Future material models must address complicated and interconnected thermal, mechanical and chemical processes that go far beyond the classical approaches. Growth and deformation of biological tissues, deformation of composite materials and shape memory alloys, flows of polymer or metal melts, flows of mixtures and geophysical materials, liquefaction of soil, transport processes in porous media and their interaction with the substrate can serve as motivations as well as targets for important **real-world applications**.

Developing descriptive and mathematically correct physical models for the materials in the complex processes they undergo represents only a starting point of the whole mathematical modelling procedure. We need to numerically simulate the **behaviour of complex materials in** various **complicated processes**. This requires new approaches guaranteeing, in particular,

**accuracy and reliability**(the simulation results must be accompanied with fully computable a posteriori error bounds),**efficiency**(the computational cost must allow us to go beyond existing horizons in order to extract qualitatively

new information), and**robustness**(the new tools must be applicable to a sufficiently large class of problems, which will justify the

effort needed for their development).

The goal of the project is to develop accurate, efficient and robust numerical methods that allow one to perform large scale simulations for models based on the novel and promising **implicit constitutive theory**. A natural part of the goal is rigorous mathematical analysis of the models.

### Challenges

The modelling challenge we are going to address has four main aspects:

- The theory that is developed to describe the macroscopic behaviour of complex bodies must be based on a continuum mechanics approach without incorporating
*ad hoc*state variables that do not have a clear physical meaning and without artificially combining microscopic and macroscopic theories. This shall bring*clarity and simplicity to the theory*. The model must obey the laws of thermodynamics and the derivation of the model shall be based on a small set of well articulated and justified fundamental assumptions. - Having a candidate for a physically relevant model satisfying all the requirements discussed above, one has to prove its mathematical consistency. We must prove the
*existence*of a solution to the governing equations. Finally, we need to understand asymptotic properties and transitional behaviour of the solution as well as its regularity. - Since solving the governing equations exactly cannot be practically achieved, one must use numerical methods and find approximate solutions. The choice and analysis of the discretisation and the subsequent solution of the resulting finite dimensional problem must be considered together in order to achieve computational efficiency.
- The finite dimensional problems can be very large. Therefore we must consider iterative algorithms for their solution (including appropriate preconditioning) with adequate stopping criteria. In the presence of singularities efficiency requires adaptive discretisation using computable
*local a posteriori error estimates*.

These four aspects will be adressed in a nonstandard way:

- The mathematical models will be based on novel implicit constitutive theory.
- The structure of the mathematical models requires the reconsideration of many classical approaches in the mathematical theory of partial differential equations and the development of new ones. In particular, this might involve revisiting the concept of a solution to the problem and the concept of
*well-posedness*. - The complexity of the mathematical models calls for the investigation on model reduction – the identification of simplified models via
*rigorous*study of singular limits. - The a posteriori error estimates shall provide a control on
*all*possible sources of error in the whole process of numerically solving the problem, including algebraic errors. - The project has a vision of addressing simultaneously all the aspects mentioned above. We believe that they are so closely interrelated that no breakthrough in mathematical modelling can be achieved without emphasizing the
*holistic approach*as the main principle.