Daniel García obtuvo el título de Ingeniero Mecánico en 2010 en la Universidad Nacional de Colombia, donde también completó un Máster en Ingeniería Mecánica. Continuó sus estudios en la Universidad de Ciencia y Tecnología King Abdullah (KAUST) en Arabia Saudita, donde formó parte del Numerical Porous Media Center (NUMPOR), mientras realizaba un segundo Máster en Ingeniería Mecánica. En 2015 se incorporó a la línea de investigación Simulación de Propagación de Ondas del Basque Center for Applied Mathematics como estudiante de doctorado.
Su tesis doctoral ha sido dirigida por el investigador UPV/EHU-BCAM e Ikerbasque Research Professor David Pardo y por el Profesor Victor M. Calo de la Universidad de Curtin.
En nombre de todo el equipo de BCAM nuestra más sincera enhorabuena a Daniel.
Isogeometric Analysis (IGA) is a computational approach frequently employed nowadays to study problems governed by partial differential equations (PDEs). This approach defines the geometry using conventional CAD functions and, in particular, NURBS. These functions represent complex geometries commonly found in engineering design and are capable of preserving exactly the geometry description under refinement as required in the analysis. Moreover, the use of NURBS as basis functions is compatible with the isoparametric concept, allowing to build algebraic systems directly from the computational domain representation based on spline functions, which arise from CAD. Therefore, it avoids to define a second space for the numerical analysis resulting in huge reductions in the total analysis time.
For the case of direct solvers, the performance strongly depends upon the employed discretization method. In particular, on IGA, the continuity of the solution spaces plays a significant role in their performance. High continuous spaces degrade the direct solver’s performance, increasing the solution times by a factor up to O(p^3) with respect to traditional finite element analysis (FEA) per unknown, being p the polynomial order.
In this work, we propose a solver-based discretization that employs highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous IGA discretization, we introduce C^0 hyperplanes, which act as separators for the direct solver, to reduce the interconnection between the degrees of freedom (DoF) in the mesh. By doing so, both the solution time and best approximation errors are simultaneously improved. We call the resulting method “refined Isogeometric analysis” (rIGA). Numerical results indicate that rIGA delivers speed-up factors proportional to p^2. For instance, in a 2D mesh with four million elements and p=5, a Laplace linear system resulting from rIGA is solved 22 times faster than the one from highly continuous IGA. In a 3D mesh with one million elements and 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. This is a hybrid solver strategy that combines a direct solver (static condensation step) to eliminate the internal macro-elements DoF, with an iterative method to solve the skeleton system. The hybrid solver strategy achieves moderate savings with respect to IGA when solving a 2D Poisson problem with a structured mesh and a uniform polynomial degree of approximation. For instance, for a mesh with four million elements and polynomial degree p=3, the iterative solver is approximately 2.6 times faster (in time) when applied to the rIGA system than to the IGA one. These savings occur because the skeleton rIGA system contains fewer non-zero entries than the IGA one. The opposite situation occurs for 3D problems, and as a result, 3D rIGA discretizations provide no gains with respect to their IGA counterparts.