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Starting from a highly continuous isogeometric analysis discretization, we introduce hyperplanes that partition the domain into subdomains and reduce the continuity of the discretization spaces at these hyperplanes. As the continuity is reduced, the number of degrees of freedom in the system grows. The resulting discretization spaces are finer than standard maximal continuity IGA spaces. Despite the increase in the number of degrees of freedom, these finer spaces deliver simulation results faster with direct solvers than both traditional finite element and isogeometric analysis for meshes with a fixed number of elements. In this work, we analyze the impact of continuity reduction on the number of Floating Point Operations (FLOPs) and computational times required to solve fluid flow and electromagnetic problems with structured meshes and uniform polynomial orders. Theoretical estimates show that for sufficiently large grids, an optimal continuity reduction decreases the computational cost by a factor of O (p 2). Numerical results confirm these theoretical estimates. In a 2D mesh with one million elements and polynomial order equal to five, the discretization including an optimal continuity pattern allows to solve the vector electric field, the scalar magnetic field, and the fluid flow problems an order of magnitude faster than when using a highly continuous IGA discretization. 3D numerical results exhibit more moderate savings due to the limited mesh sizes considered in this work. [ABSTRACT FROM AUTHOR]

We study three-dimensional isogeometric analysis (IGA) and the solution of the resulting system of linear equations via a direct solver. IGA uses highly continuous $C^{p-1}$Cp-1 basis functions, which provide multiple benefits in terms of stability and convergence properties. However, smooth basis significantly deteriorate the direct solver performance and its parallel scalability. As a partial remedy for this, refined Isogeometric Analysis (rIGA) method improves the sequential execution of direct solvers. The refinement strategy enriches traditional highly-continuous $C^{p-1}$Cp-1 IGA spaces by introducing low-continuity $C^0$C0-hyperplanes along the boundaries of certain pre-defined macro-elements. In this work, we propose a solution strategy for rIGA for parallel distributed memory machines and compare the computational costs of solving rIGA versus IGA discretizations. We verify our estimates with parallel numerical experiments. Results show that the weak parallel scalability of the direct solver improves approximately by a factor of $p^2$p2 when considering rIGA discretizations rather than highly-continuous IGA spaces. [ABSTRACT FROM AUTHOR]

We propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous Isogeometric Analysis (IGA) discretization, we introduce C 0 -separators to reduce the interconnection between degrees of freedom 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)”. To illustrate the impact of the continuity reduction, we analyze the number of Floating Point Operations (FLOPs), computational times, and memory required to solve the linear system obtained by discretizing the Laplace problem with structured meshes and uniform polynomial orders. Theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p 2 and p 3 , with p being the polynomial order of the discretization. Numerical results indicate that our proposed refined isogeometric analysis delivers a speed-up factor proportional to p 2 . In a 2 D 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 3 D mesh with one million elements and p = 3 , the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis. [ABSTRACT FROM AUTHOR]

Performance of direct solvers strongly depends upon the employed discretization method. In particular, it is possible to improve the performance of solving Isogeometric Analysis (IGA) discretizations by introducing multiple C°-continuity hyperplanes that act as separators during LU factorization [8]. In here, we further explore this venue by introducing separators of arbitrary continuity. Moreover, we develop an efficient method to obtain optimal discretizations in the sense that they minimize the time employed by the direct solver of linear equations. The search space consists of all possible discretizations obtained by enriching a given IGA mesh. Thus, the best approximation error is always reduced with respect to its IGA counterpart, while the solution time is decreased by up to a factor of 60. [ABSTRACT FROM AUTHOR]

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using C0 B-splines, which span traditional finite element spaces, and Cp-1 B-splines, which represent maximum continuity. We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size h and polynomial order of approximation p in addition to the aforementioned continuity of the basis. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most 33p²/8 times more expensive for the more continuous space, although for moderately low p, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high p. Preconditioning options can be up to p³ times more expensive to set up, although this difference significantly decreases for some popular preconditioners such as incomplete LU factorization. [ABSTRACT FROM AUTHOR]

Abstract: We study the performance of direct solvers on linear systems of equations resulting from isogeometric analysis. The problem of choice is the canonical Laplace equation in three dimensions. From this study we conclude that for a fixed number of unknowns and polynomial degree of approximation, a higher degree of continuity k drastically increases the CPU time and RAM needed to solve the problem when using a direct solver. This paper presents numerical results detailing the phenomenon as well as a theoretical analysis that explains the underlying cause. [Copyright &y& Elsevier]

We use refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems (K u = λ M u). rIGA conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) while it reduces the solution cost by adding zero-continuity basis functions, which decrease the matrix connectivity. As a result, rIGA enriches the approximation space and reduces the interconnection between degrees of freedom. We compare computational costs of rIGA versus those of IGA when employing a Lanczos eigensolver with a shift-and-invert spectral transformation. When all eigenpairs within a given interval [ λ s , λ e ] are of interest, we select several shifts σ k ∈ [ λ s , λ e ] using a spectrum slicing technique. For each shift σ k , the factorization cost of the spectral transformation matrix K − σ k M controls the total computational cost of the eigensolution. Several multiplications of the operator matrix (K − σ k M) − 1 M by vectors follow this factorization. Let p be the polynomial degree of the basis functions and assume that IGA has maximum continuity of p − 1. When using rIGA, we introduce C 0 separators at certain element interfaces to minimize the factorization cost. For this setup, our theoretical estimates predict computational savings to compute a fixed number of eigenpairs of up to O (p 2) in the asymptotic regime, that is, large problem sizes. Yet, our numerical tests show that for moderate-size eigenproblems, the total observed computational cost reduction is O (p). In addition, rIGA improves the accuracy of every eigenpair of the first N 0 eigenvalues and eigenfunctions, where N 0 is the total number of modes of the original maximum-continuity IGA discretization. • The article proposes to use rIGA discretizations to solve generalized Hermitian eigenproblems. • The shift-and-invert spectral transformation is implemented for eigenpair calculation. • Computational cost of rIGA is compared to that of high-continuity IGA discretization. • Total computational cost of eigenanalysis is reduced by up to O (p) when using rIGA instead of IGA. • A better accuracy of first N0 eigenvalues and eigenfunctions is achieved by using rIGA instead of IGA. [ABSTRACT FROM AUTHOR]

We propose a simulator for time-dependent Maxwell's equations with linear computational cost. We employ B-spline basis functions as considered in the isogeometric analysis (IGA). We focus on non-stationary Maxwell's equations defined on a regular patch of elements. We employ the idea of alternating-directions splitting (ADS) and employ a second-order accurate time-integration scheme for the time-dependent Maxwell's equations in a weak form. After discretization, the resulting stiffness matrix exhibits a Kronecker product structure. Thus, it enables linear computational cost LU factorization. Additionally, we derive a formulation for absorbing boundary conditions (ABCs) suitable for direction splitting. We perform numerical simulations of the scattering problem (traveling pulse wave) to verify the ABC. We simulate the radiation of electromagnetic (EM) waves from the dipole antenna. We verify the order of the time integration scheme using a manufactured solution problem. We then simulate magnetotelluric measurements. Our simulator is implemented in a shared memory parallel machine, with the GALOIS library supporting the parallelization. We illustrate the parallel efficiency with strong and weak scalability tests corresponding to non-stationary Maxwell simulations. [ABSTRACT FROM AUTHOR]

This paper derives theoretical estimates of the computational cost for isogeometric multi-frontal direct solver executed on parallel distributed memory machines. We show theoretically that for the C p − 1 global continuity of the isogeometric solution, both the computational cost and the communication cost of a direct solver are of order O ( log ( N ) p 2 ) for the one dimensional (1D) case, O ( N p 2 ) for the two dimensional (2D) case, and O ( N 4 / 3 p 2 ) for the three dimensional (3D) case, where N is the number of degrees of freedom and p is the polynomial order of the B-spline basis functions. The theoretical estimates are verified by numerical experiments performed with three parallel multi-frontal direct solvers: MUMPS, PaStiX and SuperLU, available through PETIGA toolkit built on top of PETSc. Numerical results confirm these theoretical estimates both in terms of p and N . For a given problem size, the strong efficiency rapidly decreases as the number of processors increases, becoming about 20% for 256 processors for a 3D example with 128 3 unknowns and linear B-splines with C 0 global continuity, and 15% for a 3D example with 64 3 unknowns and quartic B-splines with C 3 global continuity. At the same time, one cannot arbitrarily increase the problem size, since the memory required by higher order continuity spaces is large, quickly consuming all the available memory resources even in the parallel distributed memory version. Numerical results also suggest that the use of distributed parallel machines is highly beneficial when solving higher order continuity spaces, although the number of processors that one can efficiently employ is somehow limited. [ABSTRACT FROM AUTHOR]

Certain applications that analyze damping effects require the solution of quadratic eigenvalue problems (QEPs). We use refined isogeometric analysis (rIGA) to solve quadratic eigenproblems. rIGA discretization, while conserving desirable properties of maximum-continuity isogeometric analysis (IGA), reduces the interconnection between degrees of freedom by adding low-continuity basis functions. This connectivity reduction in rIGA's algebraic system results in faster matrix LU factorizations when using multifrontal direct solvers. We compare computational costs of rIGA versus those of IGA when employing Krylov eigensolvers to solve quadratic eigenproblems arising in 2D vector-valued multifield problems. For large problem sizes, the eigencomputation cost is governed by the cost of LU factorization, followed by costs of several matrix–vector and vector–vector multiplications, which correspond to Krylov projections. We minimize the computational cost by introducing C 0 and C 1 separators at specific element interfaces for our rIGA generalizations of the curl-conforming Nédélec and divergence-conforming Raviart–Thomas finite elements. Let p be the polynomial degree of basis functions; the LU factorization is up to O ( (p − 1) 2 ) times faster when using rIGA compared to IGA in the asymptotic regime. Thus, rIGA theoretically improves the total eigencomputation cost by O ( (p − 1) 2 ) for sufficiently large problem sizes. Yet, in practical cases of moderate-size eigenproblems, the improvement rate deteriorates as the number of computed eigenvalues increases because of multiple matrix–vector and vector–vector operations. Our numerical tests show that rIGA accelerates the solution of quadratic eigensystems by O (p − 1) for moderately sized problems when we seek to compute a reasonable number of eigenvalues. [ABSTRACT FROM AUTHOR]

We propose the use of machine learning techniques to find optimal quadrature rules for the construction of stiffness and mass matrices in isogeometric analysis (IGA). We initially consider 1D spline spaces of arbitrary degree spanned over uniform and non-uniform knot sequences, and then the generated optimal rules are used for integration over higher-dimensional spaces using tensor products. The quadrature rule search is posed as an optimization problem and solved by a machine learning strategy based on adaptive gradient-descent. However, since the optimization space is highly non-convex, the success of the search strongly depends on the number of quadrature points and the parameter initialization. Thus, we use a dynamic programming strategy that initializes the parameters from the optimal solution over the spline space with a lower number of knots. With this method, we found optimal quadrature rules for spline spaces when using IGA discretizations with up to 50 uniform elements and polynomial degrees up to 8, showing the generality of the approach in this scenario. For non-uniform partitions, the method also finds an optimal rule in a reasonable number of test cases. We also assess the generated optimal rules in two practical case studies, namely, the eigenvalue problem of the Laplace operator and the eigenfrequency analysis of freeform curved beams, where the latter problem shows the applicability of the method to curved geometries. In particular, the proposed method results in savings with respect to traditional Gaussian integration of up to 44% in 1D, 68% in 2D, and 82% in 3D spaces. [ABSTRACT FROM AUTHOR]

Borehole resistivity measurements are routinely inverted in real-time during geosteering operations. The inversion process can be efficiently performed with the help of advanced artificial intelligence algorithms such as deep learning. These methods require a massive dataset that relates multiple Earth models with the corresponding borehole resistivity measurements. In here, we propose to use an advanced numerical method — refined isogeometric analysis (rIGA) — to perform rapid and accurate 2.5D simulations and generate databases when considering arbitrary 2D Earth models. Numerical results show that we can generate a meaningful synthetic database composed of 100,000 Earth models with the corresponding measurements in 56 h using a workstation equipped with two CPUs. • We use rIGA discretizations for simulating 2.5D borehole electromagnetic measurements. • We generate a synthetic database as a preliminary stage for deep learning inversion. • Computational cost of rIGA is compared to that of IGA and FEA discretizations. • rIGA generates the database O (p) times faster than high-continuity IGA discretization. [ABSTRACT FROM AUTHOR]