## Refined isogeometric analysis: a solver-based discretization method

##### Resumen

Isogeometric Analysis (IGA) is a computational approach frequently employed nowadaysto study problems governed by partial differential equations (PDEs). This approach definesthe geometry using conventional CAD functions and, in particular, NURBS. Thesefunctions represent complex geometries commonly found in engineering design and arecapable of preserving exactly the geometry description under refinement as required in theanalysis. Moreover, the use of NURBS as basis functions is compatible with theisoparametric concept, allowing to build algebraic systems directly from the computationaldomain representation based on spline functions, which arise from CAD. Therefore, itavoids to define a second space for the numerical analysis resulting in huge reductions inthe total analysis time.For the case of direct solvers, the performance strongly depends upon the employeddiscretization method. In particular, on IGA, the continuity of the solution spaces plays asignificant role in their performance. High continuous spaces degrade the direct solver'sperformance, increasing the solution times by a factor up to O(p^3) with respect totraditional finite element analysis (FEA) per unknown, being p the polynomial order.In this work, we propose a solver-based discretization that employs highly continuous finiteelement spaces interconnected with low continuity hyperplanes to maximize theperformance of direct solvers. Starting from a highly continuous IGA discretization, weintroduce C^0 hyperplanes, which act as separators for the direct solver, to reduce theinterconnection between the degrees of freedom (DoF) in the mesh. By doing so, both thesolution time and best approximation errors are simultaneously improved. We call theresulting method ``refined Isogeometric analysis" (rIGA). Numerical results indicate thatrIGA delivers speed-up factors proportional to p^2. For instance, in a 2D mesh with fourmillion elements and p=5, a Laplace linear system resulting from rIGA is solved 22 timesfaster than the one from highly continuous IGA. In a 3D mesh with one million elementsand p=3, the linear rIGA system is solved 15 times faster than the IGA one.We have also designed and implemented a similar rIGA strategy for iterative solvers. Thisis a hybrid solver strategy that combines a direct solver (static condensation step) toeliminate the internal macro-elements DoF, with an iterative method to solve the skeletonsystem. The hybrid solver strategy achieves moderate savings with respect to IGA whensolving a 2D Poisson problem with a structured mesh and a uniform polynomial degree ofapproximation. For instance, for a mesh with four million elements and polynomial degreep=3, the iterative solver is approximately 2.6 times faster (in time) when applied to the rIGAsystem than to the IGA one. These savings occur because the skeleton rIGA systemcontains fewer non-zero entries than the IGA one. The opposite situation occurs for 3Dproblems, and as a result, 3D rIGA discretizations provide no gains with respect to theirIGA counterparts.Thesis director(s): David Pardo from UPV/EHU university and Victor M. Calo from Curtinuniversity