Research theme five of five
Computational mechanics and scientific computing
Exact results still earn their place. They are the only way to know whether an approximation, including a learned one, is telling the truth.
The problem
Scientific machine learning is entering structural dynamics faster than the error analysis that should accompany it. A surrogate that reproduces a training set to three significant figures can still be wrong in exactly the region a reliability calculation depends on, because failure probabilities are governed by rare events, and a training set spreads its points over the bulk.
The position taken here is to work on what the network is asked to learn. It is to supply the layer underneath: exact solutions, reference problems with published answers, and error statements that a regulator could apply.
Why it matters
An approximate method is trusted on the strength of the exact case it was checked against, so exact and semi-analytical results are the instruments that calibrate everything else. Dynamic stiffness carries this furthest: a frequency-dependent element description gives the exact response of a member at any frequency, so a structure that a mesh would need thousands of elements to resolve at high frequency is handled by a handful. The same discipline governs the machine-learning work here. A surrogate is judged by the decision it supports, so the question is where to place training points so that a failure probability moves, and a reproducible benchmark is what lets anyone else check the claim.
Current frontier
- exact and semi-analytical mechanics
- dynamic stiffness and exact member formulations
- spectral and special-function solutions
- exact stochastic operators
- stochastic computation and reduced-order modelling
- neural operators, graph networks and measure operators
- physics-informed and physics-encoded methods
- open benchmarks and reproducible examples
Signature concepts
- exact benchmarks
- dynamic stiffness
- model reduction
- operator learning
- measure-based computation
Open problems
- What error bound can be placed on a failure probability computed through a trained surrogate?
- Where should training points be placed so that they change a reliability estimate?
- Which exact stochastic results can serve as reference solutions for operator learning?
Selected papers
- Exact dynamic stiffness formulations and vibration response analysis of orthotropic viscoelastic plate built-up structures
X. Liu, X. Liu, S. Adhikari · Computers & Structures 302, 107455 · 2024 - MATLAB implementation of physics informed deep neural networks for forward and inverse structural vibration problems
T. Chatterjee, M. I. Friswell, S. Adhikari, H. H. Khodaparast · Aerospace Research Communications 2 · 2024 - Exact energy harvesting analysis of multimodal piezoelectric beams using the dynamic stiffness method
X. Liu, Y. Wang, S. Adhikari, W. Zhou · Computers & Structures 313, 107746 · 2025 - A partitioned combined computational method for multi-scale dynamic systems
P. Yuan, S. Adhikari, Y. Dong · International Journal for Numerical Methods in Engineering 124(16) · 2023 - Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms
S. Adhikari and co-workers · 2011
Foundations
Finite element analysis of damped systems, dynamic stiffness formulations, modal and reduced-order methods, and Monte Carlo and stochastic reduced models.