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Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines

  • Wozniak M.
  • Paszynski M.
  • Pardo D.
  • Dalcin L.
  • Calo V.M.
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 $\mathcal{O}(log(N)p^2)$ for the one dimensional $(1D)$ case, $\mathcal{O}(Np^2)$ for the two dimensional $(2D)$ case, and $\mathcal{O}(N^{4/3}p^2)$ 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 $643$ 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.
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Direct solvers performance on h-adapted grids

  • Paszynski M.
  • Pardo D.
  • Calo V.M.
We analyse the performance of direct solvers when applied to a system of linear equations arising from an $h$-adapted, $C^0$ finite element space. Theoretical estimates are derived for typical $h$-refinement patterns arising as a result of a point, edge, or face singularity as well as boundary layers. They are based on the elimination trees constructed specifically for the considered grids. Theoretical estimates are compared with experiments performed with MUMPS using the nested-dissection algorithm for construction of the elimination tree from METIS library. The numerical experiments provide the same performance for the cases where our trees are identical with those constructed by the nested-dissection algorithm, and worse performance for some cases where our trees are different. We also present numerical experiments for the cases with mixed singularities, where how to construct optimal elimination trees is unknown. In all analysed cases, the use of $h$-adaptive grids significantly reduces the cost of the direct solver algorithm $per$ $unknown$ as compared to uniform grids. The theoretical estimates predict and the experimental data confirm that the computational complexity is linear for various refinement patterns. In most cases, the cost of the direct solver $per$ $unknown$ is lower when employing anisotropic refinements as opposed to isotropic ones.
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Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain

  • Aboelenen T.
  • Bakr S.A.
  • El-Hawary H.M.
In this article, we first introduce a singular fractional Sturm-Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results.
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Stabilized dual-mixed method for the problem of linear elasticity with mixed boundary conditions

  • Gonzalez M.
We extend the applicability of the augmented dual-mixed method introduced recently in Gatica (2007), Gatica et al. (2009) to the problem of linear elasticity with mixed boundary conditions. The method is based on the Hellinger-Reissner principle and the symmetry of the stress tensor is imposed in a weak sense. The Neumann boundary condition is prescribed in the finite element space. Then, suitable Galerkin least-squares type terms are added in order to obtain an augmented variational formulation which is coercive in the whole space. This allows to use any finite element subspaces to approximate the displacement, the Cauchy stress tensor and the rotation.
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High-accuracy adaptive modeling of the energy distribution of a meniscus-shaped cell culture in a Petri dish

  • Gomez-Revuelto I.
  • Garcia-Castillo L.E.
  • Pardo D.
Cylindrical Petri dishes embedded in a rectangular waveguide and exposed to a polarized electromagnetic wave are often used to grow cell cultures. To guarantee the success of these cultures, it is necessary to enforce that the specific absorption rate distribution is sufficiently high and uniform over the Petri dish. Accurate numerical simulations are needed to design such systems. These simulations constitute a challenge due to the strong discontinuity of electromagnetic material properties involved, the relative low field value within the dish cultures compared with the rest of the domain, and the presence of the meniscus shape developed at the liquid boundary. The latter greatly increases the level of complexity of the model in terms of geometry and intensity of the gradients/singularities of the field solution. In here, we employ a three-dimensional (3D) hp-adaptive finite element method using isoparametric elements to obtain highly accurate simulations. We analyze the impact of the geometrical modeling of the meniscus shape cell culture in the hp-adaptivity. Numerical results showing the error convergence history indicate the numerical difficulties arisen due to the presence of a meniscus-shaped object. At the same time, the resulting energy distribution shows that to consider such meniscus shape is essential to guarantee the success of the cell culture from the biological point of view.
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Multi-objective hierarchic memetic solver for inverse parametric problems

  • Gajda-Zagórska E.
  • Smolka M.
  • Schaefer R.
  • Pardo D.
  • Alvarez-Aramberri J.
We propose a multi-objective approach for solving challenging inverse parametric problems. The objectives are misfits for several physical descriptions of a phenomenon under consideration, whereas their domain is a common set of admissible parameters. The resulting Pareto set, or parameters close to it, constitute various alternatives of minimizing individual misfits. A special type of selection applied to the memetic solution of the multi-objective problem narrows the set of alternatives to the ones that are sufficiently coherent. The proposed strategy is exemplified by solving a real-world engineering problem consisting of the magnetotelluric measurement inversion that leads to identification of oil deposits located about 3 km under the Earth's surface, where two misfit functions are related to distinct frequencies of the electric and magnetic waves.
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Semi-analytical response of acoustic logging measurements in frequency domain

  • Muga I.
  • Pardo D.
  • Matuszyk P.J.
  • Torres-Verdin C.
This work proposes a semi-analytical method for simulation of the acoustic response of multipole eccentered sources in a fluid-filled borehole. Assuming a geometry that is invariant with respect to the azimuthal and vertical directions, the solution in frequency domain is expressed in terms of a Fourier series and a Fourier integral. The proposed semi-analytical method builds upon the idea of separating singularities from the smooth part of the integrand when performing the inverse Fourier transform. The singular part is treated analytically using existing inversion formulae, while the regular part is treated with a FFT technique. As a result, a simple and effective method that can be used for simulating and understanding the main physical principles occurring in borehole-eccentered sonic measurements is obtained. Numerical results verify the proposed method and illustrate its advantages.
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The Modellers' Halting Foray into Ecological Theory: Or, What is This Thing Called 'Growth Rate'?

  • Deveau M.
  • Karsten R.
  • Teismann H.
This discussion paper describes the attempt of an imagined group of non-ecologists ("Modellers" ) to determine the population growth rate from field data. The Modellers wrestle with the multiple definitions of the growth rate available in the literature and the fact that, in their modelling, it appears to be drastically model-dependent, which seems to throw into question the very concept itself. Specifically, they observe that six representative models used to capture the data produce growth-rate values, which differ significantly. Almost ready to concede that the problem they set for themselves is ill-posed, they arrive at an alternative point of view that not only preserves the identity of the concept of the growth rate, but also helps discriminate between competing models for capturing the data. This is accomplished by assessing how robustly a given model is able to generate growth-rate values from randomized time-series data. This leads to the proposal of an iterative approach to ecological modelling in which the definition of theoretical concepts (such as the growth rate) and model selection complement each other. The paper is based on high-quality field data of mites on apple trees and may be called a “data-driven opinion piece†.
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Towards efficient 5-axis flank CNC machining of free-form surfaces via fitting envelopes of surfaces of revolution

  • Bo P.
  • Bartoň M.
  • Plakhotnik D.
  • Pottmann H.
We introduce a new method that approximates free-form surfaces by envelopes of one-parameter motions of surfaces of revolution. In the context of 5-axis computer numerically controlled (CNC) machining, we propose a flank machining methodology which is a preferable scallop-free scenario when the milling tool and the machined free-form surface meet tangentially along a smooth curve. We seek both an optimal shape of the milling tool as well as its optimal path in 3D space and propose an optimization based framework where these entities are the unknowns. We propose two initialization strategies where the first one requires a user's intervention only by setting the initial position of the milling tool while the second one enables to prescribe a preferable tool-path. We present several examples showing that the proposed method recovers exact envelopes, including semi-envelopes and incomplete data, and for general free-form objects it detects envelope sub-patches.
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Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis

  • Barton M.
  • Calo V.M.
We introduce optimal quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. Using the homotopy continuation concept (Barton and Calo, 2016) that transforms optimal quadrature rules from source spaces to target spaces, we derive optimal rules for splines defined on finite domains. Starting with the classical Gaussian quadrature for polynomials, which is an optimal rule for a discontinuous odd-degree space, we derive rules for target spaces of higher continuity. We further show how the homotopy methodology handles cases where the source and target rules require different numbers of optimal quadrature points. We demonstrate it by deriving optimal rules for various odd-degree spline spaces, particularly with non-uniform knot sequences and non-uniform multiplicities. We also discuss convergence of our rules to their asymptotic counterparts, that is, the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains. For spaces of low continuities, we numerically show that the derived rules quickly converge to their asymptotic counterparts as the weights and nodes of a few boundary elements differ from the asymptotic values.
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