Anthony Peirce

The blowup property of solutions to some diffusion equations with localized nonlinear reactions

Abstract In this paper we investigate the blowup property of solutions to the equation u t = Δu + ƒ(u(x 0 , t)) , where x0 is a fixed point in the domain. We show that under certain conditions the solution blows up in finite time. Moreover, we prove that the set of all blowup points is the whole region. Furthermore, the growth rate of solutions near the blowup time is also derived. Finally, the results are generalized to the following nonlocal reaction-diffusion equation u t = Δu + ∝ Ω ƒ(u) dx .

Control of molecular motion

In this paper we present a short introduction to the new field of control of molecular motion. Our intention is to outline how the methods and techniques of control theory play a crucial role in the development of this emerging field, and reciprocally how fundamental new problems are motivated by this interaction.

Interference Fracturing: Nonuniform Distributions of Perforation Clusters That Promote Simultaneous Growth of Multiple Hydraulic Fractures

© 2015 Society of Petroleum Engineers. One of the important hurdles in horizontal-well stimulation is the generation of hydraulic fractures (HFs) from all perforation clusters within a given stage, despite the challenges posed by stress shadowing and reservoir variability. In this paper, we use a newly developed, fully coupled, parallel-planar 3D HF model to investigate the potential to minimize the negative impact of stress shadowing and thereby to promote more-uniform fracture growth across an array of HFs by adjusting the location of the perforation clusters. In this model, the HFs are assumed to evolve in an array of parallel planes with full 3D stress coupling while the constant fluid influx into the wellbore is dynamically partitioned to each fracture so that the wellbore pressure is the same throughout the array. The model confirms the phenomenon of inner-fracture suppression because of stress shadowing when the perforation clusters are uniformly distributed. Indeed, the localization of the fracture growth to the outer fractures is so dominant that the total fractured area generated by uniform arrays is largely independent of the number of perforation clusters. However, numerical experiments indicate that certain nonuniform cluster spacings promote a profound improvement in the even development of fracture growth. Identifying this effect relies on this new models ability to capture the full hydrodynamical coupling between the simultaneously evolving HFs in their transition from radial to Perkins-Kern-Nordgren (PKN)-like geometries (Perkins and Kern 1961; Nordgren 1972).

STABILITY ANALYSIS AND DESIGN OF TIME-STEPPING SCHEMES FOR GENERAL ELASTODYNAMIC BOUNDARY ELEMENT MODELS

SUMMARY In the literature there is growing evidence of instabilities in standard time-stepping schemes to solve boundary integral elastodynamic models.1{3 However, there has been no theory to support scientists and engineers in assessing the stability of their boundary element algorithms or to help them with the design of new, more stable algorithms. In this paper we present a general framework for the analysis of the stability of any time-domain boundary element model. We illustrate how the stability theory can be used to assess the stability of existing boundary element models and how the insight gained from this analysis can be used to design more stable time-stepping schemes. In particular, we describe a new time-stepping procedure that we have developed, which has substantially enhanced stability characteristics and greater accuracy for the same computational eort. The new scheme, which we have called ‘the half-step scheme’, is shown to have substantially improved performance for the displacement discontinuity boundary element method commonly used to model dynamic fracture interaction and propagation.

A weakly nonlinear analysis of elasto-plastic-microstructure models

At certain critical values of the hardening modulus, the governing equations of elasto-plastic flow may lose their hyperbolicity and exhibit two modes of ill-posedness: shear-band and flutter ill-posedness. These modes of ill-posedness are characterized by the uncontrolled growth of modes at infinitely fine scales, which ultimately violates the continuum assumption. In previous work [L. An and A. Peirce, SIAM J. Appl. Math., 54(1994), pp. 708–730], a continuum model accounting for microscale deformations was built. Linear analysis demonstrated the regularizing effect of the microstructure and provided a relationship between the width of the localized instabilities and the microlength scale. In this paper a weakly nonlinear analysis is used to explore the immediate post-critical behavior of the solutions. For both one-dimensional and anti-plane shear models, post-critical deformations in the plastic regions are shown to be governed by the Boussinesq equation (one of the completely integrable PDEs having so...

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains

The mean first passage time (MFPT) is calculated for a Brownian particle in a bounded two-dimensional domain that contains N small nonoverlapping absorbing windows on its boundary. The reciprocal o...

An Asymptotic Framework for the Analysis of Hydraulic Fractures: The Impermeable Case

This paper presents a novel asymptotic framework to obtain detailed solutions describing the propagation of hydraulic fractures in an elastic material. The problem consists of a system of nonlinear integro-differential equations and a free boundary problem. This combination of local and nonlocal effects leads to transitions on a small scale near the crack tip, which control the behavior across the whole fracture profile. These transitions depend upon the dominant physical process(es) and are identified by simultaneously scaling the associated parameters with the distance from the tip. A smooth analytic solution incorporating several physical processes in the crucial tip region can be constructed using this new framework. In order to clarify the exposition of the new methodology, this paper is confined to considering the impermeable case in which only the two physical processes of viscous dissipation and structure energy release compete.

Uniform asymptotic approximations for accurate modeling of cracks in layered elastic media

We present uniform asymptotic solutions (UAS) for displacement discontinuities (DD) that lie within the middle layer of a three layer elastic medium. The DDs are assumed to be normal to the two parallel interfaces between the leastic media, and solutions will be presented for both 2D and 3D elastic media. Using the Fourier transform (FT) method we construct the leading term in the asymptotic expansion for the spectral coefficient functions for a DD in a three layer medium. Although a closed form solution will require an infinite series solution, we demonstrate how this UAS can be used to construct highly efficient and accurate solutions even in the case in which the DD actually touches the interface. We present an explicit UAS for elements in which the DD fields are assumed to be piecewise constant throughout a line segment in 2D and a rectangular element in 3D. We demonstrate the usefulness of this UAS by providing a number of examples in which the UAS is used to solve problems in which cracks just touch or cross an interface. The accuracy and efficiency of the algorithm is demonstrated and compared with other numerical methods such as the finite element method and the boudary integral method.

Stability of reactive flows in porous media: coupled porosity and viscosity changes

The infiltration flow of a reactive fluid in a porous medium is investigated. The reaction causes porosity/permeability changes in the porous medium as well as viscosity changes in the fluid. The coupling of the associated reaction-infiltration and Saflman-Taylor instabilities are considered. A mathematical model for this phenomenon is given in the form of a moving free-boundary problem. The morphological instability of a planar dissolution front is demonstrated using a linear stability analysis. An unexpected simplification occurs in that the resulting fourth-order equation can be solved explicitly.

The scaled flexibility matrix method for the efficient solution of boundary value problems in 2D and 3D layered elastic media

We present a method that extends the flexibility matrix method for multilayer elasticity problems to include problems with very thin layers. This method is particularly important for solving problems in which one or a number of very thin layers are juxtaposed with very thick layers. The standard flexibility matrix method suffers from round-off errors and poor scaling of the flexibility equations which occur when one of the layers in the multilayered medium becomes much smaller than the others. The method proposed in this paper makes use of power series expansions of the various components of the flexibility matrix in order to arrive at a system of equations that is appropriately scaled. The effectiveness of the scaled flexibility matrix method is demonstrated on a number of test problems.