Automatic Differentiation Using Dual Numbers: A Clear and Intuitive Understanding
In modern machine learning and scientific computing, one requirement comes up again and again: computing derivatives accurately, efficiently, and automatically.
Neural Networks as Functions: Building the Right Intuition
When we first encounter neural networks, they often seem mysterious: a web of interconnected nodes, weights, and activations. But if we strip away all the complexity,
u–p Formulation for Incompressible Solids: From Deriving Full Governing Equations to FEM Matrices
In solid mechanics, we usually solve problems by calculating displacements — how every point inside a solid moves when forces are applied.
Thermo-Piezoelectric FEM: From Deriving Full Governing Equations to Matrix Formulation
Piezoelectric materials are already powerful due to their ability to couple mechanical and electrical fields — but when we add temperature into the mix,
Piezoelectric FEM: Deriving Element Matrices for an 8-Noded Solid Element
Piezoelectric materials exhibit a two-way coupling between electrical and mechanical fields. When you apply voltage to them, they deform. When you apply mechanical stress, they generate electrical charge.