Engineering Analysis and Design Optimization

Introduction and Context

Computer-aided design (CAD) has evolved from a geometry-creation tool into the central source of product information used throughout the engineering lifecycle. A CAD model can define shape, dimensions, material assignments, assemblies, tolerances, manufacturing features, and design intent. Computer-aided engineering (CAE) tools use this information to predict how a component or system will behave under mechanical, thermal, fluid, or dynamic loading. Computer-aided manufacturing (CAM) then uses the validated design to plan toolpaths, processes, and inspection operations.

Engineering analysis is the disciplined use of physical models, numerical methods, and experimental evidence to evaluate a design. Design optimization extends this process by systematically searching for improved designs subject to objectives and constraints. The purpose is not simply to produce the lightest or cheapest part. A successful design must satisfy functional, safety, manufacturing, reliability, sustainability, and economic requirements simultaneously.

The connection between CAD and CAE is therefore an engineering workflow rather than a file-transfer operation. A geometrically perfect model may be unsuitable for analysis if it contains tiny fillets, gaps, sliver faces, or non-manifold regions. Conversely, an analysis model may be simplified for computational efficiency, but its simplifications must preserve the physical behavior relevant to the design decision.

Learning Objectives

After completing this chapter, a student should be able to:

1. Explain how CAD models are prepared and transferred to CAE systems.
2. Select suitable idealizations, element types, materials, loads, and boundary conditions.
3. Describe meshing and assess mesh quality and convergence.
4. Interpret structural, thermal, and motion-analysis results critically.
5. Distinguish verification, validation, correlation, and sensitivity analysis.
6. Formulate parametric, topology, and multi-objective optimization problems.
7. Conduct defensible design iterations and communicate engineering decisions.
8. Recognize limitations, ethical responsibilities, and emerging digital-engineering practices.

1. The CAD–CAE Engineering Workflow

A typical analysis workflow contains the following stages:

1. Define the engineering question. Specify what must be predicted: stress, displacement, temperature, natural frequency, fatigue life, mechanism motion, or another response.
2. Prepare the geometry. Remove irrelevant details, repair gaps, establish idealized surfaces or solids, and identify contact regions.
3. Assign physical properties. Define material models, thicknesses, joints, friction, damping, and other properties.
4. Apply loads and constraints. Represent forces, pressures, temperatures, prescribed motions, supports, and contacts.
5. Generate the mesh. Discretize the model into finite elements and check quality.
6. Solve the model. Select a linear, nonlinear, transient, thermal, modal, or coupled solver as appropriate.
7. Verify and interpret. Check equilibrium, convergence, units, reaction forces, deformation patterns, and sensitivity to assumptions.
8. Validate and correlate. Compare predictions with analytical solutions, benchmark problems, experiments, or field data.
9. Iterate the design. Modify dimensions, materials, topology, or operating conditions and repeat the analysis.
10. Release the design. Document assumptions, results, limitations, and approval evidence before manufacturing.

A key principle is that analysis results are predictions conditional on the model. A contour plot is not automatically evidence that the physical component will behave exactly as shown.

1.1 Geometry Preparation and Defeaturing

CAD geometry is created for design and manufacturing, whereas an analysis model is created to represent relevant physical behavior efficiently. Features may be removed when they do not materially affect the response being studied. Examples include small holes in a global stiffness study, cosmetic fillets in a preliminary thermal model, and embossed lettering in a modal analysis.

Defeaturing must be controlled. A fillet can be omitted in a global stress study but may be essential in a fatigue assessment because it affects stress concentration. Similarly, a small gap may be irrelevant to static stiffness but decisive in a contact or leakage problem.

Common geometry problems include:

• duplicate or overlapping surfaces;
• gaps between faces;
• very short edges and sliver faces;
• incorrect surface normals;
• zero-thickness regions;
• disconnected bodies that should be joined; and
• imported geometry with incorrect units or coordinate systems.

The analyst should maintain traceability between the production CAD model and the analysis idealization. A simplified model should record what was removed and why.

1.2 Data Exchange and Associativity

CAD and CAE systems may exchange geometry using native links or neutral formats such as STEP and IGES. Native associativity can allow a change in a CAD parameter to update the analysis model, mesh, and results. Neutral formats are more portable but may lose feature history, materials, assemblies, or design intent.

A robust workflow uses version control and explicit checks for:

• model revision and configuration;
• length, force, mass, and temperature units;
• material and property-library revisions;
• coordinate-system orientation;
• contact and assembly definitions; and
• solver and mesh settings.

The principle of a digital thread is useful here: requirements, CAD features, analysis assumptions, manufacturing information, test results, and revisions should remain linked sufficiently to support traceability.

2. Finite Element Modeling and Meshing

The finite element method (FEM) divides a continuous body into a finite number of elements connected at nodes. Within each element, the unknown field—often displacement, temperature, or pressure—is approximated using interpolation or shape functions. The governing differential equations are converted into algebraic equations.

For a linear structural model, the global equilibrium equation is commonly written as:

where K is the global stiffness matrix, u is the vector of unknown nodal displacements, and F is the applied nodal-force vector. This formulation assumes small deformation, linear material behavior, and linearized boundary conditions. If contact, plasticity, large deformation, or other nonlinear behavior is present, the equations must generally be solved incrementally and may be written in tangent form:

where K_t is the tangent stiffness matrix, ∆u is an incremental displacement vector, and R is the residual force vector.

2.1 Element Types

The element type should match geometry and physics. Three-dimensional tetrahedral and hexahedral solid elements are suitable for bulky parts. Shell elements efficiently represent thin-walled structures when thickness is small relative to other dimensions. Beam elements represent slender members using line geometry and section properties. Surface elements may represent membranes or thin thermal layers.

Element order also matters. Linear elements have first-order interpolation; quadratic elements have midside nodes and can represent curvature more accurately. Higher-order elements often improve accuracy for smooth fields, but they require more computation and may not solve poor geometry or inadequate boundary conditions.

2.2 Mesh Quality and Convergence

Mesh quality affects accuracy and solver robustness. Important measures include aspect ratio, skewness, warpage, Jacobian quality, and element distortion. Acceptable values depend on the solver and element formulation, so generic numerical limits should not be applied without reference to the software documentation and analysis type.

A mesh-convergence study tests whether a reported quantity changes acceptably as the mesh is refined. If a response value is q₁ on one mesh and q₂ on a refined mesh, the relative change can be reported as:

Here ε_q is the percentage change, and the denominator uses the refined result. This measure is meaningful only when the same model, loads, constraints, and result definition are used. A global displacement may converge while a sharp corner stress remains mesh-dependent. Singular stresses at idealized point loads, sharp re-entrant corners, or perfectly rigid constraints should not be interpreted as ordinary material stresses. Use stress averaging, a physically realistic radius, or a failure measure appropriate to the problem.

2.3 Boundary Conditions and Contacts

Boundary conditions must represent the physical test or operating environment. Over-constraining a model can produce artificial stiffness and unrealistic stresses; under-constraining it can produce rigid-body motion and a singular matrix. Reaction forces should be checked against applied loads. For static equilibrium:

where ∑F is the vector sum of forces and ∑M is the vector sum of moments.

Contacts may be bonded, frictionless, rough, or frictional. These choices are physical assumptions, not merely software settings. Frictional contact introduces nonlinearity because the contact status and tangential traction can change during solution.

3. Structural, Thermal, and Motion Analysis

3.1 Structural Analysis

Structural analysis predicts displacement, strain, stress, reaction forces, buckling behavior, vibration, or fatigue-related quantities. A basic normal stress relation for a uniform axial member is:

where σ is average normal stress in pascals, F is axial force in newtons, and A is cross-sectional area in square metres. This equation assumes a uniform member, concentric loading, and elastic behavior away from local discontinuities.

For bending of a slender beam in the elastic range:

where M is bending moment, y is distance from the neutral axis, and I is the second moment of area. It is not a substitute for a three-dimensional model when shear deformation, local yielding, complex geometry, or contact is important.

A commonly used ductile-material measure is von Mises equivalent stress:

where σ₁, σ₂, and σ₃ are the principal stresses. Under a simple uniaxial stress state, this reduces to the absolute value of the applied normal stress. For a ductile material under static loading, a preliminary elastic yield check often compares σ_v with yield strength σ_y. A factor of safety may be expressed as:

This comparison is inappropriate without modification for brittle fracture, fatigue, creep, temperature-dependent strength, or plastic collapse.

3.2 Thermal Analysis

Steady heat conduction in an isotropic solid is governed by:

where T is temperature in kelvin, k is thermal conductivity in W/(m·K), and q̇ is volumetric heat generation in W/m³. The equation assumes steady conditions and requires appropriate thermal boundary conditions.

For one-dimensional conduction through a plane wall with constant conductivity and no internal generation:

where Q̇ is heat-transfer rate in watts, A is area, T₁ − T₂ is temperature difference, and L is wall thickness. Convection at a surface is often represented by:

where h is the convection coefficient, T_s is surface temperature, and T_∞ is surrounding-fluid temperature. The coefficient should be justified by experiment, correlation, or a coupled fluid analysis; it should not be chosen merely to obtain a desired temperature.

Thermal expansion can create structural stress when expansion is restrained. For an unconstrained isotropic bar, the free strain is:

where α is the coefficient of thermal expansion and ∆T is temperature change. If a bar is fully restrained in a simplified one-dimensional elastic model, the induced stress is approximately:

where E is Young’s modulus. This relation assumes uniform temperature, elastic behavior, complete restraint, and temperature-independent properties.

3.3 Motion and Mechanism Analysis

Motion analysis evaluates position, velocity, acceleration, forces, and torques in assemblies. It may be kinematic, where motion is prescribed and reaction forces are calculated, or dynamic, where mass, inertia, forces, and constraints determine motion.

For a single-degree-of-freedom system, Newton’s law can be written as:

where m is mass, c is damping coefficient, k is stiffness, x is displacement, and F(t) is time-dependent force. This linear model assumes small motion, linear damping, and linear stiffness. Real mechanisms may require joint clearance, friction, nonlinear springs, flexible bodies, or control-system coupling.

Motion results should be checked against conservation laws, expected travel limits, acceleration continuity, and realistic joint reactions. A mechanism that reaches its target position but requires impossible motor torque is not a successful design.

4. Interpreting and Trusting CAE Results

4.1 Verification and Validation

Verification asks whether the equations have been solved correctly. It includes unit checks, solver residuals, mesh convergence, comparison with hand calculations, and benchmark solutions. Validation asks whether the mathematical model represents the real system adequately. It requires experimental data, field measurements, or trusted physical evidence.

A useful hierarchy of checks is:

• dimensional consistency and input review;
• free-body and equilibrium checks;
• comparison with a closed-form solution where available;
• mesh and time-step convergence;
• sensitivity to uncertain inputs;
• comparison with test data; and
• independent review of assumptions and conclusions.

4.2 Reading Results Beyond Colour Contours

Colour contours provide spatial context but can conceal numerical and modeling problems. The analyst should inspect numerical probes, deformed-shape scale, reaction forces, maximum and minimum locations, section cuts, contact status, and result averaging. Large deformation visualization may be magnified for clarity; the displayed shape is not necessarily the true scale.

Stress should be interpreted with knowledge of the failure mode. Displacement may govern alignment or sealing even when stress is acceptable. Temperature gradients may govern thermal fatigue. Natural frequency may be more important than static strength in a vibrating machine. The critical result is the one linked to the engineering requirement, not automatically the maximum value in the file.

5. Design Iteration and Optimization

A design optimization problem can be expressed generally as:

Here x is the vector of design variables, f(x) is the objective, g_i(x) are inequality constraints, h_j(x) are equality constraints, and x_L, x_U are variable bounds. The formulation is applicable to continuous, discrete, or mixed design variables, although the solution method must match the variable type.

Examples of design variables include wall thickness, hole diameter, rib height, material choice, joint position, and topology-density variables. Objectives may include mass, cost, compliance, energy consumption, peak temperature, or manufacturing time. Constraints may specify stress, displacement, frequency, temperature, factor of safety, or available material.

5.1 Parametric Optimization

Parametric optimization changes selected dimensions or parameters while preserving a feature-based CAD model. It is appropriate when the design concept is established and performance depends on identifiable dimensions. A design-of-experiments approach can explore the influence of variables before optimization. Sensitivity indicates how strongly a response changes when a parameter changes.

Parametric optimization is easy to communicate and manufacture because the resulting geometry remains recognizable. However, it may miss fundamentally different load paths because the topology and feature structure remain fixed.

5.2 Topology Optimization

Topology optimization determines where material should remain within a prescribed design domain. A common structural objective is compliance minimization subject to a volume constraint:

where C is compliance, F is load vector, u is displacement vector, V(x) is retained volume, V₀ is original design-domain volume, and V^* is the permitted volume fraction. Lower compliance generally means greater stiffness for the specified linear load case. The result depends strongly on supports, loads, manufacturing restrictions, mesh, minimum feature size, and filtering settings.

Topology results are not automatically production-ready. They often require reconstruction into smooth CAD geometry, evaluation under additional load cases, tolerance analysis, and manufacturing assessment. Additive manufacturing may permit complex organic forms, whereas machining may require draw directions, tool access, and minimum radii.

5.3 Multi-Criteria Optimization

Engineering decisions commonly involve competing objectives. A weighted-sum formulation is:

where f_r(x) are normalized objective functions and w_r are nonnegative weights whose sum is usually one. Normalization is essential because mass in kilograms and cost in currency cannot be meaningfully added without scaling. A weighted sum can identify a preferred compromise, but different weights produce different solutions.

A solution is Pareto-optimal if no objective can be improved without worsening at least one other objective. The set of such solutions is the Pareto front. Presenting a Pareto front to a design team makes trade-offs visible instead of hiding them inside one arbitrary weighting scheme.

6. Worked Example: Iterative Bracket Design

Consider an illustrative steel cantilever bracket carrying a vertical load of 2,000 N. A simplified design is modeled as a rectangular section with width 40 mm, thickness 8 mm, and an effective bending length of 120 mm. For an initial screening calculation, treat it as a beam and neglect local holes, fillets, shear deformation, and weight.

The second moment of area for bending about the relevant axis is:

With b = 0.040 m and t = 0.008 m:

The bending moment at the fixed support is:

The extreme-fibre distance is y = t/2 = 0.004 m. Therefore:

If the steel yield strength is 355 MPa, the elastic screening result exceeds yield. The design is unacceptable under the stated assumptions. The calculation also shows why a direct CAD-to-CAE workflow should begin with an order-of-magnitude check: a detailed contour plot should not be used to conceal an obviously inadequate section.

Suppose the thickness is increased to 10 mm while width and length remain unchanged. Then:

and y = 0.005 m, giving:

This remains slightly above the stated yield strength. A CAE model would now be created with realistic fillets, the mounting condition, load application area, material properties, and contact assumptions. Candidate iterations might add a triangular rib, increase the support fillet, or change the section shape rather than simply increasing thickness.

The engineering decision cannot be based on stress alone. The team should check displacement, mass, manufacturability, bolt loads, fatigue under repeated loading, and sensitivity to load direction. If a rib reduces mass while maintaining stiffness, its geometry can be optimized parametrically. If the load path is uncertain, several load cases should be included before topology optimization.

7. Applied Case: Lightweight Robotic End-Effector Bracket

The following is an illustrative case based on a common industrial situation.

Background

A robotic end-effector bracket must carry a 1.5 kg tool and withstand acceleration during rapid pick-and-place motion. The existing machined aluminium bracket is reliable but heavy, increasing robot energy consumption and reducing allowable acceleration.

Problem

The design team seeks a 25% mass reduction while maintaining a minimum first natural frequency of 180 Hz, a displacement below 0.20 mm under the service load, and a factor of safety of at least 2 against yield for the specified static load cases. The bracket must remain machinable on a three-axis milling machine.

Analysis and Approach

The CAD assembly was simplified by removing cosmetic grooves but retaining bolt holes, bearing seats, and fillets near the load path. A shell model was first used for rapid concept screening, followed by a solid model for the selected design. Three load cases represented vertical acceleration, horizontal acceleration, and a combined off-axis condition. Bolt interfaces were represented with connector elements and distributed bearing constraints rather than fully fixed faces, because fully fixing the faces artificially stiffened the bracket.

A mesh-convergence study refined the critical region around the fillet and bolt pattern. The team checked reactions against the applied inertial loads and compared the first bending mode with a simplified hand estimate. The initial design showed a first frequency of 205 Hz, a peak non-singular von Mises stress of 132 MPa, and a mass of 0.80 kg. The material yield strength was 275 MPa at the relevant temperature, giving a static yield ratio of approximately 2.08 before other safety factors and uncertainty allowances.

Parametric optimization varied rib thickness, rib height, fillet radius, and pocket depth. Topology optimization was used only inside a restricted design space, with preserved bolt interfaces, machining directions, minimum member size, and symmetry constraints. The resulting organic concept was remodeled as machinable ribs and pockets.

Outcome

The final design mass was 0.59 kg, a reduction of:

The final CAE model predicted a first natural frequency of 191 Hz, displacement of 0.17 mm, and a peak non-singular stress of 121 MPa for the governing load case. A prototype was manufactured and tested. Measured frequency was 187 Hz and static displacement was 0.18 mm. The difference in frequency was approximately 2.1%, which was judged acceptable after accounting for joint compliance and material-property variation.

Learning Points

The successful result did not come from topology optimization alone. It depended on realistic interfaces, multiple load cases, convergence checks, manufacturability restrictions, redesign of the raw optimization output, and physical correlation. The test also revealed that joint compliance influenced dynamic behavior more than predicted by the initial idealized model.

8. Limitations, Risk, and Professional Responsibility

CAE models contain uncertainty from loads, material properties, dimensions, friction, damping, manufacturing variation, and boundary conditions. Uncertainty should be addressed through sensitivity studies, conservative design values, tolerancing, and testing. A highly precise numerical result can still be inaccurate if the input model is wrong.

Engineers must not select assumptions solely because they produce a favourable result. Analysis reports should state model purpose, geometry revision, element type, mesh statistics, material data, load cases, constraints, solver settings, convergence evidence, validation data, and known limitations. Results that affect public safety require competent review and compliance with applicable standards, regulations, and organizational approval procedures.

AI-assisted design and generative-design tools can explore large design spaces, produce surrogate models, and identify patterns in simulation data. They do not remove the need for engineering judgment. Training data may contain hidden bias, optimization objectives may omit safety requirements, and generated geometry may be impossible to inspect or manufacture. Human responsibility for assumptions, verification, validation, and release remains essential.

9. Best Practices for an Integrated CAD–CAE Process

• Begin with requirements and failure modes, not software commands.
• Use hand calculations to establish expected magnitudes.
• Preserve design intent and analysis traceability.
• Simplify geometry deliberately and document simplifications.
• Use realistic interfaces, contacts, and load distributions.
• Check units, coordinate systems, material data, and revision status.
• Perform mesh or time-step convergence for decision-critical outputs.
• Inspect reactions, energy balance, deformation, and result locations.
• Separate numerical singularities from physically meaningful stresses.
• Validate important predictions with tests or trusted benchmarks.
• Optimize only after defining manufacturability and lifecycle constraints.
• Examine multiple load cases and uncertainty, not only the nominal case.
• Communicate trade-offs through Pareto solutions where appropriate.
• Retain an auditable record of model, solver, assumptions, and decisions.

Chapter Summary

CAD–CAE integration connects geometric design with prediction, verification, testing, and optimization. Effective analysis begins by defining the engineering question and preparing a physically meaningful model. Meshing requires appropriate element types, quality checks, and convergence studies. Structural analysis evaluates stiffness, stress, deformation, vibration, and failure-related responses; thermal analysis evaluates conduction, convection, generation, and thermally induced stress; motion analysis evaluates kinematics, dynamics, inertia, and joint behavior.

CAE results must be verified computationally and validated physically. Colour contours alone are insufficient evidence. Optimization can vary parameters, redistribute topology, or balance multiple objectives. Parametric optimization is effective for established feature-based designs, topology optimization can reveal new load paths within a defined domain, and multi-criteria methods expose trade-offs among mass, stiffness, cost, frequency, temperature, and manufacturability. Professional practice requires traceability, realistic assumptions, uncertainty awareness, ethical reporting, and confirmation through testing where consequences justify it.

Key Terms

CAD model: Digital geometric and product-definition representation of a component or assembly.

CAE: Computer-aided engineering used to simulate and evaluate physical behaviour.

Finite element method: Numerical method that discretizes a continuous physical domain into elements.

Node: Discrete point at which finite-element unknowns are stored.

Element: Finite region within which field variables are approximated.

Mesh convergence: Reduction of result changes as the discretization is refined.

Boundary condition: Prescribed physical restriction, input, or interaction at a model boundary.

Contact: Mechanical interaction between bodies or surfaces.

Verification: Demonstration that the numerical model and solution process are correct.

Validation: Demonstration that the model represents the physical system adequately.

Stress singularity: Mathematical result that becomes unbounded or mesh-dependent at an idealized feature.

Factor of safety: Ratio comparing an allowable or strength value with a predicted demand.

Design variable: Adjustable parameter that defines a candidate design.

Objective function: Quantity to be minimized or maximized in optimization.

Constraint: Requirement that a design must satisfy.

Parametric optimization: Optimization of selected dimensions or feature parameters.

Topology optimization: Redistribution of material within a defined design domain.

Pareto-optimal solution: Solution for which no objective can improve without worsening another.

Digital thread: Traceable connection among requirements, models, analyses, tests, and manufacturing information.

Review Questions

Short Answer Questions

1. Why must CAD geometry often be simplified before finite element analysis?
2. Distinguish verification from validation.
3. What is the purpose of a mesh-convergence study?
4. Why can a maximum stress in a contour plot be misleading?
5. State one important difference between parametric and topology optimization.

Long Answer Questions

1. Describe a complete CAD–CAE workflow from requirements definition to design release.
2. Explain how element type, mesh quality, contacts, and boundary conditions affect structural-analysis reliability.
3. Discuss the assumptions and limitations of using beam equations as a preliminary design check.
4. Explain how thermal expansion can generate stress in a restrained component.
5. Compare weighted-sum multi-objective optimization with presentation of a Pareto front.

Analytical / Application Questions

1. A cantilever carries a known end load. Explain how you would combine a hand calculation, a finite element model, mesh refinement, and an experimental test to establish confidence in the design.
2. A topology-optimization result is lightweight but contains thin organic members that cannot be machined. Propose a redesign and verification strategy.
3. A bracket meets its static stress requirement but fails its vibration requirement. Identify possible design variables, additional analyses, and trade-offs that should be considered.

Multiple Choice Questions with Answers

Question 1

Which activity is primarily associated with verification?

A. Comparing predicted displacement with a prototype measurement

B. Checking mesh convergence and solver residuals

C. Interviewing a machine operator about usability

D. Selecting a lower-cost material without analysis

Correct Answer: B. Checking mesh convergence and solver residuals

Question 2

For a linear structural finite element model, what does K u = F represent?

A. A thermal energy balance

B. A global stiffness equilibrium equation

C. A fatigue damage equation

D. A fluid continuity equation

Correct Answer: B. A global stiffness equilibrium equation

Question 3

A mesh refinement changes a reported displacement from 0.200 mm to 0.196 mm. Using the refined value as denominator, the percentage change is closest to:

A. 0.4%

B. 2.0%

C. 4.0%

D. 20.0%

Correct Answer: B. 2.0%

Question 4

Which optimization method is most directly suited to changing rib height, pocket depth, and fillet radius while retaining the CAD feature structure?

A. Parametric optimization

B. Topology optimization

C. Random material removal

D. Modal testing

Correct Answer: A. Parametric optimization

Question 5

In the simplified restrained-bar relation σ_th = E α ∆T, complete restraint means that:

A. The bar is free to expand in every direction

B. Thermal expansion is prevented in the modeled direction

C. The material has zero thermal conductivity

D. The temperature is necessarily zero

Correct Answer: B. Thermal expansion is prevented in the modeled direction

Question 6

What is the principal purpose of a topology-optimization volume constraint?

A. To prescribe the colour scale of the result

B. To limit the amount of material retained

C. To eliminate all boundary conditions

D. To guarantee machinability

Correct Answer: B. To limit the amount of material retained

Question 7

A stress peak occurs at an idealized sharp re-entrant corner and increases as the mesh is refined. This is most likely:

A. A converged physical stress

B. A stress singularity

C. A material phase transformation

D. A rigid-body mode

Correct Answer: B. A stress singularity

Question 8

Which result is most directly associated with a mechanism’s dynamic feasibility?

A. Required actuator torque during motion

B. CAD file size

C. Number of sketch constraints

D. Surface colour

Correct Answer: A. Required actuator torque during motion

Question 9

Why must objectives be normalized before using a weighted-sum formulation involving mass and cost?

A. To make all geometry two-dimensional

B. To place quantities with different units on a comparable scale

C. To remove all design constraints

D. To guarantee a unique global optimum

Correct Answer: B. To place quantities with different units on a comparable scale

Question 10

A design is Pareto-optimal when:

A. Every objective has its individual absolute minimum simultaneously

B. No objective can improve without worsening at least one other objective

C. The design has the lowest mass regardless of constraints

D. The design has the highest stress

Correct Answer: B. No objective can improve without worsening at least one other objective