Abstract: Refined Isogeometric Analysis (rIGA) is a discretization method used to solve numerical problems governed by partial differential equations (PDEs). Starting from a highly continuous Isogeometric Analysis (IGA) discretization, rIGA reduces the continuity over certain hyperplanes that split the mesh into subdomains. This method maximizes the performance of direct solvers by reducing the continuity until C^0 over selected hyperplanes, which act as separators during the elimination of the degrees of freedom (DoF). By doing so, the solution time and best approximation error are simultaneously improved. In particular, D. Garcia et al. show that rIGA delivers a speedup factor with respect to IGA that is proportional to $p^2$ when solving Laplace based problems in 2D and 3D, with p being the polynomial degree. In this work, we extend rIGA method to solve multi-field problems. We consider incompressible fluid flow problems on bounded domains. They include the pressure and the vectorial velocity of the fluid. We use a spline-based generalization of the Raviart-Thomas finite element spaces to approximate the velocity field. We show that rIGA delivers a reduction in the computational cost when solving incompressible fluid flow problems that asymptotically reaches O(p^2), and it provides better accuracy than C^(p-1) IGA. For multi-field problems, however, we require larger problems to arrive at the asymptotic limit and reach the maximum possible savings since the system involves more equations. In our numerical 2D results, we observe a reduction factor in the computational cost of up to p^2. In 3D, the maximum reproducible problems are in the pre-asymptotic regime, and the maximum observed gain factors are of O(p).

Abstract: We focus on a fast explicit solver for solution of non-stationary problems using L2 projections with isogeometric finite element method. The algorithm has been implemented in Fortran 2008 with MPI and OpenMP frameworks. It enables for parallel multi-core hybrid memory simulations of different time-dependent problems in 3D. We have prepared the solwer framework in a way that enables for direct implementation of the selected Partial Differential Equations and corresponding boundary conditions. The presented package generates output suitable for interfacing with ParaView visualisation software. Finally new implementation is complementary to previously released GALOIS based shared memory code written in C++. Our library manages most of computations, while user has to provide subroutine with equations for Right Hand Side.

Abstract: Mistakes in pedestrian infrastructure design in modern cities decrease transfer comfort for people, impact greenery due to appearance of desire paths, and thus increase the amount of dust in the air because of open ground. These mistakes can be avoided if optimal path networks are created considering behavioral aspects of pedestrian traffic, which is a challenge. In this article, we introduce Ant Road Planner, a new method of computer simulation for estimation and creation of optimal path networks which not only considers pedestrians' behavior but also helps minimize the total length of the paths so that the area is used more efficiently. The method, which includes a modeling algorithm and its software implementation with a user-friendly web interface, makes it possible to predict pedestrian networks for new territories with high precision and detect problematic areas in existing networks. The algorithm was successfully tested on real territories and proved its potential as a decision making support system for urban planners.