## REFINED ISOGEOMETRIC ANALYSIS: A SOLVER-BASED DISCRETIZATION METHOD

##### Abstract

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 computer-aided design (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.
To perform the numerical analysis, we can use either direct or iterative solvers. Direct solvers are preferred to study stiff linear problems that are not solvable with iterative solvers, as well as another type of problems, e.g., problems with multiple right-hand side (RHS). Moreover, when the problem size and the floating point operations (FLOPs) required to solve the problem are enormous, and the direct solvers become excessively expensive, the iterative solver turns into a more suitable alternative.
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 $\mathcal{O}(p^3)$ 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)".
While this method can be applied to a variety of problems, in this Dissertation we focus on analyzing the impact of the continuity reduction when solving a Laplace problem with structured meshes and uniform polynomial orders in both 2D and 3D. Numerical results indicate that rIGA delivers speed-up factors proportional to $p^2$. For instance, in a 2D mesh with four million elements and $p=5$, the 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 then develop a version of the rIGA strategy that introduces hyperplanes of arbitrary continuity ($C^k$ hyperplanes with $0\leq k\leq p-1$). This strategy, called Optimally refined Isogeometric Analysis (OrIGA), leads to more efficient discretization than those obtained with the original version of rIGA. By using separators of arbitrary continuity degree, we achieve a performance boost of up to 25$\%$ in the direct solvers with respect to rIGA. Thus, the savings with respect to IGA are larger than in rIGA.
We have also designed and implemented a similar rIGA strategy for iterative solvers. This strategy splits the mesh into subdomains using $C^0$-hyperplanes and constructs the Schur complements of the subdomains (macro-elements) using a direct solver. We then assemble those Schur complements into a global skeleton system, which is only composed of the DoF located along the boundaries of all the subdomains. Subsequently, we solve this system iteratively using conjugate gradient method (CG) with an incomplete LU (ILU) preconditioner. Lastly, a backward substitution is performed to recover the eliminated DoF and obtain the solution of the original system. Thus, rIGA for iterative solvers 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.
In this work, we also apply rIGA to solve incompressible fluid flow problems on an enclosed domain. To satisfy the inf-sup stability condition and guarantee divergence-free discrete solutions, the implemented rIGA employs a combination of $C^0$ and $C^1$ hyperplanes to reduce the continuity on the solution spaces. Numerical results show that the $L^2$ norm of the discretization error improves as we reduce the continuity. Therefore, rIGA delivers smaller errors than $C^{p-1}$ IGA discretizations. In terms of the computational savings, rIGA provides a reduction in the computational cost of the direct solvers by a factor of $\mathcal{O}(p^2)$ in both 2D and 3D.