Saulo P. Oliveira

Optimal blended spectral-element operators for acoustic wave modeling

Spectral-element methods, based on high-order polynomials, are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media. However, to retain computational efficiency, very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided. In the present work, we devise an approach for reducing such errors by considering modified discrete wave operators. We analyze consistent and lumped operators together with blended operators (weighted averages of consistent and lumped operators). Furthermore, using the operator-blending approach and a novel dispersion analysis method, we develop optimal spectral-element operators that have increased numerical accuracy, without resorting to very high order operators. The new operators are faster and computationally more efficient than consistent operators. Our approach is based on the tensor product decomposition of the element matrices into 1D f...

Pressure bubbles stabilization features in the Stokes problem

Abstract The standard finite element approximation using equal-order-linear-continuous velocity–pressure variables is enriched with velocity and pressure bubble functions to model the Stokes problem. We show by static condensation that these bubble functions give rise to a stabilized method involving least-squares forms of the momentum and of the continuity equations. In particular, pressure bubbles play a key role in explaining the addition of the least-squares form of the continuity equation in a stabilized method for Stokes.

Spectral element approximation of Fredholm integral eigenvalue problems

We consider the numerical approximation of homogeneous Fredholm integral equations of second kind, with emphasis on computing truncated Karhunen-Loeve expansions. We employ the spectral element method with Gauss-Lobatto-Legendre (GLL) collocation points. Similar to the piecewise-constant finite elements, this approach is simple to implement and does not lead to generalized discrete eigenvalue problems. Numerical experiments confirm the expected convergence rates for some classical kernels and illustrate how this approach can improve the finite element solution of partial differential equations with random input data.

DFT MODAL ANALYSIS OF SPECTRAL ELEMENT METHODS FOR ACOUSTIC WAVE PROPAGATION

Spectral element methods are now widely used for wave propagation simulations. They are appreciated for their high order of accuracy, but are still used on a heuristic basis. In this work we present the numerical dispersion of spectral elements, which allows us to assess their error limits and to devise efficient numerical simulations, particularly for 2D and 3D problems. We propose a novel approach based on a discrete Fourier transform of both the probing plane waves and the discrete wave operators. The underlying dispersion relation is estimated by the Rayleigh quotients of the plane waves with respect to the discrete operator. Together with the Kronecker product properties, this approach delivers numerical dispersion estimates for 1D operators as well as for 2D and 3D operators, and is well suited for spectral element methods, which use nonequidistant collocation points such as Gauss–Lobatto–Chebyshev and Gauss–Lobatto–Legendre points. We illustrate this methodology with dispersion and anisotropy graphs for spectral elements with polynomial degrees from 4 to 12. These graphs confirm the rule of thumb that spectral element methods reach a safe level of accuracy at about four grid points per wavelength.

DFT modal analysis of spectral element methods for the 2D elastic wave equation

The DFT modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete Fourier transform domain sampled in the mesh nodes. This technique provides a natural matching of exact and approximate modes of propagation. We extend this technique to spectral element methods for the 2D isotropic elastic wave equation, by using a Rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation, taking full advantage of the tensor product representation of the spectral element matrices. Numerical experiments illustrate the dependence of dispersion errors on the grid resolution, polynomial degree, and discretization in time. We consider spectral element methods with Chebyshev and Legendre collocation points.

Numerical approximation of 2D Fredholm integral eigenvalue problems by orthogonal wavelets

We investigate the numerical approximation of two-dimensional, second kind Fredholm integral eigenvalue problems by the Galerkin method with the Cohen-Daubechies-Vial (CDV) wavelet family. This choice provides us orthogonal bases for bounded domains, avoiding the need of periodization or domain truncation. The CDV family is indexed by the number of vanishing moments, which drives the regularity of the basis. We generate the Galerkin basis from tensorized scaling functions and employ weighted Gaussian quadratures derived from refinement equations. Numerical experiments address the relative computational cost of this approach with respect to the Haar basis and the relationship between convergence rate and number of vanishing moments.

CONTINUOUS Q1–Q1 STOKES ELEMENTS STABILIZED WITH NON-CONFORMING NULL EDGE AVERAGE VELOCITY FUNCTIONS

We present a stabilized finite element method for Stokes equations with piecewise continuous bilinear approximations for both velocity and pressure variables. The velocity field is enriched with piecewise polynomial bubble functions with null average at element edges. These functions are statically condensed at the element level and therefore they can be viewed as a continuous Q1–Q1 stabilized finite element method. The enriched velocity-pressure pair satisfies optimal inf–sup conditions and approximation properties. Numerical experiments show that the proposed discretization outperforms the Galerkin least-squares method.

EdgeDetectPFI: An algorithm for automatic edge detection in potential field anomaly images – application to dike-like magnetic structures

Abstract We propose an algorithm to automatically locate the spatial position of anomalies in potential field images, which can be used to estimate the depth and width of causative sources. The magnetic anomaly is firstly enhanced using an edge detection filter based on a simple transformation (the Signum transform) which retains only the signs of the anomalous field. The theoretical edge positions can be recognized from the locations where one of the spatial field derivatives (or a function of them) change its sign: the zero crossover points. These points are easily identified from the Signum transformed spatial derivatives. The actual sources depths and widths are then estimated using the widths of the positive plateaus obtained from two different Signum transformed data: one based on the vertical derivative and the other using the vertical derivative minus the absolute value of the horizontal derivative. Our algorithm finds these widths in an automatic fashion by computing the radius of the largest circles inside the positive plateaus. Numerical experiments with synthetic data show that the proposed approach provides reliable estimates for the target parameters. Additional testing is carried out with aeromagnetic data from Santa Catarina, Southern Brazil, and the resulting parameter maps are compared with Euler deconvolution.

A new methodology for computing ionic profiles and disjoining pressure in swelling porous media

A new two-scale computational model is proposed to construct the constitutive law of the swelling pressure which appears in the modified form of the macroscopic effective stress principle for expansive clays saturated by an aqueous electrolyte solution containing multivalent ionic species. The microscopic non-local nanoscale model is constructed based on a coupled Poisson-Fredholm integral equation arising from the thermodynamics of inhomogeneous fluids in nanopores (Density Functional Theory), which governs the local electric double layer potential profile coupled with the ion-particle correlation function in an electrolytic solution of finite size ions. The local problem is discretized by invoking the eigenvalue expansion of the convolution kernel in conjunction with the Galerkin method for the Gauss-Poisson equation. The discretization of the Fredholm equation is accomplished by a collocation scheme employing eigenfunction basis. Numerical simulations of the local ionic profiles in rectangular cell geometries are obtained showing considerable discrepancies with those computed with Poisson-Boltzmann based models for point charges, particularly for divalent ions in calcium montmorillonite. The constitutive law for the disjoining pressure is reconstructed numerically by invoking the contact theorem within a post-processing approach. The resultant computational model is capable of capturing ranges of particle attraction characterized by negative values of the disjoining pressure overlooked by the classical electric double layer theory. Such results provide further insight in the role the swelling pressure plays in the modified macroscopic effective stress principle for expansive porous media.

A note on the alternate trapezoidal quadrature method for Fredholm integral eigenvalue problems

We consider an approximate method based on the alternate trapezoidal quadrature for the eigenvalue problem given by a periodic singular Fredholm integral equation of second kind. For some convolution-type integral kernels, the eigenvalues of the discrete eigenvalue problem provided by the alternate trapezoidal quadrature method have multiplicity at least two, except up to two eigenvalues of multiplicity one. In general, these eigenvalues exhibit some symmetry properties that are not necessarily observed in the eigenvalues of the continuous problem. For a class of Hilbert-type kernels, we provide error estimates that are valid for a subset of the discrete spectrum. This subset is further enlarged in an improved quadrature method presented herein. The results are illustrated through numerical examples.