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The goal of this paper is to develop a numerical algorithm that solves a two-dimensional elliptic partial differential equation in a polygonal domain using tensor methods and ideas from isogeometric analysis. The proposed algorithm is based on the Finite Element (FE) approximation with Quantized Tensor Train decomposition (QTT) used for matrix representation and solution approximation. In this paper we propose a special discretisation scheme that allows to construct the global stiffness matrix in the QTT-format. The algorithm has $O(log n)$ complexity, where $n=2^d$ is the number of nodes per quadrangle side. A new operation called z-kron is introduced for QTT-format. It makes it possible to build a matrix in z-order if the matrix can be expressed in terms of Kronecker products and sums. An algorithm for building a QTT coefficient matrix for FEM in z-order on the fly, as opposed to the transformation of a calculated matrix into QTT, is presented. This algorithm has $O(log n)$ complexity for $n$ as above.
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Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs. The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.
This work is motivated by the difficulty in assembling the Galerkin matrix when solving Partial Differential Equations (PDEs) with Isogeometric Analysis (IGA) using B-splines of moderate-to-high polynomial degree. To mitigate this problem, we propose a novel methodology named CossIGA (COmpreSSive IsoGeometric Analysis), which combines the IGA principle with CORSING, a recently introduced sparse recovery approach for PDEs based on compressive sensing. CossIGA assembles only a small portion of a suitable IGA Petrov-Galerkin discretization and is effective whenever the PDE solution is sufficiently sparse or compressible, i.e., when most of its coefficients are zero or negligible. The sparsity of the solution is promoted by employing a multilevel dictionary of B-splines as opposed to a basis. Thanks to sparsity and the fact that only a fraction of the full discretization matrix is assembled, the proposed technique has the potential to lead to significant computational savings. We show the effectiveness of CossIGA for the solution of the 2D and 3D Poisson equation over nontrivial geometries by means of an extensive numerical investigation.
The fast assembling of stiffness and mass matrices is a key issue in isogeometric analysis, particularly if the spline degree is increased. We present two algorithms based on the idea of sum factorization, one for matrix assembling and one for matrix-free methods, and study the behavior of their computational complexity in terms of the spline order $p$. Opposed to the standard approach, these algorithms do not apply the idea element-wise, but globally or on macro-elements. If this approach is applied to Gauss quadrature, the computational complexity grows as $p^{d+2}$ instead of $p^{2d+1}$ as previously achieved.
We propose a numerical method for the solution of electromagnetic problems on axisymmetric domains, based on a combination of a spectral Fourier approximation in the azimuthal direction with an IsoGeometric Analysis (IGA) approach in the radial and axial directions. This combination allows to blend the flexibility and accuracy of IGA approaches with the advantages of a Fourier representation on axisymmetric domains. It also allows to reduce significantly the computational cost by decoupling of the computations required for each Fourier mode. We prove that the discrete approximation spaces employed functional space constitute a closed and exact de Rham sequence. Numerical simulations of relevant benchmarks confirm the high order convergence and other computational advantages of the proposed method.
The construction of volumetric parametrizations for computational domains is a key step in the pipeline of isogeometric analysis. Here, we investigate a solution to this problem based on the mesh deformation approach. The desired domain is modeled as a deformed configuration of an initial simple geometry. Assuming that the parametrization of the initial domain is bijective and that it is possible to find a locally invertible displacement field, the method yields a bijective parametrization of the target domain. We compute the displacement field by solving the equations of nonlinear elasticity with the neo-Hookean material law, and we show an efficient variation of the incremental loading algorithm tuned specifically to this application. In order to construct the initial domain, we simplify the target domains boundary by means of an L2-projection onto a coarse basis and then apply the Coons patch approach. The proposed methodology is not restricted to a single patch scenario but can be utilized to construct multi-patch parametrizations with naturally looking boundaries between neighboring patches. We illustrate its performance and compare the result to other established parametrization approaches on a range of two-dimensional and three-dimensional examples.
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