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iFEM — Inverse Finite Element Methods

Interactive study notes — from continuum mechanics to learning-based inverse methods

Topic Map

Foundations Core iFEM Inverse-FEM Family Modern Developments Prerequisite Shared concepts

Foundations

1. Continuum Mechanics Primer — Strain, stress, elasticity, hyperelasticity 2. The Forward Finite Element Method — Weak form, shape functions, assembly, beams & plates 3. Inverse Problems & Regularization — Ill-posedness, least squares, SVD, Tikhonov

Core iFEM — Shape Sensing

4. The iFEM Principle — Tessler–Spangler least-squares shape sensing 5. iFEM Elements & Formulations — Inverse beam, iMIN3, iQS4, RZT composites 6. Sensors, Placement & Deployment — Strain gauges, FBG fibers, sparse sensing

The Wider Inverse-FEM Family

7. Material Identification — FEMU, Virtual Fields Method, DIC 8. Load & Contact Force Reconstruction — Influence matrices, tactile sensing 9. Reference Configuration & Prestress — Inverse elastostatics, unloaded shape, residual stress

Modern Developments

10. Adjoints & Differentiable FEM — Sensitivities, adjoint method, implicit AD, differentiable solvers 11. Learning-Enhanced Inversion — Virtual strains, surrogates, PINNs, neural operators 12. Digital Twins & Frontiers — Assimilation, uncertainty, real-time iFEM, soft-robot proprioception