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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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DOI: http://hdl.handle.net/20.500.11824/101
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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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DOI: http://hdl.handle.net/20.500.11824/102
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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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DOI: http://hdl.handle.net/20.500.11824/103
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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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DOI: http://hdl.handle.net/20.500.11824/104
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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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DOI: http://hdl.handle.net/20.500.11824/107
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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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DOI: http://hdl.handle.net/20.500.11824/109
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NON-CONFORMING CURVED FINITE ELEMENT SCHEMES FOR TIME-DEPENDENT ELASTIC-ACOUSTIC COUPLED PROBLEMS

  • Rodríguez-Rozas, A.
  • Diaz, J.
High-order numerical methods for solving time-dependent acoustic-elastic coupled problems are introduced. These methods, based on Finite Element techniques, allow for a flexible coupling between the fluid and the solid domain by using non-conforming meshes and curved elements. Since characteristic waves travel at different speeds through different media, specific levels of granularity for the mesh discretization are required on each domain, making impractical a possible conforming coupling in between. Advantageously, physical domains may be independently discretized in our framework due to the non-conforming feature. Consequently, an important increase in computational efficiency may be achieved compared to other implementations based on conforming techniques, namely by reducing the total number of degrees of freedom. Differently from other non-conforming approaches proposed so far, our technique is relatively simpler and requires only a geometrical adjustment at the coupling interface at a preprocessing stage, so that no extra computations are necessary during the time evolution of the simulation. On the other hand, as an advantage of using curvilinear elements, the geometry of the coupling interface between the two media of interest is faithfully represented up to the order of the scheme used. In other words, higher order schemes are in consonance with higher order approximations of the geometry. Concerning the time discretization, we analyze both explicit and implicit schemes. These schemes are energy conserving and, for the explicit case, the stability is guaranteed by a CFL condition.In order to illustrate the accuracy and convergence of these methods, a set of representative numerical tests are presented.

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DOI: http://hdl.handle.net/20.500.11824/110
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RED REFINEMENTS OF SIMPLICES INTO CONGRUENT SUBSIMPLICES

  • Korotov, S.
  • Krizek, M.
We show that in dimensions higher than two, the popular "red refinement" technique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original one.

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DOI: http://hdl.handle.net/20.500.11824/111
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ON NUMERICAL REGULARITY OF THE FACE-TO-FACE LONGEST-EDGE BISECTION ALGORITHM FOR TETRAHEDRAL PARTITIONS

  • Hannukainen, A.
  • Korotov, S.
  • Krizek, M.
The finite element method usually requires regular or strongly regular families of partitions in order to get guaranteed a priori or a posteriori error estimates. In this paper we examine the recently invented longest-edge bisection algorithm that always produces only face-to-face simplicial partitions. First, we prove that the regularity of the family of partitions generated by this algorithm is equivalent to its strong regularity in any dimension. Second, we present a number of 3d numerical tests, which demonstrate that the technique seems to produce regular (and therefore strongly regular) families of tetrahedral partitions. However, a mathematical proof of this statement is still an open problem.

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DOI: http://hdl.handle.net/20.500.11824/113
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ANALYSIS OF EXTREME WAVE EVENTS ON THE SOUTHERN COAST OF BRAZIL

  • Guimaraes, P.V.
  • Farina, L.
  • Toldo, E.E.
Using the wave model SWAN (simulating waves nearshore), high waves on the southwestern Atlantic generated by extra-tropical cyclones are simulated from 2000 to 2010, and their impact on the Rio Grande do Sul (RS) coast is studied. The modeled waves are compared with buoy data and good agreement is found. The six extreme events in the period that presented significant wave heights above 5 m, on a particular point of interest, are investigated in detail. It is found that the cyclogenetic pattern between the latitudes 31.5 and 34 S is the most favorable for developing high waves. Hovmöller diagrams for deep water show that the region between the south of Rio Grande do Sul up to a latitude of 31.5 S is the most energetic during a cyclone's passage, although the event of May 2008 indicates that the location of this region can vary, depending on the cyclone's displacement. On the other hand, the Hovmöller diagrams for shallow water show that the different shoreface morphologies were responsible for focusing or dissipating the waves' energy; the regions found are in agreement with the observations of erosion and progradation regions. It can be concluded that some of the urban areas of the beaches of Hermenegildo, Cidreira, Pinhal, Tramandaí, Imbé and Torres have been more exposed during the extreme wave events on the Rio Grande do Sul coast, and are more vulnerable to this natural hazard.

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DOI: http://hdl.handle.net/20.500.11824/114
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