Michael E. Henderson

A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

MULTIPLE PARAMETER CONTINUATION: COMPUTING IMPLICITLY DEFINED k-MANIFOLDS

We present a new continuation method for computing implicitly defined manifolds. The manifold is represented as a set of overlapping neighborhoods, and extended by an added neighborhood of a bounda...

CLASSIFICATION OF THE SPATIAL EQUILIBRIA OF THE CLAMPED ELASTICA: NUMERICAL CONTINUATION OF THE SOLUTION SET

We consider equilibrium configurations of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods when subject to end loads and clamped boundary conditions. In a first paper [Neukirch & Henderson, 2002], we discussed symmetry properties of the equilibrium configurations of the center line of the rod. Here, we are interested in the set of all parameter values that yield equilibrium configurations that fulfill clamped boundary conditions. We call this set the solution manifold and we compute it using a recently introduced continuation algorithm. We then describe the topology of this manifold and how it comprises different interconnected layers. We show that the border set of the different layers is the well-known solution set of buckled rings.

Classification of the Spatial Equilibria of the Clamped Elastica: Symmetries and Zoology of Solutions

We investigate the configurations of twisted elastic rods under applied end loads and clamped boundary conditions. We classify all the possible equilibrium states of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods. We show that all solutions of the clamped boundary value problem exhibit a π-flip symmetry. The Kirchhoff equations which describe the equilibria of these rods are integrated in a formal way which enable us to describe the boundary conditions in terms of 2 closed form equations involving 4 free parameters. We show that the flip symmetry property is equivalent to a reversibility property of the solutions of the Kirchhoff differential equations. We sort these solutions according to their period in the phase plane. We show how planar untwisted configurations as well as circularly closed configurations play an important role in the classification.

Planning Singularity-Free Paths on Closed-Chain Manipulators

This paper provides an algorithm for computing singularity-free paths on closed-chain manipulators. Given two nonsingular configurations of the manipulator, the method attempts to connect them through a path that maintains a minimum clearance with respect to the singularity locus at all points, which guarantees the controllability of the manipulator everywhere along the path. The method can be applied to nonredundant manipulators of general architecture, and it is resolution complete. It always returns a path whenever one exists at a given resolution or determines path nonexistence otherwise. The strategy relies on defining a smooth manifold that maintains a one-to-one correspondence with the singularity-free C-space of the manipulator, and on using a higher dimensional continuation technique to explore this manifold systematically from one configuration, until the second configuration is found. If desired, the method can also be used to compute an exhaustive atlas of the whole singularity-free component reachable from a given configuration, which is useful to rapidly resolve subsequent planning queries within such component, or to visualize the singularity-free workspace of any of the manipulator coordinates. Examples are included that demonstrate the performance of the method on illustrative situations.

MULTIPARAMETER PARALLEL SEARCH BRANCH SWITCHING

A continuation method (sometimes called path following) is a way to compute solution curves of a nonlinear system of equations with a parameter. We derive a simple algorithm for branch switching at...

A singularity-free path planner for closed-chain manipulators

This paper provides an algorithm for computing singularity-free paths on non-redundant closed-chain manipulators. Given two non-singular configurations of the manipulator, the method attempts to connect them through a configuration space path that maintains a minimum clearance with respect to the singularity locus at all points. The method is resolution-complete, in the sense that it always returns a path if one exists at a given resolution, or returns “failure” otherwise. The path is computed by defining a new manifold that maintains a one-to-one correspondence with the singularity-free configuration space of the manipulator, and then using a higher-dimensional continuation technique to explore this manifold systematically from one configuration, until the second configuration is found. Examples are included that demonstrate the performance of the method on illustrative situations.

Techniques for including large deformations associated with salt and fault motion in basin modeling

Abstract Modeling large deformations such as non-vertical fault displacement and salt motion have been a major obstacle for the improvement of regional basin modeling studies. Because salt has a large thermal conductivity and is practically impervious, and faults can act as conduits or seals during the evolution of sedimentary basins, they are critical in making accurate predictions of the generation, migration and accumulation of hydrocarbons within salt-bearing basins. To model numerically the evolution of salt structures is not a trivial task and one of the major difficulties in modeling the motion of salt and faults is the management of numerical meshes that are severely corrupted with large deformations. In this study, we use a topological framework for the representation of complex geological structures that makes it possible to model geological processes with large deformation within sedimentary basins and the lithosphere. This framework greatly facilitates the automatic meshing and remeshing required during modeling because meshes, like lithology and physical properties, are treated as attributes of subregions of the model. In this context, we developed a series of techniques to classify fault blocks in order to model the displacement of multiple faults simultaneously in the correct order. In addition, we show that this framework allows the decomposition of the basin model along geological discontinuities and makes it suitable for parallel computation of the solution of differential equations governing generation and migration of hydrocarbons.

Synergy in parallel algorithms

A property of algorithms called synergy is introduced, and a quantitys, of synergy is defined. When synergized, both parallel and serial algorithms run faster, the parallel algorithms benefiting from a cooperation between processors. Examples show that synergy is a useful concept in the design of parallel algorithms.

Assisted seismic matching: Joint inversion of seismic, rock physics, and basin modeling

In this study, we apply advanced numerical optimization techniques to extend the work of Imhof and Sharma (2005, 2006) to integrate geological and geophysical data and infer the sedimentary parameters that produce a match to seismic data. In particular, we seek to match not just event timing (phase) but also reflection strength (amplitude). This inverse problem of quantitatively matching present-day measurements of structure, stratigraphy, petrology and/or fluids is inherently ill-posed and computationally difficult. In our approach we automatically adjust parameters, which control numerical forward models such as numerical basin models, petrophysical models, and seismic acoustic models, to match observed seismic data and observed stratigraphy. Note that the problem we tackle is more complex than traditional seismic inversion, which “just” estimates the velocity and reflectivity model that fits the seismic data. Here we also estimate the geological layer composition, as well as rock physics parameters controlling the relationships defining the bulk rock density and velocity.