Creation and Analysis of Isogeometric and Adaptive Solver Algorithms
The research that is being conducted in the group (led by Prof. Maciej Paszyński) focuses on the development and analysis of algorithms for simulations using modern isogeometric and adaptive finite element methods.
The finite element method is one of the most widely used numerical methods for simulating various engineering processes. It is used for analyzing the structural integrity of buildings, simulating car collisions, calculating the acoustics of automobiles, assessing the resilience of aircraft structures regarding turbulence or high-speed effects, and evaluating the mechanical durability of mobile phones, among many other applications. The finite element method has also been applied in geophysics; for instance, to simulate the propagation of electromagnetic and acoustic waves from antennas that are placed in boreholes in order to identify subsurface layers (including oil and gas reserves). In addition, it has been employed in biomechanics as well, such as simulating blood flow through a bypass model, the impact of electromagnetic waves on the human head, simulating the propagation of acoustic waves in order to optimize hearing aid designs, and numerous other applications.
The primary goals of the developed algorithms are to reduce the computational complexity, stabilize the numerical simulations, and efficiently parallelize the simulations on modern parallel machines, including Linux clusters, GPU graphics cards, and hybrid machines.
The research includes the following:
designing fast and accurate solvers for isogeometric finite element methods (both direct and iterative solvers);
algorithms based on graph grammars;
developing direct solvers with linear and logarithmic complexity in order to solve equation systems that are generated during adaptive mesh computations;
applying algorithmics to conduct isogeometric simulations in such areas as tumor growth and pollution propagation;
stabilizing numerical simulations using residual minimization methods, DPG methods, and hybrid mixed finite element method.
keywords: finite element method, isogeometric method, adaptive algorithms, graph grammars, direct solvers, variable-direction solvers, iterative solvers, H-matrix solvers, stabilization of numerical simulations
Key publications related to research area:
1. Łoś M., Kłusek A., Hassaan M., Pingali K., Dzwinel W., Paszyński M.; Parallel fast isogeometric L2 projection solver with GALOIS system for 3D tumor growth simulations, Computer Methods in Applied Mechanics and Engineering, Available online 1 September 2018 In Press, Accepted Manuscript, DOI: 10.1016/j.cma.2018.08.036
2. Łoś M., Paszyński M., Kłusek A., Dzwinel W., Application of fast isogeometric L2 projection solver for tumor growth simulations, Computer Methods in Applied Mechanics and Engineering, 2017 vol. 316 spec. iss.: Isogeometric Analysis: Progress and Challenges, s. 1257–1269
3. Woźniak M., Kuźnik K., Paszyński M., Calo V. M., D. Pardo, Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers, Computers and Mathematics with Applications 2014, vol. 67 iss. 10, s. 1864–1883.
4. Woźniak M., Paszyński M., Pardo D., Dalcin L., Calo V.; Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines, Computer Methods in Applied Mechanics and Engineering, Volume 284, 2015, pp. 971-987, DOI: 10.1016/j.cma.2014.11.020
5. Łoś M., Woźniak M,. Paszyński M., A. Lenharth, M. A. Hassaan, K. Pingali , IGA-ADS: isogeometric analysis FEM using ADS solver, Computer Physics Communications, 2017 vol. 217, s. 99–116.