Constrained Engineering Optimization Trade Study Project

This project produces a constrained engineering optimization trade-study package. The deliverable is not only an optimum number. It is a decision file that states the design variables, objective, constraints, model assumptions, candidate solutions, sensitivity, uncertainty, validation evidence, and release recommendation.

Optimization is useful only when the optimized design is feasible, robust, explainable, and connected to engineering evidence. A solver can find a mathematically attractive point that violates manufacturing limits, depends on an invalid surrogate model, has no margin to uncertainty, or cannot be defended in a design review. This project shows how to avoid that failure.

Project Objective

Choose a support-bracket design that minimizes mass while meeting strength, stiffness, dynamic, envelope, and manufacturing constraints.

The final package must include:

• decision boundary and requirement table;
• design variables and fixed assumptions;
• scaled objective function;
• constraint definitions and sign convention;
• candidate-screening calculations;
• sensitivity and Pareto interpretation;
• robustness checks for model uncertainty;
• validation evidence required before release;
• final recommendation and open actions.

The bracket scenario is intentionally simple. The workflow transfers to pumps, heat exchangers, power converters, control tuning, production schedules, network capacity, materials selection, and maintenance planning whenever an engineering team must optimize under constraints.

Engineering Scenario

A compact aluminum support bracket carries an instrument module on a machine frame. The design team can change two geometric variables before releasing drawings:

Variable	Meaning	Allowed range
t	base plate thickness	5.0 to 8.0\ \text{mm}
h	rib height	8.0 to 18.0\ \text{mm}

The bracket must satisfy:

Requirement	Limit
maximum von Mises stress	\sigma \le 120\ \text{MPa}
tip deflection	\delta \le 0.50\ \text{mm}
first natural frequency	f_1 \ge 75\ \text{Hz}
rib height envelope	h \le 18\ \text{mm}
manufacturing preference	avoid unnecessary mass and tall ribs

The model uses a fast surrogate fitted from finite element runs and a small number of prototype checks. The surrogate is adequate for trade-study screening, not for final drawing release by itself.

Model Boundary

The optimization boundary is:

• fixed material grade and heat treatment;
• fixed bolt pattern and load direction;
• fixed instrument module mass and interface geometry;
• static load plus first-mode frequency screen;
• no fatigue, corrosion, thermal distortion, or bolted-joint slip in the surrogate;
• final release requires detailed finite element verification and prototype measurement.

This boundary matters. If the load path, bolt preload, material, or operating vibration changes, the optimization evidence must be reopened.

Surrogate Equations

Use the following fitted screening models:

Mass estimate:

where m is in kg when t and h are in mm.

Stress estimate:

where \sigma is in MPa.

Deflection estimate:

where \delta is in mm.

First-mode frequency estimate:

where f_1 is in Hz.

These equations are not universal bracket design formulas. They are local surrogate models for this bracket family. Their validity range is only the stated t and h window.

Scaled Objective

Use a weighted objective that penalizes mass and rib height:

with:

The objective is dimensionless. Scaling prevents kg and mm from being combined as if they had the same meaning. The weights state the engineering preference: mass matters more than rib height, but a tall rib is still undesirable because it consumes envelope and complicates handling.

Candidate Screening

Evaluate five candidate designs.

Candidate	t	h	Mass m	Stress \sigma	Deflection \delta	Frequency f_1
A	5.5	10	1.43\ \text{kg}	146.9\ \text{MPa}	0.673\ \text{mm}	66.2\ \text{Hz}
B	6.0	12	1.58\ \text{kg}	130.4\ \text{MPa}	0.529\ \text{mm}	69.7\ \text{Hz}
C	6.5	14	1.73\ \text{kg}	116.7\ \text{MPa}	0.425\ \text{mm}	76.9\ \text{Hz}
D	7.0	14	1.81\ \text{kg}	108.3\ \text{MPa}	0.366\ \text{mm}	81.0\ \text{Hz}
E	6.5	16	1.80\ \text{kg}	113.7\ \text{MPa}	0.404\ \text{mm}	79.8\ \text{Hz}

Worked Check for Candidate C

Mass:

Stress:

Deflection:

Frequency:

Candidate C passes the nominal screens:

Engineering Comment

Candidate C is the lightest nominal feasible design in the candidate table. A weak trade study would stop here. A stronger trade study asks whether this pass is robust to surrogate error, measurement uncertainty, load uncertainty, and unmodelled details.

Objective Values

Compute the scaled objective for the feasible nominal candidates.

For candidate C:

For candidate D:

For candidate E:

Nominally, candidate C has the best objective because it is lighter and no taller than the reference.

Robustness Check

Apply conservative uncertainty guards to the surrogate responses:

Response	Guard
stress	multiply by 1.08
deflection	multiply by 1.10
frequency	multiply by 0.96

These guards represent model-form error, mesh sensitivity, material-property scatter, load uncertainty, and measurement uncertainty in a simplified way. They are not a substitute for final validation.

Candidate C guarded responses:

Candidate C fails the guarded stress and frequency checks.

Candidate D guarded responses:

Candidate D passes the guarded checks.

Engineering Comment

The nominal optimum is not the release recommendation. Candidate C is attractive mathematically but fragile. Candidate D is slightly heavier, but it preserves margin under the uncertainty guards. This is the central engineering lesson: optimization should not erase the margin needed to make the decision credible.

Constraint Sensitivity

Use finite differences around candidate D to understand which variables control margin.

Stress at candidate D:

Stress at t=7.5, h=14:

Approximate thickness sensitivity:

Stress at t=7.0, h=16:

Approximate rib-height sensitivity:

Engineering Comment

Near candidate D, thickness is much more effective than rib height for reducing stress. If the stress constraint tightens, adding thickness is the efficient response. If the frequency constraint tightens, rib height may still be useful because it raises stiffness without increasing base thickness as much. Sensitivity turns the optimization result into an engineering explanation.

Pareto Interpretation

The feasible candidates show a trade-off:

• candidate C is the lightest nominal feasible design, but it is not robust;
• candidate D is heavier but robust under the guards;
• candidate E improves frequency and deflection with more rib height, but it has weaker envelope preference than D;
• candidates A and B are dominated because they fail constraints.

A Pareto view is useful because the final decision may change if the business priority changes. If the project later values envelope height more strongly, D remains attractive. If vibration separation becomes more important, E may deserve a second look. If mass becomes critical, the team should improve the model and test candidate C rather than release it on weak margin.

Validation Evidence

Before releasing candidate D, collect evidence that the surrogate is valid enough:

Evidence	Required check
finite element verification	detailed model at candidate D with mesh convergence on stress and deflection
boundary-condition review	bolt preload, contact, load direction and support stiffness match test setup
prototype static test	measured deflection within agreed tolerance of prediction
modal test	first mode above the guarded requirement
material certificate	elastic modulus and yield basis match model assumptions
manufacturing review	rib height, tool access and tolerance stack-up acceptable
sensitivity record	documented effect of load, thickness, rib height and material variation
release traceability	model version, solver settings, input data and decision owner recorded

Optimization without validation is only a suggestion. The release package must show that the chosen design remains acceptable when the model meets the physical bracket.

Final Recommendation

Recommend candidate D for detailed verification:

The recommendation is conditional:

• do not release drawings until detailed finite element and prototype checks confirm stress, deflection and modal margins;
• keep candidate C as a weight-reduction option only if additional validation reduces the uncertainty guards or if a local design change restores robust frequency margin;
• reject candidates A and B because they fail nominal constraints;
• reject candidate E as the baseline because it is taller than D without enough benefit to justify the envelope penalty.

Final Deliverable

The completed trade-study package should contain:

1. requirement and decision boundary;
2. design variables, limits and fixed assumptions;
3. surrogate equations and validity range;
4. scaled objective with documented weights;
5. candidate table and feasibility checks;
6. nominal optimum and robust recommendation;
7. sensitivity and Pareto interpretation;
8. validation evidence plan;
9. final decision, open actions and owner sign-off.

Limitations

This project uses a small candidate table and fitted surrogate equations. A production optimization may require a larger design space, formal optimizer settings, constraint activity checks, nonlinear finite element analysis, fatigue review, bolted-joint review, manufacturing tolerance analysis, cost modeling, uncertainty propagation, and independent design review.

The result is also local. It is valid for the stated bracket family and requirement set. A new load case, material, bolt pattern, manufacturing route, vibration environment, or acceptance limit can change the feasible region and the recommended design.

Common Mistakes

Common optimization trade-study errors include:

• optimizing before the constraints are complete;
• combining units in an unscaled objective;
• reporting the solver result without feasibility checks;
• treating a surrogate model as a physical truth;
• ignoring constraint margins when uncertainty is comparable to the margin;
• selecting the lightest nominal design even when it is fragile;
• hiding business preferences inside unexplained weights;
• failing to record model version, data source and solver settings;
• releasing an optimized design without validation evidence.

The engineering standard is not “the optimizer converged.” The standard is: the chosen design is feasible, robust enough for the decision, validated against evidence, and explainable to the engineers who must own it.