### MSc/PhD Research Topics for Prospective
Postgraduate Students in Computational Mathematics, Applied Analysis and Solid
Mechanics

· Analysis
of Linear and Nonlinear Localised Boundary-Domain Integral and
Integro-Differential Equations.

Aim: To investigate existence and uniqueness
of solution and spectral properties of localised boundary-domain integral and
integro-differential equations needed for development of a new family of fast
convergent and robust iteration methods for numerical solution of linear and
nonlinear boundary value problems for partial differential equations with
variable coefficients.

· Computational
Implementation of Linear and Nonlinear Localised Boundary-Domain Integral and
Integro-Differential Equations.

Aim: To develop a new family of fast
convergent and robust mesh-based and mesh-less iteration methods and computer
codes for numerical solution of linear and nonlinear boundary value problems
for partial differential equations with variable coefficients reduced to
localised boundary-domain integral and integro-differential equations.

·
**Advanced
Numerical Methods and Computer Modelling of Crack Nucleation, Initiation and
Propagation under Fatigue, Creep and Dynamic Loading. **

Aim:** **Application of normalised equivalent
stress functionals to modelling crack nucleation/ initiation in structure
elements with stress concentrators and the crack propagation through damaged
material as a united process. This leads to analysis and numerical solution of
non-linear Volterra integral equation with analytically or numerically
calculated kernels.

·
**Free Boundary
Problem of (Possibly Degenerate) PDEs of Elasticity with Damage**

Aim: Modelling
rock percussive drilling as a stationary-periodic process. The unknown boundary
consists of traction-free and contact parts with or without boundary rupture.
This reduces the model to a free-boundary problem of (possibly degenerate) PDEs
of elasticity with damage. Analysis of the problem and developing numerical
iterative solution algorithms based on the boundary (-domain) integral equation
method is to be done in the project.** **

·
**Mathematical
Problems of Damage Mechanics.**

Aim:
The material micro-damage under static, dynamic or fatigue mechanical loading
decreases material elastic macro-module, that is, causes damage softening. This
manifests itself in non-monotonicity of the stress-strain diagram and the
material response appears to be highly nonlinear even for originally linear
elastic materials. This generally may lead to degeneration of the partial
differential equation type for the damaged materials. The boundary value
problems for such nonlinear equations are to be analysed and relations of
possible solution localisation with the cohesive zone model are to be
investigated.

If you are interested in one of the topics,
contact Prof. Sergey E. Mikhailov
for more information