Isogeometric analysis on non-matching segmentation: discontinuous Galerkin techniques and efficient solvers

Abstract

The Isogeometric Analysis (IgA) of boundary value problems in complex domains often requires a decomposition of the computational domain into patches such that each of which can be parametrized by the so-called geometrical mapping. In this paper, we develop discontinuous Galerkin (dG) IgA techniques for solving elliptic diffusion problems on decompositions that can include non-matching parametrizations of the interfaces, i.e., the interfaces of the adjacent patches may be not identical. The lack of the exact parametrization of the patches leads to the creation of gap and overlapping regions between the patches. This does not allow the immediate use of the classical numerical fluxes that are known in the literature. The unknown normal fluxes of the solution on the non-matching interfaces are approximated by Taylor expansions using the values of the solution computed on the boundary of the patches. These approximations are used in order to build up the numerical fluxes of the final dG IgA scheme and to couple the local patch-wise discrete problems. The resulting linear systems are solved by using efficient domain decomposition methods based on the tearing and interconnecting technology. We present numerical results of a series of test problems that validate the theoretical estimates presented.

Discontinuous Galerkin isogeometric analysis for segmentations generating overlapping regions

“…We also are constructing domaindecomposition methods, [13], on these type of multipatch representations and we are discussing the influence of the size of the overlapping region on the performance of the proposed methods. The first results of this work are included in [15]. Finally, we mention that during the investigation of the proposed methodology in Section 3, we considered simple interior penalty fluxes on ∂Ω o21 .…”

“…The bilinear form in ( 49 ) is bounded and elliptic on i.e. there are positive constants and such that the estimates hold for all provided that η is sufficiently large, see [ 25 ].…”

“…In the same work, we also discuss issues related to the construction of domain-decomposition methods on these multipatch representations and provide several numerical tests for evaluating their performance. The first results in this direction can be found in [ 25 ].…”

Biorthogonal splines for optimal weak patch-coupling in isogeometric analysis with applications to finite deformation elasticity

“…Typically, multivariate splines are defined based on a tensor-product structure and a flexible coupling between the patches is important to gain some flexibility of the local meshes. Different approaches are considered, e.g., Nitsche's method [6,7], penalty based methods [8] and mortar methods [9,10], and there is a recent interest in higher-order couplings, see, e.g. [11,12].…”

Analysis of discontinuous Galerkin dual-primal isogeometric tearing and interconnecting methods

“…However, this procedure may introduce small gaps and overlapps at the patch interfaces, leading to so called segmentation crimes, see [25], [31] and [23] for a more comprehensive analysis. In order to solve PDEs on such domains, numerical schemes based on the discontinuous Galerkin (dG) method for elliptic PDEs were developed and analysed in [19], [21] and [20]. Moreover, the dG formulation is used when considering different B-Splines spaces across interfaces, e.g., non-matching grids or different spline degrees.…”