Browsing by Author "Sanchez, Manuel A."

Now showing 1 - 9 of 9
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    An O(p3) hp-version FEM in two dimensions: Preconditioning and post-processing
    (2019) Ainsworth, Mark; Jiang, Shuai; Sanchez, Manuel A.
    The Bernstein polynomials have been known for over a century and are widely used in the spline literature, computer aided geometric design, and computer graphics. However, the realization that the Bernstein basis has favorable properties allowing the efficient implementation of high order methods for the approximation of partial differential equations is a relatively recent development. For instance, it is known (Ainsworth et al., 2011) that the Bernstein basis can be exploited to compute all of the entries in the load vector in O(p(3)) operations even in the case of non-linear problems on curvilinear elements for a degree p approximation. Moreover, the element matrices can be assembled in O(1) operations per entry. We show that properties of the Bernstein polynomials can also be exploited to obtain O(p(3)) complexity procedures for all of the main components needed to implement a high order finite element code including: computation of the residuals needed for an iterative solution method; evaluating the action of a preconditioner for the global mass matrices; and, visualization and post-processing of the resulting finite element approximations.
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    Corrected finite element methods on unfitted meshes for Stokes moving interface problem
    (2022) Laymuns, Genaro; Sanchez, Manuel A.
    We propose an explicit in time corrected finite element method for a problem with a moving elastic interface in Stokes flow, using unfitted meshes and standard finite element approximation spaces. Piecewise polynomial correction functions are computed to accurately capture the discontinuities of the solution resulting from the elastic force applied to the fluid and are used to correct the load vector. The method is proven to be optimal for the steady state problem. Additionally, we propose a trigonometric interpolant to approximate and evolve the interface and explicit Euler scheme to advance in time. Numerical examples demonstrate the third and second order accuracy of the method for the velocity and pressure, respectively, and the stability of the numerical scheme.
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    Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics
    (2022) Cockburn, Bernardo; Du, Shukai; Sanchez, Manuel A.
    We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces. These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy. We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy. Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time. The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system. The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time. The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables.
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    Experiments, Modelling, and Simulations for a Gel Bonded to a Rigid Substrate
    (2023) Song, Sichen; Siegel, Ronald A.; Sanchez, Manuel A.; Carme Calderer, M.; Henao, Duvan
    In preparation for a more thorough study based on our own experimental work of the debonding of a thin film gel by stress concentration on the interface with a rigid substrate, in this article we revisit, from the viewpoint of the synergy between mathematics, experiments, and finite element simulations, the problem of the swelling of a thin rectangular polyacrylamide gel covalently bonded on the bottom surface to a glass slide. With methods of the calculus of variations and perturbation theory we show that the solution to the corresponding zero-displacement boundary value problem converges, in the thin film limit, to a uniquely defined uniform uniaxial extension on the direction normal to the substrate. Both the experiments and the finite element simulations that we perform confirm that the amount of lateral swelling is very small, with a very good quantitative agreement between the two approaches. The proposed model of minimizing an energy functional comprising both a term for the elastic distortion and the Flory-Huggins expression for the entropy of mixing is thus experimentally and numerically validated, with parameters coming from experimental measurements, including the initial polymer volume fraction of the hydrogel synthesized in the laboratory (which is taken as the reference configuration instead of the dry polymer).
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    Gels: Energetics, Singularities, and Cavitation
    (2024) Calderer, M. Carme; Henao, Duvan; Sanchez, Manuel A.; Siegel, Ronald A.; Song, Sichen
    This article studies equilibrium singular configurations of gels and addresses open questions concerning gel energetics. We model a gel as an incompressible, immiscible and saturated mixture of a solid polymer and a solvent that sustain chemical interactions at the molecular level. We assume that the energy of the gel consists of the elastic energy of its polymer network plus the Flory-Huggins energy of mixing. The latter involves the entropic energies of the individual components plus that of interaction between polymer and solvent, with the temperature dependent Flory parameter, ?, encoding properties of the solvent. In particular, a good solvent promoting the mixing regime, is found below the threshold value ? = 0.5, whereas the phase separating regime develops above that critical value. We show that cavities and singularities develop in the latter regime. We find two main classes of singularities: (i) drying out of the solvent, with water possibly exiting the gel domain through the boundary, leaving behind a core of exposed polymer at the centre of the gel; (ii) cavitation, in response to traction on the boundary or some form of negative pressure, with a cavity that can be either void or flooded by the solvent. The straightforward and unified mathematical approach to treat all such singularities is based on the construction of appropriate test functions, inspired by the particular states of uniform swelling or compression. The last topic of the article addresses a statistical mechanics rooted controversy in the research community, providing an experimental and analytic study in support of the phantom elastic energy versus the affine one.
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    A priori error analysis of new semidiscrete, Hamiltonian HDG methods for the time-dependent Maxwell's equations
    (2023) Cockburn, Bernardo; Du, Shukai; Sanchez, Manuel A.
    We present the first a priori error analysis of a class of space-discretizations by Hybridizable Discontinuous Galerkin (HDG) methods for the time-dependent Maxwell's equations introduced in Sanchez et al. [Comput. Methods Appl. Mech. Eng. 396 (2022) 114969]. The distinctive feature of these discretizations is that they display a discrete version of the Hamiltonian structure of the original Maxwell's equations. This is why they are called ``Hamiltonian'' HDG methods. Because of this, when combined with symplectic time-marching methods, the resulting methods display an energy that does not drift in time. We provide a single analysis for several of these methods by exploiting the fact that they only differ by the choice of the approximation spaces and the stabilization functions. We also introduce a new way of discretizing the static Maxwell's equations in order to define the initial condition in a manner consistent with our technique of analysis. Finally, we present numerical tests to validate our theoretical orders of convergence and to explore the convergence properties of the method in situations not covered by our analysis.
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    REFLECTIONLESS DISCRETE PERFECTLY MATCHED LAYERS FOR HIGHER-ORDER FINITE DIFFERENCE SCHEMES
    (2024) Hojas, Vicente A.; Perez-Arancib, Carlos; Sanchez, Manuel A.
    This paper introduces discrete holomorphic perfectly matched layers (PMLs) specifically designed for high-order finite difference (FD) discretizations of the scalar wave equation. In contrast to standard PDE-based PMLs, the proposed method achieves the remarkable outcome of completely eliminating numerical reflections at the PML interface, in practice achieving errors at the level of machine precision. Our approach builds upon the ideas put forth in a recent publication [A. Chern, J. Comput. Phys., 381 (2019), pp. 91--109] expanding the scope from the standard second- order FD method to arbitrarily high-order schemes. This generalization uses additional localized PML variables to accommodate the larger stencils employed. We establish that the numerical solutions generated by our proposed schemes exhibit a geometric decay rate as they propagate within the PML domain. To showcase the effectiveness of our method, we present a variety of numerical examples, including waveguide problems. These examples highlight the importance of employing high-order schemes to effectively address and minimize undesired numerical dispersion errors, emphasizing the practical advantages and applicability of our approach.
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    Symplectic Hamiltonian finite element methods for electromagnetics
    (2022) Sanchez, Manuel A.; Du, Shukai; Cockburn, Bernardo; Nguyen, Ngoc-Cuong; Peraire, Jaime
    We present several high-order accurate finite element methods for the Maxwell's equations which provide time-invariant, non-drifting approximations to the total electric and magnetic charges, and to the total energy. We devise these methods by taking advantage of the Hamiltonian structures of the Maxwell's equations as follows. First, we introduce spatial discretizations of the Maxwell's equations using mixed finite element, discontinuous Galerkin, and hybridizable discontinuous Galerkin methods to obtain a semi-discrete system of equations which display discrete versions of the Hamiltonian structure of the Maxwell's equations. Then we discretize the resulting semi-discrete system in time by using a symplectic integrator. This ensures the conservation properties of the fully discrete system of equations. For the Symplectic Hamiltonian HDG method, we present numerical experiments which confirm its optimal orders of convergence for all variables and its conservation properties for the total linear and angular momenta, as well as the total energy. Finally, we discuss the extension of our results to other boundary conditions and to numerical schemes defined by different weak formulations.(c) 2022 Elsevier B.V. All rights reserved.
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    Symplectic Hamiltonian Finite Element Methods for Semilinear Wave Propagation
    (2024) Sanchez, Manuel A.; Valenzuela, Joaquin
    This paper presents Hamiltonian finite element methods for approximating semilinear wave propagation problems, including the nonlinear Klein-Gordon and sine-Gordon equations. The aim is to obtain accurate high-order approximations while conserving physical quantities of interest such as energy. To achieve conservation properties at a discrete level, we propose semidiscrete schemes based on two Hamiltonian structures of the equation. These include Mixed finite element methods, discontinuous Galerkin methods, and hybridizable discontinuous Galerkin methods (HDG). In particular, we propose a new class of DG methods using time operators to define the numerical traces, ultimately leading to an energy-conserving scheme. Time discretization uses Symplectic explicit-partitioned and diagonally-implicit Runge-Kutta schemes. Furthermore, the paper showcases several numerical examples that demonstrate the accuracy and energy conservation properties of the approximations, along with the simulation of soliton cloning.