Horacio Florez

By Horacio Florez

Most engineering applications estimate the deformation induced by loads by using the linear elasticity theory. The discretization process starts with the equilibrium equation and then develops a displacement formulation that employs the Hooke’s law. Problems of practical interest encompass designing of large structures, buildings, subsurface deformation, etc. These applications require determining stresses to compare them with a given failure criteria. One often tackles this way a design or material strength type of problems. For instance, Geomechanics applications in the oil and gas industry assess the induced stresses changes that hydrocarbon production or the injection of fluids, i.e., artificial lift, in a reservoir produce in the surrounding rock mass. These studies often include reservoir compaction and subsidence that pose harmful and costly effects such as in wells casing, cap-rock stability, faults reactivation, and environmental issues as well. Estimating these stress-induced changes and their consequences require accurate elasticity simulations that are usually carried out through finite element (FE) simulations. Geomechanics implies that the flow in porous media simulation must be coupled with mechanics, which causes a substantial increase in CPU time and memory requirements.

Part of the book: Finite Element Method

By Horacio Florez and Miguel Argáez

A global regularized Gauss-Newton (GN) method is proposed to obtain a zero residual for square nonlinear problems on an affine subspace built by wavelets, which allows reducing systems that arise from the discretization of nonlinear elliptic partial differential equations (PDEs) without performing a priori simulations. This chapter introduces a Petrov-Galerkin (PG) GN approach together with its standard assumptions that ensure retaining the q-quadratic rate of convergence. It also proposes a regularization strategy, which maintains the fast pace of convergence, to avoid singularities and high nonlinearities. It also includes a line-search method for achieving global convergence. The numerical results manifest the capability of the algorithm for reproducing the full-order model (FOM) essential features while decreasing the runtime by a significant magnitude. This chapter refers to a wavelet-based reduced-order model (ROM) as WROM, while PROM is the proper orthogonal decomposition (POD)-based counterpart. The authors also implemented the combination of WROM and PROM as a hybrid method referred herein as (HROM). Preliminary results with Bratu?s problem show that if the WROM could correctly reproduce the FOM behavior, then HROM can also reproduce that FOM accurately.

Part of the book: Nonlinear Systems

By Horacio Florez and Belsay Borges

A common engineering task consists of interpolating a set of discrete points that arise from measurements and experiments. Another traditional requirement implies creating a curve that mimics a given array of points, namely, a polyline. Any of these problems require building an analytical representation of the given discrete set of points. If the geometrical shape represented by the input polyline is complicated, then we may expect that a global interpolant or polynomial will be of a high degree, to honor all imposed constraints, which makes its use prohibited. Indeed, a global interpolant often experiences inflection points and sudden changes in curvature. To avoid these drawbacks, we often seek solving the interpolation/approximation problem using piecewise polynomial functions called “splines.”

Part of the book: Topics in Splines and Applications