Final optimized simply supported beam topology using 50% of the design-domain material with a 5 × 5 minimum-thickness filter.
Simply Supported Beam Topology Optimization
Role: Lead Developer
Tools: MATLAB, finite element analysis, topology optimization, level-set methods, numerical simulation, engineering visualization
Project Overview
This project developed a MATLAB-based topology optimization tool for a simply supported beam, a structural model commonly used to represent bridges, machine frames, brackets, supports, and other load-bearing mechanical components. The model applies a downward load near the center of the beam and supports the structure with a pin on one end and a roller on the other, which matches a standard real-world simply supported beam condition.
The goal was to determine how material should be distributed inside a rectangular design domain so the beam can carry the applied load efficiently while using only a limited amount of material. Instead of manually designing ribs, cutouts, or internal supports, the optimization process automatically identifies efficient load paths between the center load and the two supports.
The final simulation used a 120 × 45 finite element mesh and targeted a 50% material volume fraction. The optimized result reached a final volume fraction of 0.4996, with only 0.0803% error from the material constraint. A 5 × 5 minimum-thickness filter was added to reduce unrealistic thin members and produce a more practical structural layout.
Methods
The beam was modeled as a two-dimensional finite element design domain. Each finite element represents a small region of material that can remain solid, become void, or evolve near the material boundary during optimization. The structural objective was to minimize compliance, which measures how flexible the beam is under load. Lower compliance means the structure is stiffer and more efficient.
The compliance objective was: θ = FᵀU
where θ is compliance, F is the applied load vector, and U is the displacement vector.
The finite element equilibrium equation was: K(ρ)U = F
where K(ρ) is the stiffness matrix that depends on the material distribution ρ.
The optimization problem was formulated as: Minimize θ(ρ) = FᵀU
Subject to: V / V₀ ≤ 0.50, K(ρ)U = F, 0 ≤ ρ ≤ 1
The volume constraint limits the optimized structure to 50% of the original design-domain material. This represents a practical design restriction similar to mass, cost, or material usage limits in real engineering applications.
The simply supported boundary conditions were:
Left support: ux = 0, uy = 0
Right support: uy = 0
Center load: downward force applied near midspan
These constraints allow the beam to behave like a real simply supported structure: one end prevents horizontal and vertical motion, while the other end prevents vertical motion but allows horizontal expansion.
Results and Interpretation
The final optimized structure reached:
Mesh = 120 × 45 elements
Target volume fraction = 0.5000
Final volume fraction = 0.4996
Volume error = 0.0803%
Minimum thickness filter = 5 × 5
Final compliance = 189,720.43
Iterations = 162
The volume result shows that the optimizer matched the 50% material target very closely. The final volume fraction of 0.4996 means the design used nearly exactly half of the original material domain while still forming a connected and efficient structure.
Compliance and volume history showing convergence toward the 50% material target.
The optimized topology forms structural members that connect the load point to the two supports. This is physically meaningful because the applied load must travel through the material to reach the support reactions. The resulting geometry shows where stiffness is most valuable: material is preserved along the main load-transfer paths and removed from regions that contribute less to carrying the load.
The 5 × 5 minimum-thickness filter improved the final design by reducing overly thin members. Without this type of control, topology optimization can generate very slender features that may be difficult to manufacture or unrealistic for physical structures. The filter encourages thicker, smoother members while preserving the overall optimized load path.
Optimized material layout showing the main load paths between the center load and the two supports.
Validation, Skills, and Significance
This project demonstrates how topology optimization can support early-stage mechanical and structural design. In real engineering workflows, designers often need to reduce weight, material usage, or cost while preserving stiffness and load-carrying performance. This tool automates that process by using finite element analysis to evaluate structural performance and optimization logic to evolve the material layout.
Validation focused on confirming that the beam setup, boundary conditions, material constraint, and optimization behavior were physically consistent. The final design maintained a volume fraction of 0.4996 against a 0.5000 target, giving a volume error of only 0.0803%. The pinned and roller supports created a standard simply supported beam condition, and the resulting topology formed a clear load path from the center load to both supports.
This project demonstrates finite element analysis, topology optimization, level-set methods, stiffness-based design, numerical optimization, MATLAB software development, boundary condition implementation, minimum-thickness control, and engineering interpretation of optimized structures. The workflow is directly connected to real-world design problems involving lightweight structures, brackets, supports, bridges, frames, and mechanical components where material must be placed efficiently.