RP2 (Principal Developer: IICT-BAS)


The finite element method (FEM) is the most powerful representative of mesh methods for numerical solution of differential equations which describe multiscale and multiphysics problems in science and engineering. The state-of-the-art FEMs are determined by the development of mixed FEM, discontinuous Galerkin methods, nonconforming finite elements, isoparametric analysis. The robust methods for strongly heterogeneous media with strong coefficient jumps are to be noticed. An important feature of the mesh methods is the existence of local basis of the discrete space where the approximate solution is defined. This provides opportunities to construct efficient parallel algorithms based on partitioning of the mesh based graph representation of computational domain. The recent achievements in this field lead to scalable algorithms for multiprocessor systems and MPI implementations. There are also efficient implementations for GPUs (Open MP is used, e.g.) for certain classes of problems. The development of new approaches for hybridization the numerical methods and algorithms is expected to provide a basis for future scalable algorithms for high performance systems with hybrid architecture. The recent advanced applications of FEM concern computer modelling of more and more complex high tech processes and materials. The approximation of such problems leads to huge size of the discrete problems requiring Big Data analysis. 

  1. Advanced Finite Element Methods for Strongly Heterogeneous Problems

The modern FEM is a computational technology for numerical solution of multiscale and multiphysics problems in frontiers of science and engineering. The planned research activities will address such challenging topics as: (i) robust discretization of problems with extremely heterogeneous coefficients; (ii) stable splittings and adaptive refinement and/or time stepping for strongly coupled problems; (iii) numerical methods for advanced nonlinear phenomena, including in particular beyond state-of-the-art models with fractional derivatives. New advanced methods for µFEM analysis will be developed. A key topic in this area is the voxel data processing based on computed tomography (CT) images of microstructures. The addressed applications will include 3D digitalization and prototyping, fabless technologies for design of new composite and/or porous materials, quality control of technological processes and nondestructive fault detection, high-tech innovations in biomedical engineering. A new direction of image segmentation will be developed, where mass-conservation properties of the CT scanned specimens are incorporated in the segmentation process as constraints. This task is based on intensive use of the unique capabilities of the Laboratory for 3D digitalization and   microstructural analysis.

2. Scalable Algorithms and Software Implementations on Heterogeneous High Performance Computer Architectures

This task has a special focus on scalability for advanced high-performance and distributed systems, including Big Data, Grid and Cloud Technologies. Unlike most of the Big Data applications, which make statistics on sets consisting of huge amount of uncorrelated point data, (for instance) in microstructure simulations there is a demand for statistics on sets each object of which has large, very large or extreme dimension. The following challenging parallel methods, algorithms and software implementations will be addressed: (i) fully algebraic multilevel solvers and their efficient mapping on the graph representation of heterogeneous computing systems, including multicore nodes and accelerators; (ii) multiscale methods and algorithms for multiphysics applications in strongly heterogeneous media and uncertain data; (iii) new hybrid parallel methods and algorithms integrating direct-iterative, sparse-dense, etc. hybridization approaches; (iv) algorithms that fully avoid global communications-redesign of existing software implementations avoiding such operations like FFT, dot products, etc.; (v) new methods and algorithms reducing/avoiding synchronizations; (vi) new fault-tolerance algorithms with robust self-correcting mechanisms providing guaranteed stability and global accuracy/convergence.

3. Large-Scale Scientific Computing:  Advanced Applications .

Microstructure analysis: The µFEM analysis of advanced composite and/or porous materials and biology tissues is based on CT image processing. In many cases the numerical upscaling (numerical homogenization) is the unique opportunity for determining of the effective material characteristics. Let us note that for some advanced applications the full resolution of micro CT scanner is needed, going beyond the limit of O(109) degrees of freedom of the µFEM  model. This fully concerns the advanced equipment of the planned laboratory for 3D digitalization. The hierarchical modelling includes cluster analysis of nano particles as well as molecular dynamics simulations. The Data Analytics of uncertain Big Data includes advanced characterization techniques based on anisotropy indexing.

FEM Simulation of coupled problems: The large-scale computer simulation of coupled hydro-electro-thermo-mechanical interactions in strongly heterogeneous media is often a problem beyond the scope of nowadays commercial software systems. The foreseen engineering applications include: industrial flows, filtration and environmental remediation; parametric analysis of nonlinear dynamical systems, including analysis of vibrations; biomedical engineering, including advanced radio frequency electrosurgery and electromagnetic bio-resonance therapy instruments and technologies.

The planned applications are especially dedicated to IT support of innovative SMEs.