Jacobian Terms Research Articles

A stereographic projection path integral study of the coupling between the orientation and the bending degrees of freedom of water

A Monte Carlo path integral method to study the coupling between the rotation and bending degrees of freedom for water is developed. It is demonstrated that soft internal degrees of freedom that are not stretching in nature can be mapped with stereographic projection coordinates. For water, the bending coordinate is orthogonal to the stereographic projection coordinates used to map its orientation. Methods are developed to compute the classical and quantum Jacobian terms so that the proper infinitely stiff spring constant limit is recovered in the classical limit, and so that the nonconstant nature of the Riemann Cartan curvature scalar is properly accounted in the quantum simulations. The theory is used to investigate the effects of the geometric coupling between the bending and the rotating degrees of freedom for the water monomer in an external field in the 250 to 500 K range. We detect no evidence of geometric coupling between the bending degree of freedom and the orientations.

Modelling non-linear heat conduction via a fast matrix-free implicit unstructured-hybrid algorithm

A matrix-free implicit algorithm is developed for the fast and efficient solution of the non-linear diffusion equation on unstructured-hybrid meshes. The aforementioned captures the physics prevalent in plasticity (structural mechanics), moisture transport in porous materials and non-linear heat conduction. This paper will be restricted to the latter. The boundary condition types supported are prescribed temperature, convection and radiation type Neumann conditions. A compact edge-based vertex-centered finite volume technique is employed for spatial discretisation purposes, while implicit temporal discretisation is effected. The resulting system of equations is Newton-linearized, where approximate analytical expressions for the Jacobian terms are developed. The discrete system is solved via a preconditioned Generalised Minimal Residual (GMRES) algorithm, where Lower–Upper Symmetric Gauss–Seidel (LU-SGS) is used as preconditioner. The computational performance of the developed algorithm is demonstrated by application to the modelling of steady 2D non-linear heat conduction problems. The calculated Jacobian terms are stored in the interest of computational speed, with an associated additional overall memory cost of less than 10%. Even for small problems, the developed solver outperforms Jacobi with local time-stepping in terms of computational time by a factor ranging between 30 and 1000, while having a total memory requirement of 1.9 times the aforementioned. Further, as compared to the LU-SGS and GMRES methods, the proposed solver methodology offers a speed-up in solution time of between one and two orders of magnitude.

Modeling fractured and faulted regions: Local grid refinement methods for implicit solvers

In order to give a full 3D description of reservoir fault zones, we use a non-overlapping, non-matching local grid refinement (LGR) technique in the multi-phase, multi-component hydrocarbon migration simulator Som (Secondary Oil Migration). The resulting small spatial scales severely restrict the time steps if the mass transport equations are solved explicitly. Thus, an implicit formulation is derived where the Jacobian terms are analytical expressions based on binary mixture thermodynamics. We will present details of the implementation of this method in the LGR framework using Newton-GMRES and a multigrid-like, two-level iterative solver.

A Lie group formulation of the dynamics of cooperating robot systems

In this article we formulate the dynamics of cooperating robot systems using standard ideas and notation from the theory of Lie groups. Utilizing the geometric framework introduced in Park et al. (1995), we develop the equations of motion for a system of N cooperating robots manipulating a common workpiece. In the resulting dynamic equations the Jacobian, mass matrix, Coriolis, and gravity terms of the closed chain system admit succinct block-triangular factorizations in terms of the basic linear operators on se(3), the Lie algebra of the Euclidean group SE(3). The resulting closed-form equations provide a high-level description of the equations of motion that reduces the symbolic complexity without sacrificing computational efficiency.

Calculating the two-dimensional magnetotelluric Jacobian in finite elements using reciprocity

SUMMARY To speed up the calculation of the field Jacobian for 2-D magnetotelluric inversion using finite elements, the principle of electromagnetic reciprocity is applied. The governing relationship for the Jacobian of the field along strike is obtained by differentiating the Helmholtz equation with respect to the resistivity of each region in the finite-element mesh. The result is a similar Helmholtz equation for the Jacobian, with new sources distributed over all nodes within the parameter medium. However, according to the principle of electromagnetic reciprocity, the roles of sources and receivers are interchangeable. Utilizing reciprocity, the field values obtained from the original forward problem and for new unit sources imposed at the receivers are then utilized in the calculation of the Jacobian by simple multiplication and summation with finite-element terms at each rectangle in the mesh. For the auxiliary (across-strike) fields, the Jacobian terms are obtained by solving source vectors loaded with parabola coefficients used in the approximation to Maxwell's equations. Jacobian terms for the apparent resistivity (pa), the impedance phase (4) and the vertical magnetic field &) are then calculated utilizing the parallel- and auxiliary-field Jacobians. Comparison of Jacobian values obtained from reciprocity calculations and by differencing two forward solutions show that the reciprocity method is accurate and can be used to decrease the number of calculations required to obtain sensitivities by one to two orders of magnitude.

Soliton physics and the periodic inverse scattering transform

Abstract I focus on the mathematical and physical structure of inverse scattering transform solutions of particular integrable nonlinear wave equations which have periodic boundary conditions. Emphasis is placed on the so-called θ-function representation of the Korteweg-de Vries (KdV) equation for which the nonlinear wave motion is describable as a linear superposition of Jacobian elliptic functions (cnoidal waves) plus additional terms which include nonlinear interactions among the waves; the elliptic functions generally take the form not only of solitons but also of sine waves and Stokes waves. It is suggested that the formulation is reducible to a power spectral representation for which statistical mechanical and stochastic solutions of KdV may be computed in terms of random soliton modes interacting with a random radiation sea. A number of other aspects of the θ-function representation, including numerical computation of solutions to KdV and the analysis of experimental data, are also discussed.

A Recursive Formulation for Constrained Mechanical System Dynamics: Part II. Closed Loop Systems

ABSTRACT ABSTRACT A recursive formulation of the equations of motion of constrained mechanical systems with closed loops is derived, using tools of variational and vector calculus. Kinematic couplings between pairs of contiguous bodies presented in Part 1 of this paper are generalized. Lagrange multipliers are introduced to account for the effects of joints that are cut to define a tree structure. Constraint Jacobian terms are added to the reduced variational equations derived in Part I. Cut-joint constraint acceleration equations are derived, to complete the reduced equations of motion. Lagrange multipliers associated with each cut-joint are eliminated at the first junction body encountered that permits closing the loop that constraints in cut joint. The inductive algorithm developed in Part I is used to calculate accelerations for the system. A multi-loop compressor is analyzed to illustrate use of the method.

Nonlinear HVDC link model reduction in multi AC-HVDC power system state estimation

Attempting to reduce the order of the HVDC link subspace in a multi AC-HVDC state estimator from 11 states to 4 increases the complexity and nonlinearity of the HVDC link measurement equations, decreases the HVDC link estimate accuracy, and increases the density of nonzero HVDC link Jacobian partial derivative terms.