To ensure absolute precision before numerical substitution, I utilized SymPy for the symbolic differentiation and integration of the shape functions. This allowed for an exact derivation of the strain-displacement matrix and the element stiffness matrix for a four-node quadrilateral element. Once the local matrices were verified, I applied a local-to-global mapping algorithm (edofMat) to assemble the global structural stiffness matrix under plane stress conditions. 

Topology optimization is prone to numerical anomalies like alternating solid and void pixels. To solve this, I engineered a radius-based sensitivity filter that weights the structural compliance gradient of neighboring elements, ensuring mesh independence. Finally, to strictly enforce the structural volume constraint, I utilized the Optimality Criteria (OC) method. I implemented a bisection root-finding loop to dynamically calculate the optimal Lagrange multiplier, ensuring the algorithm smoothly converges on a design that minimizes compliance while strictly adhering to the maximum weight limit. 

Developing this topology optimization solver from scratch was a rigorous exercise in translating theoretical solid mechanics into functional code. By manually deriving the global stiffness matrices and programming the SIMP penalization logic, I gained a fundamental understanding of how commercial FEA software operates. This project reinforced my ability to leverage Python and numerical methods to solve complex, computationally heavy mechanical design constraints, ensuring structures that are both functional and optimized for efficiency.