in Applied Mathematics, Applied Analysis and Solid Mechanics
Analysis of Linear and Nonlinear Localised Boundary-Domain Integral and Integro-Differential Equations. Investigation of 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.Developing 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. 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 DamageModelling 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.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.