Constrained Optimisation with fmincon for layered sound absorber design

Project info

• Field of studyAcoustics and Audio
• LevelMaster's thesis (S2)
• Method typeOptimisation
• Estimated build time± 5 working days
• Toolsfmincon, optimoptions (Optimization Toolbox)
• PriceUS$28 - US$169

Research goal

Find the decision variables that optimise the objective in layered sound absorber design with fmincon under constraints.

Variables

Example result

• 0.849Optimum x
• 0.72Optimum y
• 0.023Objective f*
• 13Iterations
• 66Function evaluations
• 0Constraint value

fmincon finds the optimum First layer thickness=0.849 and Second layer thickness=0.720 with objective Sound absorption coefficient=0.023 after 13 iterations; the circular constraint of radius 1.11 is active (constraint value 0.0e+00).

Computed from illustrative synthetic data, not from a real study.

About Constrained Optimisation with fmincon

fmincon minimises a nonlinear objective subject to linear and nonlinear constraints and variable bounds. It is used in design optimisation, model parameter calibration, and constrained engineering problems.

Steps

1. 1Formulate the objective and all constraints
2. 2Write the objective and nonlinear constraint functions
3. 3Set the starting point and variable bounds
4. 4Set options with optimoptions (algorithm, tolerances)
5. 5Run fmincon and check exitflag
6. 6Review active constraints and Lagrange multipliers

Assumptions to check

• Functions and constraints are smooth and differentiable
• The result is a local optimum so different starting points should be tried
• A feasible point satisfying all constraints exists

Reading and reporting the results

Report the solution, objective value, exitflag, and active constraints. Explain multipliers as the sensitivity of the objective to the constraints.

Example MATLAB code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

sound_absorption_coefficient = @(x) (x(1) - 3)^2 + (x(2) - 2)^2;           % objective
nonlcon = @(x) deal(x(1)^2 + x(2)^2 - 9, []);       % x1^2 + x2^2 <= 9
A = [1 1];  b = 4;                                  % x1 + x2 <= 4
lb = [0 0];  ub = [5 5];
first_layer_thickness = [0.5 0.5];                                 % starting point

opts = optimoptions('fmincon', 'Algorithm', 'sqp', 'Display', 'off', ...
    'OptimalityTolerance', 1e-10);
[x, fval, exitflag, output, lambda] = fmincon(sound_absorption_coefficient, first_layer_thickness, A, b, [], [], lb, ub, nonlcon, opts);

fprintf('Solution: x1 = %.4f, x2 = %.4f\n', x(1), x(2));
fprintf('Objective: %.5f, exitflag = %d, iterations = %d\n', fval, exitflag, output.iterations);
fprintf('Multiplier (linear): %.4f, (nonlinear): %.4f\n', lambda.ineqlin, lambda.ineqnonlin);

[g1, g2] = meshgrid(linspace(0, 5, 200));
Z = arrayfun(@(a, b) sound_absorption_coefficient([a b]), g1, g2);
figure; contour(g1, g2, Z, 25); hold on;
th = linspace(0, pi/2, 100);
plot(3*cos(th), 3*sin(th), 'r', 'LineWidth', 1.2);
plot([0 4], [4 0], 'k--', x(1), x(2), 'm*', 'MarkerSize', 10);
xlabel('x_1'); ylabel('x_2'); legend('Objective', 'Circle', 'Line', 'Linear', 'Optimum');

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions