complex systems has evolved over centuries, leading to the existence of remarkable structures like buildings, bridges, automobiles, and airplanes. However, this evolution has often been slow, time-consuming, and costly, frequently resulting in systems that were merely adequate rather than optimal. The procedure was often to design, fabricate, and use a system regardless of whether it was the best one, with improvements only being considered after substantial investments had been recouped.

Design optimization offers a systematic and organized approach to improving this process. It frames the design of a system as a problem of optimization, where a specific performance measure is maximized or minimized, while ensuring that all other design requirements and constraints are satisfied. Essentially, it helps engineers find the best possible system given a set of criteria and limitations.

Numerical methods of optimization have been extensively developed and are widely used to achieve these better, more efficient designs. This course will focus on the practical design process, emphasizing the applicability of optimization concepts and methods to real-world engineering challenges, rather than delving into rigorous theoretical proofs.

Why is Design Optimization So Important?

The importance of design optimization has grown rapidly across various industries. In today’s competitive landscape, engineers are constantly challenged to create systems that are efficient and cost-effective without compromising their integrity.

Competitive Advantage: Optimizing designs allows companies to beat the competition by producing superior products or services at a lower cost.
Improved Bottom Line: By minimizing costs (e.g., material, manufacturing, energy expenditure) or maximizing desirable outcomes (e.g., profit, performance, reliability, ride quality), optimization directly contributes to financial success.
Resource Efficiency: It helps in making the best use of available resources, preventing waste and ensuring sustainability.
Enhanced Performance: Optimization leads to systems that are more efficient, reliable, durable, and perform better under various conditions.
Handling Complexity: Modern engineering systems are increasingly complex. Optimization provides a structured way to manage this complexity and find solutions that would be difficult or impossible to achieve through trial-and-error.

The Overall Process of Designing Systems

Designing engineering systems is often a complex, interdisciplinary endeavor. It begins with the identification of a need and progresses through several organized steps:

Define Specifications: The first crucial step is to precisely define the specifications for the system. This often requires considerable interaction between the engineer and the project sponsor to quantify all system requirements.
Analysis of Options: The design process starts by analyzing various options. Subsystems and their components are identified, designed, and tested.
Detailed Design with Optimization: This is where optimization methods significantly accelerate the process. For all promising design concepts, a detailed design for all subsystems is performed using an iterative process. Design parameters for the subsystems are identified, and system performance requirements are formulated. The goal is to design subsystems to maximize system worth or minimize a measure of cost. The outcome is a detailed description of the final design, presented in reports and drawings.
Fabrication and Use: The system is then fabricated and put into use.

It is important to understand that design is an iterative process. This means analyzing several trial designs one after another until an acceptable (or, in optimization, the best) design is obtained. Designers typically start with an initial trial design based on experience, intuition, or simple analyses. This trial design is then analyzed. In conventional design, the process terminates if the design is acceptable. In optimum design, the trial design is analyzed to determine if it is the “best,” implying it is cost-effective, efficient, reliable, and durable.

Example: Designing a High-Rise Building or Passenger Car Consider the design of a high-rise building. This project involves architects, structural engineers, mechanical engineers, electrical engineers, environmental engineers, and construction management experts. Similarly, designing a passenger car requires cooperation among structural, mechanical, automotive, electrical, chemical, hydraulics design, and human factors engineers. In such interdisciplinary environments, the entire design project must often be broken down into several subproblems, each of which can be posed as a problem of optimum design.

Engineering Design vs. Engineering Analysis

It’s crucial to distinguish between engineering design and engineering analysis:

Engineering Analysis: This activity involves determining the response of a given system to a given input. For example, if you have a bridge of known dimensions and materials, analysis would calculate the stresses, deflections, and natural frequencies under a specified load. This typically involves applying fundamental principles of physics and mathematics.
Engineering Design: This activity focuses on determining the parameters of a system to achieve desired performance under given inputs. Using the bridge example, design would involve determining the dimensions (e.g., beam depths, column widths) and materials such as to safely support a given load, minimize material cost, and satisfy all relevant codes. Design is inherently a synthesis process, where choices are made to meet objectives.

Understanding analysis methods is a prerequisite for design optimization. In this course, we will assume students understand basic analysis methods; equations for system analysis will be provided where necessary.

Conventional Design Process vs. Optimum Design Process

Both conventional and optimum design methods are iterative. However, a key difference lies in how they approach the goal:

Conventional Design Process:
- Begins with data and an initial design estimate.
- The system is analyzed.
- Performance requirements are checked. If they are met, the design is accepted and the process stops.
- If requirements are not met, the design is modified based on engineering judgment or simple calculations, and the process loops back to analysis.
- This approach aims for an acceptable design.
Optimum Design Process:
- Includes an initial step (Block 0) where the problem is formulated as one of optimization. This means defining an objective function that precisely measures the merits of different designs (e.g., minimize cost, maximize efficiency).
- Similar to conventional design, it requires data and an initial design estimate, followed by system analysis.
- Instead of just checking for acceptability, the design is systematically improved by an optimization algorithm to move towards the optimum value of the objective function while satisfying all constraints.
- The process continues refining the design until the objective function can no longer be improved, leading to the “best” possible design.

Key takeaway: The optimum design method formalizes the improvement process, leading to superior results compared to the trial-and-error nature of conventional design.

Optimum Design vs. Optimal Control Problems

While both optimum design and optimal control involve making systems “optimal,” they address different aspects:

Optimum Design: Focuses on determining the best physical parameters or configuration of a system to achieve a desired performance. The design variables (e.g., dimensions, material properties) are typically fixed values once determined.
- Example: Determining the optimal dimensions (radius, thickness) of a tubular column to minimize its mass while ensuring it doesn’t buckle or overstress under a given load. The goal is to find the best structure.
Optimal Control: Deals with determining the best way to operate or control an existing system over time to achieve a desired output. The control variables are functions of time or state variables.
- Example: A cruise control mechanism in a passenger car. The system’s output (vehicle’s cruising speed) is known. The control mechanism’s job is to sense fluctuations in speed due to road conditions and adjust fuel injection (the control input) accordingly to maintain that constant speed. The goal is to find the best operation strategy.

It is important to note that while they are separate activities, some optimal control problems can be effectively transformed into optimum design problems and solved using the methods we will cover in this course. This highlights the broad power and applicability of optimum design techniques.

Basic Terminology and Notation

To effectively understand and apply the methods of optimum design, familiarity with some basic mathematical terminology and notation is essential. This includes concepts from linear algebra (vectors, matrices, and their operations) and basic calculus (functions of single and multiple variables, derivatives).

We will use standard terminology throughout this course. Here are some key concepts you should be familiar with:

Design Variables (x): These are the parameters of the system that can be changed by the designer. They are usually represented as an n-dimensional vector x = (x1, x2, …, xn). For example, in designing a beam, the width and depth could be design variables.
Objective Function (f(x)): This is the single scalar measure of performance that you want to optimize (minimize or maximize). It depends on the design variables.
Constraints: These are limitations or restrictions on the design variables or system performance. They arise naturally in engineering problems (e.g., material strength limits, available space, budget).
- Equality Constraints (hj(x) = 0): These must be satisfied exactly.
- Inequality Constraints (gi(x) ≤ 0): These must be satisfied such that a certain value is less than or equal to a limit. Sometimes they are written as g_i(x) >= 0, but for standardization, we often convert them to g_i(x) <= 0.
Feasible Set (S): This is the collection of all design points (vectors x) that satisfy all the constraints. A design is feasible if it belongs to this set.
**Optimum Solution (x*): This is the design variable vector x** that yields the best value of the objective function (minimum or maximum) while satisfying all constraints. The corresponding objective function value is f(x*).
Gradient Vector (∇f(x)): For a function f(x) of multiple variables, the gradient vector is a vector of its partial derivatives with respect to each variable. It points in the direction of the steepest increase of the function.
Hessian Matrix (H(x)): This is a square matrix of second-order partial derivatives of a function. It is important for determining the nature of an optimum point (e.g., local minimum, maximum, or saddle point).
Continuous Functions/Variables: Variables that can take any real value within a range, and functions whose graphs can be drawn without lifting the pen.
Differentiable Functions: Functions for which derivatives can be calculated. Note that a function can be continuous everywhere but not differentiable everywhere (e.g., f(x) = |x| is continuous at x = 0 but not differentiable there).

We will define these terms in more detail and review relevant mathematical operations as we encounter them in the context of solving optimization problems. It’s important to understand and internalize this terminology as it forms the language of optimum design.

This concludes our introduction to Design Optimization. In the next lecture, we will dive into the critical step of formulating an optimum design problem mathematically.