Abstract

In this work I develop numerical algorithms that can be applied directly to differential equations of the general form f (t, x, x') = 0, without the need to cleared x'. My methods are hybrid algorithms between standard methods of solving differential equations and methods of solving algebraic equations, with which the variable x' is numerically cleared. The application of these methods ranges from the ordinary differential equations of order one, to the more general case of systems of m equations of order n. These algorithms are applied to the solution of different physical-mathematical equations. Finally, the corresponding numerical analysis of existence, uniqueness, stability, consistency and convergence is made, mainly for the simplest case of a single ordinary differential equation of the first order.

Abstract

We study three “incompressibility flavors” of linearly-elastic anisotropic solids that exhibit volumetric constraints: isochoric, hydroisochoric and rigidtropic. An isochoric material deforms without volume change under any stress system. An hydroisochoric material does so under hydrostatic stress. A rigidtropic material undergoes zero deformations under a certain stress pattern. Whereas the three models coalesce for isotropic materials, important differences appear for anisotropic behavior. We find that isochoric and hydroisochoric models under certain conditions may be hampered by unstable physical behavior. Rigidtropic models can represent semistable physical materials of arbitrary anisotropy while including isochoric and hydroisochoric behavior as special cases.

Abstract

We study three “incompressibility flavors” of linearly-elastic anisotropic solids that exhibit volumetric constraints: isochoric, hydroisochoric and rigidtropic. An isochoric material deforms without volume change under any stress system. An hydroisochoric material does so under hydrostatic stress. A rigidtropic material undergoes zero deformations under a certain stress pattern. Whereas the three models coalesce for isotropic materials, important differences appear for anisotropic behavior. We find that isochoric and hydroisochoric models under certain conditions may be hampered by unstable physical behavior. Rigidtropic models can represent semistable physical materials of arbitrary anisotropy while including isochoric and hydroisochoric behavior as special cases.

Abstract

In this work we explore a velocity correction method that introduces the splitting at the discrete level. In order to do so, the algebraic continuity equation is transformed into a discrete pressure Poisson equation and a velocity extrapolation is used. In Badia et al. (IJNMF, 2008, p. 351), where the method was introduced, the discrete Laplacian that appears in the pressure Poisson equation is approximated by a continuous one using an extrapolation for the pressure. In this work we explore the possibility of actually solving the discrete Laplacian. This introduces significant differences because the pressure extrapolation is avoided and only a velocity extrapolation is needed. Our numerical results indicate that it is the second‐order pressure extrapolation which makes third‐order methods unstable. Instead, second‐order velocity extrapolations do not lead to instabilities. Avoiding the pressure extrapolation allows to obtain stable solutions in problems that become unstable when the Laplacian is approximated. A comparison with a pressure correction scheme is also presented to verify the well‐known fact that the use of a second order pressure extrapolation leads to instabilities. Therefore we conclude that it is the combination of a velocity correction scheme with a discrete Laplacian that allows to obtain a stable third‐order scheme by avoiding the pressure extrapolation.

Abstract

In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least‐squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method. 

Abstract

Durante la época de lluvias se presentan en el estado de Tabasco (México)
deslizamientos en los bordos de los ríos que atraviesan la entidad, siendo
los de la zona conocida como La Manga de los más afectados. Este artículo
presenta el análisis de la evolución de la estabilidad de uno de estos bordos
debido a los cambios en el grado de saturación del suelo (Sr) que lo
conforma. El objetivo fue determinar el Sr del suelo para el cual el bordo
pasa de una condición de seguridad a una de falla. Para lograrlo, primero
se realizó una campaña de exploración de campo y ensayos de laboratorio,
lo que permitió obtener las características geométricas del bordo, así como
las propiedades del suelo que lo constituye. De las muestras inalteradas
obtenidas en campo se realizaron ensayos de compresión triaxial variando
solo el grado de saturación del suelo. Con los datos obtenidos y mediante
el software GeoSlope 2016 se realizaron diversos análisis de la estabilidad.
Los resultados muestran que conforme el grado de saturación crece la
estabilidad del bordo disminuye. Se comprobó que para el bordo en estudio
sobrepasar un grado de saturación de 70 % implica una condición de peligro
de deslizamiento. Por lo tanto, el monitoreo del grado de saturación del
suelo que constituye a los bordos permite anticipar su falla y, en consecuencia,
hace posible establecer acciones de prevención.

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